Subgroups — #26115619

Order 12032.

406 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	#20241886	C₂	1	2	yes	maximal
4	#20241790	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	2	yes	—
7	#15689528	C₄	1	1	yes	characteristic, complemented
8	#15689528	C₄	1	1	yes	characteristic, complemented
9	#15689525	—	2	1	no	—
10	#15689525	—	2	1	no	—
11	#15689521	C₄	1	2	yes	complemented
12	#15689176	C₂²	1	2	yes	—
13	#15689161	—	2	1	no	—
14	#15689161	—	2	1	no	—
15	#15688876	—	2	1	no	—
16	#15688874	—	2	1	no	—
17	#15688859	—	2	4	no	—
18	#15688853	—	2	4	no	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#336	—	2	1	no	—
24	#13094024	C₂×C₄	1	2	yes	—
25	#13094024	C₂×C₄	1	2	yes	—
26	#13094016	—	2	2	no	—
27	#13094003	—	2	2	no	—
28	#13094003	—	2	2	no	—
29	C₄×Dic₉₄	—	2	2	no	—
30	C₄×Dic₉₄	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093958	—	2	1	no	—
33	#13093958	—	2	1	no	—
34	#13093958	—	2	1	no	—
35	#13093958	—	2	1	no	—
36	#13093956	—	2	2	no	—
37	#13093956	—	2	2	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	1	no	—
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₇	—	2	1	no	—
48	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
49	C₄×C₈×C₄₇	—	4	2	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
54	#95×C₄₇	D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂×Dic₉₄	—	2	2	no	—
60	C₂×Dic₉₄	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	—	2	1	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	4	1	no	—
65	C₂×C₈×C₄₇	—	4	1	no	—
66	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
71	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	4	1	no	—
74	C₄²×C₄₇	—	4	1	no	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	#41	1	1	yes	characteristic
79	C₂²⋊Dic₄₇	—	4	2	no	—
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	Dic₄₇⋊C₄	—	4	2	no	—
85	Dic₄₇⋊C₄	—	4	2	no	—
86	C₄×Dic₄₇	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	4	1	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	Dic₉₄	—	4	4	no	—
107	Dic₉₄	—	8	2	no	—
108	C₈×C₄₇	—	2	1	no	—
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	—	4	2	no	—
111	C₈×C₄₇	#103	1	2	yes	—
112	C₈×C₄₇	#103	1	2	yes	—
113	C₄₇⋊C₈	—	8	2	no	—
114	#9749	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	#390	1	1	yes	characteristic
118	C₂²×C₄₇	#461	1	1	yes	characteristic
119	C₂²×C₄₇	#462	1	1	yes	characteristic
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	4	4	no	—
124	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
125	C₄×C₄₇	#419	1	1	yes	characteristic
126	C₄×C₄₇	#443	1	2	yes	—
127	C₄×C₄₇	#461	1	1	yes	characteristic
128	C₄×C₄₇	#462	1	1	yes	characteristic
129	Dic₄₇	—	8	2	no	—
130	Dic₄₇	—	8	2	no	—
131	#3160	—	47	1	no	—
132	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
133	#1564	—	47	1	no	—
134	#1503	—	47	2	no	—
135	#1481	—	47	2	no	—
136	C₂×C₄₇	—	2	1	no	—
137	C₂×C₄₇	—	2	1	no	—
138	C₂×C₄₇	#1974	1	1	yes	characteristic
139	C₂×C₄₇	#1976	1	1	yes	characteristic
140	C₂×C₄₇	#1981	1	1	yes	characteristic
141	#533	—	47	2	no	—
142	#500	—	47	1	no	—
143	#500	—	47	1	no	—
144	#499	—	94	1	no	—
145	#499	—	94	1	no	—
146	#498	—	47	2	no	—
147	#431	—	47	2	no	—
148	#427	—	94	1	no	—
149	#427	—	94	1	no	—
150	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₄×C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#335	—	94	1	no	—
156	#334	—	94	1	no	—
157	#328	—	94	4	no	—
158	Q₈⋊C₈	—	94	4	no	—
159	C₄₇	#9749	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	47	2	no	—
164	C₄⋊Q₈	—	47	2	no	—
165	#125	—	94	2	no	—
166	C₂²⋊Q₈	—	94	2	no	—
167	C₂²⋊Q₈	—	94	2	no	—
168	C₄×Q₈	—	94	2	no	—
169	C₄×Q₈	—	94	2	no	—
170	#117	—	94	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#107	—	94	1	no	—
175	#107	—	94	1	no	—
176	#107	—	94	1	no	—
177	#107	—	94	1	no	—
178	#106	—	94	2	no	—
179	#106	—	94	2	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₄⋊C₈	D₁₈₈	1	2	yes	—
183	C₂²⋊C₈	—	94	2	no	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	94	2	no	—
186	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
187	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
188	C₄×C₈	D₁₈₈	1	1	yes	characteristic
189	C₄×C₈	D₁₈₈	1	1	yes	characteristic
190	#95	D₁₈₈	1	2	yes	—
191	C₂×Q₈	—	94	2	no	—
192	C₂×Q₈	—	94	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
195	C₂²×C₄	#10625350	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	188	2	no	—
203	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	#10625335	1	2	yes	—
208	C₂×C₈	#10625337	1	2	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	94	2	no	—
211	C₄⋊C₄	—	94	2	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	—	94	2	no	—
214	C₄⋊C₄	—	188	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₂²⋊C₄	—	188	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	188	2	no	—
220	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
222	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
223	C₄²	#10625350	1	1	yes	characteristic
224	C₂³	#13093998	1	1	yes	characteristic
225	Q₈	—	376	2	no	—
226	Q₈	—	188	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	188	2	no	—
233	C₂×C₄	—	188	2	no	—
234	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
236	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
237	C₂×C₄	#13093998	1	1	yes	characteristic
238	C₂×C₄	#13094012	1	1	yes	characteristic
239	C₂×C₄	#13094021	1	1	yes	characteristic
240	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
241	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
242	C₂×C₄	#13094054	1	1	yes	characteristic
243	C₂×C₄	#13094054	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	376	2	no	—
248	C₈	#13093947	1	2	yes	—
249	C₈	#13093947	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#15689025	1	1	yes	characteristic
253	C₂²	#15689328	1	1	yes	characteristic
254	C₂²	#15689333	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	4	no	—
259	C₄	—	376	2	no	—
260	C₄	—	376	2	no	—
261	C₄	C₂×#13093947	1	1	yes	characteristic
262	C₄	#15689131	1	1	yes	characteristic
263	C₄	#15689241	1	2	yes	—
264	C₄	#15689328	1	1	yes	characteristic
265	C₄	#15689333	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#20244276	1	1	yes	characteristic
269	C₂	#20244286	1	1	yes	characteristic
270	C₂	#20244311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#9749
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #15689528	C₄	semidirect
8. #15689528	C₄	semidirect
11. #15689521	C₄	semidirect
159. C₄₇	#9749	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20242136	2
#20241886	2
#20241790	2
C₈×C₄⋊C₄×C₄₇	2
#9749	47