Subgroups — #26247237

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246132	C₂	1	2	yes	maximal
2	#20244875	C₂	1	2	yes	maximal
3	#20244873	C₂	1	2	yes	maximal
4	#20244719	C₂	1	2	yes	maximal
5	#20244715	C₂	1	2	yes	maximal
6	#20243286	C₂	1	2	yes	maximal
7	C₄×#13093958	C₂	1	1	yes	maximal, characteristic
8	#20243139	C₂	1	1	yes	maximal, characteristic
9	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094023	C₂²	1	2	yes	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093957	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	1	yes	characteristic
15	#15689029	—	2	2	no	—
16	#15689027	—	2	2	no	—
17	#15689022	C₂²	1	2	yes	—
18	#15689019	C₂²	1	2	yes	—
19	#15689010	—	2	2	no	—
20	#15689007	—	2	2	no	—
21	#15688988	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	#15688969	C₂²	1	1	yes	characteristic
24	#15688886	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	1	yes	characteristic
26	#15688885	C₂²	1	2	yes	—
27	#15688885	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	C₂×#106×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
33	C₄₇×#340	C₂²	1	1	yes	characteristic
34	C₄₇×#340	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094023	—	2	4	no	—
38	#13094023	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	2	no	—
43	C₂×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	#13093958	—	2	4	no	—
47	#13093957	—	2	4	no	—
48	#13093955	—	2	4	no	—
49	#13093935	Q₈	1	4	yes	complemented
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	C₄₇⋊#95	D₄	1	1	yes	characteristic
56	C₄₇⋊#95	D₄	1	2	yes	—
57	C₄₇⋊#95	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#106×C₄₇	—	2	4	no	—
65	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
66	#95×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
75	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
76	C₂×Dic₉₄	—	2	8	no	—
77	C₂×Dic₉₄	—	8	2	no	—
78	C₂×Dic₉₄	—	8	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	#41	1	4	yes	—
87	C₄²×C₄₇	—	8	2	no	—
88	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	8	2	no	—
94	C₄⋊Dic₄₇	#41	1	4	yes	—
95	Dic₄₇⋊C₄	—	2	8	no	—
96	Dic₄₇⋊C₄	—	8	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	8	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	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	Dic₉₄	—	8	8	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₄₇⋊C₈	—	2	4	no	—
135	#23842	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	#525	1	2	yes	—
137	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	#576	1	1	yes	characteristic
139	C₂²×C₄₇	#577	1	2	yes	—
140	C₂²×C₄₇	#578	1	1	yes	characteristic
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	—	8	2	no	—
145	C₄×C₄₇	#533	1	1	yes	characteristic
146	C₄×C₄₇	#533	1	2	yes	—
147	C₄×C₄₇	#533	1	1	yes	characteristic
148	Dic₄₇	—	4	2	no	—
149	Dic₄₇	—	4	4	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	8	2	no	—
152	#2331	—	47	2	no	—
153	#2085	—	47	2	no	—
154	#2085	—	47	2	no	—
155	#2058	—	47	2	no	—
156	#2057	—	47	2	no	—
157	#1776	—	47	2	no	—
158	C₄×#107	—	47	1	no	—
159	C₄×#106	D₄₇	1	1	yes	characteristic
160	#1746	—	47	1	no	—
161	C₂×C₄₇	#2399	1	1	yes	characteristic
162	C₂×C₄₇	#3118	1	2	yes	—
163	C₂×C₄₇	#3121	1	2	yes	—
164	C₂×C₄₇	#3163	1	1	yes	characteristic
165	C₂×C₄₇	#3164	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	—	47	2	no	—
167	C₂×#107	—	47	2	no	—
168	C₂×#107	—	47	2	no	—
169	C₂×#106	—	47	2	no	—
170	C₂×#106	C₂×D₄₇	1	2	yes	—
171	C₂×#97	—	47	1	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	#391	—	94	2	no	—
174	#391	—	94	2	no	—
175	#389	—	47	2	no	—
176	#389	—	47	2	no	—
177	#387	—	94	2	no	—
178	#387	—	94	2	no	—
179	#382	—	47	2	no	—
180	C₄×C₄⋊C₄	—	47	2	no	—
181	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
182	#376	—	47	1	no	—
183	#340	—	47	1	no	—
184	#340	—	47	1	no	—
185	#340	—	47	2	no	—
186	#340	—	47	2	no	—
187	#340	C₂×D₄₇	1	1	yes	characteristic
188	#340	C₂×D₄₇	1	1	yes	characteristic
189	#336	—	47	2	no	—
190	#336	—	47	2	no	—
191	C₄₇	#23842	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	94	2	no	—
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊Q₈	—	94	4	no	—
197	C₄⋊Q₈	—	94	4	no	—
198	C₂×C₄⋊C₄	—	47	2	no	—
199	C₂×C₄⋊C₄	—	47	2	no	—
200	C₂×C₄⋊C₄	—	94	2	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄⋊C₄	—	94	2	no	—
203	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	#107	—	94	4	no	—
210	#107	—	94	4	no	—
211	#106	—	2	4	no	—
212	#106	—	94	4	no	—
213	#97	—	47	4	no	—
214	C₄×C₈	Dic₉₄	1	4	yes	complemented
215	#95	—	2	2	no	—
216	#95	—	47	1	no	—
217	#95	—	47	2	no	—
218	#95	—	47	1	no	—
219	#95	—	94	2	no	—
220	#95	—	94	2	no	—
221	#95	—	94	2	no	—
222	#95	—	94	2	no	—
223	#95	—	94	2	no	—
224	C₂×Q₈	—	94	8	no	—
225	C₂×Q₈	—	376	2	no	—
226	C₂×Q₈	—	376	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	94	2	no	—
241	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
242	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	47	4	no	—
247	C₄⋊C₄	—	94	8	no	—
248	C₄⋊C₄	—	94	2	no	—
249	C₄⋊C₄	—	94	2	no	—
250	C₄⋊C₄	—	94	2	no	—
251	C₄⋊C₄	—	376	2	no	—
252	C₄⋊C₄	—	376	2	no	—
253	C₄⋊C₄	#10625350	1	4	yes	—
254	C₄²	—	8	2	no	—
255	C₄²	—	94	2	no	—
256	C₄²	—	94	2	no	—
257	C₄²	—	376	2	no	—
258	C₄²	C₂×Dic₉₄	1	2	yes	—
259	C₄²	Q₈×D₄₇	1	2	yes	—
260	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
261	Q₈	—	376	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	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	#13094023	1	2	yes	—
284	C₂×C₄	#13094023	1	2	yes	—
285	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
287	C₂×C₄	#41×D₄₇	1	2	yes	—
288	C₂×C₄	C₂×#10625350	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	—	94	4	no	—
292	C₂²	#15689632	1	2	yes	—
293	C₂²	C₂×#13094023	1	1	yes	characteristic
294	C₂²	#15689925	1	1	yes	characteristic
295	C₂²	#15689929	1	2	yes	—
296	C₂²	#15689937	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	2	no	—
301	C₄	—	376	2	no	—
302	C₄	—	188	2	no	—
303	C₄	—	188	4	no	—
304	C₄	—	188	2	no	—
305	C₄	#15689676	1	1	yes	characteristic
306	C₄	#15689677	1	2	yes	—
307	C₄	#15689680	1	1	yes	characteristic
308	C₂	#20246682	1	1	yes	characteristic
309	C₂	#20253578	1	2	yes	—
310	C₂	#20253613	1	2	yes	—
311	C₂	#20253916	1	1	yes	characteristic
312	C₂	#20253928	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#23842
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
49. #13093935	Q₈	semidirect
65. C₄×C₈×C₄₇	Q₈	semidirect
191. C₄₇	#23842	semidirect
214. C₄×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20246132	2
#20244875	2
#20244873	2
#20244719	2
#20244715	2
#20243286	2
C₄×#13093958	2
#20243139	2
C₄×#106×C₄₇	2
#23842	47