Subgroups — #26204288

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254315	C₂	1	2	yes	maximal, complemented
2	#20253967	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253782	C₂	1	2	yes	maximal, complemented
5	#20243041	C₂	1	4	yes	maximal, complemented
6	#20242998	C₂	1	2	yes	maximal, complemented
7	#20242980	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689673	C₂²	1	2	yes	—
10	C₂×#13094024	C₂²	1	2	yes	—
11	#15689528	—	2	2	no	—
12	#15689528	C₂²	1	1	yes	characteristic
13	#15689528	C₂²	1	2	yes	—
14	#15689528	C₂²	1	2	yes	—
15	#15689500	C₂²	1	1	yes	characteristic
16	#15689496	C₂²	1	4	yes	—
17	#15689483	—	2	4	no	—
18	#15689474	C₂²	1	4	yes	—
19	#15689474	C₂²	1	4	yes	—
20	#15689470	C₂²	1	2	yes	—
21	#15689470	C₂²	1	4	yes	—
22	#15689359	—	2	2	no	—
23	#15689329	—	2	4	no	—
24	#15689264	—	2	4	no	—
25	C₄×D₃₇₆	—	2	2	no	—
26	C₂×#13093958	—	2	2	no	—
27	#15689125	—	2	4	no	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₈²×C₄₇	C₂²	1	2	yes	—
33	C₂×#10625350	—	2	2	no	—
34	C₂²×Dic₉₄	—	2	2	no	—
35	C₂×#10625325	—	2	8	no	—
36	C₂×D₃₇₆	—	2	2	no	—
37	#13094024	—	2	2	no	—
38	#13094024	—	2	2	no	—
39	#13094024	—	2	2	no	—
40	#13094024	C₂³	1	2	yes	—
41	#13094024	C₂³	1	4	yes	—
42	#13094021	C₂³	1	2	yes	—
43	#13094012	—	2	2	no	—
44	#13093998	—	2	2	no	—
45	#13093998	—	2	2	no	—
46	C₄×D₁₈₈	—	2	2	no	—
47	C₄×D₁₈₈	—	2	2	no	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	C₂×C₄⋊Dic₄₇	—	2	2	no	—
50	#13093958	—	2	4	no	—
51	#13093958	—	2	4	no	—
52	#13093958	—	2	2	no	—
53	#13093949	—	2	8	no	—
54	#13093949	—	2	8	no	—
55	#13093946	—	2	8	no	—
56	#13093946	—	2	4	no	—
57	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
58	C₂²×C₈×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
61	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
62	#97×C₄₇	D₄	1	2	yes	—
63	C₄×C₈×C₄₇	D₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	#10625350	—	4	8	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	—	2	2	no	—
77	C₂×C₄×D₄₇	—	4	2	no	—
78	C₂×Dic₉₄	—	2	2	no	—
79	C₂×Dic₉₄	—	2	4	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	C₂×Dic₉₄	—	8	4	no	—
82	M₁₆×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	4	2	no	—
93	(C₂×D₄₇)⋊C₄	—	4	2	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	4	2	no	—
99	C₄⋊Dic₄₇	—	4	2	no	—
100	D₃₇₆	—	4	4	no	—
101	#10625325	—	8	8	no	—
102	C₂³×C₄₇	#127	1	1	yes	characteristic
103	C₂²×D₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#135	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₄₇	—	8	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	8	4	no	—
117	D₁₈₈	—	4	4	no	—
118	D₁₈₈	—	4	4	no	—
119	D₁₈₈	—	4	4	no	—
120	C₄×D₄₇	—	8	4	no	—
121	Dic₉₄	—	4	4	no	—
122	Dic₉₄	—	4	8	no	—
123	Dic₉₄	—	8	4	no	—
124	C₈×C₄₇	—	2	4	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	#137	1	4	yes	—
128	#19638	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	#495	1	2	yes	—
132	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
133	C₂×D₄₇	—	8	2	no	—
134	C₂×D₄₇	—	8	2	no	—
135	C₂×D₄₇	—	8	2	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	#495	1	2	yes	—
139	C₄×C₄₇	C₂×#135	1	2	yes	—
140	C₄×C₄₇	C₂×#137	1	2	yes	—
141	C₄×C₄₇	#576	1	2	yes	—
142	Dic₄₇	—	8	2	no	—
143	Dic₄₇	—	8	4	no	—
144	#3206	—	47	2	no	—
145	#3169	—	47	1	no	—
146	C₂×#500	—	47	1	no	—
147	#3141	—	47	2	no	—
148	#1719	—	47	4	no	—
149	#1710	—	47	2	no	—
150	#1706	—	47	2	no	—
151	C₈×M₁₆	D₄₇	1	1	yes	characteristic, complemented
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
154	C₂×C₄₇	#3152	1	1	yes	characteristic
155	C₂×C₄₇	#3155	1	1	yes	characteristic
156	D₄₇	—	16	2	no	—
157	#532	—	47	2	no	—
158	C₂×C₄⋊Q₈	—	47	2	no	—
159	#500	—	47	2	no	—
160	#500	—	47	2	no	—
161	#500	—	47	1	no	—
162	#500	—	94	2	no	—
163	#493	—	47	1	no	—
164	#492	—	47	4	no	—
165	#489	—	94	4	no	—
166	#487	—	47	4	no	—
167	#487	—	47	4	no	—
168	#486	—	47	4	no	—
169	#486	—	47	2	no	—
170	#466	—	94	2	no	—
171	#461	—	94	4	no	—
172	#448	—	94	4	no	—
173	C₄×D₈	—	94	2	no	—
174	#431	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#107	—	94	2	no	—
176	#417	—	94	4	no	—
177	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
179	#322	C₂×D₄₇	1	2	yes	—
180	C₈²	C₂×D₄₇	1	2	yes	—
181	C₄₇	#19638	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	94	2	no	—
183	C₂²×Q₈	—	94	2	no	—
184	C₂×SD₁₆	—	94	8	no	—
185	C₂×D₈	—	94	2	no	—
186	C₂×M₁₆	D₁₈₈	1	2	yes	—
187	C₂²×C₈	D₁₈₈	1	2	yes	—
188	C₄⋊Q₈	—	47	4	no	—
189	C₄⋊Q₈	—	47	2	no	—
190	C₄⋊Q₈	—	94	2	no	—
191	C₄⋊Q₈	—	94	2	no	—
192	C₄⋊Q₈	—	94	2	no	—
193	#127	—	47	2	no	—
194	#124	—	94	2	no	—
195	#121	—	94	2	no	—
196	#121	—	94	2	no	—
197	C₄×D₄	—	94	2	no	—
198	C₄×D₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	94	2	no	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	#107	—	94	4	no	—
203	#107	—	94	4	no	—
204	#107	—	94	2	no	—
205	C₄⋊C₈	D₁₈₈	1	2	yes	—
206	#103	—	94	8	no	—
207	#103	—	94	8	no	—
208	#102	—	94	8	no	—
209	#102	—	94	4	no	—
210	C₂²⋊C₈	D₁₈₈	1	2	yes	—
211	#97	D₁₈₈	1	2	yes	—
212	C₄×C₈	D₁₈₈	1	2	yes	—
213	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₄₇	1	2	yes	—
215	C₄×C₈	C₂²×D₄₇	1	2	yes	—
216	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	#41	—	188	8	no	—
218	C₂×Q₈	—	94	2	no	—
219	C₂×Q₈	—	94	4	no	—
220	C₂×Q₈	—	376	4	no	—
221	C₂×Q₈	—	188	2	no	—
222	C₂×D₄	—	94	2	no	—
223	C₂×D₄	—	94	2	no	—
224	C₂×D₄	—	94	2	no	—
225	C₂×D₄	—	188	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	—	188	2	no	—
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
231	SD₁₆	—	376	8	no	—
232	D₈	—	188	4	no	—
233	M₁₆	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
236	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
237	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
238	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	94	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₄⋊C₄	—	188	2	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₂²⋊C₄	—	188	2	no	—
246	C₂²⋊C₄	—	188	2	no	—
247	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
248	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
249	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
250	C₄²	C₂³×D₄₇	1	1	yes	characteristic
251	C₂³	—	188	2	no	—
252	C₂³	#13094021	1	1	yes	characteristic
253	Q₈	—	376	4	no	—
254	Q₈	—	188	4	no	—
255	Q₈	—	188	8	no	—
256	D₄	—	376	4	no	—
257	D₄	—	188	4	no	—
258	D₄	—	188	4	no	—
259	D₄	—	188	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	376	4	no	—
263	C₂×C₄	—	376	4	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	#13094021	1	1	yes	characteristic
267	C₂×C₄	#13094021	1	1	yes	characteristic
268	C₂×C₄	#13094021	1	1	yes	characteristic
269	C₂×C₄	#13094046	1	4	yes	—
270	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	#13094057	1	4	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	376	2	no	—
280	C₂²	—	376	2	no	—
281	C₂²	—	376	2	no	—
282	C₂²	#15689507	1	2	yes	—
283	C₂²	C₂×#13094021	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	376	2	no	—
287	C₄	—	376	4	no	—
288	C₄	#15689507	1	2	yes	—
289	C₄	C₂×#13094046	1	2	yes	—
290	C₄	C₂×#13094057	1	2	yes	—
291	C₄	#15689928	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	752	2	no	—
294	C₂	C₂×#15689507	1	1	yes	characteristic
295	C₂	#20253847	1	1	yes	characteristic
296	C₂	#20253874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253967	C₂
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#19638
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20254315	C₂	semidirect
3. C₂×#15689528	C₂	semidirect
4. #20253782	C₂	semidirect
5. #20243041	C₂	semidirect
6. #20242998	C₂	semidirect
7. #20242980	C₂	semidirect
8. C₈×M₁₆×C₄₇	C₂	semidirect
151. C₈×M₁₆	D₄₇	semidirect
181. C₄₇	#19638	semidirect

Maximal subgroups

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

subgroup	index
#20254315	2
#20253967	2
C₂×#15689528	2
#20253782	2
#20243041	2
#20242998	2
#20242980	2
C₈×M₁₆×C₄₇	2
#19638	47