Subgroups — #26198919

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20253427	C₂	1	2	yes	maximal
3	#20242916	C₂	1	2	yes	maximal, complemented
4	#20242906	C₂	1	2	yes	maximal, complemented
5	#20242894	C₂	1	4	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic
7	#20242046	C₂	1	2	yes	maximal
8	C₂×C₄₇×#446	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	C₂²	1	1	yes	characteristic
11	#15689528	C₂²	1	1	yes	characteristic
12	#15689525	—	2	1	no	—
13	#15689525	—	2	1	no	—
14	#15689521	C₂²	1	2	yes	—
15	#15689458	—	2	2	no	—
16	#15689418	C₂²	1	4	yes	—
17	#15689333	C₂²	1	4	yes	—
18	#15689279	—	2	2	no	—
19	C₄×#10625337	—	2	4	no	—
20	#15689176	C₂²	1	2	yes	—
21	#15689161	—	2	1	no	—
22	#15689161	—	2	1	no	—
23	#15688876	C₂²	1	2	yes	—
24	#15688874	C₂²	1	2	yes	—
25	#15688871	C₂²	1	4	yes	—
26	#15688868	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#446	—	2	1	no	—
29	C₄₇×#446	—	2	1	no	—
30	C₄₇×#446	C₂²	1	2	yes	—
31	C₄₇×#446	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#10625337	—	2	4	no	—
37	#13094024	C₂³	1	2	yes	—
38	#13094024	C₂³	1	2	yes	—
39	#13094016	—	2	2	no	—
40	#13094003	—	2	2	no	—
41	#13094003	—	2	2	no	—
42	C₄×Dic₉₄	—	2	2	no	—
43	C₄×Dic₉₄	—	2	2	no	—
44	#13093983	—	2	2	no	—
45	#13093958	—	2	1	no	—
46	#13093958	—	2	1	no	—
47	#13093958	—	2	1	no	—
48	#13093958	—	2	1	no	—
49	#13093957	—	2	4	no	—
50	#13093956	—	2	2	no	—
51	#13093956	—	2	2	no	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
53	#13093948	—	2	4	no	—
54	#13093947	—	2	4	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
56	C₄×C₄₇⋊C₈	—	2	2	no	—
57	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₄₇	—	2	1	no	—
62	C₄×Q₈×C₄₇	—	2	1	no	—
63	C₄×Q₈×C₄₇	D₄	1	2	yes	—
64	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	—	2	1	no	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	2	no	—
70	C₄⋊C₈×C₄₇	—	2	2	no	—
71	C₄⋊C₈×C₄₇	—	2	1	no	—
72	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
73	#97×C₄₇	—	2	2	no	—
74	#97×C₄₇	—	2	2	no	—
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	2	1	no	—
80	C₂×Q₈×C₄₇	—	4	2	no	—
81	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	4	2	no	—
92	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
93	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	1	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
101	C₄²×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	#41	1	1	yes	characteristic
106	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₄₇	—	4	2	no	—
108	#10625337	—	4	8	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	Dic₄₇⋊C₄	—	4	2	no	—
114	Dic₄₇⋊C₄	—	4	2	no	—
115	C₄×Dic₄₇	—	4	2	no	—
116	C₂×C₄₇⋊C₈	—	2	4	no	—
117	C₂³×C₄₇	#121	1	1	yes	characteristic
118	Q₈×C₄₇	—	2	4	no	—
119	Q₈×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₄₇	#137	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	Dic₉₄	—	4	4	no	—
137	Dic₉₄	—	8	2	no	—
138	C₈×C₄₇	—	2	1	no	—
139	C₈×C₄₇	—	2	1	no	—
140	C₈×C₄₇	—	4	2	no	—
141	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
142	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
143	C₄₇⋊C₈	—	4	4	no	—
144	#19150	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	#462	1	2	yes	—
148	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	#462	1	2	yes	—
155	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
157	C₄×C₄₇	#575	1	2	yes	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	—	8	2	no	—
160	#3160	—	47	1	no	—
161	#3104	—	47	2	no	—
162	C₂×#446	D₄₇	1	1	yes	characteristic
163	#1696	—	47	2	no	—
164	#1694	—	47	2	no	—
165	#1692	—	47	4	no	—
166	#1564	—	47	1	no	—
167	#1542	—	47	2	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
170	C₂×C₄₇	#3054	1	1	yes	characteristic
171	C₂×C₄₇	#3067	1	1	yes	characteristic
172	#533	—	47	2	no	—
173	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
174	#500	—	47	1	no	—
175	#500	—	47	1	no	—
176	#499	—	94	1	no	—
177	#499	—	94	1	no	—
178	#498	—	47	2	no	—
179	#484	—	94	2	no	—
180	#477	—	47	4	no	—
181	#462	—	47	4	no	—
182	#451	—	94	2	no	—
183	#446	—	2	1	no	—
184	#446	—	2	1	no	—
185	#446	C₂×D₄₇	1	2	yes	—
186	#446	C₂×D₄₇	1	2	yes	—
187	C₄×Q₁₆	—	94	4	no	—
188	#431	—	47	2	no	—
189	#427	—	94	1	no	—
190	#427	—	94	1	no	—
191	C₂×C₄⋊C₈	—	2	1	no	—
192	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#97	—	2	1	no	—
194	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
195	#335	—	47	2	no	—
196	#334	—	47	2	no	—
197	#333	—	47	4	no	—
198	#332	—	47	4	no	—
199	C₄₇	#19150	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
201	C₂×Q₁₆	—	94	4	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	47	2	no	—
206	C₄⋊Q₈	—	47	2	no	—
207	#125	—	94	2	no	—
208	C₂²⋊Q₈	—	94	2	no	—
209	C₂²⋊Q₈	—	94	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	94	2	no	—
213	C₄×Q₈	—	94	2	no	—
214	C₄×Q₈	D₁₈₈	1	2	yes	—
215	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
216	#117	—	94	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	#107	—	94	1	no	—
222	#107	—	94	1	no	—
223	#107	—	94	1	no	—
224	#107	—	94	1	no	—
225	#107	—	94	4	no	—
226	#106	—	94	2	no	—
227	#106	—	94	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	47	4	no	—
232	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
233	#103	—	94	4	no	—
234	#103	—	94	4	no	—
235	C₂²⋊C₈	—	94	2	no	—
236	#97	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	—	94	2	no	—
239	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	94	2	no	—
246	C₂×Q₈	—	94	2	no	—
247	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂²×C₄	#10625350	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	Q₁₆	—	188	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	94	4	no	—
258	C₂×C₈	#10625337	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₄⋊C₄	—	94	2	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
273	C₂²⋊C₄	—	188	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	188	2	no	—
277	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
278	C₄²	#10625350	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₄²	C₂³×D₄₇	1	1	yes	characteristic
281	C₂³	#13093998	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	376	2	no	—
285	Q₈	—	188	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	#13093998	1	1	yes	characteristic
295	C₂×C₄	#13093998	1	1	yes	characteristic
296	C₂×C₄	#13093998	1	1	yes	characteristic
297	C₂×C₄	C₂×#10625337	1	2	yes	—
298	C₂×C₄	#13094054	1	2	yes	—
299	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
301	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	2	no	—
305	C₈	—	188	4	no	—
306	C₈	C₂×#10625337	1	2	yes	—
307	C₈	C₂×#10625337	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#15689333	1	2	yes	—
311	C₂²	C₂×#13093998	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	376	2	no	—
318	C₄	—	376	2	no	—
319	C₄	#15689333	1	2	yes	—
320	C₄	C₂²×#10625337	1	1	yes	characteristic
321	C₄	C₂×#13094054	1	1	yes	characteristic
322	C₄	#15689923	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	C₂×#15689333	1	1	yes	characteristic
325	C₂	#20252997	1	1	yes	characteristic
326	C₂	#20253108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#446	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂³×C₄₇	#121
hypercenter	C₂×#446	D₄₇
hypocenter	C₄₇	#19150
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #20242916	C₂	semidirect
4. #20242906	C₂	semidirect
5. #20242894	C₂	semidirect
92. C₂×C₈×C₄₇	Q₁₆	semidirect
199. C₄₇	#19150	semidirect
258. C₂×C₈	#10625337	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20253427	2
#20242916	2
#20242906	2
#20242894	2
#20242136	2
#20242046	2
C₂×C₄₇×#446	2
#19150	47