Subgroups — C₄₇×#21235 #26060933

Order 12032.

1678 conjugacy classes of subgroups.

The subgroup lattice has 608 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
2	C₂×C₄₇×#428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₇×#416	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#102×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2311	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1973	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1972	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1970	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1968	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
16	C₂×AGL(1,ℤ/8)×C₄₇	C₄	1	8	yes	complemented
17	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#428	—	2	4	no	—
25	C₂×#107×C₄₇	C₂²	1	2	yes	complemented
26	C₂×#106×C₄₇	C₂²	1	2	yes	complemented
27	C₄₇×#416	—	2	4	no	—
28	C₂×#103×C₄₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#103×C₄₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#102×C₄₇	—	2	4	no	—
31	C₂×#102×C₄₇	C₂²	1	2	yes	—
32	C₂×#102×C₄₇	C₂²	1	2	yes	—
33	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic, complemented
35	C₄₇×#405	—	2	4	no	—
36	C₂×#97×C₄₇	C₂²	1	2	yes	complemented
37	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	complemented
38	C₄₇×#391	C₂²	1	2	yes	complemented
39	C₄₇×#390	—	2	2	no	—
40	C₄₇×#390	C₂²	1	2	yes	—
41	C₄₇×#390	C₂²	1	2	yes	—
42	C₄₇×#390	C₂²	1	2	yes	complemented
43	C₄₇×#386	—	2	2	no	—
44	C₄₇×#380	—	2	2	no	—
45	C₄₇×#379	—	2	2	no	—
46	C₂⁵×C₄₇	—	2	1	no	—
47	C₂×#41×C₄₇	—	2	2	no	—
48	C₂×#41×C₄₇	—	2	2	no	—
49	C₂×#41×C₄₇	—	2	2	no	—
50	C₂×#41×C₄₇	—	2	2	no	—
51	C₂×#41×C₄₇	—	2	2	no	—
52	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
53	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₄₇	—	2	1	no	—
55	C₂²×D₄×C₄₇	—	2	2	no	—
56	C₂²×D₄×C₄₇	—	2	2	no	—
57	C₂²×D₄×C₄₇	—	2	2	no	—
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂²×D₄×C₄₇	—	2	2	no	—
60	C₂²×D₄×C₄₇	—	2	2	no	—
61	C₂²×D₄×C₄₇	C₂×C₄	1	4	yes	—
62	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₄₇	—	2	1	no	—
65	C₂³×C₄×C₄₇	—	2	1	no	—
66	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	AGL(1,ℤ/8)×C₄₇	—	2	32	no	—
68	C₂×SD₁₆×C₄₇	—	2	4	no	—
69	C₂×SD₁₆×C₄₇	—	2	4	no	—
70	C₂×SD₁₆×C₄₇	C₂×C₄	1	8	yes	complemented
71	C₂×D₈×C₄₇	—	2	4	no	—
72	C₂×D₈×C₄₇	—	2	4	no	—
73	C₂×D₈×C₄₇	C₂×C₄	1	8	yes	complemented
74	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
75	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
76	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
77	C₂²×C₈×C₄₇	—	2	2	no	—
78	C₂²×C₈×C₄₇	—	2	2	no	—
79	C₂²×C₈×C₄₇	D₄	1	4	yes	—
80	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
83	#117×C₄₇	—	2	2	no	—
84	#117×C₄₇	—	2	2	no	—
85	#117×C₄₇	—	2	2	no	—
86	#117×C₄₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
99	C₂×C₄²×C₄₇	—	2	1	no	—
100	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
101	#107×C₄₇	—	2	4	no	—
102	#106×C₄₇	—	2	4	no	—
103	#103×C₄₇	—	2	2	no	—
104	#103×C₄₇	—	2	2	no	—
105	#102×C₄₇	—	2	4	no	—
106	#102×C₄₇	—	2	4	no	—
107	#102×C₄₇	—	2	2	no	—
108	#102×C₄₇	—	2	2	no	—
109	#97×C₄₇	—	2	4	no	—
110	C₄×C₈×C₄₇	—	2	4	no	—
111	#95×C₄₇	—	2	4	no	—
112	C₂⁴×C₄₇	—	2	1	no	—
113	C₂⁴×C₄₇	—	2	2	no	—
114	C₂⁴×C₄₇	—	2	4	no	—
115	C₂⁴×C₄₇	—	4	2	no	—
116	C₂⁴×C₄₇	—	4	2	no	—
117	C₂⁴×C₄₇	—	8	2	no	—
118	C₂⁴×C₄₇	—	8	2	no	—
119	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
120	#41×C₄₇	—	2	8	no	—
121	#41×C₄₇	—	2	8	no	—
122	#41×C₄₇	—	4	4	no	—
123	#41×C₄₇	—	4	4	no	—
124	C₂×Q₈×C₄₇	—	2	1	no	—
125	C₂×Q₈×C₄₇	—	2	1	no	—
126	C₂×Q₈×C₄₇	—	2	1	no	—
127	C₂×Q₈×C₄₇	—	2	1	no	—
128	C₂×Q₈×C₄₇	—	2	1	no	—
129	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₄₇	—	2	4	no	—
131	C₂×D₄×C₄₇	—	2	8	no	—
132	C₂×D₄×C₄₇	—	2	8	no	—
133	C₂×D₄×C₄₇	—	2	4	no	—
134	C₂×D₄×C₄₇	—	2	1	no	—
135	C₂×D₄×C₄₇	—	2	1	no	—
136	C₂×D₄×C₄₇	—	2	2	no	—
137	C₂×D₄×C₄₇	—	2	2	no	—
138	C₂×D₄×C₄₇	—	2	2	no	—
139	C₂×D₄×C₄₇	—	2	1	no	—
140	C₂×D₄×C₄₇	—	2	2	no	—
141	C₂×D₄×C₄₇	—	2	1	no	—
142	C₂×D₄×C₄₇	—	2	2	no	—
143	C₂×D₄×C₄₇	—	2	1	no	—
144	C₂×D₄×C₄₇	—	4	2	no	—
145	C₂×D₄×C₄₇	—	4	2	no	—
146	C₂×D₄×C₄₇	—	4	4	no	—
147	C₂×D₄×C₄₇	—	4	2	no	—
148	C₂×D₄×C₄₇	—	4	4	no	—
149	C₂×D₄×C₄₇	—	4	2	no	—
150	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
151	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
152	C₂²×C₄×C₄₇	—	2	1	no	—
153	C₂²×C₄×C₄₇	—	2	4	no	—
154	C₂²×C₄×C₄₇	—	2	1	no	—
155	C₂²×C₄×C₄₇	—	2	1	no	—
156	C₂²×C₄×C₄₇	—	2	2	no	—
157	C₂²×C₄×C₄₇	—	4	2	no	—
158	C₂²×C₄×C₄₇	—	4	2	no	—
159	C₂²×C₄×C₄₇	—	4	2	no	—
160	C₂²×C₄×C₄₇	—	8	2	no	—
161	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
162	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₄₇	#41	1	2	yes	—
168	C₂²×C₄×C₄₇	#41	1	2	yes	—
169	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₄₇	—	2	16	no	—
171	D₈×C₄₇	—	2	16	no	—
172	M₁₆×C₄₇	C₂²⋊C₄	1	16	yes	complemented
173	C₂×C₈×C₄₇	—	2	8	no	—
174	C₂×C₈×C₄₇	—	2	1	no	—
175	C₂×C₈×C₄₇	—	2	1	no	—
176	C₂×C₈×C₄₇	—	2	1	no	—
177	C₂×C₈×C₄₇	—	2	1	no	—
178	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
179	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
180	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
181	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
182	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₄₇	—	2	4	no	—
184	C₄⋊C₄×C₄₇	—	2	1	no	—
185	C₄⋊C₄×C₄₇	—	2	1	no	—
186	C₄⋊C₄×C₄₇	—	2	1	no	—
187	C₄⋊C₄×C₄₇	—	2	2	no	—
188	C₄⋊C₄×C₄₇	—	2	1	no	—
189	C₄⋊C₄×C₄₇	—	2	2	no	—
190	C₄⋊C₄×C₄₇	—	4	2	no	—
191	C₂²⋊C₄×C₄₇	—	2	4	no	—
192	C₂²⋊C₄×C₄₇	—	4	2	no	—
193	C₂²⋊C₄×C₄₇	—	8	4	no	—
194	C₂²⋊C₄×C₄₇	—	8	4	no	—
195	C₂²⋊C₄×C₄₇	—	8	4	no	—
196	C₄²×C₄₇	—	2	2	no	—
197	C₄²×C₄₇	—	2	2	no	—
198	C₄²×C₄₇	—	4	2	no	—
199	C₂³×C₄₇	—	2	2	no	—
200	C₂³×C₄₇	—	2	2	no	—
201	C₂³×C₄₇	—	2	2	no	—
202	C₂³×C₄₇	—	2	4	no	—
203	C₂³×C₄₇	—	4	1	no	—
204	C₂³×C₄₇	—	4	2	no	—
205	C₂³×C₄₇	—	4	1	no	—
206	C₂³×C₄₇	—	4	2	no	—
207	C₂³×C₄₇	—	4	2	no	—
208	C₂³×C₄₇	—	4	4	no	—
209	C₂³×C₄₇	—	4	4	no	—
210	C₂³×C₄₇	—	4	2	no	—
211	C₂³×C₄₇	—	4	8	no	—
212	C₂³×C₄₇	—	8	8	no	—
213	C₂³×C₄₇	—	8	4	no	—
214	C₂³×C₄₇	—	8	4	no	—
215	C₂³×C₄₇	—	8	8	no	—
216	C₂³×C₄₇	C₄×D₄	1	2	yes	—
217	C₂³×C₄₇	#121	1	2	yes	—
218	C₂³×C₄₇	#127	1	2	yes	—
219	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₄₇	—	2	2	no	—
221	Q₈×C₄₇	—	2	2	no	—
222	Q₈×C₄₇	—	4	1	no	—
223	Q₈×C₄₇	—	4	1	no	—
224	D₄×C₄₇	—	2	2	no	—
225	D₄×C₄₇	—	2	4	no	—
226	D₄×C₄₇	—	2	4	no	—
227	D₄×C₄₇	—	2	2	no	—
228	D₄×C₄₇	—	4	8	no	—
229	D₄×C₄₇	—	4	2	no	—
230	D₄×C₄₇	—	4	1	no	—
231	D₄×C₄₇	—	4	2	no	—
232	D₄×C₄₇	—	4	1	no	—
233	D₄×C₄₇	—	4	8	no	—
234	C₂×C₄×C₄₇	—	2	2	no	—
235	C₂×C₄×C₄₇	—	2	4	no	—
236	C₂×C₄×C₄₇	—	2	4	no	—
237	C₂×C₄×C₄₇	—	2	4	no	—
238	C₂×C₄×C₄₇	—	4	1	no	—
239	C₂×C₄×C₄₇	—	4	1	no	—
240	C₂×C₄×C₄₇	—	4	8	no	—
241	C₂×C₄×C₄₇	—	4	1	no	—
242	C₂×C₄×C₄₇	—	4	1	no	—
243	C₂×C₄×C₄₇	—	4	2	no	—
244	C₂×C₄×C₄₇	—	4	2	no	—
245	C₂×C₄×C₄₇	—	4	4	no	—
246	C₂×C₄×C₄₇	—	4	2	no	—
247	C₂×C₄×C₄₇	—	8	4	no	—
248	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
249	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
250	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
251	C₂×C₄×C₄₇	#121	1	4	yes	—
252	C₂×C₄×C₄₇	#121	1	2	yes	—
253	C₂×C₄×C₄₇	#124	1	4	yes	—
254	C₂×C₄×C₄₇	#127	1	2	yes	—
255	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
261	C₈×C₄₇	—	2	2	no	—
262	C₈×C₄₇	—	2	2	no	—
263	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
264	#21235	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂²×C₄₇	—	2	2	no	—
266	C₂²×C₄₇	—	2	2	no	—
267	C₂²×C₄₇	—	2	4	no	—
268	C₂²×C₄₇	—	2	4	no	—
269	C₂²×C₄₇	—	4	4	no	—
270	C₂²×C₄₇	—	4	8	no	—
271	C₂²×C₄₇	—	4	2	no	—
272	C₂²×C₄₇	—	4	4	no	—
273	C₂²×C₄₇	—	8	16	no	—
274	C₂²×C₄₇	—	8	1	no	—
275	C₂²×C₄₇	—	8	2	no	—
276	C₂²×C₄₇	—	8	1	no	—
277	C₂²×C₄₇	—	8	2	no	—
278	C₂²×C₄₇	—	8	2	no	—
279	C₂²×C₄₇	—	8	4	no	—
280	C₂²×C₄₇	#390	1	4	yes	—
281	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
282	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
283	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
284	C₂²×C₄₇	#576	1	2	yes	—
285	C₂²×C₄₇	#577	1	2	yes	—
286	C₄×C₄₇	—	4	2	no	—
287	C₄×C₄₇	—	4	4	no	—
288	C₄×C₄₇	—	8	1	no	—
289	C₄×C₄₇	—	8	1	no	—
290	C₄×C₄₇	—	8	2	no	—
291	C₄×C₄₇	#390	1	4	yes	—
292	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
293	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
294	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
295	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
296	C₂²×AGL(1,ℤ/8)	C₂×C₄₇	1	1	yes	characteristic
297	C₂×#428	C₂×C₄₇	1	1	yes	characteristic, complemented
298	C₂×#416	C₂×C₄₇	1	1	yes	characteristic, complemented
299	C₂²×#102	C₂×C₄₇	1	1	yes	characteristic, complemented
300	C₂×#405	C₂×C₄₇	1	1	yes	characteristic, complemented
301	#2311	C₂×C₄₇	1	1	yes	characteristic, complemented
302	#2309	C₂×C₄₇	1	1	yes	characteristic, complemented
303	#1973	C₂×C₄₇	1	2	yes	complemented
304	#1972	C₂×C₄₇	1	2	yes	complemented
305	#1970	C₂×C₄₇	1	2	yes	complemented
306	#1968	C₂×C₄₇	1	2	yes	complemented
307	C₂×C₄₇	—	2	4	no	—
308	C₂×C₄₇	—	8	2	no	—
309	C₂×C₄₇	—	8	4	no	—
310	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
311	C₂×C₄₇	#2946	1	2	yes	—
312	C₂×C₄₇	#3062	1	2	yes	—
313	C₂×C₄₇	#3154	1	1	yes	characteristic
314	C₂×C₄₇	#3155	1	1	yes	characteristic
315	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
316	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
317	C₂×AGL(1,ℤ/8)	—	2	4	no	—
318	C₂×AGL(1,ℤ/8)	—	2	4	no	—
319	C₂×AGL(1,ℤ/8)	C₄×C₄₇	1	8	yes	complemented
320	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
321	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
322	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
323	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
324	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
325	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
326	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
327	#428	—	2	4	no	—
328	C₂×#107	C₂²×C₄₇	1	2	yes	complemented
329	C₂×#106	C₂²×C₄₇	1	2	yes	complemented
330	#416	—	2	4	no	—
331	C₂×#103	C₂²×C₄₇	1	1	yes	characteristic, complemented
332	C₂×#103	C₂²×C₄₇	1	1	yes	characteristic, complemented
333	C₂×#102	—	2	4	no	—
334	C₂×#102	C₂²×C₄₇	1	2	yes	—
335	C₂×#102	C₂²×C₄₇	1	2	yes	—
336	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic, complemented
337	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic, complemented
338	#405	—	2	4	no	—
339	C₂×#97	C₂²×C₄₇	1	2	yes	complemented
340	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	complemented
341	#391	C₂²×C₄₇	1	2	yes	complemented
342	#390	—	2	2	no	—
343	#390	C₂²×C₄₇	1	2	yes	—
344	#390	C₂²×C₄₇	1	2	yes	—
345	#390	C₂²×C₄₇	1	2	yes	complemented
346	#386	—	2	2	no	—
347	#380	—	2	2	no	—
348	#379	—	2	2	no	—
349	C₄₇	#21235	1	1	yes	characteristic, direct product factor, Sylow
350	C₂⁵	—	2	1	no	—
351	C₂×#41	—	2	2	no	—
352	C₂×#41	—	2	2	no	—
353	C₂×#41	—	2	2	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	C₂×C₄×C₄₇	1	4	yes	—
357	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	2	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	2	no	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	C₂×C₄×C₄₇	1	4	yes	—
366	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
367	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
371	AGL(1,ℤ/8)	—	2	32	no	—
372	C₂×SD₁₆	—	2	4	no	—
373	C₂×SD₁₆	—	2	4	no	—
374	C₂×SD₁₆	C₂×C₄×C₄₇	1	8	yes	complemented
375	C₂×D₈	—	2	4	no	—
376	C₂×D₈	—	2	4	no	—
377	C₂×D₈	C₂×C₄×C₄₇	1	8	yes	complemented
378	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
379	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
380	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
381	C₂²×C₈	—	2	2	no	—
382	C₂²×C₈	—	2	2	no	—
383	C₂²×C₈	D₄×C₄₇	1	4	yes	—
384	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
387	#117	—	2	2	no	—
388	#117	—	2	2	no	—
389	#117	—	2	2	no	—
390	#117	—	2	2	no	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	D₄×C₄₇	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	4	no	—
400	C₂×C₂²⋊C₄	—	2	4	no	—
401	C₂×C₂²⋊C₄	—	2	4	no	—
402	C₂×C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
403	C₂×C₄²	—	2	1	no	—
404	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
405	#107	—	2	4	no	—
406	#106	—	2	4	no	—
407	#103	—	2	2	no	—
408	#103	—	2	2	no	—
409	#102	—	2	4	no	—
410	#102	—	2	4	no	—
411	#102	—	2	2	no	—
412	#102	—	2	2	no	—
413	#97	—	2	4	no	—
414	C₄×C₈	—	2	4	no	—
415	#95	—	2	4	no	—
416	C₂⁴	—	2	1	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	2	no	—
420	C₂⁴	—	4	2	no	—
421	C₂⁴	—	8	2	no	—
422	C₂⁴	—	8	2	no	—
423	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
424	#41	—	2	8	no	—
425	#41	—	2	8	no	—
426	#41	—	4	4	no	—
427	#41	—	4	4	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	C₂²×C₄×C₄₇	1	2	yes	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	—	4	2	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	C₂²×C₄×C₄₇	1	2	yes	—
455	C₂×D₄	C₂²×C₄×C₄₇	1	4	yes	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	2	no	—
464	C₂²×C₄	—	8	2	no	—
465	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
467	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
471	C₂²×C₄	#41×C₄₇	1	2	yes	—
472	C₂²×C₄	#41×C₄₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
474	SD₁₆	—	2	16	no	—
475	D₈	—	2	16	no	—
476	M₁₆	C₂²⋊C₄×C₄₇	1	16	yes	complemented
477	C₂×C₈	—	2	8	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
483	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
484	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
485	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
486	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	4	2	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	4	2	no	—
497	C₂²⋊C₄	—	8	4	no	—
498	C₂²⋊C₄	—	8	4	no	—
499	C₂²⋊C₄	—	8	4	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	2	no	—
502	C₄²	—	4	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	4	8	no	—
516	C₂³	—	8	8	no	—
517	C₂³	—	8	8	no	—
518	C₂³	—	8	4	no	—
519	C₂³	—	8	4	no	—
520	C₂³	C₄×D₄×C₄₇	1	2	yes	—
521	C₂³	#121×C₄₇	1	2	yes	—
522	C₂³	#127×C₄₇	1	2	yes	—
523	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
524	Q₈	—	2	2	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	4	1	no	—
527	Q₈	—	4	1	no	—
528	D₄	—	2	2	no	—
529	D₄	—	2	4	no	—
530	D₄	—	2	4	no	—
531	D₄	—	2	2	no	—
532	D₄	—	4	8	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	1	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	8	4	no	—
552	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
553	C₂×C₄	C₄×D₄×C₄₇	1	4	yes	—
554	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
555	C₂×C₄	#121×C₄₇	1	4	yes	—
556	C₂×C₄	#121×C₄₇	1	2	yes	—
557	C₂×C₄	#124×C₄₇	1	4	yes	—
558	C₂×C₄	#127×C₄₇	1	2	yes	—
559	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
560	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
563	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
568	C₂²	—	2	4	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	8	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	8	16	no	—
577	C₂²	—	8	4	no	—
578	C₂²	—	8	2	no	—
579	C₂²	—	8	2	no	—
580	C₂²	—	8	1	no	—
581	C₂²	—	8	2	no	—
582	C₂²	—	8	1	no	—
583	C₂²	C₄₇×#390	1	4	yes	—
584	C₂²	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
585	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
586	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
587	C₂²	C₄₇×#576	1	2	yes	—
588	C₂²	C₄₇×#577	1	2	yes	—
589	C₄	—	4	2	no	—
590	C₄	—	4	4	no	—
591	C₄	—	8	1	no	—
592	C₄	—	8	2	no	—
593	C₄	—	8	1	no	—
594	C₄	C₄₇×#390	1	4	yes	—
595	C₄	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
596	C₄	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
597	C₄	C₂×#121×C₄₇	1	1	yes	characteristic
598	C₄	C₂×#124×C₄₇	1	1	yes	characteristic
599	C₂	—	2	4	no	—
600	C₂	—	8	4	no	—
601	C₂	—	8	2	no	—
602	C₂	C₂×C₄₇×#390	1	1	yes	characteristic
603	C₂	C₄₇×#2946	1	2	yes	—
604	C₂	C₄₇×#3062	1	2	yes	—
605	C₂	C₄₇×#3154	1	1	yes	characteristic
606	C₂	C₄₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂²×AGL(1,ℤ/8)×C₄₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×#117×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#428	C₂	semidirect
3. C₂×C₄₇×#416	C₂	semidirect
4. C₂²×#102×C₄₇	C₂	semidirect
5. C₂×C₄₇×#405	C₂	semidirect
6. C₄₇×#2311	C₂	semidirect
7. C₄₇×#2309	C₂	semidirect
8. C₄₇×#1973	C₂	semidirect
9. C₄₇×#1972	C₂	semidirect
10. C₄₇×#1970	C₂	semidirect
11. C₄₇×#1968	C₂	semidirect
16. C₂×AGL(1,ℤ/8)×C₄₇	C₄	semidirect
25. C₂×#107×C₄₇	C₂²	semidirect
26. C₂×#106×C₄₇	C₂²	semidirect
28. C₂×#103×C₄₇	C₂²	semidirect
29. C₂×#103×C₄₇	C₂²	semidirect
33. C₂×#102×C₄₇	C₂²	semidirect
34. C₂×#102×C₄₇	C₂²	semidirect
36. C₂×#97×C₄₇	C₂²	semidirect
37. C₂×C₄×C₈×C₄₇	C₂²	semidirect
38. C₄₇×#391	C₂²	semidirect
42. C₄₇×#390	C₂²	semidirect
70. C₂×SD₁₆×C₄₇	C₂×C₄	semidirect
73. C₂×D₈×C₄₇	C₂×C₄	semidirect
172. M₁₆×C₄₇	C₂²⋊C₄	semidirect
264. #21235	C₄₇	direct (N × H)
297. C₂×#428	C₂×C₄₇	semidirect
298. C₂×#416	C₂×C₄₇	semidirect
299. C₂²×#102	C₂×C₄₇	semidirect
300. C₂×#405	C₂×C₄₇	semidirect
301. #2311	C₂×C₄₇	semidirect
302. #2309	C₂×C₄₇	semidirect
303. #1973	C₂×C₄₇	semidirect
304. #1972	C₂×C₄₇	semidirect
305. #1970	C₂×C₄₇	semidirect
306. #1968	C₂×C₄₇	semidirect
319. C₂×AGL(1,ℤ/8)	C₄×C₄₇	semidirect
328. C₂×#107	C₂²×C₄₇	semidirect
329. C₂×#106	C₂²×C₄₇	semidirect
331. C₂×#103	C₂²×C₄₇	semidirect
332. C₂×#103	C₂²×C₄₇	semidirect
336. C₂×#102	C₂²×C₄₇	semidirect
337. C₂×#102	C₂²×C₄₇	semidirect
339. C₂×#97	C₂²×C₄₇	semidirect
340. C₂×C₄×C₈	C₂²×C₄₇	semidirect
341. #391	C₂²×C₄₇	semidirect
345. #390	C₂²×C₄₇	semidirect
349. C₄₇	#21235	direct (N × H)
374. C₂×SD₁₆	C₂×C₄×C₄₇	semidirect
377. C₂×D₈	C₂×C₄×C₄₇	semidirect
476. M₁₆	C₂²⋊C₄×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×#428	2
C₂×C₄₇×#416	2
C₂²×#102×C₄₇	2
C₂×C₄₇×#405	2
C₄₇×#2311	2
C₄₇×#2309	2
C₄₇×#1973	2
C₄₇×#1972	2
C₄₇×#1970	2
C₄₇×#1968	2
#21235	47

Subgroups — C47×#21235 #26060933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#21235 #26060933