Subgroups — #26125229

Order 12032.

842 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	#20256793	C₂	1	1	yes	maximal, characteristic
2	#20243126	C₂	1	2	yes	maximal
3	#20243086	C₂	1	2	yes	maximal
4	#20243085	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	#15689754	—	2	2	no	—
9	#15689751	—	2	4	no	—
10	#15689749	C₄	1	4	yes	—
11	#15689680	—	2	2	no	—
12	#15689676	—	2	4	no	—
13	C₂×#13093983	C₂²	1	2	yes	—
14	#15688891	—	2	4	no	—
15	#15688878	—	2	2	no	—
16	#15688877	—	2	4	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#343	—	2	2	no	—
20	C₂×#10625329	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	Q₈	1	2	yes	complemented
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	complemented
30	#13094024	—	2	4	no	—
31	#13094023	—	2	8	no	—
32	#13094016	—	2	4	no	—
33	#13094015	—	2	8	no	—
34	#13094003	—	2	8	no	—
35	#13094001	—	2	16	no	—
36	C₄×Dic₉₄	—	2	8	no	—
37	#13093983	—	2	2	no	—
38	#13093983	—	2	2	no	—
39	#13093983	—	2	2	no	—
40	#13093983	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	D₄	1	4	yes	—
42	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	—	2	2	no	—
50	C₂×M₁₆×C₄₇	Q₈	1	4	yes	complemented
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	#117×C₄₇	—	2	2	no	—
53	#117×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂²⋊C₄	1	8	yes	—
70	C₂×Dic₉₄	—	2	8	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	4	2	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₂²⋊Dic₄₇	—	2	4	no	—
83	C₂²⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	8	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	Dic₄₇⋊C₄	—	2	16	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	4	1	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	4	1	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
104	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	#120	1	2	yes	—
109	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#124	1	2	yes	—
119	C₂×C₄×C₄₇	#124	1	2	yes	—
120	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	4	8	no	—
128	Dic₉₄	—	8	8	no	—
129	C₈×C₄₇	—	4	4	no	—
130	C₄₇⋊C₈	—	4	2	no	—
131	C₄₇⋊C₈	—	4	2	no	—
132	#11391	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	#379	1	1	yes	characteristic
139	C₂²×C₄₇	#391	1	2	yes	—
140	C₂²×C₄₇	#391	1	2	yes	—
141	C₂²×C₄₇	#391	1	1	yes	characteristic
142	C₂²×C₄₇	#391	1	1	yes	characteristic
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	#393	1	2	yes	—
145	C₄×C₄₇	#393	1	1	yes	characteristic
146	C₄×C₄₇	#393	1	1	yes	characteristic
147	C₄×C₄₇	#395	1	2	yes	—
148	C₄×C₄₇	#395	1	1	yes	characteristic
149	C₄×C₄₇	#395	1	1	yes	characteristic
150	Dic₄₇	—	4	8	no	—
151	#3458	—	47	1	no	—
152	#1742	—	47	2	no	—
153	#1742	D₄₇	1	1	yes	characteristic
154	#1729	—	47	2	no	—
155	#1729	—	47	1	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	#1446	1	1	yes	characteristic
161	C₂×C₄₇	#1791	1	2	yes	—
162	C₂²×M₁₆	—	47	1	no	—
163	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
164	C₂³×C₈	—	47	1	no	—
165	#544	—	94	4	no	—
166	#544	—	94	2	no	—
167	#543	—	47	4	no	—
168	#533	—	94	4	no	—
169	#533	—	94	2	no	—
170	C₂×#117	—	47	2	no	—
171	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
172	#343	—	2	2	no	—
173	#343	—	94	4	no	—
174	#336	—	94	4	no	—
175	#336	—	94	2	no	—
176	C₄₇	#11391	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	47	2	no	—
184	C₂×M₁₆	—	94	1	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	Dic₉₄	1	4	yes	complemented
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	94	1	no	—
191	C₂²×C₈	—	94	1	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	—	94	2	no	—
194	C₂²×C₈	—	94	1	no	—
195	C₄⋊Q₈	—	94	4	no	—
196	C₄⋊Q₈	—	94	8	no	—
197	#125	—	94	4	no	—
198	#125	—	94	8	no	—
199	C₂²⋊Q₈	—	94	16	no	—
200	C₂²⋊Q₈	—	94	8	no	—
201	C₄×Q₈	—	94	8	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	94	2	no	—
205	#117	—	94	2	no	—
206	#117	—	94	2	no	—
207	#117	—	94	2	no	—
208	C₂×C₄⋊C₄	—	47	4	no	—
209	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
210	C₂×C₂²⋊C₄	—	47	4	no	—
211	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
212	C₂×C₄²	—	47	4	no	—
213	C₂×C₄²	D₁₈₈	1	2	yes	—
214	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂×Q₈	—	94	8	no	—
216	C₂²×C₄	—	47	8	no	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
218	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
219	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
220	C₂²×C₄	#10625350	1	2	yes	—
221	C₂²×C₄	D₄×D₄₇	1	2	yes	—
222	C₂²×C₄	D₄×D₄₇	1	2	yes	—
223	C₂²×C₄	#10625352	1	1	yes	characteristic
224	C₂²×C₄	#10625352	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	94	4	no	—
233	M₁₆	—	188	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	188	1	no	—
241	C₂×C₈	—	188	1	no	—
242	C₂×C₈	—	188	2	no	—
243	C₂×C₈	—	188	1	no	—
244	C₂×C₈	—	188	1	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	94	16	no	—
247	C₄⋊C₄	—	94	8	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	94	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	94	4	no	—
252	C₂²⋊C₄	—	94	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂²⋊C₄×D₄₇	1	2	yes	—
259	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
260	C₂³	#13093989	1	1	yes	characteristic
261	C₂³	#13093989	1	1	yes	characteristic
262	C₂³	C₄×Dic₉₄	1	2	yes	—
263	Q₈	—	376	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	188	8	no	—
268	C₂×C₄	#13093992	1	2	yes	—
269	C₂×C₄	#13093994	1	1	yes	characteristic
270	C₂×C₄	#13093994	1	1	yes	characteristic
271	C₂×C₄	#13094001	1	2	yes	—
272	C₂×C₄	#13094001	1	2	yes	—
273	C₂×C₄	#13094002	1	2	yes	—
274	C₂×C₄	#13094002	1	2	yes	—
275	C₂×C₄	#13094003	1	2	yes	—
276	C₂×C₄	#13094004	1	1	yes	characteristic
277	C₂×C₄	#13094004	1	1	yes	characteristic
278	C₂×C₄	#13094010	1	2	yes	—
279	C₂×C₄	#13094010	1	2	yes	—
280	C₂×C₄	#13094011	1	1	yes	characteristic
281	C₂×C₄	#13094012	1	1	yes	characteristic
282	C₂×C₄	#13094022	1	1	yes	characteristic
283	C₂×C₄	#13094023	1	2	yes	—
284	C₂×C₄	#13094023	1	2	yes	—
285	C₂×C₄	#13094024	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₈	—	188	2	no	—
288	C₈	—	188	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#15688978	1	1	yes	characteristic
295	C₂²	#15689027	1	2	yes	—
296	C₂²	#15689027	1	2	yes	—
297	C₂²	#15689028	1	1	yes	characteristic
298	C₂²	#15689029	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	—	188	8	no	—
301	C₄	#15689035	1	2	yes	—
302	C₄	#15689036	1	1	yes	characteristic
303	C₄	#15689036	1	1	yes	characteristic
304	C₄	#15689043	1	2	yes	—
305	C₄	#15689044	1	1	yes	characteristic
306	C₄	#15689044	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	#20241658	1	1	yes	characteristic
312	C₂	#20243337	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093994
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1742	C₂
socle	C₂²×C₄₇	#379
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1742	D₄₇
hypocenter	C₄₇	#11391
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
23. C₂×#10625329	Q₈	semidirect
29. C₂²×C₄₇⋊C₈	Q₈	semidirect
50. C₂×M₁₆×C₄₇	Q₈	semidirect
176. C₄₇	#11391	semidirect
187. C₂×M₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20256793	2
#20243126	2
#20243086	2
#20243085	2
C₄₇×#1742	2
#11391	47