Subgroups — #26213562

Order 12032.

542 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256193	C₂	1	2	yes	maximal
2	#20256189	C₂	1	2	yes	maximal
3	#20242998	C₂	1	1	yes	maximal, characteristic
4	#20242943	C₂	1	2	yes	maximal
5	#20242942	C₂	1	2	yes	maximal
6	#20242633	C₂	1	2	yes	maximal
7	#20242632	C₂	1	2	yes	maximal
8	#20242286	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1591	C₂	1	1	yes	maximal, characteristic
10	#15689852	C₂²	1	2	yes	—
11	#15689851	C₂²	1	2	yes	—
12	#15689528	C₂²	1	1	yes	characteristic
13	#15689528	C₂²	1	1	yes	characteristic
14	#15689492	—	2	2	no	—
15	#15689490	—	2	2	no	—
16	#15689474	C₂²	1	2	yes	—
17	#15689474	C₂²	1	2	yes	—
18	#15689431	—	2	2	no	—
19	#15689429	—	2	2	no	—
20	#15689409	C₂²	1	2	yes	—
21	#15689408	C₂²	1	2	yes	—
22	#15689334	—	2	2	no	—
23	#15689334	C₂²	1	2	yes	—
24	#15689333	C₂²	1	2	yes	—
25	#15689331	—	2	2	no	—
26	#15689253	C₂²	1	2	yes	—
27	#15689231	—	2	2	no	—
28	#15689227	—	2	2	no	—
29	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
30	#15688876	C₂²	1	1	yes	characteristic
31	#15688870	C₂²	1	2	yes	—
32	#15688868	C₂²	1	2	yes	—
33	#15688855	C₂²	1	2	yes	—
34	C₄₇×#446	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#335	C₂²	1	1	yes	characteristic
37	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₈₈	—	2	2	no	—
40	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
41	C₂×#10625337	—	2	2	no	—
42	#13094024	—	2	2	no	—
43	#13094024	C₂³	1	2	yes	—
44	#13094024	C₂³	1	2	yes	—
45	#13094022	—	2	2	no	—
46	#13094016	—	2	2	no	—
47	#13094015	—	2	2	no	—
48	#13094015	—	4	2	no	—
49	#13094014	—	4	2	no	—
50	Q₈×Dic₄₇	—	2	2	no	—
51	C₄×Dic₉₄	—	2	2	no	—
52	C₄×Dic₉₄	—	2	2	no	—
53	C₄×Dic₉₄	C₂³	1	2	yes	—
54	#13093989	—	2	2	no	—
55	#13093958	—	4	1	no	—
56	#13093958	—	4	1	no	—
57	#13093958	D₄	1	1	yes	characteristic
58	#13093956	—	2	2	no	—
59	#13093955	—	2	2	no	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	#13093949	—	2	2	no	—
63	#13093949	—	2	2	no	—
64	#13093949	—	2	2	no	—
65	#13093949	—	2	2	no	—
66	#13093949	—	2	2	no	—
67	#13093949	—	2	2	no	—
68	#13093949	D₄	1	2	yes	—
69	#13093948	—	2	2	no	—
70	#13093947	—	2	2	no	—
71	#13093934	—	2	2	no	—
72	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
74	#107×C₄₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	—	2	2	no	—
76	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
77	#103×C₄₇	D₄	1	2	yes	—
78	#97×C₄₇	—	2	2	no	—
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×Dic₉₄	—	2	2	no	—
84	C₂×Dic₉₄	—	2	2	no	—
85	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
86	Q₁₆×C₄₇	—	4	1	no	—
87	C₂×C₈×C₄₇	—	2	1	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
97	#10625337	—	4	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	4	2	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
105	Dic₄₇⋊C₄	—	2	2	no	—
106	Dic₄₇⋊C₄	—	4	2	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	Dic₄₇⋊C₄	—	4	2	no	—
109	Dic₄₇⋊C₄	—	4	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	4	2	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	Dic₁₈₈	—	4	1	no	—
115	Dic₁₈₈	—	4	4	no	—
116	Q₈×C₄₇	—	2	2	no	—
117	Q₈×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	Dic₉₄	—	2	2	no	—
128	Dic₉₄	—	2	2	no	—
129	Dic₉₄	—	4	2	no	—
130	Dic₉₄	—	4	2	no	—
131	Dic₉₄	—	4	2	no	—
132	Dic₉₄	—	4	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	—	4	2	no	—
139	C₄₇⋊C₈	—	4	2	no	—
140	#20321	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	D₄²	1	1	yes	characteristic
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	#575	1	1	yes	characteristic
146	C₄×C₄₇	#575	1	1	yes	characteristic
147	C₄×C₄₇	#578	1	1	yes	characteristic
148	C₄×C₄₇	#578	1	1	yes	characteristic
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	8	2	no	—
152	Dic₄₇	—	8	2	no	—
153	#3374	—	47	2	no	—
154	#3374	—	47	2	no	—
155	#1710	—	47	1	no	—
156	#1700	—	47	2	no	—
157	#1700	—	47	2	no	—
158	#1648	—	47	2	no	—
159	#1648	—	47	2	no	—
160	#1591	—	47	1	no	—
161	#1591	D₄₇	1	1	yes	characteristic
162	C₂×C₄₇	#3277	1	1	yes	characteristic
163	C₂×C₄₇	#3277	1	1	yes	characteristic
164	C₂×C₄₇	#3293	1	1	yes	characteristic
165	#557	—	47	2	no	—
166	#557	—	47	2	no	—
167	#500	—	47	1	no	—
168	#500	—	47	1	no	—
169	#491	—	94	2	no	—
170	#491	—	94	2	no	—
171	#487	—	47	2	no	—
172	#487	—	47	2	no	—
173	#479	—	94	2	no	—
174	#479	—	94	2	no	—
175	#475	—	47	2	no	—
176	#475	—	47	2	no	—
177	#462	—	47	2	no	—
178	#462	—	47	2	no	—
179	#462	—	94	2	no	—
180	#462	—	94	2	no	—
181	#446	—	47	2	no	—
182	#446	C₂×D₄₇	1	2	yes	—
183	#441	—	94	2	no	—
184	#441	—	94	2	no	—
185	C₄×Q₁₆	—	47	1	no	—
186	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
187	#335	—	47	1	no	—
188	#335	C₂×D₄₇	1	1	yes	characteristic
189	#332	—	47	2	no	—
190	#332	—	47	2	no	—
191	Q₈⋊C₈	—	47	2	no	—
192	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
193	C₈²	C₂×D₄₇	1	1	yes	characteristic
194	C₄₇	#20321	1	1	yes	characteristic, complemented, Sylow
195	C₂×Q₁₆	—	47	1	no	—
196	C₂×Q₁₆	—	94	2	no	—
197	C₂×Q₁₆	—	94	2	no	—
198	C₂×Q₁₆	D₁₈₈	1	1	yes	characteristic
199	C₄⋊Q₈	—	47	2	no	—
200	C₄⋊Q₈	—	47	2	no	—
201	C₄⋊Q₈	—	94	2	no	—
202	C₄⋊Q₈	—	94	2	no	—
203	#125	—	94	2	no	—
204	#125	—	94	2	no	—
205	#125	—	188	2	no	—
206	#125	—	188	2	no	—
207	C₄×Q₈	—	47	2	no	—
208	C₄×Q₈	—	94	2	no	—
209	C₄×Q₈	—	94	2	no	—
210	C₄×Q₈	—	94	2	no	—
211	C₄×Q₈	—	94	2	no	—
212	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
213	#107	—	47	1	no	—
214	#107	—	188	1	no	—
215	#107	—	188	1	no	—
216	#107	D₁₈₈	1	1	yes	characteristic
217	#106	—	94	2	no	—
218	#106	—	94	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	47	2	no	—
221	C₄⋊C₈	—	94	2	no	—
222	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
223	#103	—	47	2	no	—
224	#103	—	94	2	no	—
225	#103	—	94	2	no	—
226	#103	—	94	2	no	—
227	#103	—	94	2	no	—
228	#103	—	94	2	no	—
229	#103	—	94	2	no	—
230	#103	—	94	2	no	—
231	#103	—	94	2	no	—
232	#103	D₁₈₈	1	2	yes	—
233	#97	—	2	2	no	—
234	#97	—	94	2	no	—
235	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×Q₈	—	47	2	no	—
239	C₂×Q₈	—	94	2	no	—
240	C₂×Q₈	—	94	2	no	—
241	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
242	Q₁₆	—	4	1	no	—
243	Q₁₆	—	188	1	no	—
244	Q₁₆	—	188	4	no	—
245	Q₁₆	—	188	4	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
252	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	47	2	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄⋊C₄	—	94	2	no	—
258	C₄⋊C₄	—	94	2	no	—
259	C₄⋊C₄	—	94	2	no	—
260	C₄⋊C₄	—	188	2	no	—
261	C₄⋊C₄	—	188	2	no	—
262	C₄⋊C₄	—	188	2	no	—
263	C₄⋊C₄	—	188	2	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	2	no	—
266	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	94	2	no	—
269	C₄²	—	188	2	no	—
270	C₄²	—	188	2	no	—
271	C₄²	C₂³×D₄₇	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	94	2	no	—
275	Q₈	—	94	2	no	—
276	Q₈	—	188	2	no	—
277	Q₈	—	188	2	no	—
278	Q₈	—	188	2	no	—
279	Q₈	—	188	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
288	C₂×C₄	#13094070	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	4	2	no	—
295	C₈	—	188	2	no	—
296	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	376	2	no	—
301	C₄	—	376	2	no	—
302	C₄	—	188	2	no	—
303	C₄	—	188	2	no	—
304	C₄	#15689920	1	1	yes	characteristic
305	C₄	#15689923	1	1	yes	characteristic
306	C₄	#15689936	1	1	yes	characteristic
307	C₄	#15689940	1	1	yes	characteristic
308	C₂	#20255044	1	1	yes	characteristic
309	C₂	#20255046	1	1	yes	characteristic
310	C₂	#20255193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1591	C₂
socle, prime radical	C₂²×C₄₇	D₄²
hypercenter	#1591	D₄₇
hypocenter	C₄₇	#20321
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
194. C₄₇	#20321	semidirect

Maximal subgroups

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

subgroup	index
#20256193	2
#20256189	2
#20242998	2
#20242943	2
#20242942	2
#20242633	2
#20242632	2
#20242286	2
C₄₇×#1591	2
#20321	47