Subgroups — C₄₇×#18834 #26058532

Order 12032.

1036 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3514	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3205	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3038	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1628	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1627	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1620	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1619	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1557	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1488	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₇	—	2	1	no	—
13	C₄₇×#559	C₂²	1	1	yes	characteristic, complemented
14	C₄₇×#556	C₂²	1	1	yes	characteristic, complemented
15	C₄₇×#555	—	2	1	no	—
16	C₄₇×#551	—	2	2	no	—
17	D₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
18	C₄₇×#548	—	2	1	no	—
19	C₄₇×#546	—	2	1	no	—
20	C₄₇×#546	—	2	1	no	—
21	D₄²×C₄₇	C₂²	1	1	yes	characteristic, complemented
22	C₄₇×#542	—	2	1	no	—
23	C₄₇×#538	—	2	2	no	—
24	C₄₇×#532	C₂²	1	1	yes	characteristic
25	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic, complemented
26	C₄₇×#529	—	2	1	no	—
27	C₄₇×#529	C₂²	1	1	yes	characteristic
28	C₂×#124×C₄₇	—	2	1	no	—
29	C₄₇×#518	—	2	1	no	—
30	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#496	—	2	1	no	—
32	C₄₇×#493	C₂²	1	2	yes	—
33	C₄₇×#490	—	2	1	no	—
34	C₄₇×#486	C₂²	1	2	yes	complemented
35	C₄₇×#460	—	2	1	no	—
36	C₄₇×#460	C₂²	1	2	yes	complemented
37	C₄₇×#460	C₂²	1	2	yes	complemented
38	C₄₇×#459	—	2	1	no	—
39	C₄₇×#459	—	2	2	no	—
40	C₄₇×#459	C₂²	1	2	yes	—
41	C₄₇×#459	C₂²	1	2	yes	—
42	C₄₇×#450	—	2	1	no	—
43	C₄₇×#447	—	2	2	no	—
44	C₄₇×#447	—	2	1	no	—
45	C₄₇×#432	C₂²	1	1	yes	characteristic, complemented
46	C₄₇×#431	C₂²	1	1	yes	characteristic, complemented
47	C₄₇×#423	C₂²	1	1	yes	characteristic, complemented
48	C₄₇×#418	—	2	1	no	—
49	C₄₇×#332	C₂²	1	2	yes	complemented
50	C₄₇×#331	C₂²	1	2	yes	complemented
51	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
52	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
53	C₄₇×D₄⋊C₈	C₂²	1	2	yes	complemented
54	C₄₇×D₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×#41×C₄₇	—	2	1	no	—
56	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₄₇	—	2	1	no	—
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂²×D₄×C₄₇	—	2	1	no	—
60	C₂²×D₄×C₄₇	—	2	1	no	—
61	C₂²×D₄×C₄₇	—	2	1	no	—
62	C₂²×D₄×C₄₇	—	2	1	no	—
63	AGL(1,ℤ/8)×C₄₇	—	4	4	no	—
64	C₂×SD₁₆×C₄₇	—	2	2	no	—
65	C₂×SD₁₆×C₄₇	—	2	2	no	—
66	C₂×D₈×C₄₇	—	2	2	no	—
67	C₂×D₈×C₄₇	—	2	4	no	—
68	C₂×D₈×C₄₇	—	2	2	no	—
69	C₂×M₁₆×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	1	no	—
71	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
72	#127×C₄₇	—	2	1	no	—
73	#127×C₄₇	—	2	1	no	—
74	#127×C₄₇	—	2	1	no	—
75	#127×C₄₇	C₂³	1	2	yes	—
76	#127×C₄₇	C₂³	1	1	yes	characteristic
77	#126×C₄₇	—	2	2	no	—
78	#126×C₄₇	—	2	2	no	—
79	#126×C₄₇	—	2	1	no	—
80	#125×C₄₇	D₄	1	1	yes	characteristic
81	#124×C₄₇	—	2	2	no	—
82	#124×C₄₇	—	2	2	no	—
83	#124×C₄₇	—	2	2	no	—
84	#124×C₄₇	—	2	1	no	—
85	#124×C₄₇	—	2	1	no	—
86	#124×C₄₇	—	2	1	no	—
87	#124×C₄₇	—	2	1	no	—
88	#124×C₄₇	D₄	1	1	yes	characteristic
89	#123×C₄₇	—	2	2	no	—
90	#123×C₄₇	—	2	2	no	—
91	#123×C₄₇	—	2	2	no	—
92	#123×C₄₇	—	2	1	no	—
93	C₂²⋊Q₈×C₄₇	—	2	2	no	—
94	C₂²⋊Q₈×C₄₇	—	2	1	no	—
95	C₂²⋊Q₈×C₄₇	—	2	1	no	—
96	C₂²⋊Q₈×C₄₇	—	2	2	no	—
97	C₂²⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
98	#121×C₄₇	—	2	2	no	—
99	#121×C₄₇	—	2	2	no	—
100	#121×C₄₇	—	2	1	no	—
101	#121×C₄₇	—	2	1	no	—
102	#121×C₄₇	—	2	1	no	—
103	#121×C₄₇	D₄	1	1	yes	characteristic
104	#121×C₄₇	D₄	1	2	yes	—
105	#120×C₄₇	—	2	2	no	—
106	#120×C₄₇	—	2	2	no	—
107	C₄×Q₈×C₄₇	—	2	1	no	—
108	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
109	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
110	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
111	C₄×D₄×C₄₇	—	2	2	no	—
112	C₄×D₄×C₄₇	—	2	2	no	—
113	C₄×D₄×C₄₇	—	2	1	no	—
114	C₄×D₄×C₄₇	—	2	1	no	—
115	C₄×D₄×C₄₇	—	2	1	no	—
116	C₄×D₄×C₄₇	—	2	1	no	—
117	C₄×D₄×C₄₇	—	2	1	no	—
118	C₄×D₄×C₄₇	—	2	1	no	—
119	C₄×D₄×C₄₇	—	2	1	no	—
120	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
121	C₄×D₄×C₄₇	D₄	1	2	yes	—
122	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
123	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
124	#117×C₄₇	—	2	2	no	—
125	#117×C₄₇	—	2	1	no	—
126	#117×C₄₇	—	2	1	no	—
127	#117×C₄₇	—	2	1	no	—
128	#117×C₄₇	D₄	1	1	yes	characteristic
129	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
131	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
132	C₂×C₄²×C₄₇	—	2	1	no	—
133	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
134	C₄⋊C₈×C₄₇	—	2	1	no	—
135	C₄⋊C₈×C₄₇	—	2	1	no	—
136	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
137	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
138	#102×C₄₇	—	2	2	no	—
139	#102×C₄₇	—	2	2	no	—
140	#102×C₄₇	—	2	4	no	—
141	#102×C₄₇	—	2	2	no	—
142	#102×C₄₇	—	2	2	no	—
143	C₂²⋊C₈×C₄₇	—	2	1	no	—
144	C₂²⋊C₈×C₄₇	—	2	1	no	—
145	#97×C₄₇	—	2	1	no	—
146	C₄×C₈×C₄₇	C₂³	1	2	yes	—
147	C₂⁴×C₄₇	—	2	2	no	—
148	C₂⁴×C₄₇	—	4	1	no	—
149	#41×C₄₇	—	2	2	no	—
150	#41×C₄₇	—	4	1	no	—
151	#41×C₄₇	—	8	2	no	—
152	C₂×Q₈×C₄₇	—	2	2	no	—
153	C₂×Q₈×C₄₇	—	2	1	no	—
154	C₂×Q₈×C₄₇	—	8	2	no	—
155	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₄₇	—	2	2	no	—
158	C₂×D₄×C₄₇	—	2	1	no	—
159	C₂×D₄×C₄₇	—	2	2	no	—
160	C₂×D₄×C₄₇	—	2	1	no	—
161	C₂×D₄×C₄₇	—	2	2	no	—
162	C₂×D₄×C₄₇	—	2	1	no	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	2	no	—
166	C₂×D₄×C₄₇	—	2	2	no	—
167	C₂×D₄×C₄₇	—	4	2	no	—
168	C₂×D₄×C₄₇	—	4	2	no	—
169	C₂×D₄×C₄₇	—	4	2	no	—
170	C₂×D₄×C₄₇	—	4	2	no	—
171	C₂×D₄×C₄₇	—	4	1	no	—
172	C₂×D₄×C₄₇	—	4	1	no	—
173	C₂×D₄×C₄₇	—	4	1	no	—
174	C₂×D₄×C₄₇	—	8	2	no	—
175	C₂×D₄×C₄₇	—	8	2	no	—
176	C₂×D₄×C₄₇	—	8	2	no	—
177	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
179	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₄₇	—	2	2	no	—
181	C₂²×C₄×C₄₇	—	2	2	no	—
182	C₂²×C₄×C₄₇	—	2	1	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	2	no	—
187	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
190	SD₁₆×C₄₇	—	4	4	no	—
191	SD₁₆×C₄₇	—	8	2	no	—
192	D₈×C₄₇	—	4	4	no	—
193	D₈×C₄₇	—	8	2	no	—
194	D₈×C₄₇	—	8	4	no	—
195	M₁₆×C₄₇	—	4	2	no	—
196	C₂×C₈×C₄₇	—	2	2	no	—
197	C₂×C₈×C₄₇	—	2	2	no	—
198	C₂×C₈×C₄₇	—	2	2	no	—
199	C₄⋊C₄×C₄₇	—	2	2	no	—
200	C₄⋊C₄×C₄₇	—	2	2	no	—
201	C₄⋊C₄×C₄₇	—	2	1	no	—
202	C₄⋊C₄×C₄₇	—	2	1	no	—
203	C₄⋊C₄×C₄₇	—	2	1	no	—
204	C₄⋊C₄×C₄₇	—	2	1	no	—
205	C₄⋊C₄×C₄₇	—	2	1	no	—
206	C₄⋊C₄×C₄₇	—	2	1	no	—
207	C₄⋊C₄×C₄₇	—	2	1	no	—
208	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
209	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
210	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
211	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
212	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
213	C₂²⋊C₄×C₄₇	—	2	2	no	—
214	C₂²⋊C₄×C₄₇	—	2	2	no	—
215	C₂²⋊C₄×C₄₇	—	2	2	no	—
216	C₂²⋊C₄×C₄₇	—	2	2	no	—
217	C₂²⋊C₄×C₄₇	—	2	2	no	—
218	C₂²⋊C₄×C₄₇	—	2	2	no	—
219	C₂²⋊C₄×C₄₇	—	2	2	no	—
220	C₂²⋊C₄×C₄₇	—	2	1	no	—
221	C₂²⋊C₄×C₄₇	—	2	1	no	—
222	C₂²⋊C₄×C₄₇	—	2	1	no	—
223	C₂²⋊C₄×C₄₇	—	2	1	no	—
224	C₂²⋊C₄×C₄₇	—	2	1	no	—
225	C₂²⋊C₄×C₄₇	—	2	1	no	—
226	C₂²⋊C₄×C₄₇	—	2	2	no	—
227	C₂²⋊C₄×C₄₇	—	2	2	no	—
228	C₂²⋊C₄×C₄₇	—	2	1	no	—
229	C₂²⋊C₄×C₄₇	—	2	1	no	—
230	C₄²×C₄₇	—	2	2	no	—
231	C₄²×C₄₇	—	2	1	no	—
232	C₄²×C₄₇	—	2	1	no	—
233	C₄²×C₄₇	—	2	1	no	—
234	C₄²×C₄₇	—	2	1	no	—
235	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
239	C₂³×C₄₇	—	2	1	no	—
240	C₂³×C₄₇	—	2	1	no	—
241	C₂³×C₄₇	—	2	2	no	—
242	C₂³×C₄₇	—	4	4	no	—
243	C₂³×C₄₇	—	4	2	no	—
244	C₂³×C₄₇	—	8	2	no	—
245	C₂³×C₄₇	—	8	2	no	—
246	C₂³×C₄₇	—	8	2	no	—
247	C₂³×C₄₇	—	8	2	no	—
248	C₂³×C₄₇	—	8	2	no	—
249	C₂³×C₄₇	#120	1	1	yes	characteristic
250	Q₈×C₄₇	—	2	1	no	—
251	Q₈×C₄₇	—	2	1	no	—
252	Q₈×C₄₇	—	2	2	no	—
253	Q₈×C₄₇	—	4	2	no	—
254	Q₈×C₄₇	—	8	2	no	—
255	D₄×C₄₇	—	2	1	no	—
256	D₄×C₄₇	—	2	2	no	—
257	D₄×C₄₇	—	4	2	no	—
258	D₄×C₄₇	—	4	4	no	—
259	D₄×C₄₇	—	4	2	no	—
260	D₄×C₄₇	—	4	4	no	—
261	D₄×C₄₇	—	4	1	no	—
262	D₄×C₄₇	—	4	2	no	—
263	D₄×C₄₇	—	4	1	no	—
264	D₄×C₄₇	—	4	2	no	—
265	D₄×C₄₇	—	4	2	no	—
266	D₄×C₄₇	—	4	2	no	—
267	D₄×C₄₇	—	4	2	no	—
268	D₄×C₄₇	—	8	2	no	—
269	D₄×C₄₇	—	8	2	no	—
270	D₄×C₄₇	—	8	2	no	—
271	D₄×C₄₇	—	8	1	no	—
272	D₄×C₄₇	—	8	2	no	—
273	D₄×C₄₇	—	8	2	no	—
274	D₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₄×C₄₇	—	2	1	no	—
276	C₂×C₄×C₄₇	—	2	1	no	—
277	C₂×C₄×C₄₇	—	2	2	no	—
278	C₂×C₄×C₄₇	—	2	1	no	—
279	C₂×C₄×C₄₇	—	2	1	no	—
280	C₂×C₄×C₄₇	—	2	1	no	—
281	C₂×C₄×C₄₇	—	2	1	no	—
282	C₂×C₄×C₄₇	—	2	1	no	—
283	C₂×C₄×C₄₇	—	2	1	no	—
284	C₂×C₄×C₄₇	—	2	2	no	—
285	C₂×C₄×C₄₇	—	4	1	no	—
286	C₂×C₄×C₄₇	—	4	2	no	—
287	C₂×C₄×C₄₇	—	4	1	no	—
288	C₂×C₄×C₄₇	—	4	1	no	—
289	C₂×C₄×C₄₇	—	4	2	no	—
290	C₂×C₄×C₄₇	—	8	1	no	—
291	C₂×C₄×C₄₇	—	8	2	no	—
292	C₂×C₄×C₄₇	—	8	2	no	—
293	C₂×C₄×C₄₇	—	8	2	no	—
294	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
295	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
296	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
297	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
300	C₈×C₄₇	—	4	2	no	—
301	C₈×C₄₇	—	4	2	no	—
302	C₈×C₄₇	—	4	2	no	—
303	#18834	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂²×C₄₇	—	2	1	no	—
305	C₂²×C₄₇	—	2	2	no	—
306	C₂²×C₄₇	—	2	1	no	—
307	C₂²×C₄₇	—	2	1	no	—
308	C₂²×C₄₇	—	4	1	no	—
309	C₂²×C₄₇	—	4	1	no	—
310	C₂²×C₄₇	—	4	1	no	—
311	C₂²×C₄₇	—	4	2	no	—
312	C₂²×C₄₇	—	4	2	no	—
313	C₂²×C₄₇	—	4	1	no	—
314	C₂²×C₄₇	—	4	2	no	—
315	C₂²×C₄₇	—	4	1	no	—
316	C₂²×C₄₇	—	8	2	no	—
317	C₂²×C₄₇	—	8	2	no	—
318	C₂²×C₄₇	—	8	2	no	—
319	C₂²×C₄₇	—	16	4	no	—
320	C₂²×C₄₇	—	16	4	no	—
321	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
322	C₄×C₄₇	—	2	1	no	—
323	C₄×C₄₇	—	2	1	no	—
324	C₄×C₄₇	—	2	1	no	—
325	C₄×C₄₇	—	2	1	no	—
326	C₄×C₄₇	—	4	1	no	—
327	C₄×C₄₇	—	4	1	no	—
328	C₄×C₄₇	—	4	1	no	—
329	C₄×C₄₇	—	4	2	no	—
330	C₄×C₄₇	—	4	1	no	—
331	C₄×C₄₇	—	4	1	no	—
332	C₄×C₄₇	—	4	1	no	—
333	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₄×C₄₇	#575	1	1	yes	characteristic
335	C₄×C₄₇	#576	1	2	yes	—
336	#3514	C₂×C₄₇	1	1	yes	characteristic, complemented
337	#3217	C₂×C₄₇	1	1	yes	characteristic, complemented
338	#3205	C₂×C₄₇	1	1	yes	characteristic, complemented
339	#3038	C₂×C₄₇	1	1	yes	characteristic, complemented
340	#1628	C₂×C₄₇	1	2	yes	complemented
341	#1627	C₂×C₄₇	1	2	yes	complemented
342	#1620	C₂×C₄₇	1	2	yes	complemented
343	#1619	C₂×C₄₇	1	2	yes	complemented
344	#1557	C₂×C₄₇	1	1	yes	characteristic, complemented
345	#1495	C₂×C₄₇	1	1	yes	characteristic, complemented
346	#1488	C₂×C₄₇	1	1	yes	characteristic, complemented
347	C₂×C₄₇	—	4	1	no	—
348	C₂×C₄₇	—	4	2	no	—
349	C₂×C₄₇	—	8	2	no	—
350	C₂×C₄₇	—	8	1	no	—
351	C₂×C₄₇	—	16	2	no	—
352	C₂×C₄₇	#3006	1	1	yes	characteristic
353	C₂×C₄₇	#3008	1	1	yes	characteristic
354	C₂×C₄₇	#3021	1	1	yes	characteristic
355	C₂×AGL(1,ℤ/8)	—	2	1	no	—
356	#559	C₂²×C₄₇	1	1	yes	characteristic, complemented
357	#556	C₂²×C₄₇	1	1	yes	characteristic, complemented
358	#555	—	2	1	no	—
359	#551	—	2	2	no	—
360	D₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic, complemented
361	#548	—	2	1	no	—
362	#546	—	2	1	no	—
363	#546	—	2	1	no	—
364	D₄²	C₂²×C₄₇	1	1	yes	characteristic, complemented
365	#542	—	2	1	no	—
366	#538	—	2	2	no	—
367	#532	C₂²×C₄₇	1	1	yes	characteristic
368	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic, complemented
369	#529	—	2	1	no	—
370	#529	C₂²×C₄₇	1	1	yes	characteristic
371	C₂×#124	—	2	1	no	—
372	#518	—	2	1	no	—
373	C₄×#41	C₂²×C₄₇	1	1	yes	characteristic
374	#496	—	2	1	no	—
375	#493	C₂²×C₄₇	1	2	yes	—
376	#490	—	2	1	no	—
377	#486	C₂²×C₄₇	1	2	yes	complemented
378	#460	—	2	1	no	—
379	#460	C₂²×C₄₇	1	2	yes	complemented
380	#460	C₂²×C₄₇	1	2	yes	complemented
381	#459	—	2	2	no	—
382	#459	—	2	1	no	—
383	#459	C₂²×C₄₇	1	2	yes	—
384	#459	C₂²×C₄₇	1	2	yes	—
385	#450	—	2	1	no	—
386	#447	—	2	2	no	—
387	#447	—	2	1	no	—
388	#432	C₂²×C₄₇	1	1	yes	characteristic, complemented
389	#431	C₂²×C₄₇	1	1	yes	characteristic, complemented
390	#423	C₂²×C₄₇	1	1	yes	characteristic, complemented
391	#418	—	2	1	no	—
392	#332	C₂²×C₄₇	1	2	yes	complemented
393	#331	C₂²×C₄₇	1	2	yes	complemented
394	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
395	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
396	D₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
397	D₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
398	C₄₇	#18834	1	1	yes	characteristic, direct product factor, Sylow
399	C₂×#41	—	2	1	no	—
400	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
401	C₂²×Q₈	—	2	1	no	—
402	C₂²×D₄	—	2	1	no	—
403	C₂²×D₄	—	2	1	no	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	—	2	1	no	—
406	C₂²×D₄	—	2	1	no	—
407	AGL(1,ℤ/8)	—	4	4	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂×D₈	—	2	2	no	—
411	C₂×D₈	—	2	4	no	—
412	C₂×D₈	—	2	2	no	—
413	C₂×M₁₆	—	2	1	no	—
414	C₄⋊Q₈	—	2	1	no	—
415	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
416	#127	—	2	1	no	—
417	#127	—	2	1	no	—
418	#127	—	2	1	no	—
419	#127	C₂³×C₄₇	1	2	yes	—
420	#127	C₂³×C₄₇	1	1	yes	characteristic
421	#126	—	2	2	no	—
422	#126	—	2	2	no	—
423	#126	—	2	1	no	—
424	#125	D₄×C₄₇	1	1	yes	characteristic
425	#124	—	2	2	no	—
426	#124	—	2	2	no	—
427	#124	—	2	2	no	—
428	#124	—	2	1	no	—
429	#124	—	2	1	no	—
430	#124	—	2	1	no	—
431	#124	—	2	1	no	—
432	#124	D₄×C₄₇	1	1	yes	characteristic
433	#123	—	2	2	no	—
434	#123	—	2	2	no	—
435	#123	—	2	2	no	—
436	#123	—	2	1	no	—
437	C₂²⋊Q₈	—	2	2	no	—
438	C₂²⋊Q₈	—	2	1	no	—
439	C₂²⋊Q₈	—	2	1	no	—
440	C₂²⋊Q₈	—	2	2	no	—
441	C₂²⋊Q₈	D₄×C₄₇	1	1	yes	characteristic
442	#121	—	2	2	no	—
443	#121	—	2	2	no	—
444	#121	—	2	1	no	—
445	#121	—	2	1	no	—
446	#121	—	2	1	no	—
447	#121	D₄×C₄₇	1	1	yes	characteristic
448	#121	D₄×C₄₇	1	2	yes	—
449	#120	—	2	2	no	—
450	#120	—	2	2	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
453	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
454	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	—	2	1	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	—	2	1	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	D₄×C₄₇	1	1	yes	characteristic
465	C₄×D₄	D₄×C₄₇	1	2	yes	—
466	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
467	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
468	#117	—	2	2	no	—
469	#117	—	2	1	no	—
470	#117	—	2	1	no	—
471	#117	—	2	1	no	—
472	#117	D₄×C₄₇	1	1	yes	characteristic
473	C₂×C₂²⋊C₄	—	2	2	no	—
474	C₂×C₂²⋊C₄	—	2	2	no	—
475	C₂×C₂²⋊C₄	—	2	2	no	—
476	C₂×C₄²	—	2	1	no	—
477	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
478	C₄⋊C₈	—	2	1	no	—
479	C₄⋊C₈	—	2	1	no	—
480	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
481	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
482	#102	—	2	2	no	—
483	#102	—	2	2	no	—
484	#102	—	2	4	no	—
485	#102	—	2	2	no	—
486	#102	—	2	2	no	—
487	C₂²⋊C₈	—	2	1	no	—
488	C₂²⋊C₈	—	2	1	no	—
489	#97	—	2	1	no	—
490	C₄×C₈	C₂³×C₄₇	1	2	yes	—
491	C₂⁴	—	2	2	no	—
492	C₂⁴	—	4	1	no	—
493	#41	—	2	2	no	—
494	#41	—	4	1	no	—
495	#41	—	8	2	no	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	8	2	no	—
499	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
500	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	2	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	4	2	no	—
512	C₂×D₄	—	4	2	no	—
513	C₂×D₄	—	4	2	no	—
514	C₂×D₄	—	4	2	no	—
515	C₂×D₄	—	4	1	no	—
516	C₂×D₄	—	4	1	no	—
517	C₂×D₄	—	4	1	no	—
518	C₂×D₄	—	8	2	no	—
519	C₂×D₄	—	8	2	no	—
520	C₂×D₄	—	8	2	no	—
521	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
522	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
523	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
534	SD₁₆	—	4	4	no	—
535	SD₁₆	—	8	2	no	—
536	D₈	—	4	4	no	—
537	D₈	—	8	2	no	—
538	D₈	—	8	4	no	—
539	M₁₆	—	4	2	no	—
540	C₂×C₈	—	2	2	no	—
541	C₂×C₈	—	2	2	no	—
542	C₂×C₈	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
553	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
554	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
555	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
556	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	—	2	2	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	1	no	—
567	C₂²⋊C₄	—	2	1	no	—
568	C₂²⋊C₄	—	2	1	no	—
569	C₂²⋊C₄	—	2	1	no	—
570	C₂²⋊C₄	—	2	2	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	1	no	—
573	C₂²⋊C₄	—	2	1	no	—
574	C₄²	—	2	2	no	—
575	C₄²	—	2	1	no	—
576	C₄²	—	2	1	no	—
577	C₄²	—	2	1	no	—
578	C₄²	—	2	1	no	—
579	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
580	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
581	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
582	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
583	C₂³	—	2	1	no	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	4	4	no	—
587	C₂³	—	4	2	no	—
588	C₂³	—	8	2	no	—
589	C₂³	—	8	2	no	—
590	C₂³	—	8	2	no	—
591	C₂³	—	8	2	no	—
592	C₂³	—	8	2	no	—
593	C₂³	#120×C₄₇	1	1	yes	characteristic
594	Q₈	—	2	1	no	—
595	Q₈	—	2	1	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	4	2	no	—
598	Q₈	—	8	2	no	—
599	D₄	—	2	1	no	—
600	D₄	—	2	2	no	—
601	D₄	—	4	2	no	—
602	D₄	—	4	4	no	—
603	D₄	—	4	2	no	—
604	D₄	—	4	4	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	2	no	—
607	D₄	—	4	1	no	—
608	D₄	—	4	2	no	—
609	D₄	—	4	2	no	—
610	D₄	—	4	2	no	—
611	D₄	—	4	2	no	—
612	D₄	—	8	2	no	—
613	D₄	—	8	2	no	—
614	D₄	—	8	2	no	—
615	D₄	—	8	1	no	—
616	D₄	—	8	2	no	—
617	D₄	—	8	2	no	—
618	D₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	2	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	8	1	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	2	no	—
637	C₂×C₄	—	8	2	no	—
638	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
639	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
640	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
641	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
642	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
643	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
644	C₈	—	4	2	no	—
645	C₈	—	4	2	no	—
646	C₈	—	4	2	no	—
647	C₂²	—	2	1	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	1	no	—
650	C₂²	—	2	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	4	1	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	1	no	—
659	C₂²	—	8	2	no	—
660	C₂²	—	8	2	no	—
661	C₂²	—	8	2	no	—
662	C₂²	—	16	4	no	—
663	C₂²	—	16	4	no	—
664	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
665	C₄	—	2	1	no	—
666	C₄	—	2	1	no	—
667	C₄	—	2	1	no	—
668	C₄	—	2	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	2	no	—
673	C₄	—	4	1	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	1	no	—
676	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
677	C₄	C₄₇×#575	1	1	yes	characteristic
678	C₄	C₄₇×#576	1	2	yes	—
679	C₂	—	4	1	no	—
680	C₂	—	4	2	no	—
681	C₂	—	8	1	no	—
682	C₂	—	8	2	no	—
683	C₂	—	16	2	no	—
684	C₂	C₄₇×#3006	1	1	yes	characteristic
685	C₂	C₄₇×#3008	1	1	yes	characteristic
686	C₂	C₄₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3514	C₂	semidirect
2. C₄₇×#3217	C₂	semidirect
3. C₄₇×#3205	C₂	semidirect
4. C₄₇×#3038	C₂	semidirect
5. C₄₇×#1628	C₂	semidirect
6. C₄₇×#1627	C₂	semidirect
7. C₄₇×#1620	C₂	semidirect
8. C₄₇×#1619	C₂	semidirect
9. C₄₇×#1557	C₂	semidirect
10. C₄₇×#1495	C₂	semidirect
11. C₄₇×#1488	C₂	semidirect
13. C₄₇×#559	C₂²	semidirect
14. C₄₇×#556	C₂²	semidirect
17. D₄×Q₈×C₄₇	C₂²	semidirect
21. D₄²×C₄₇	C₂²	semidirect
25. C₂×#127×C₄₇	C₂²	semidirect
34. C₄₇×#486	C₂²	semidirect
36. C₄₇×#460	C₂²	semidirect
37. C₄₇×#460	C₂²	semidirect
45. C₄₇×#432	C₂²	semidirect
46. C₄₇×#431	C₂²	semidirect
47. C₄₇×#423	C₂²	semidirect
49. C₄₇×#332	C₂²	semidirect
50. C₄₇×#331	C₂²	semidirect
51. C₄₇×Q₈⋊C₈	C₂²	semidirect
52. C₄₇×Q₈⋊C₈	C₂²	semidirect
53. C₄₇×D₄⋊C₈	C₂²	semidirect
54. C₄₇×D₄⋊C₈	C₂²	semidirect
303. #18834	C₄₇	direct (N × H)
336. #3514	C₂×C₄₇	semidirect
337. #3217	C₂×C₄₇	semidirect
338. #3205	C₂×C₄₇	semidirect
339. #3038	C₂×C₄₇	semidirect
340. #1628	C₂×C₄₇	semidirect
341. #1627	C₂×C₄₇	semidirect
342. #1620	C₂×C₄₇	semidirect
343. #1619	C₂×C₄₇	semidirect
344. #1557	C₂×C₄₇	semidirect
345. #1495	C₂×C₄₇	semidirect
346. #1488	C₂×C₄₇	semidirect
356. #559	C₂²×C₄₇	semidirect
357. #556	C₂²×C₄₇	semidirect
360. D₄×Q₈	C₂²×C₄₇	semidirect
364. D₄²	C₂²×C₄₇	semidirect
368. C₂×#127	C₂²×C₄₇	semidirect
377. #486	C₂²×C₄₇	semidirect
379. #460	C₂²×C₄₇	semidirect
380. #460	C₂²×C₄₇	semidirect
388. #432	C₂²×C₄₇	semidirect
389. #431	C₂²×C₄₇	semidirect
390. #423	C₂²×C₄₇	semidirect
392. #332	C₂²×C₄₇	semidirect
393. #331	C₂²×C₄₇	semidirect
394. Q₈⋊C₈	C₂²×C₄₇	semidirect
395. Q₈⋊C₈	C₂²×C₄₇	semidirect
396. D₄⋊C₈	C₂²×C₄₇	semidirect
397. D₄⋊C₈	C₂²×C₄₇	semidirect
398. C₄₇	#18834	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#3514	2
C₄₇×#3217	2
C₄₇×#3205	2
C₄₇×#3038	2
C₄₇×#1628	2
C₄₇×#1627	2
C₄₇×#1620	2
C₄₇×#1619	2
C₄₇×#1557	2
C₄₇×#1495	2
C₄₇×#1488	2
#18834	47

Subgroups — C47×#18834 #26058532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18834 #26058532