Subgroups — #26235267

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246345	C₂	1	2	yes	maximal
2	#20244158	C₂	1	2	yes	maximal
3	#20244158	C₂	1	2	yes	maximal
4	#20244147	C₂	1	2	yes	maximal
5	#20243522	C₂	1	2	yes	maximal
6	#20243519	C₂	1	1	yes	maximal, characteristic
7	#20243285	C₂	1	2	yes	maximal
8	C₄×#13093958	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#446	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093956	C₂²	1	2	yes	—
13	#15689030	—	2	2	no	—
14	#15689009	—	2	2	no	—
15	#15689009	—	2	2	no	—
16	#15688993	C₂²	1	2	yes	—
17	#15688989	C₂²	1	2	yes	—
18	#15688989	C₂²	1	2	yes	—
19	C₄⋊C₄×Dic₄₇	—	2	2	no	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	#15688886	C₂²	1	2	yes	—
24	#15688886	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	1	yes	characteristic
26	#15688886	C₂²	1	1	yes	characteristic
27	#15688886	C₂²	1	2	yes	—
28	#15688886	C₂²	1	2	yes	—
29	#15688886	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#446	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	#13093958	—	2	4	no	—
46	#13093958	—	2	4	no	—
47	#13093956	—	2	4	no	—
48	C₄₇⋊#95	—	2	2	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₄₇⋊#95	—	2	2	no	—
55	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₄×Q₈×C₄₇	Q₈	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
65	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
67	#97×C₄₇	C₂×C₄	1	8	yes	—
68	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
69	C₂×Q₈×C₄₇	—	4	2	no	—
70	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
88	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	4	4	no	—
101	C₄⋊Dic₄₇	—	8	2	no	—
102	C₄⋊Dic₄₇	#41	1	4	yes	—
103	C₄×Dic₄₇	—	4	4	no	—
104	C₄×Dic₄₇	—	8	2	no	—
105	C₄×Dic₄₇	—	8	2	no	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₈×C₄₇	—	2	4	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	—	2	2	no	—
138	#22744	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#548	1	2	yes	—
141	C₂²×C₄₇	#575	1	2	yes	—
142	C₂²×C₄₇	#576	1	1	yes	characteristic
143	C₂²×C₄₇	#578	1	1	yes	characteristic
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
149	C₄×C₄₇	#519	1	1	yes	characteristic
150	C₄×C₄₇	#557	1	2	yes	—
151	Dic₄₇	—	4	4	no	—
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	Dic₄₇	—	8	2	no	—
155	C₂×#446	D₄₇	1	1	yes	characteristic
156	#2355	—	47	2	no	—
157	#1949	—	47	2	no	—
158	#1949	—	47	2	no	—
159	#1947	—	47	2	no	—
160	#1836	—	47	2	no	—
161	#1835	—	47	1	no	—
162	#1776	—	47	2	no	—
163	C₄×#107	—	47	1	no	—
164	C₂×C₄₇	#2351	1	1	yes	characteristic
165	C₂×C₄₇	#2911	1	1	yes	characteristic
166	C₂×C₄₇	#2913	1	1	yes	characteristic
167	C₂×C₄₇	#3370	1	2	yes	—
168	C₂×C₄₇	#3377	1	2	yes	—
169	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
170	#446	Dic₄₇	1	8	yes	complemented
171	C₂×#107	—	47	2	no	—
172	C₂×#107	—	47	2	no	—
173	C₂×#106	—	47	2	no	—
174	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
175	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#97	C₂×D₄₇	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#391	—	94	2	no	—
179	#387	—	94	2	no	—
180	#387	—	94	2	no	—
181	#384	—	47	2	no	—
182	#382	—	47	2	no	—
183	#382	—	47	2	no	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	47	2	no	—
187	C₄×C₄⋊C₄	—	94	2	no	—
188	#340	—	47	2	no	—
189	#340	—	47	1	no	—
190	#340	—	47	1	no	—
191	#340	—	47	1	no	—
192	#340	—	47	2	no	—
193	#340	—	47	2	no	—
194	#340	—	47	1	no	—
195	C₄₇	#22744	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	C₄×Q₈	Dic₉₄	1	4	yes	—
201	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	47	2	no	—
204	C₂×C₄⋊C₄	—	47	2	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
209	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
210	C₂×C₄²	—	94	2	no	—
211	C₂×C₄²	—	94	2	no	—
212	C₂×C₄²	—	94	2	no	—
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
215	#107	—	94	4	no	—
216	#107	—	94	4	no	—
217	#106	—	94	4	no	—
218	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
219	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
220	#97	C₂×Dic₄₇	1	8	yes	—
221	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
222	#95	—	94	2	no	—
223	#95	—	94	2	no	—
224	#95	—	94	2	no	—
225	#95	—	94	2	no	—
226	#95	—	94	2	no	—
227	#95	—	94	2	no	—
228	#95	—	94	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	2	no	—
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
238	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
245	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
246	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
247	C₄⋊C₄	—	47	4	no	—
248	C₄⋊C₄	—	94	2	no	—
249	C₄⋊C₄	—	94	2	no	—
250	C₄⋊C₄	—	94	2	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	376	2	no	—
256	C₄⋊C₄	—	188	4	no	—
257	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
258	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
259	C₄²	—	376	2	no	—
260	C₄²	—	376	2	no	—
261	C₄²	—	188	4	no	—
262	C₄²	C₂×Dic₉₄	1	2	yes	—
263	C₄²	C₂×Dic₉₄	1	4	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₄²	C₂²×Dic₄₇	1	4	yes	—
266	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
289	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
291	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
293	C₂×C₄	#41×D₄₇	1	2	yes	—
294	C₂×C₄	#13094072	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
299	C₂²	#15689785	1	2	yes	—
300	C₂²	#15689923	1	2	yes	—
301	C₂²	#15689928	1	1	yes	characteristic
302	C₂²	#15689940	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	376	2	no	—
308	C₄	—	376	2	no	—
309	C₄	—	376	2	no	—
310	C₄	—	188	4	no	—
311	C₄	#41×Dic₄₇	1	1	yes	characteristic
312	C₄	#15689601	1	1	yes	characteristic
313	C₄	#15689849	1	2	yes	—
314	C₂	#20246309	1	1	yes	characteristic
315	C₂	#20251978	1	1	yes	characteristic
316	C₂	#20251988	1	1	yes	characteristic
317	C₂	#20256143	1	2	yes	—
318	C₂	#20256232	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#446	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#446	D₄₇
hypocenter	C₄₇	#22744
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
31. C₄₇×#446	C₄	semidirect
170. #446	Dic₄₇	semidirect
195. C₄₇	#22744	semidirect

Maximal subgroups

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

subgroup	index
#20246345	2
#20244158	2
#20244147	2
#20243522	2
#20243519	2
#20243285	2
C₄×#13093958	2
C₂×C₄₇×#446	2
#22744	47