Subgroups — #26368889

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689532	C₂	1	2	yes	maximal
2	C₂×#15689525	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689521	C₂	1	1	yes	maximal, characteristic
4	#20249507	C₂	1	2	yes	maximal
5	#20243556	C₂	1	4	yes	maximal
6	#20243552	C₂	1	2	yes	maximal
7	#20243545	C₂	1	2	yes	maximal
8	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×#13094024	C₂²	1	2	yes	—
10	C₂×#13094016	C₂²	1	2	yes	—
11	#15689532	—	2	8	no	—
12	#15689525	—	2	4	no	—
13	#15689521	—	2	4	no	—
14	C₂×#13093958	C₂²	1	4	yes	—
15	C₂×#13093958	C₂²	1	2	yes	—
16	C₂×#13093956	C₂²	1	1	yes	characteristic
17	C₂×#13093956	C₂²	1	2	yes	—
18	C₂×#13093956	C₂²	1	4	yes	—
19	C₂×#13093956	C₂²	1	1	yes	characteristic
20	C₂×#13093956	C₂²	1	2	yes	—
21	#15689055	—	2	4	no	—
22	#15689044	—	2	4	no	—
23	#15688993	C₂²	1	4	yes	—
24	#15688989	C₂²	1	4	yes	—
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#376	C₂²	1	1	yes	characteristic
30	C₂²×Dic₉₄	—	2	2	no	—
31	#13094024	—	2	4	no	—
32	#13094024	—	2	4	no	—
33	#13094016	—	2	4	no	—
34	#13094016	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	8	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×Dic₄₇⋊C₄	—	2	2	no	—
41	C₂×Dic₄₇⋊C₄	—	2	2	no	—
42	#13093958	—	2	4	no	—
43	#13093958	—	2	8	no	—
44	#13093956	—	2	4	no	—
45	#13093956	—	2	4	no	—
46	#13093956	—	2	2	no	—
47	#13093956	—	2	2	no	—
48	#13093956	—	2	8	no	—
49	C₄₇⋊#95	—	2	8	no	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	#97×C₄₇	D₄	1	4	yes	—
56	#97×C₄₇	Q₈	1	8	yes	complemented
57	C₄×C₈×C₄₇	D₄	1	4	yes	—
58	C₄×C₈×C₄₇	Q₈	1	8	yes	complemented
59	#95×C₄₇	D₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂×Dic₉₄	—	4	4	no	—
67	C₂×Dic₉₄	—	8	4	no	—
68	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
75	C₄²×C₄₇	C₂×D₄	1	2	yes	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
78	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	8	8	no	—
87	Dic₄₇⋊C₄	—	4	4	no	—
88	Dic₄₇⋊C₄	—	4	4	no	—
89	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	#127	1	4	yes	—
93	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
98	C₂×Dic₄₇	—	4	2	no	—
99	C₂×Dic₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	C₂×Dic₄₇	—	4	4	no	—
102	C₂×Dic₄₇	—	4	8	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	Dic₉₄	—	8	8	no	—
105	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
106	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
107	C₈×C₄₇	C₄⋊Q₈	1	8	yes	—
108	#33318	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₄₇	#577	1	4	yes	—
112	C₄×C₄₇	—	2	4	no	—
113	C₄×C₄₇	—	2	8	no	—
114	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
117	Dic₄₇	—	8	4	no	—
118	Dic₄₇	—	8	2	no	—
119	Dic₄₇	—	8	2	no	—
120	C₂×#501	—	47	2	no	—
121	C₂×#499	—	47	1	no	—
122	C₂×#498	—	47	1	no	—
123	#2660	—	47	2	no	—
124	#1847	—	47	4	no	—
125	#1846	—	47	2	no	—
126	#1844	—	47	2	no	—
127	#1726	D₄₇	1	1	yes	characteristic
128	C₂×C₄₇	#2868	1	1	yes	characteristic
129	C₂×C₄₇	#3154	1	2	yes	—
130	C₂×C₄₇	#3163	1	4	yes	—
131	C₂×C₄⋊Q₈	—	47	2	no	—
132	C₂×#125	—	47	2	no	—
133	#501	—	94	8	no	—
134	#499	—	94	4	no	—
135	#498	—	94	4	no	—
136	C₂×#107	—	47	4	no	—
137	C₂×#107	—	47	2	no	—
138	C₂×#106	—	47	1	no	—
139	C₂×#106	—	47	2	no	—
140	C₂×#106	—	47	4	no	—
141	C₂×#106	—	47	1	no	—
142	C₂×#106	—	47	2	no	—
143	C₂×#97	C₂×D₄₇	1	2	yes	—
144	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
147	#398	—	94	4	no	—
148	#395	—	94	4	no	—
149	#384	—	47	4	no	—
150	#382	—	47	4	no	—
151	#376	C₂×D₄₇	1	1	yes	characteristic
152	C₄₇	#33318	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	94	2	no	—
154	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
156	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	C₄⋊Q₈	—	94	4	no	—
158	C₄⋊Q₈	—	94	4	no	—
159	#125	—	94	4	no	—
160	#125	—	94	4	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	—	47	4	no	—
164	C₂×C₄⋊C₄	—	94	8	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄⋊C₄	—	94	2	no	—
167	C₂×C₄⋊C₄	—	94	2	no	—
168	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#107	—	94	4	no	—
171	#107	—	94	8	no	—
172	#106	—	94	4	no	—
173	#106	—	94	4	no	—
174	#106	—	94	2	no	—
175	#106	—	94	2	no	—
176	#106	—	94	8	no	—
177	#97	Dic₉₄	1	8	yes	complemented
178	#97	D₁₈₈	1	4	yes	—
179	C₄×C₈	Dic₉₄	1	8	yes	complemented
180	C₄×C₈	D₁₈₈	1	4	yes	—
181	#95	—	94	8	no	—
182	#95	D₁₈₈	1	4	yes	—
183	C₂×Q₈	—	376	4	no	—
184	C₂×Q₈	—	188	4	no	—
185	C₂²×C₄	—	94	2	no	—
186	C₂²×C₄	—	94	4	no	—
187	C₂²×C₄	—	94	2	no	—
188	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
189	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
190	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
191	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
192	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
193	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
194	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
195	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
196	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
197	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
198	C₄⋊C₄	—	94	2	no	—
199	C₄⋊C₄	—	94	4	no	—
200	C₄⋊C₄	—	94	2	no	—
201	C₄⋊C₄	—	94	2	no	—
202	C₄⋊C₄	—	94	4	no	—
203	C₄⋊C₄	—	94	2	no	—
204	C₄⋊C₄	—	376	8	no	—
205	C₄⋊C₄	—	188	4	no	—
206	C₄⋊C₄	—	188	4	no	—
207	C₄⋊C₄	—	188	4	no	—
208	C₄²	C₂×Dic₉₄	1	4	yes	—
209	C₄²	C₂×Dic₉₄	1	4	yes	—
210	C₄²	C₂×D₁₈₈	1	2	yes	—
211	C₄²	C₂×D₁₈₈	1	2	yes	—
212	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
213	Q₈	—	376	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	188	2	no	—
217	C₂×C₄	—	188	2	no	—
218	C₂×C₄	—	188	4	no	—
219	C₂×C₄	—	188	4	no	—
220	C₂×C₄	—	188	8	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	#13094021	1	4	yes	—
223	C₂×C₄	#13094024	1	8	yes	—
224	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
227	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
228	C₈	#13094024	1	4	yes	—
229	C₈	#13094024	1	8	yes	—
230	C₈	#13094024	1	4	yes	—
231	C₂²	C₂×#13094021	1	1	yes	characteristic
232	C₂²	C₂×#13094024	1	2	yes	—
233	C₂²	#15689935	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	376	4	no	—
237	C₄	—	376	2	no	—
238	C₄	—	376	2	no	—
239	C₄	C₂×#13094024	1	1	yes	characteristic
240	C₄	C₂×#13094024	1	2	yes	—
241	C₄	C₂×#13094024	1	1	yes	characteristic
242	C₂	#20251786	1	1	yes	characteristic
243	C₂	#20253864	1	2	yes	—
244	C₂	#20253919	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#33318
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
56. #97×C₄₇	Q₈	semidirect
58. C₄×C₈×C₄₇	Q₈	semidirect
152. C₄₇	#33318	semidirect
177. #97	Dic₉₄	semidirect
179. 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
C₂×#15689532	2
C₂×#15689525	2
C₂×#15689521	2
#20249507	2
#20243556	2
#20243552	2
#20243545	2
C₄₇×#1726	2
#33318	47