Subgroups — #26114909

Order 12032.

382 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244146	C₂	1	2	yes	maximal
2	#20243545	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
4	#20241272	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093957	—	2	2	no	—
7	C₂×#13093956	—	2	1	no	—
8	C₂×#13093956	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#15688993	C₂²	1	2	yes	—
11	#15688885	—	2	2	no	—
12	#15688874	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	—	2	1	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	#13093957	—	4	4	no	—
23	#13093956	—	4	2	no	—
24	#13093956	—	4	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₄₇	—	2	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	—	2	1	no	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	—	2	1	no	—
34	C₄⋊C₈×C₄₇	Q₈	1	2	yes	—
35	C₄×C₈×C₄₇	—	4	2	no	—
36	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	#95×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₇	D₈	1	2	yes	—
52	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	SD₁₆	1	4	yes	—
55	C₄²×C₄₇	—	4	1	no	—
56	C₄²×C₄₇	—	4	1	no	—
57	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₄₇	#41	1	1	yes	characteristic
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	8	2	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	—	4	2	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₂³×C₄₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	—	4	2	no	—
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#135	1	2	yes	—
84	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
86	C₂×Dic₄₇	—	8	2	no	—
87	C₂×Dic₄₇	—	8	2	no	—
88	C₂×Dic₄₇	—	8	2	no	—
89	C₈×C₄₇	—	2	2	no	—
90	C₈×C₄₇	—	4	2	no	—
91	C₈×C₄₇	#107	1	4	yes	complemented
92	C₄₇⋊C₈	—	4	4	no	—
93	#9611	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	#389	1	1	yes	characteristic
95	C₂²×C₄₇	#459	1	1	yes	characteristic
96	C₂²×C₄₇	#462	1	1	yes	characteristic
97	C₂²×C₄₇	#465	1	2	yes	—
98	C₂²×C₄₇	#471	1	2	yes	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	4	4	no	—
102	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
103	C₄×C₄₇	#428	1	1	yes	characteristic
104	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
105	C₄×C₄₇	#474	1	1	yes	characteristic
106	C₄×C₄₇	#477	1	1	yes	characteristic
107	C₄×C₄₇	#498	1	1	yes	characteristic
108	C₄×C₄₇	#499	1	1	yes	characteristic
109	Dic₄₇	—	16	2	no	—
110	#1947	—	47	2	no	—
111	#1844	—	47	1	no	—
112	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
113	C₂×#334	—	47	1	no	—
114	#1296	—	47	2	no	—
115	C₂×C₄₇	#1665	1	2	yes	—
116	C₂×C₄₇	#1672	1	2	yes	—
117	C₂×C₄₇	#1941	1	1	yes	characteristic
118	C₂×C₄₇	#1947	1	1	yes	characteristic
119	C₂×C₄₇	#1951	1	1	yes	characteristic
120	C₂×#107	—	94	2	no	—
121	C₂×#106	—	94	1	no	—
122	C₂×#106	—	94	1	no	—
123	C₂×C₄⋊C₈	—	47	2	no	—
124	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
125	C₂×C₄×C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
127	#384	—	47	2	no	—
128	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
129	#340	—	94	2	no	—
130	#336	—	2	1	no	—
131	#334	—	47	4	no	—
132	C₄₇	#9611	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	94	2	no	—
135	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄⋊C₄	—	94	2	no	—
139	C₂×C₄⋊C₄	—	94	2	no	—
140	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	#107	—	188	4	no	—
144	#106	—	188	2	no	—
145	#106	—	188	2	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	47	4	no	—
148	C₄⋊C₈	—	94	2	no	—
149	C₄⋊C₈	Dic₉₄	1	2	yes	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	Dic₉₄	1	2	yes	—
152	C₄×C₈	C₄×D₄₇	1	2	yes	—
153	#95	—	2	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	188	2	no	—
156	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
157	C₂²×C₄	#10625350	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	94	4	no	—
165	C₂×C₈	—	188	2	no	—
166	C₂×C₈	—	188	2	no	—
167	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
168	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
169	C₂×C₈	#10625334	1	2	yes	—
170	C₂×C₈	#10625337	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	376	2	no	—
173	C₄⋊C₄	—	188	2	no	—
174	C₄⋊C₄	—	188	2	no	—
175	C₄⋊C₄	—	188	2	no	—
176	C₄⋊C₄	—	188	2	no	—
177	C₄⋊C₄	#10625325	1	4	yes	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
181	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
182	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
183	C₄²	#10625350	1	1	yes	characteristic
184	C₂³	#13093998	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	376	2	no	—
190	C₂×C₄	—	376	2	no	—
191	C₂×C₄	—	376	2	no	—
192	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
193	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
194	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
195	C₂×C₄	#13094003	1	1	yes	characteristic
196	C₂×C₄	#13094016	1	1	yes	characteristic
197	C₂×C₄	#13094024	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
199	C₂×C₄	C₂×#10625325	1	2	yes	—
200	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
201	C₂×C₄	#13094046	1	2	yes	—
202	C₂×C₄	#13094049	1	1	yes	characteristic
203	C₂×C₄	#13094054	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	188	4	no	—
207	C₈	#13093957	1	4	yes	complemented
208	C₂²	#15689021	1	1	yes	characteristic
209	C₂²	#15689318	1	1	yes	characteristic
210	C₂²	#15689333	1	1	yes	characteristic
211	C₂²	#15689352	1	2	yes	—
212	C₂²	#15689386	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	752	2	no	—
217	C₄	C₂×#13093957	1	1	yes	characteristic
218	C₄	#15689166	1	1	yes	characteristic
219	C₄	C₄×#10625325	1	2	yes	—
220	C₄	#15689403	1	1	yes	characteristic
221	C₄	#15689418	1	1	yes	characteristic
222	C₄	#15689521	1	1	yes	characteristic
223	C₄	#15689525	1	1	yes	characteristic
224	C₂	#20242729	1	2	yes	—
225	C₂	#20242770	1	2	yes	—
226	C₂	#20244116	1	1	yes	characteristic
227	C₂	#20244146	1	1	yes	characteristic
228	C₂	#20244168	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#9611
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #15688874	C₄	semidirect
91. C₈×C₄₇	#107	semidirect
132. C₄₇	#9611	semidirect
207. C₈	#13093957	semidirect

Maximal subgroups

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

subgroup	index
#20244146	2
#20243545	2
C₂×#15688874	2
#20241272	2
C₈×C₄⋊C₄×C₄₇	2
#9611	47