Subgroups — C₄₇×#23864 #26063562

Order 12032.

1950 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₇×#449	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₇×#450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2013	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2000	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1894	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1879	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1870	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
16	C₂×AGL(1,ℤ/8)×C₄₇	—	2	8	no	—
17	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#547	—	2	4	no	—
22	C₄₇×#547	—	2	4	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#123×C₄₇	—	2	1	no	—
25	C₂×#123×C₄₇	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#121×C₄₇	—	2	1	no	—
29	C₂×#121×C₄₇	C₂²	1	2	yes	—
30	C₂×#120×C₄₇	—	2	1	no	—
31	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄₇×#450	—	2	4	no	—
35	C₄₇×#449	—	2	8	no	—
36	C₂×#103×C₄₇	C₂²	1	2	yes	complemented
37	C₂×#103×C₄₇	C₂²	1	2	yes	complemented
38	C₂×#102×C₄₇	C₂²	1	2	yes	—
39	C₂×#102×C₄₇	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	complemented
42	C₄₇×#394	—	2	1	no	—
43	C₄₇×#392	—	2	1	no	—
44	C₄₇×#390	C₂²	1	2	yes	—
45	C₄₇×#380	C₂²	1	1	yes	characteristic
46	C₄₇×#379	—	2	1	no	—
47	C₄₇×#340	C₂²	1	2	yes	complemented
48	C₄₇×#336	C₂²	1	2	yes	complemented
49	C₂⁵×C₄₇	—	2	1	no	—
50	C₂×#41×C₄₇	—	2	2	no	—
51	C₂×#41×C₄₇	—	2	2	no	—
52	C₂×#41×C₄₇	—	2	2	no	—
53	C₂×#41×C₄₇	—	2	4	no	—
54	C₂×#41×C₄₇	—	2	4	no	—
55	C₂×#41×C₄₇	—	4	2	no	—
56	C₂×#41×C₄₇	—	4	2	no	—
57	C₂×#41×C₄₇	D₄	1	4	yes	complemented
58	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	—	2	4	no	—
61	C₂²×D₄×C₄₇	—	2	1	no	—
62	C₂²×D₄×C₄₇	—	2	1	no	—
63	C₂²×D₄×C₄₇	—	2	2	no	—
64	C₂²×D₄×C₄₇	—	2	2	no	—
65	C₂²×D₄×C₄₇	—	2	2	no	—
66	C₂²×D₄×C₄₇	—	2	1	no	—
67	C₂²×D₄×C₄₇	—	4	2	no	—
68	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
70	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₄₇	—	2	1	no	—
72	C₂³×C₄×C₄₇	—	2	1	no	—
73	C₂³×C₄×C₄₇	—	2	1	no	—
74	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
76	AGL(1,ℤ/8)×C₄₇	—	2	32	no	—
77	C₂×SD₁₆×C₄₇	—	2	4	no	—
78	C₂×SD₁₆×C₄₇	—	2	4	no	—
79	C₂×SD₁₆×C₄₇	—	2	4	no	—
80	C₂×SD₁₆×C₄₇	—	2	4	no	—
81	C₂×SD₁₆×C₄₇	—	2	4	no	—
82	C₂×SD₁₆×C₄₇	—	4	2	no	—
83	C₂×D₈×C₄₇	—	2	4	no	—
84	C₂×D₈×C₄₇	—	2	4	no	—
85	C₂×D₈×C₄₇	D₄	1	8	yes	complemented
86	C₂×M₁₆×C₄₇	—	2	2	no	—
87	C₂×M₁₆×C₄₇	—	2	4	no	—
88	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
89	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₄₇	—	2	2	no	—
91	C₄⋊Q₈×C₄₇	—	2	2	no	—
92	#123×C₄₇	—	2	8	no	—
93	#123×C₄₇	—	2	8	no	—
94	C₂²⋊Q₈×C₄₇	—	2	2	no	—
95	C₂²⋊Q₈×C₄₇	—	2	2	no	—
96	C₂²⋊Q₈×C₄₇	—	2	2	no	—
97	C₂²⋊Q₈×C₄₇	D₄	1	4	yes	complemented
98	#121×C₄₇	—	2	8	no	—
99	#121×C₄₇	—	2	4	no	—
100	#121×C₄₇	—	4	2	no	—
101	#120×C₄₇	—	4	4	no	—
102	C₄×D₄×C₄₇	—	2	2	no	—
103	C₄×D₄×C₄₇	—	2	2	no	—
104	C₄×D₄×C₄₇	—	2	2	no	—
105	C₄×D₄×C₄₇	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
107	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
109	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
111	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
122	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
124	#103×C₄₇	—	2	4	no	—
125	#103×C₄₇	—	4	2	no	—
126	#102×C₄₇	—	2	4	no	—
127	#102×C₄₇	—	2	4	no	—
128	C₂²⋊C₈×C₄₇	—	2	2	no	—
129	C₂²⋊C₈×C₄₇	—	2	4	no	—
130	#95×C₄₇	—	2	1	no	—
131	C₂⁴×C₄₇	—	2	1	no	—
132	C₂⁴×C₄₇	—	2	2	no	—
133	C₂⁴×C₄₇	—	4	4	no	—
134	C₂⁴×C₄₇	—	4	2	no	—
135	C₂⁴×C₄₇	—	4	2	no	—
136	C₂⁴×C₄₇	—	4	2	no	—
137	C₂⁴×C₄₇	—	8	2	no	—
138	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
139	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
140	#41×C₄₇	—	2	8	no	—
141	#41×C₄₇	—	4	4	no	—
142	#41×C₄₇	—	4	8	no	—
143	#41×C₄₇	—	4	8	no	—
144	#41×C₄₇	—	4	4	no	—
145	#41×C₄₇	—	4	4	no	—
146	C₂×Q₈×C₄₇	—	2	1	no	—
147	C₂×Q₈×C₄₇	—	2	1	no	—
148	C₂×Q₈×C₄₇	—	2	2	no	—
149	C₂×Q₈×C₄₇	—	2	2	no	—
150	C₂×Q₈×C₄₇	—	2	2	no	—
151	C₂×Q₈×C₄₇	—	2	2	no	—
152	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
153	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₄₇	—	2	8	no	—
155	C₂×D₄×C₄₇	—	2	4	no	—
156	C₂×D₄×C₄₇	—	2	2	no	—
157	C₂×D₄×C₄₇	—	2	4	no	—
158	C₂×D₄×C₄₇	—	2	2	no	—
159	C₂×D₄×C₄₇	—	2	1	no	—
160	C₂×D₄×C₄₇	—	2	2	no	—
161	C₂×D₄×C₄₇	—	2	2	no	—
162	C₂×D₄×C₄₇	—	2	2	no	—
163	C₂×D₄×C₄₇	—	2	2	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	2	no	—
166	C₂×D₄×C₄₇	—	4	8	no	—
167	C₂×D₄×C₄₇	—	4	1	no	—
168	C₂×D₄×C₄₇	—	4	4	no	—
169	C₂×D₄×C₄₇	—	4	4	no	—
170	C₂×D₄×C₄₇	—	4	4	no	—
171	C₂×D₄×C₄₇	—	4	4	no	—
172	C₂×D₄×C₄₇	—	4	4	no	—
173	C₂×D₄×C₄₇	—	4	4	no	—
174	C₂×D₄×C₄₇	—	4	2	no	—
175	C₂×D₄×C₄₇	—	8	2	no	—
176	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
177	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
178	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₄₇	—	2	2	no	—
180	C₂²×C₄×C₄₇	—	2	2	no	—
181	C₂²×C₄×C₄₇	—	2	4	no	—
182	C₂²×C₄×C₄₇	—	2	4	no	—
183	C₂²×C₄×C₄₇	—	2	1	no	—
184	C₂²×C₄×C₄₇	—	2	1	no	—
185	C₂²×C₄×C₄₇	—	2	1	no	—
186	C₂²×C₄×C₄₇	—	2	4	no	—
187	C₂²×C₄×C₄₇	—	2	1	no	—
188	C₂²×C₄×C₄₇	—	2	1	no	—
189	C₂²×C₄×C₄₇	—	2	2	no	—
190	C₂²×C₄×C₄₇	—	2	2	no	—
191	C₂²×C₄×C₄₇	—	2	2	no	—
192	C₂²×C₄×C₄₇	—	4	4	no	—
193	C₂²×C₄×C₄₇	—	4	4	no	—
194	C₂²×C₄×C₄₇	—	4	2	no	—
195	C₂²×C₄×C₄₇	—	4	2	no	—
196	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
200	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
203	SD₁₆×C₄₇	—	2	16	no	—
204	SD₁₆×C₄₇	—	4	8	no	—
205	D₈×C₄₇	—	2	16	no	—
206	M₁₆×C₄₇	—	2	8	no	—
207	C₂×C₈×C₄₇	—	2	2	no	—
208	C₂×C₈×C₄₇	—	2	2	no	—
209	C₂×C₈×C₄₇	—	2	2	no	—
210	C₂×C₈×C₄₇	—	2	2	no	—
211	C₂×C₈×C₄₇	—	2	2	no	—
212	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
213	C₄⋊C₄×C₄₇	—	2	4	no	—
214	C₄⋊C₄×C₄₇	—	2	4	no	—
215	C₄⋊C₄×C₄₇	—	2	4	no	—
216	C₄⋊C₄×C₄₇	—	2	2	no	—
217	C₄⋊C₄×C₄₇	—	2	4	no	—
218	C₄⋊C₄×C₄₇	—	2	1	no	—
219	C₄⋊C₄×C₄₇	—	2	2	no	—
220	C₄⋊C₄×C₄₇	—	2	2	no	—
221	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₄₇	—	2	4	no	—
223	C₂²⋊C₄×C₄₇	—	2	4	no	—
224	C₂²⋊C₄×C₄₇	—	2	4	no	—
225	C₂²⋊C₄×C₄₇	—	2	2	no	—
226	C₂²⋊C₄×C₄₇	—	2	2	no	—
227	C₂²⋊C₄×C₄₇	—	4	4	no	—
228	C₂²⋊C₄×C₄₇	—	4	2	no	—
229	C₂²⋊C₄×C₄₇	—	4	4	no	—
230	C₂²⋊C₄×C₄₇	—	8	2	no	—
231	C₄²×C₄₇	—	2	1	no	—
232	C₄²×C₄₇	—	2	1	no	—
233	C₂³×C₄₇	—	2	2	no	—
234	C₂³×C₄₇	—	2	2	no	—
235	C₂³×C₄₇	—	2	4	no	—
236	C₂³×C₄₇	—	2	1	no	—
237	C₂³×C₄₇	—	2	1	no	—
238	C₂³×C₄₇	—	2	2	no	—
239	C₂³×C₄₇	—	2	1	no	—
240	C₂³×C₄₇	—	2	4	no	—
241	C₂³×C₄₇	—	2	2	no	—
242	C₂³×C₄₇	—	2	1	no	—
243	C₂³×C₄₇	—	2	1	no	—
244	C₂³×C₄₇	—	2	2	no	—
245	C₂³×C₄₇	—	4	8	no	—
246	C₂³×C₄₇	—	4	1	no	—
247	C₂³×C₄₇	—	4	1	no	—
248	C₂³×C₄₇	—	4	2	no	—
249	C₂³×C₄₇	—	4	2	no	—
250	C₂³×C₄₇	—	4	2	no	—
251	C₂³×C₄₇	—	4	2	no	—
252	C₂³×C₄₇	—	4	2	no	—
253	C₂³×C₄₇	—	4	2	no	—
254	C₂³×C₄₇	—	4	2	no	—
255	C₂³×C₄₇	—	8	4	no	—
256	C₂³×C₄₇	—	8	4	no	—
257	C₂³×C₄₇	—	8	8	no	—
258	C₂³×C₄₇	—	8	8	no	—
259	C₂³×C₄₇	#120	1	2	yes	—
260	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
261	Q₈×C₄₇	—	2	2	no	—
262	Q₈×C₄₇	—	2	2	no	—
263	Q₈×C₄₇	—	2	4	no	—
264	Q₈×C₄₇	—	4	1	no	—
265	Q₈×C₄₇	—	4	1	no	—
266	Q₈×C₄₇	—	4	2	no	—
267	D₄×C₄₇	—	2	4	no	—
268	D₄×C₄₇	—	2	2	no	—
269	D₄×C₄₇	—	2	4	no	—
270	D₄×C₄₇	—	2	4	no	—
271	D₄×C₄₇	—	4	4	no	—
272	D₄×C₄₇	—	4	8	no	—
273	D₄×C₄₇	—	4	8	no	—
274	D₄×C₄₇	—	4	8	no	—
275	D₄×C₄₇	—	4	2	no	—
276	D₄×C₄₇	—	4	2	no	—
277	D₄×C₄₇	—	8	8	no	—
278	D₄×C₄₇	#120	1	4	yes	complemented
279	C₂×C₄×C₄₇	—	2	1	no	—
280	C₂×C₄×C₄₇	—	2	2	no	—
281	C₂×C₄×C₄₇	—	2	1	no	—
282	C₂×C₄×C₄₇	—	2	2	no	—
283	C₂×C₄×C₄₇	—	2	2	no	—
284	C₂×C₄×C₄₇	—	2	4	no	—
285	C₂×C₄×C₄₇	—	2	1	no	—
286	C₂×C₄×C₄₇	—	2	2	no	—
287	C₂×C₄×C₄₇	—	2	1	no	—
288	C₂×C₄×C₄₇	—	2	2	no	—
289	C₂×C₄×C₄₇	—	2	2	no	—
290	C₂×C₄×C₄₇	—	2	2	no	—
291	C₂×C₄×C₄₇	—	2	2	no	—
292	C₂×C₄×C₄₇	—	4	8	no	—
293	C₂×C₄×C₄₇	—	4	8	no	—
294	C₂×C₄×C₄₇	—	4	8	no	—
295	C₂×C₄×C₄₇	—	4	2	no	—
296	C₂×C₄×C₄₇	—	4	2	no	—
297	C₂×C₄×C₄₇	—	4	1	no	—
298	C₂×C₄×C₄₇	—	4	1	no	—
299	C₂×C₄×C₄₇	—	4	1	no	—
300	C₂×C₄×C₄₇	—	4	1	no	—
301	C₂×C₄×C₄₇	—	4	2	no	—
302	C₂×C₄×C₄₇	—	4	2	no	—
303	C₂×C₄×C₄₇	—	4	2	no	—
304	C₂×C₄×C₄₇	—	4	2	no	—
305	C₂×C₄×C₄₇	#120	1	2	yes	—
306	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
307	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
309	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
311	C₈×C₄₇	—	2	4	no	—
312	C₈×C₄₇	—	4	2	no	—
313	#23864	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
314	C₂²×C₄₇	—	2	2	no	—
315	C₂²×C₄₇	—	2	4	no	—
316	C₂²×C₄₇	—	2	2	no	—
317	C₂²×C₄₇	—	2	2	no	—
318	C₂²×C₄₇	—	4	2	no	—
319	C₂²×C₄₇	—	4	2	no	—
320	C₂²×C₄₇	—	4	2	no	—
321	C₂²×C₄₇	—	4	4	no	—
322	C₂²×C₄₇	—	4	2	no	—
323	C₂²×C₄₇	—	4	4	no	—
324	C₂²×C₄₇	—	4	1	no	—
325	C₂²×C₄₇	—	4	4	no	—
326	C₂²×C₄₇	—	4	1	no	—
327	C₂²×C₄₇	—	4	1	no	—
328	C₂²×C₄₇	—	4	2	no	—
329	C₂²×C₄₇	—	4	4	no	—
330	C₂²×C₄₇	—	4	2	no	—
331	C₂²×C₄₇	—	4	4	no	—
332	C₂²×C₄₇	—	8	16	no	—
333	C₂²×C₄₇	—	8	2	no	—
334	C₂²×C₄₇	—	8	2	no	—
335	C₂²×C₄₇	—	8	2	no	—
336	C₂²×C₄₇	—	8	2	no	—
337	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
338	C₂²×C₄₇	D₄²	1	2	yes	—
339	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
340	C₂²×C₄₇	#575	1	1	yes	characteristic
341	C₂²×C₄₇	#577	1	2	yes	—
342	C₄×C₄₇	—	2	2	no	—
343	C₄×C₄₇	—	2	2	no	—
344	C₄×C₄₇	—	4	1	no	—
345	C₄×C₄₇	—	4	2	no	—
346	C₄×C₄₇	—	4	2	no	—
347	C₄×C₄₇	—	4	2	no	—
348	C₄×C₄₇	—	4	1	no	—
349	C₄×C₄₇	—	8	2	no	—
350	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
351	C₄×C₄₇	#534	1	1	yes	characteristic
352	C₄×C₄₇	D₄²	1	2	yes	—
353	C₂²×AGL(1,ℤ/8)	C₂×C₄₇	1	1	yes	characteristic, complemented
354	C₂×#547	C₂×C₄₇	1	1	yes	characteristic, complemented
355	C₂×#449	C₂×C₄₇	1	2	yes	complemented
356	C₂×#450	C₂×C₄₇	1	1	yes	characteristic, complemented
357	#2408	C₂×C₄₇	1	1	yes	characteristic, complemented
358	#2013	C₂×C₄₇	1	2	yes	complemented
359	#2000	C₂×C₄₇	1	2	yes	complemented
360	#1894	C₂×C₄₇	1	1	yes	characteristic, complemented
361	#1879	C₂×C₄₇	1	2	yes	complemented
362	#1870	C₂×C₄₇	1	1	yes	characteristic, complemented
363	#1788	C₂×C₄₇	1	1	yes	characteristic, complemented
364	C₂×C₄₇	—	4	2	no	—
365	C₂×C₄₇	—	4	4	no	—
366	C₂×C₄₇	—	8	1	no	—
367	C₂×C₄₇	—	8	4	no	—
368	C₂×C₄₇	#2403	1	1	yes	characteristic
369	C₂×C₄₇	#3006	1	1	yes	characteristic
370	C₂×C₄₇	#3010	1	1	yes	characteristic
371	C₂×C₄₇	#3267	1	2	yes	—
372	C₂×C₄₇	#3272	1	2	yes	—
373	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
374	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic, complemented
375	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
376	C₂×AGL(1,ℤ/8)	—	2	4	no	—
377	C₂×AGL(1,ℤ/8)	—	2	8	no	—
378	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
379	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
380	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
381	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
382	#547	—	2	4	no	—
383	#547	—	2	4	no	—
384	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic, complemented
385	C₂×#123	—	2	1	no	—
386	C₂×#123	—	2	1	no	—
387	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
388	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic, complemented
389	C₂×#121	—	2	1	no	—
390	C₂×#121	C₂²×C₄₇	1	2	yes	—
391	C₂×#120	—	2	1	no	—
392	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
393	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic, complemented
394	C₂²×C₄⋊C₄	—	2	1	no	—
395	#450	—	2	4	no	—
396	#449	—	2	8	no	—
397	C₂×#103	C₂²×C₄₇	1	2	yes	complemented
398	C₂×#103	C₂²×C₄₇	1	2	yes	complemented
399	C₂×#102	C₂²×C₄₇	1	2	yes	—
400	C₂×#102	C₂²×C₄₇	1	2	yes	complemented
401	C₂×C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
402	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	complemented
403	#394	—	2	1	no	—
404	#392	—	2	1	no	—
405	#390	C₂²×C₄₇	1	2	yes	—
406	#380	C₂²×C₄₇	1	1	yes	characteristic
407	#379	—	2	1	no	—
408	#340	C₂²×C₄₇	1	2	yes	complemented
409	#336	C₂²×C₄₇	1	2	yes	complemented
410	C₄₇	#23864	1	1	yes	characteristic, direct product factor, Sylow
411	C₂⁵	—	2	1	no	—
412	C₂×#41	—	2	2	no	—
413	C₂×#41	—	2	2	no	—
414	C₂×#41	—	2	2	no	—
415	C₂×#41	—	2	4	no	—
416	C₂×#41	—	2	4	no	—
417	C₂×#41	—	4	2	no	—
418	C₂×#41	—	4	2	no	—
419	C₂×#41	D₄×C₄₇	1	4	yes	complemented
420	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
421	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
422	C₂²×D₄	—	2	4	no	—
423	C₂²×D₄	—	2	1	no	—
424	C₂²×D₄	—	2	1	no	—
425	C₂²×D₄	—	2	2	no	—
426	C₂²×D₄	—	2	2	no	—
427	C₂²×D₄	—	2	2	no	—
428	C₂²×D₄	—	2	1	no	—
429	C₂²×D₄	—	4	2	no	—
430	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
431	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
432	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
433	C₂³×C₄	—	2	1	no	—
434	C₂³×C₄	—	2	1	no	—
435	C₂³×C₄	—	2	1	no	—
436	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
437	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
438	AGL(1,ℤ/8)	—	2	32	no	—
439	C₂×SD₁₆	—	2	4	no	—
440	C₂×SD₁₆	—	2	4	no	—
441	C₂×SD₁₆	—	2	4	no	—
442	C₂×SD₁₆	—	2	4	no	—
443	C₂×SD₁₆	—	2	4	no	—
444	C₂×SD₁₆	—	4	2	no	—
445	C₂×D₈	—	2	4	no	—
446	C₂×D₈	—	2	4	no	—
447	C₂×D₈	D₄×C₄₇	1	8	yes	complemented
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	—	2	4	no	—
450	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
451	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
452	C₄⋊Q₈	—	2	2	no	—
453	C₄⋊Q₈	—	2	2	no	—
454	#123	—	2	8	no	—
455	#123	—	2	8	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	D₄×C₄₇	1	4	yes	complemented
460	#121	—	2	8	no	—
461	#121	—	2	4	no	—
462	#121	—	4	2	no	—
463	#120	—	4	4	no	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₂×C₄⋊C₄	—	2	8	no	—
469	C₂×C₄⋊C₄	—	2	1	no	—
470	C₂×C₄⋊C₄	—	2	1	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	1	no	—
473	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
474	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
475	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
476	C₂×C₂²⋊C₄	—	2	1	no	—
477	C₂×C₂²⋊C₄	—	2	1	no	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	—	2	1	no	—
480	C₂×C₂²⋊C₄	—	2	2	no	—
481	C₂×C₂²⋊C₄	—	2	2	no	—
482	C₂×C₂²⋊C₄	—	2	2	no	—
483	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
484	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
486	#103	—	2	4	no	—
487	#103	—	4	2	no	—
488	#102	—	2	4	no	—
489	#102	—	2	4	no	—
490	C₂²⋊C₈	—	2	2	no	—
491	C₂²⋊C₈	—	2	4	no	—
492	#95	—	2	1	no	—
493	C₂⁴	—	2	1	no	—
494	C₂⁴	—	2	2	no	—
495	C₂⁴	—	4	4	no	—
496	C₂⁴	—	4	2	no	—
497	C₂⁴	—	4	2	no	—
498	C₂⁴	—	4	2	no	—
499	C₂⁴	—	8	2	no	—
500	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
501	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
502	#41	—	2	8	no	—
503	#41	—	4	4	no	—
504	#41	—	4	8	no	—
505	#41	—	4	8	no	—
506	#41	—	4	4	no	—
507	#41	—	4	4	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	2	no	—
511	C₂×Q₈	—	2	2	no	—
512	C₂×Q₈	—	2	2	no	—
513	C₂×Q₈	—	2	2	no	—
514	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
515	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
516	C₂×D₄	—	2	8	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	—	2	4	no	—
520	C₂×D₄	—	2	2	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	2	2	no	—
525	C₂×D₄	—	2	2	no	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	—	2	2	no	—
528	C₂×D₄	—	4	8	no	—
529	C₂×D₄	—	4	1	no	—
530	C₂×D₄	—	4	4	no	—
531	C₂×D₄	—	4	4	no	—
532	C₂×D₄	—	4	4	no	—
533	C₂×D₄	—	4	4	no	—
534	C₂×D₄	—	4	4	no	—
535	C₂×D₄	—	4	4	no	—
536	C₂×D₄	—	4	2	no	—
537	C₂×D₄	—	8	2	no	—
538	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
539	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
540	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	2	4	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	2	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	4	4	no	—
555	C₂²×C₄	—	4	4	no	—
556	C₂²×C₄	—	4	2	no	—
557	C₂²×C₄	—	4	2	no	—
558	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
560	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
561	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
562	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
563	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
564	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
565	SD₁₆	—	2	16	no	—
566	SD₁₆	—	4	8	no	—
567	D₈	—	2	16	no	—
568	M₁₆	—	2	8	no	—
569	C₂×C₈	—	2	2	no	—
570	C₂×C₈	—	2	2	no	—
571	C₂×C₈	—	2	2	no	—
572	C₂×C₈	—	2	2	no	—
573	C₂×C₈	—	2	2	no	—
574	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	2	no	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	4	4	no	—
590	C₂²⋊C₄	—	4	2	no	—
591	C₂²⋊C₄	—	4	4	no	—
592	C₂²⋊C₄	—	8	2	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	2	1	no	—
595	C₂³	—	2	2	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	4	no	—
598	C₂³	—	2	1	no	—
599	C₂³	—	2	1	no	—
600	C₂³	—	2	2	no	—
601	C₂³	—	2	1	no	—
602	C₂³	—	2	4	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	2	1	no	—
605	C₂³	—	2	1	no	—
606	C₂³	—	2	2	no	—
607	C₂³	—	4	8	no	—
608	C₂³	—	4	1	no	—
609	C₂³	—	4	1	no	—
610	C₂³	—	4	2	no	—
611	C₂³	—	4	2	no	—
612	C₂³	—	4	2	no	—
613	C₂³	—	4	2	no	—
614	C₂³	—	4	2	no	—
615	C₂³	—	4	2	no	—
616	C₂³	—	4	2	no	—
617	C₂³	—	8	4	no	—
618	C₂³	—	8	4	no	—
619	C₂³	—	8	8	no	—
620	C₂³	—	8	8	no	—
621	C₂³	#120×C₄₇	1	2	yes	—
622	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
623	Q₈	—	2	2	no	—
624	Q₈	—	2	2	no	—
625	Q₈	—	2	4	no	—
626	Q₈	—	4	1	no	—
627	Q₈	—	4	1	no	—
628	Q₈	—	4	2	no	—
629	D₄	—	2	4	no	—
630	D₄	—	2	2	no	—
631	D₄	—	2	4	no	—
632	D₄	—	2	4	no	—
633	D₄	—	4	4	no	—
634	D₄	—	4	8	no	—
635	D₄	—	4	8	no	—
636	D₄	—	4	8	no	—
637	D₄	—	4	2	no	—
638	D₄	—	4	2	no	—
639	D₄	—	8	8	no	—
640	D₄	#120×C₄₇	1	4	yes	complemented
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	4	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	4	8	no	—
655	C₂×C₄	—	4	8	no	—
656	C₂×C₄	—	4	8	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	4	2	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	2	no	—
664	C₂×C₄	—	4	2	no	—
665	C₂×C₄	—	4	2	no	—
666	C₂×C₄	—	4	2	no	—
667	C₂×C₄	#120×C₄₇	1	2	yes	—
668	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	complemented
669	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
670	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
671	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
672	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
673	C₈	—	2	4	no	—
674	C₈	—	4	2	no	—
675	C₂²	—	2	2	no	—
676	C₂²	—	2	4	no	—
677	C₂²	—	2	2	no	—
678	C₂²	—	2	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	4	2	no	—
682	C₂²	—	4	4	no	—
683	C₂²	—	4	2	no	—
684	C₂²	—	4	4	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	4	4	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	4	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	4	no	—
693	C₂²	—	8	16	no	—
694	C₂²	—	8	2	no	—
695	C₂²	—	8	2	no	—
696	C₂²	—	8	2	no	—
697	C₂²	—	8	2	no	—
698	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
699	C₂²	D₄²×C₄₇	1	2	yes	—
700	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
701	C₂²	C₄₇×#575	1	1	yes	characteristic
702	C₂²	C₄₇×#577	1	2	yes	—
703	C₄	—	2	2	no	—
704	C₄	—	2	2	no	—
705	C₄	—	4	1	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	2	no	—
708	C₄	—	4	2	no	—
709	C₄	—	4	1	no	—
710	C₄	—	8	2	no	—
711	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
712	C₄	C₄₇×#534	1	1	yes	characteristic
713	C₄	D₄²×C₄₇	1	2	yes	—
714	C₂	—	4	2	no	—
715	C₂	—	4	4	no	—
716	C₂	—	8	1	no	—
717	C₂	—	8	4	no	—
718	C₂	C₄₇×#2403	1	1	yes	characteristic
719	C₂	C₄₇×#3006	1	1	yes	characteristic
720	C₂	C₄₇×#3010	1	1	yes	characteristic
721	C₂	C₄₇×#3267	1	2	yes	—
722	C₂	C₄₇×#3272	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄×D₄×C₄₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₄₇	C₂	semidirect
2. C₂×C₄₇×#547	C₂	semidirect
3. C₂×C₄₇×#449	C₂	semidirect
4. C₂×C₄₇×#450	C₂	semidirect
5. C₄₇×#2408	C₂	semidirect
6. C₄₇×#2013	C₂	semidirect
7. C₄₇×#2000	C₂	semidirect
8. C₄₇×#1894	C₂	semidirect
9. C₄₇×#1879	C₂	semidirect
10. C₄₇×#1870	C₂	semidirect
11. C₄₇×#1788	C₂	semidirect
13. C₂²×#41×C₄₇	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
27. C₂×C₂²⋊Q₈×C₄₇	C₂²	semidirect
32. C₂×C₄×D₄×C₄₇	C₂²	semidirect
36. C₂×#103×C₄₇	C₂²	semidirect
37. C₂×#103×C₄₇	C₂²	semidirect
39. C₂×#102×C₄₇	C₂²	semidirect
41. C₂×C₂²⋊C₈×C₄₇	C₂²	semidirect
47. C₄₇×#340	C₂²	semidirect
48. C₄₇×#336	C₂²	semidirect
57. C₂×#41×C₄₇	D₄	semidirect
85. C₂×D₈×C₄₇	D₄	semidirect
97. C₂²⋊Q₈×C₄₇	D₄	semidirect
278. D₄×C₄₇	#120	semidirect
306. C₂×C₄×C₄₇	AGL(1,ℤ/8)	semidirect
313. #23864	C₄₇	direct (N × H)
353. C₂²×AGL(1,ℤ/8)	C₂×C₄₇	semidirect
354. C₂×#547	C₂×C₄₇	semidirect
355. C₂×#449	C₂×C₄₇	semidirect
356. C₂×#450	C₂×C₄₇	semidirect
357. #2408	C₂×C₄₇	semidirect
358. #2013	C₂×C₄₇	semidirect
359. #2000	C₂×C₄₇	semidirect
360. #1894	C₂×C₄₇	semidirect
361. #1879	C₂×C₄₇	semidirect
362. #1870	C₂×C₄₇	semidirect
363. #1788	C₂×C₄₇	semidirect
374. C₂²×#41	C₂²×C₄₇	semidirect
384. C₂×C₄⋊Q₈	C₂²×C₄₇	semidirect
388. C₂×C₂²⋊Q₈	C₂²×C₄₇	semidirect
393. C₂×C₄×D₄	C₂²×C₄₇	semidirect
397. C₂×#103	C₂²×C₄₇	semidirect
398. C₂×#103	C₂²×C₄₇	semidirect
400. C₂×#102	C₂²×C₄₇	semidirect
402. C₂×C₂²⋊C₈	C₂²×C₄₇	semidirect
408. #340	C₂²×C₄₇	semidirect
409. #336	C₂²×C₄₇	semidirect
410. C₄₇	#23864	direct (N × H)
419. C₂×#41	D₄×C₄₇	semidirect
447. C₂×D₈	D₄×C₄₇	semidirect
459. C₂²⋊Q₈	D₄×C₄₇	semidirect
640. D₄	#120×C₄₇	semidirect
668. C₂×C₄	AGL(1,ℤ/8)×C₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×AGL(1,ℤ/8)×C₄₇	2
C₂×C₄₇×#547	2
C₂×C₄₇×#449	2
C₂×C₄₇×#450	2
C₄₇×#2408	2
C₄₇×#2013	2
C₄₇×#2000	2
C₄₇×#1894	2
C₄₇×#1879	2
C₄₇×#1870	2
C₄₇×#1788	2
#23864	47

Subgroups — C47×#23864 #26063562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23864 #26063562