Subgroups — #26247258

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246136	C₂	1	1	yes	maximal, characteristic
2	#20244869	C₂	1	2	yes	maximal
3	#20244865	C₂	1	2	yes	maximal
4	#20244723	C₂	1	2	yes	maximal
5	#20244721	C₂	1	2	yes	maximal
6	#20243286	C₂	1	2	yes	maximal
7	C₄×#13093955	C₂	1	2	yes	maximal
8	#20243140	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2330	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13093957	C₂²	1	2	yes	—
12	C₂×#13093955	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	2	yes	—
14	C₂×#13093955	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689027	—	2	2	no	—
18	#15689021	C₂²	1	2	yes	—
19	#15689006	—	2	2	no	—
20	#15688989	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688885	C₂²	1	2	yes	—
23	#15688885	C₂²	1	2	yes	—
24	#15688885	C₂²	1	2	yes	—
25	#15688885	C₂²	1	2	yes	—
26	#15688878	C₂²	1	2	yes	—
27	#15688878	C₂²	1	2	yes	—
28	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#389	—	2	2	no	—
30	C₄₇×#389	C₂²	1	2	yes	—
31	C₄₇×#387	—	2	2	no	—
32	C₄₇×#384	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
34	C₄₇×#376	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	#13094024	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	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×C₄×Dic₄₇	—	2	2	no	—
44	#13093957	—	2	4	no	—
45	#13093955	—	2	4	no	—
46	#13093955	—	2	4	no	—
47	#13093955	—	2	4	no	—
48	#13093935	Q₈	1	4	yes	complemented
49	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	1	no	—
52	C₄₇⋊#95	—	2	1	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	#125×C₄₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₇	—	2	2	no	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#95×C₄₇	—	2	2	no	—
65	#95×C₄₇	—	2	2	no	—
66	#95×C₄₇	D₄	1	2	yes	—
67	#95×C₄₇	D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂×Dic₉₄	—	2	4	no	—
78	C₂×Dic₉₄	—	8	2	no	—
79	C₄⋊C₄×C₄₇	—	2	8	no	—
80	C₄⋊C₄×C₄₇	—	2	2	no	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	—	2	2	no	—
83	C₄⋊C₄×C₄₇	—	8	2	no	—
84	C₄⋊C₄×C₄₇	—	8	2	no	—
85	C₄⋊C₄×C₄₇	#41	1	4	yes	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	8	2	no	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	#41	1	4	yes	—
96	Dic₄₇⋊C₄	—	8	2	no	—
97	C₄×Dic₄₇	—	8	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	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	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₂×Dic₄₇	—	2	4	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	4	no	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	#23845	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	#525	1	1	yes	characteristic
136	C₂²×C₄₇	#525	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	#577	1	2	yes	—
139	C₂²×C₄₇	#577	1	2	yes	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	—	4	1	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	—	4	1	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	#533	1	2	yes	—
147	C₄×C₄₇	#533	1	2	yes	—
148	Dic₄₇	—	4	4	no	—
149	Dic₄₇	—	8	2	no	—
150	#2331	—	47	1	no	—
151	#2330	D₄₇	1	1	yes	characteristic
152	#2085	—	47	2	no	—
153	#2084	—	47	2	no	—
154	#2059	—	47	2	no	—
155	#2058	—	47	2	no	—
156	#1776	—	47	2	no	—
157	C₄×#106	—	47	2	no	—
158	#1746	—	47	1	no	—
159	C₂×C₄₇	#2399	1	1	yes	characteristic
160	C₂×C₄₇	#3123	1	1	yes	characteristic
161	C₂×C₄₇	#3125	1	2	yes	—
162	C₂×C₄₇	#3126	1	1	yes	characteristic
163	C₂×C₄₇	#3163	1	2	yes	—
164	C₂×C₄⋊Q₈	—	47	1	no	—
165	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
166	C₂×#107	—	47	2	no	—
167	C₂×#106	—	47	2	no	—
168	C₂×#106	—	47	2	no	—
169	C₂×#106	—	47	2	no	—
170	C₂×#97	—	47	1	no	—
171	C₂×C₄×C₈	—	47	1	no	—
172	#391	—	94	2	no	—
173	#389	—	2	2	no	—
174	#389	—	47	2	no	—
175	#389	C₂×D₄₇	1	2	yes	—
176	#387	—	2	2	no	—
177	#387	—	94	2	no	—
178	#384	C₂×D₄₇	1	1	yes	characteristic
179	#382	—	47	1	no	—
180	C₄×C₄⋊C₄	—	47	2	no	—
181	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
182	#376	C₂×D₄₇	1	1	yes	characteristic
183	#340	—	47	2	no	—
184	#340	—	47	2	no	—
185	#340	—	47	2	no	—
186	#340	—	47	2	no	—
187	#336	—	47	2	no	—
188	#336	—	47	2	no	—
189	C₄₇	#23845	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	94	1	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	2	no	—
193	C₄⋊Q₈	—	94	4	no	—
194	#125	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	47	2	no	—
199	C₂×C₄⋊C₄	—	47	2	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	#107	—	94	4	no	—
210	#106	—	94	4	no	—
211	#106	—	94	4	no	—
212	#106	—	94	4	no	—
213	#97	—	47	4	no	—
214	C₄×C₈	—	47	4	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	94	2	no	—
218	#95	—	94	1	no	—
219	#95	—	94	1	no	—
220	#95	—	94	2	no	—
221	#95	C₂²⋊D₄₇	1	2	yes	—
222	#95	C₂²⋊D₄₇	1	2	yes	—
223	C₂×Q₈	—	94	4	no	—
224	C₂×Q₈	—	376	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	47	4	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	8	2	no	—
245	C₄⋊C₄	—	8	2	no	—
246	C₄⋊C₄	—	47	4	no	—
247	C₄⋊C₄	—	94	4	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	#10625350	1	4	yes	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	8	2	no	—
256	C₄²	—	376	2	no	—
257	C₄²	Q₈×D₄₇	1	2	yes	—
258	C₄²	Q₈×D₄₇	1	2	yes	—
259	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
260	Q₈	—	376	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	94	4	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	#13094022	1	4	yes	—
283	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
285	C₂×C₄	#41×D₄₇	1	2	yes	—
286	C₂×C₄	C₂×#10625350	1	2	yes	—
287	C₈	—	94	4	no	—
288	C₈	—	94	4	no	—
289	C₂²	#15689630	1	1	yes	characteristic
290	C₂²	#15689634	1	1	yes	characteristic
291	C₂²	C₂×#13094022	1	1	yes	characteristic
292	C₂²	#15689931	1	2	yes	—
293	C₂²	#15689934	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	376	2	no	—
301	C₄	—	188	4	no	—
302	C₄	#15689677	1	2	yes	—
303	C₄	#15689677	1	2	yes	—
304	C₂	#20246679	1	1	yes	characteristic
305	C₂	#20253635	1	1	yes	characteristic
306	C₂	#20253665	1	2	yes	—
307	C₂	#20253674	1	1	yes	characteristic
308	C₂	#20253914	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2330	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2330	D₄₇
hypocenter	C₄₇	#23845
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2330

Complements

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

N	H ≅ G/N	type
48. #13093935	Q₈	semidirect
49. C₄×C₄₇⋊C₈	Q₈	semidirect
189. C₄₇	#23845	semidirect

Maximal subgroups

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

subgroup	index
#20246136	2
#20244869	2
#20244865	2
#20244723	2
#20244721	2
#20243286	2
C₄×#13093955	2
#20243140	2
C₄₇×#2330	2
#23845	47