Subgroups — C₄₇×#24355 #26064053

Order 12032.

1742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#463	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₇×#450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2435	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2054	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2012	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1892	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1879	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1850	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1786	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₈×C₄₇	—	2	4	no	—
17	C₂²×D₈×C₄₇	—	2	8	no	—
18	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
19	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#124×C₄₇	C₂²	1	2	yes	complemented
22	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	complemented
24	C₂²×C₂²⋊C₄×C₄₇	—	2	1	no	—
25	C₄₇×#463	—	2	8	no	—
26	C₄₇×#450	—	2	4	no	—
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	complemented
28	C₂×#103×C₄₇	C₂²	1	2	yes	complemented
29	C₂×#102×C₄₇	C₂²	1	2	yes	complemented
30	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic, complemented
31	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×#102×C₄₇	C₂²	1	2	yes	complemented
33	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#394	—	2	2	no	—
35	C₄₇×#390	C₂²	1	2	yes	complemented
36	C₄₇×#390	C₂²	1	2	yes	complemented
37	C₄₇×#386	—	2	2	no	—
38	C₄₇×#380	C₂²	1	1	yes	characteristic
39	C₄₇×#379	—	2	1	no	—
40	C₄₇×#379	—	2	1	no	—
41	C₄₇×#379	—	2	1	no	—
42	C₄×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
43	C₄₇×#340	C₂²	1	2	yes	complemented
44	C₄₇×#336	C₂²	1	2	yes	complemented
45	C₂⁵×C₄₇	—	2	1	no	—
46	C₂⁵×C₄₇	—	2	1	no	—
47	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₄₇	—	2	4	no	—
49	C₂²×D₄×C₄₇	—	2	4	no	—
50	C₂²×D₄×C₄₇	—	2	2	no	—
51	C₂²×D₄×C₄₇	—	2	1	no	—
52	C₂²×D₄×C₄₇	—	2	1	no	—
53	C₂²×D₄×C₄₇	—	2	2	no	—
54	C₂²×D₄×C₄₇	—	2	2	no	—
55	C₂²×D₄×C₄₇	—	4	2	no	—
56	C₂²×D₄×C₄₇	—	4	2	no	—
57	C₂²×D₄×C₄₇	D₄	1	4	yes	—
58	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	—	2	2	no	—
62	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×SD₁₆×C₄₇	—	2	2	no	—
64	C₂×SD₁₆×C₄₇	—	2	2	no	—
65	C₂×SD₁₆×C₄₇	—	2	4	no	—
66	C₂×SD₁₆×C₄₇	—	4	2	no	—
67	C₂×D₈×C₄₇	—	2	32	no	—
68	C₂×D₈×C₄₇	—	2	4	no	—
69	C₂×D₈×C₄₇	—	2	4	no	—
70	C₂×D₈×C₄₇	—	2	4	no	—
71	C₂×D₈×C₄₇	—	2	4	no	—
72	C₂×D₈×C₄₇	—	2	4	no	—
73	C₂×D₈×C₄₇	D₄	1	8	yes	—
74	C₂²×C₈×C₄₇	—	2	2	no	—
75	C₂²×C₈×C₄₇	—	2	4	no	—
76	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
79	#124×C₄₇	—	2	4	no	—
80	#124×C₄₇	—	4	2	no	—
81	C₂²⋊Q₈×C₄₇	—	2	2	no	—
82	C₂²⋊Q₈×C₄₇	—	2	2	no	—
83	C₄×D₄×C₄₇	—	2	4	no	—
84	C₄×D₄×C₄₇	—	4	2	no	—
85	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₄₇	—	4	4	no	—
98	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
101	C₄⋊C₈×C₄₇	—	2	4	no	—
102	#103×C₄₇	—	2	4	no	—
103	#102×C₄₇	—	2	2	no	—
104	#102×C₄₇	—	2	2	no	—
105	#102×C₄₇	—	2	4	no	—
106	#102×C₄₇	—	4	2	no	—
107	C₂²⋊C₈×C₄₇	—	2	2	no	—
108	#95×C₄₇	—	2	1	no	—
109	#95×C₄₇	D₄	1	2	yes	—
110	C₂⁴×C₄₇	—	2	1	no	—
111	C₂⁴×C₄₇	—	2	1	no	—
112	C₂⁴×C₄₇	—	2	2	no	—
113	C₂⁴×C₄₇	—	2	4	no	—
114	C₂⁴×C₄₇	—	4	4	no	—
115	C₂⁴×C₄₇	—	4	4	no	—
116	C₂⁴×C₄₇	—	4	2	no	—
117	C₂⁴×C₄₇	—	4	2	no	—
118	C₂⁴×C₄₇	—	4	2	no	—
119	C₂⁴×C₄₇	—	4	2	no	—
120	C₂⁴×C₄₇	—	4	2	no	—
121	C₂⁴×C₄₇	—	8	2	no	—
122	C₂⁴×C₄₇	—	8	2	no	—
123	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Q₈×C₄₇	—	2	2	no	—
125	C₂×Q₈×C₄₇	—	2	1	no	—
126	C₂×Q₈×C₄₇	—	2	1	no	—
127	C₂×Q₈×C₄₇	—	2	2	no	—
128	C₂×D₄×C₄₇	—	2	8	no	—
129	C₂×D₄×C₄₇	—	2	8	no	—
130	C₂×D₄×C₄₇	—	2	4	no	—
131	C₂×D₄×C₄₇	—	2	2	no	—
132	C₂×D₄×C₄₇	—	2	2	no	—
133	C₂×D₄×C₄₇	—	2	2	no	—
134	C₂×D₄×C₄₇	—	2	1	no	—
135	C₂×D₄×C₄₇	—	2	2	no	—
136	C₂×D₄×C₄₇	—	2	1	no	—
137	C₂×D₄×C₄₇	—	2	2	no	—
138	C₂×D₄×C₄₇	—	2	2	no	—
139	C₂×D₄×C₄₇	—	2	2	no	—
140	C₂×D₄×C₄₇	—	2	2	no	—
141	C₂×D₄×C₄₇	—	4	4	no	—
142	C₂×D₄×C₄₇	—	4	8	no	—
143	C₂×D₄×C₄₇	—	4	8	no	—
144	C₂×D₄×C₄₇	—	4	4	no	—
145	C₂×D₄×C₄₇	—	4	4	no	—
146	C₂×D₄×C₄₇	—	4	4	no	—
147	C₂×D₄×C₄₇	—	4	2	no	—
148	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
151	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₄₇	—	2	2	no	—
153	C₂²×C₄×C₄₇	—	2	2	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	4	no	—
158	C₂²×C₄×C₄₇	—	4	4	no	—
159	C₂²×C₄×C₄₇	—	8	2	no	—
160	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₄₇	#41	1	2	yes	—
165	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₄₇	—	4	4	no	—
167	SD₁₆×C₄₇	—	4	4	no	—
168	D₈×C₄₇	—	2	16	no	—
169	D₈×C₄₇	—	2	16	no	—
170	C₂×C₈×C₄₇	—	2	8	no	—
171	C₂×C₈×C₄₇	—	2	2	no	—
172	C₂×C₈×C₄₇	—	2	2	no	—
173	C₂×C₈×C₄₇	—	2	2	no	—
174	C₂×C₈×C₄₇	—	2	1	no	—
175	C₂×C₈×C₄₇	—	2	1	no	—
176	C₂×C₈×C₄₇	—	2	2	no	—
177	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₄₇	—	2	2	no	—
179	C₄⋊C₄×C₄₇	—	2	1	no	—
180	C₄⋊C₄×C₄₇	—	2	1	no	—
181	C₄⋊C₄×C₄₇	—	2	2	no	—
182	C₄⋊C₄×C₄₇	—	4	2	no	—
183	C₂²⋊C₄×C₄₇	—	2	2	no	—
184	C₂²⋊C₄×C₄₇	—	2	2	no	—
185	C₂²⋊C₄×C₄₇	—	4	4	no	—
186	C₂²⋊C₄×C₄₇	—	4	4	no	—
187	C₂²⋊C₄×C₄₇	—	4	4	no	—
188	C₂²⋊C₄×C₄₇	—	4	4	no	—
189	C₂²⋊C₄×C₄₇	—	4	2	no	—
190	C₂²⋊C₄×C₄₇	—	4	2	no	—
191	C₂²⋊C₄×C₄₇	—	8	2	no	—
192	C₂²⋊C₄×C₄₇	—	8	2	no	—
193	C₂²⋊C₄×C₄₇	—	8	2	no	—
194	C₄²×C₄₇	—	2	2	no	—
195	C₄²×C₄₇	—	2	2	no	—
196	C₂³×C₄₇	—	2	2	no	—
197	C₂³×C₄₇	—	2	2	no	—
198	C₂³×C₄₇	—	2	2	no	—
199	C₂³×C₄₇	—	2	4	no	—
200	C₂³×C₄₇	—	2	2	no	—
201	C₂³×C₄₇	—	2	4	no	—
202	C₂³×C₄₇	—	4	8	no	—
203	C₂³×C₄₇	—	4	8	no	—
204	C₂³×C₄₇	—	4	1	no	—
205	C₂³×C₄₇	—	4	1	no	—
206	C₂³×C₄₇	—	4	2	no	—
207	C₂³×C₄₇	—	4	2	no	—
208	C₂³×C₄₇	—	4	2	no	—
209	C₂³×C₄₇	—	4	2	no	—
210	C₂³×C₄₇	—	4	2	no	—
211	C₂³×C₄₇	—	4	1	no	—
212	C₂³×C₄₇	—	4	1	no	—
213	C₂³×C₄₇	—	4	2	no	—
214	C₂³×C₄₇	—	4	2	no	—
215	C₂³×C₄₇	—	4	2	no	—
216	C₂³×C₄₇	—	4	2	no	—
217	C₂³×C₄₇	—	4	8	no	—
218	C₂³×C₄₇	—	8	4	no	—
219	C₂³×C₄₇	—	8	8	no	—
220	C₂³×C₄₇	—	8	8	no	—
221	C₂³×C₄₇	—	8	8	no	—
222	C₂³×C₄₇	—	8	8	no	—
223	C₂³×C₄₇	—	8	4	no	—
224	C₂³×C₄₇	—	8	4	no	—
225	C₂³×C₄₇	#121	1	2	yes	—
226	C₂³×C₄₇	#135	1	2	yes	—
227	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
228	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₄₇	—	4	1	no	—
230	Q₈×C₄₇	—	4	1	no	—
231	Q₈×C₄₇	—	4	2	no	—
232	D₄×C₄₇	—	2	2	no	—
233	D₄×C₄₇	—	2	4	no	—
234	D₄×C₄₇	—	2	2	no	—
235	D₄×C₄₇	—	2	2	no	—
236	D₄×C₄₇	—	2	4	no	—
237	D₄×C₄₇	—	2	4	no	—
238	D₄×C₄₇	—	4	8	no	—
239	D₄×C₄₇	—	4	8	no	—
240	D₄×C₄₇	—	4	2	no	—
241	D₄×C₄₇	—	4	2	no	—
242	D₄×C₄₇	—	4	2	no	—
243	D₄×C₄₇	#120	1	4	yes	—
244	C₂×C₄×C₄₇	—	2	2	no	—
245	C₂×C₄×C₄₇	—	2	2	no	—
246	C₂×C₄×C₄₇	—	2	2	no	—
247	C₂×C₄×C₄₇	—	2	4	no	—
248	C₂×C₄×C₄₇	—	2	2	no	—
249	C₂×C₄×C₄₇	—	2	2	no	—
250	C₂×C₄×C₄₇	—	2	2	no	—
251	C₂×C₄×C₄₇	—	2	2	no	—
252	C₂×C₄×C₄₇	—	2	2	no	—
253	C₂×C₄×C₄₇	—	2	2	no	—
254	C₂×C₄×C₄₇	—	4	1	no	—
255	C₂×C₄×C₄₇	—	4	1	no	—
256	C₂×C₄×C₄₇	—	4	2	no	—
257	C₂×C₄×C₄₇	—	4	2	no	—
258	C₂×C₄×C₄₇	—	4	2	no	—
259	C₂×C₄×C₄₇	—	4	2	no	—
260	C₂×C₄×C₄₇	—	4	2	no	—
261	C₂×C₄×C₄₇	—	8	8	no	—
262	C₂×C₄×C₄₇	#121	1	2	yes	—
263	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
267	C₈×C₄₇	—	2	4	no	—
268	C₈×C₄₇	—	4	1	no	—
269	C₈×C₄₇	—	4	1	no	—
270	#24355	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂²×C₄₇	—	2	2	no	—
272	C₂²×C₄₇	—	2	4	no	—
273	C₂²×C₄₇	—	2	4	no	—
274	C₂²×C₄₇	—	2	2	no	—
275	C₂²×C₄₇	—	2	4	no	—
276	C₂²×C₄₇	—	4	4	no	—
277	C₂²×C₄₇	—	4	4	no	—
278	C₂²×C₄₇	—	4	2	no	—
279	C₂²×C₄₇	—	4	2	no	—
280	C₂²×C₄₇	—	4	2	no	—
281	C₂²×C₄₇	—	4	4	no	—
282	C₂²×C₄₇	—	4	4	no	—
283	C₂²×C₄₇	—	4	4	no	—
284	C₂²×C₄₇	—	4	2	no	—
285	C₂²×C₄₇	—	4	2	no	—
286	C₂²×C₄₇	—	8	16	no	—
287	C₂²×C₄₇	—	8	16	no	—
288	C₂²×C₄₇	—	8	2	no	—
289	C₂²×C₄₇	—	8	2	no	—
290	C₂²×C₄₇	—	8	2	no	—
291	C₂²×C₄₇	—	8	2	no	—
292	C₂²×C₄₇	—	8	2	no	—
293	C₂²×C₄₇	—	8	2	no	—
294	C₂²×C₄₇	#463	1	4	yes	complemented
295	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
296	C₂²×C₄₇	D₄²	1	2	yes	—
297	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
298	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂²×C₄₇	#576	1	2	yes	—
300	C₄×C₄₇	—	2	2	no	—
301	C₄×C₄₇	—	4	2	no	—
302	C₄×C₄₇	—	4	2	no	—
303	C₄×C₄₇	—	8	1	no	—
304	C₄×C₄₇	—	8	2	no	—
305	C₄×C₄₇	—	8	1	no	—
306	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
307	C₄×C₄₇	#534	1	1	yes	characteristic
308	C₄×C₄₇	#546	1	2	yes	—
309	C₂³×D₈	C₂×C₄₇	1	1	yes	characteristic
310	C₂×#463	C₂×C₄₇	1	2	yes	complemented
311	C₂×#450	C₂×C₄₇	1	1	yes	characteristic, complemented
312	#2435	C₂×C₄₇	1	1	yes	characteristic, complemented
313	D₄×C₂²⋊C₄	C₂×C₄₇	1	1	yes	characteristic, complemented
314	#2054	C₂×C₄₇	1	2	yes	complemented
315	#2012	C₂×C₄₇	1	2	yes	complemented
316	#1892	C₂×C₄₇	1	1	yes	characteristic, complemented
317	#1879	C₂×C₄₇	1	2	yes	complemented
318	#1850	C₂×C₄₇	1	1	yes	characteristic, complemented
319	#1786	C₂×C₄₇	1	1	yes	characteristic, complemented
320	C₂×C₄₇	—	2	4	no	—
321	C₂×C₄₇	—	4	4	no	—
322	C₂×C₄₇	—	8	4	no	—
323	C₂×C₄₇	—	8	2	no	—
324	C₂×C₄₇	#2410	1	1	yes	characteristic
325	C₂×C₄₇	C₂×#463	1	1	yes	characteristic
326	C₂×C₄₇	#3041	1	1	yes	characteristic
327	C₂×C₄₇	#3264	1	2	yes	—
328	C₂×C₄₇	#3280	1	2	yes	—
329	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
330	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
331	C₂²×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
332	C₂²×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
333	C₂²×D₈	—	2	4	no	—
334	C₂²×D₈	—	2	8	no	—
335	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
336	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
337	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
338	C₂×#124	C₂²×C₄₇	1	2	yes	complemented
339	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₄₇	1	2	yes	complemented
341	C₂²×C₂²⋊C₄	—	2	1	no	—
342	#463	—	2	8	no	—
343	#450	—	2	4	no	—
344	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
345	C₂×#103	C₂²×C₄₇	1	2	yes	complemented
346	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic, complemented
347	C₂×#102	C₂²×C₄₇	1	2	yes	complemented
348	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic, complemented
349	C₂×#102	C₂²×C₄₇	1	2	yes	complemented
350	C₂×C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
351	#394	—	2	2	no	—
352	#390	C₂²×C₄₇	1	2	yes	complemented
353	#390	C₂²×C₄₇	1	2	yes	complemented
354	#386	—	2	2	no	—
355	#380	C₂²×C₄₇	1	1	yes	characteristic
356	#379	—	2	1	no	—
357	#379	—	2	1	no	—
358	#379	—	2	1	no	—
359	C₄×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
360	#340	C₂²×C₄₇	1	2	yes	complemented
361	#336	C₂²×C₄₇	1	2	yes	complemented
362	C₄₇	#24355	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁵	—	2	1	no	—
364	C₂⁵	—	2	1	no	—
365	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
366	C₂²×D₄	—	2	4	no	—
367	C₂²×D₄	—	2	4	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	—	2	1	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	4	2	no	—
374	C₂²×D₄	—	4	2	no	—
375	C₂²×D₄	D₄×C₄₇	1	4	yes	—
376	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
377	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
378	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
379	C₂³×C₄	—	2	2	no	—
380	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
381	C₂×SD₁₆	—	2	2	no	—
382	C₂×SD₁₆	—	2	2	no	—
383	C₂×SD₁₆	—	2	4	no	—
384	C₂×SD₁₆	—	4	2	no	—
385	C₂×D₈	—	2	32	no	—
386	C₂×D₈	—	2	4	no	—
387	C₂×D₈	—	2	4	no	—
388	C₂×D₈	—	2	4	no	—
389	C₂×D₈	—	2	4	no	—
390	C₂×D₈	—	2	4	no	—
391	C₂×D₈	D₄×C₄₇	1	8	yes	—
392	C₂²×C₈	—	2	2	no	—
393	C₂²×C₈	—	2	4	no	—
394	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
397	#124	—	2	4	no	—
398	#124	—	4	2	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	2	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	4	2	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₂²⋊C₄	—	4	4	no	—
416	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
417	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
419	C₄⋊C₈	—	2	4	no	—
420	#103	—	2	4	no	—
421	#102	—	2	2	no	—
422	#102	—	2	2	no	—
423	#102	—	2	4	no	—
424	#102	—	4	2	no	—
425	C₂²⋊C₈	—	2	2	no	—
426	#95	—	2	1	no	—
427	#95	D₄×C₄₇	1	2	yes	—
428	C₂⁴	—	2	1	no	—
429	C₂⁴	—	2	1	no	—
430	C₂⁴	—	2	2	no	—
431	C₂⁴	—	2	4	no	—
432	C₂⁴	—	4	4	no	—
433	C₂⁴	—	4	4	no	—
434	C₂⁴	—	4	2	no	—
435	C₂⁴	—	4	2	no	—
436	C₂⁴	—	4	2	no	—
437	C₂⁴	—	4	2	no	—
438	C₂⁴	—	4	2	no	—
439	C₂⁴	—	8	2	no	—
440	C₂⁴	—	8	2	no	—
441	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	8	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	4	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	2	no	—
466	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
467	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
468	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
469	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	8	2	no	—
478	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
482	C₂²×C₄	#41×C₄₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
484	SD₁₆	—	4	4	no	—
485	SD₁₆	—	4	4	no	—
486	D₈	—	2	16	no	—
487	D₈	—	2	16	no	—
488	C₂×C₈	—	2	8	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	2	2	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	4	4	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	—	4	4	no	—
507	C₂²⋊C₄	—	4	2	no	—
508	C₂²⋊C₄	—	4	2	no	—
509	C₂²⋊C₄	—	8	2	no	—
510	C₂²⋊C₄	—	8	2	no	—
511	C₂²⋊C₄	—	8	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	4	no	—
520	C₂³	—	4	8	no	—
521	C₂³	—	4	8	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	4	1	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	8	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	2	no	—
529	C₂³	—	4	1	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	1	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	4	2	no	—
534	C₂³	—	4	2	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	8	4	no	—
537	C₂³	—	8	8	no	—
538	C₂³	—	8	8	no	—
539	C₂³	—	8	8	no	—
540	C₂³	—	8	8	no	—
541	C₂³	—	8	4	no	—
542	C₂³	—	8	4	no	—
543	C₂³	#121×C₄₇	1	2	yes	—
544	C₂³	#135×C₄₇	1	2	yes	—
545	C₂³	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
546	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
547	Q₈	—	4	1	no	—
548	Q₈	—	4	1	no	—
549	Q₈	—	4	2	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	4	no	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	2	4	no	—
555	D₄	—	2	4	no	—
556	D₄	—	4	8	no	—
557	D₄	—	4	8	no	—
558	D₄	—	4	2	no	—
559	D₄	—	4	2	no	—
560	D₄	—	4	2	no	—
561	D₄	#120×C₄₇	1	4	yes	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	8	8	no	—
580	C₂×C₄	#121×C₄₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
583	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
584	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
585	C₈	—	2	4	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	4	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	8	16	no	—
604	C₂²	—	8	16	no	—
605	C₂²	—	8	2	no	—
606	C₂²	—	8	2	no	—
607	C₂²	—	8	2	no	—
608	C₂²	—	8	2	no	—
609	C₂²	—	8	2	no	—
610	C₂²	—	8	2	no	—
611	C₂²	C₄₇×#463	1	4	yes	complemented
612	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
613	C₂²	D₄²×C₄₇	1	2	yes	—
614	C₂²	C₂×#135×C₄₇	1	1	yes	characteristic
615	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
616	C₂²	C₄₇×#576	1	2	yes	—
617	C₄	—	2	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	8	1	no	—
621	C₄	—	8	1	no	—
622	C₄	—	8	2	no	—
623	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
624	C₄	C₄₇×#534	1	1	yes	characteristic
625	C₄	C₄₇×#546	1	2	yes	—
626	C₂	—	2	4	no	—
627	C₂	—	4	4	no	—
628	C₂	—	8	2	no	—
629	C₂	—	8	4	no	—
630	C₂	C₄₇×#2410	1	1	yes	characteristic
631	C₂	C₂×C₄₇×#463	1	1	yes	characteristic
632	C₂	C₄₇×#3041	1	1	yes	characteristic
633	C₂	C₄₇×#3264	1	2	yes	—
634	C₂	C₄₇×#3280	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂³×D₈×C₄₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₄×C₂²⋊C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#463	C₂	semidirect
3. C₂×C₄₇×#450	C₂	semidirect
4. C₄₇×#2435	C₂	semidirect
5. D₄×C₂²⋊C₄×C₄₇	C₂	semidirect
6. C₄₇×#2054	C₂	semidirect
7. C₄₇×#2012	C₂	semidirect
8. C₄₇×#1892	C₂	semidirect
9. C₄₇×#1879	C₂	semidirect
10. C₄₇×#1850	C₂	semidirect
11. C₄₇×#1786	C₂	semidirect
14. C₂²×SD₁₆×C₄₇	C₂²	semidirect
15. C₂²×SD₁₆×C₄₇	C₂²	semidirect
21. C₂×#124×C₄₇	C₂²	semidirect
23. C₂×C₄×D₄×C₄₇	C₂²	semidirect
27. C₂×C₄⋊C₈×C₄₇	C₂²	semidirect
28. C₂×#103×C₄₇	C₂²	semidirect
29. C₂×#102×C₄₇	C₂²	semidirect
30. C₂×#102×C₄₇	C₂²	semidirect
31. C₂×#102×C₄₇	C₂²	semidirect
32. C₂×#102×C₄₇	C₂²	semidirect
35. C₄₇×#390	C₂²	semidirect
36. C₄₇×#390	C₂²	semidirect
43. C₄₇×#340	C₂²	semidirect
44. C₄₇×#336	C₂²	semidirect
270. #24355	C₄₇	direct (N × H)
294. C₂²×C₄₇	#463	semidirect
310. C₂×#463	C₂×C₄₇	semidirect
311. C₂×#450	C₂×C₄₇	semidirect
312. #2435	C₂×C₄₇	semidirect
313. D₄×C₂²⋊C₄	C₂×C₄₇	semidirect
314. #2054	C₂×C₄₇	semidirect
315. #2012	C₂×C₄₇	semidirect
316. #1892	C₂×C₄₇	semidirect
317. #1879	C₂×C₄₇	semidirect
318. #1850	C₂×C₄₇	semidirect
319. #1786	C₂×C₄₇	semidirect
331. C₂²×SD₁₆	C₂²×C₄₇	semidirect
332. C₂²×SD₁₆	C₂²×C₄₇	semidirect
338. C₂×#124	C₂²×C₄₇	semidirect
340. C₂×C₄×D₄	C₂²×C₄₇	semidirect
344. C₂×C₄⋊C₈	C₂²×C₄₇	semidirect
345. C₂×#103	C₂²×C₄₇	semidirect
346. C₂×#102	C₂²×C₄₇	semidirect
347. C₂×#102	C₂²×C₄₇	semidirect
348. C₂×#102	C₂²×C₄₇	semidirect
349. C₂×#102	C₂²×C₄₇	semidirect
352. #390	C₂²×C₄₇	semidirect
353. #390	C₂²×C₄₇	semidirect
360. #340	C₂²×C₄₇	semidirect
361. #336	C₂²×C₄₇	semidirect
362. C₄₇	#24355	direct (N × H)
611. C₂²	C₄₇×#463	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₄₇	2
C₂×C₄₇×#463	2
C₂×C₄₇×#450	2
C₄₇×#2435	2
D₄×C₂²⋊C₄×C₄₇	2
C₄₇×#2054	2
C₄₇×#2012	2
C₄₇×#1892	2
C₄₇×#1879	2
C₄₇×#1850	2
C₄₇×#1786	2
#24355	47

Subgroups — C47×#24355 #26064053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#24355 #26064053