Subgroups — #26110049

Order 12032.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243548	C₂	1	2	yes	maximal
2	#20243545	C₂	1	2	yes	maximal
3	#20243516	C₂	1	2	yes	maximal
4	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093958	—	2	4	no	—
6	C₂×#13093956	—	2	4	no	—
7	#15688993	C₂²	1	4	yes	—
8	#15688886	—	2	4	no	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₈²×C₄₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₄₇	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	4	no	—
16	#13093958	—	4	8	no	—
17	#13093956	—	4	8	no	—
18	C₂²×C₈×C₄₇	D₄	1	2	yes	—
19	C₂²×C₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₈×C₄₇	D₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₄×C₈×C₄₇	Q₈	1	2	yes	—
27	C₄×C₈×C₄₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₄₇	—	4	4	no	—
32	C₂×C₈×C₄₇	—	2	4	no	—
33	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₄₇	D₈	1	4	yes	—
37	C₂×C₈×C₄₇	SD₁₆	1	8	yes	—
38	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
39	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₄⋊Dic₄₇	—	4	8	no	—
44	C₄⋊Dic₄₇	—	4	8	no	—
45	C₄⋊Dic₄₇	—	4	8	no	—
46	C₂³×C₄₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
55	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
56	C₂×C₄×C₄₇	#135	1	4	yes	—
57	C₂×Dic₄₇	—	8	4	no	—
58	C₂×Dic₄₇	—	8	4	no	—
59	C₂×Dic₄₇	—	8	4	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₈×C₄₇	#106	1	8	yes	complemented
62	C₈×C₄₇	#107	1	8	yes	complemented
63	#8814	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#384	1	1	yes	characteristic
65	C₂²×C₄₇	#492	1	2	yes	—
66	C₂²×C₄₇	#493	1	1	yes	characteristic
67	C₂²×C₄₇	#494	1	1	yes	characteristic
68	C₂²×C₄₇	#495	1	2	yes	—
69	C₄×C₄₇	C₂×#106	1	2	yes	—
70	C₄×C₄₇	C₂×#107	1	2	yes	—
71	C₄×C₄₇	#427	1	2	yes	—
72	C₄×C₄₇	#498	1	2	yes	—
73	C₄×C₄₇	#499	1	2	yes	—
74	C₄×C₄₇	#500	1	2	yes	—
75	Dic₄₇	—	16	4	no	—
76	#1845	—	47	2	no	—
77	#1844	—	47	2	no	—
78	#1834	—	47	2	no	—
79	C₂×C₈²	D₄₇	1	1	yes	characteristic
80	C₂×C₄₇	#1706	1	2	yes	—
81	C₂×C₄₇	#1707	1	2	yes	—
82	C₂×C₄₇	#1844	1	1	yes	characteristic
83	C₂×C₄₇	#1845	1	1	yes	characteristic
84	C₂×C₄₇	#1846	1	1	yes	characteristic
85	C₂×#107	—	94	4	no	—
86	C₂×#106	—	94	4	no	—
87	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
90	#384	—	47	4	no	—
91	#340	—	94	4	no	—
92	C₈²	Dic₄₇	1	4	yes	complemented
93	C₄₇	#8814	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	D₁₈₈	1	2	yes	—
95	C₂²×C₈	D₁₈₈	1	2	yes	—
96	C₂²×C₈	D₁₈₈	1	2	yes	—
97	C₂×C₄⋊C₄	—	94	4	no	—
98	C₂×C₄⋊C₄	—	94	4	no	—
99	C₂×C₄⋊C₄	—	94	4	no	—
100	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
101	#107	—	188	8	no	—
102	#106	—	188	8	no	—
103	C₄×C₈	Dic₉₄	1	2	yes	—
104	C₄×C₈	Dic₉₄	1	2	yes	—
105	C₄×C₈	Dic₉₄	1	2	yes	—
106	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
107	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
108	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
109	C₂²×C₄	—	188	4	no	—
110	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	#10625325	1	8	yes	—
115	C₂×C₈	D₃₇₆	1	4	yes	—
116	C₂×C₈	Dic₁₈₈	1	4	yes	—
117	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
118	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
119	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
120	C₄⋊C₄	—	188	8	no	—
121	C₄⋊C₄	—	188	8	no	—
122	C₄⋊C₄	—	188	8	no	—
123	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
124	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
125	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
126	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
127	C₂³	#13094021	1	1	yes	characteristic
128	C₂×C₄	—	376	4	no	—
129	C₂×C₄	—	376	4	no	—
130	C₂×C₄	—	376	4	no	—
131	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₄	#13094024	1	1	yes	characteristic
135	C₂×C₄	#13094024	1	1	yes	characteristic
136	C₂×C₄	#13094024	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
138	C₂×C₄	C₂×#10625325	1	4	yes	—
139	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
140	C₂×C₄	#13094046	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	#13093956	1	8	yes	complemented
143	C₈	#13093958	1	8	yes	complemented
144	C₂²	#15688993	1	1	yes	characteristic
145	C₂²	#15689496	1	2	yes	—
146	C₂²	#15689500	1	1	yes	characteristic
147	C₂²	#15689503	1	1	yes	characteristic
148	C₂²	#15689507	1	2	yes	—
149	C₄	—	752	4	no	—
150	C₄	C₂×#13093956	1	2	yes	—
151	C₄	C₂×#13093958	1	2	yes	—
152	C₄	#15689161	1	2	yes	—
153	C₄	#15689521	1	2	yes	—
154	C₄	#15689525	1	2	yes	—
155	C₄	#15689528	1	2	yes	—
156	C₂	#20242980	1	2	yes	—
157	C₂	#20242985	1	2	yes	—
158	C₂	#20243545	1	1	yes	characteristic
159	C₂	#20243548	1	1	yes	characteristic
160	C₂	#20243552	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#8814
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. C₈²×C₄₇	C₄	semidirect
61. C₈×C₄₇	#106	semidirect
62. C₈×C₄₇	#107	semidirect
92. C₈²	Dic₄₇	semidirect
93. C₄₇	#8814	semidirect
142. C₈	#13093956	semidirect
143. C₈	#13093958	semidirect

Maximal subgroups

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

subgroup	index
#20243548	2
#20243545	2
#20243516	2
C₂×C₈²×C₄₇	2
#8814	47