Subgroups — #11928577

Order 1248.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64604	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#88194	—	3	1	no	maximal, Hall
6	C₂²×D₃₉	C₂²	1	2	yes	—
7	C₂²×D₃×C₁₃	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	D₁₅₆	—	2	2	no	—
10	C₁₃×D₁₂	—	2	2	no	—
11	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
12	#64604	—	2	8	no	—
13	C₂×D₄×C₁₃	—	3	1	no	—
14	C₂×C₂²⋊D₁₃	—	3	4	no	—
15	C₂×D₅₂	—	3	1	no	—
16	C₄×Dic₁₃	D₃	1	1	yes	characteristic, complemented
17	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×D₃₉	—	2	4	no	—
19	C₂×D₃₉	—	2	2	no	—
20	C₂×D₃×C₁₃	—	2	4	no	—
21	C₂×D₃×C₁₃	—	2	2	no	—
22	C₃×C₄×C₁₃	D₄	1	2	yes	complemented
23	C₃×Dic₁₃	D₄	1	4	yes	complemented
24	C₂³×C₁₃	—	3	2	no	—
25	C₂²×D₁₃	—	3	2	no	—
26	D₄×C₁₃	—	6	2	no	—
27	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
28	C₂²⋊D₁₃	—	6	8	no	—
29	C₂×Dic₁₃	C₂×D₃	1	2	yes	—
30	D₅₂	—	6	2	no	—
31	#874	—	13	1	no	maximal, Hall
32	C₂×C₃×C₁₃	C₂×D₄	1	2	yes	—
33	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
34	D₃₉	—	4	2	no	—
35	D₃×C₁₃	—	4	2	no	—
36	C₂²×C₁₃	—	6	4	no	—
37	C₂²×C₁₃	—	6	2	no	—
38	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
39	C₂×D₁₃	—	6	2	no	—
40	C₂×D₁₃	—	6	4	no	—
41	C₄×C₁₃	D₁₂	1	2	yes	complemented
42	Dic₁₃	D₁₂	1	4	yes	complemented
43	C₂×D₁₂	—	13	4	no	—
44	C₂×D₁₂	—	13	1	no	—
45	C₂×D₁₂	D₁₃	1	1	yes	characteristic, complemented
46	C₃×C₄²	—	13	1	no	—
47	C₃×C₁₃	#127	1	1	yes	characteristic, complemented, Hall
48	#127	—	39	1	no	Sylow
49	C₂×C₁₃	—	12	2	no	—
50	C₂×C₁₃	C₂×D₁₂	1	2	yes	—
51	C₂×C₁₃	C₂×D₁₂	1	1	yes	characteristic
52	D₁₃	—	12	2	no	—
53	C₂²×D₃	—	13	2	no	—
54	C₂²×D₃	C₂×D₁₃	1	2	yes	—
55	C₂×C₃×C₄	—	13	2	no	—
56	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
57	D₁₂	—	2	2	no	—
58	D₁₂	—	26	8	no	—
59	D₁₂	—	26	2	no	—
60	C₂×D₄	—	3	1	no	—
61	C₂×D₄	—	39	4	no	—
62	C₂×D₄	—	39	1	no	—
63	C₄²	—	13	1	no	—
64	C₁₃	#874	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
66	C₂×D₃	—	2	4	no	—
67	C₂×D₃	—	2	2	no	—
68	C₂×D₃	—	26	4	no	—
69	C₂×D₃	—	26	2	no	—
70	C₃×C₄	—	13	4	no	—
71	C₃×C₄	C₂²⋊D₁₃	1	2	yes	complemented
72	C₂³	—	3	2	no	—
73	C₂³	—	39	2	no	—
74	D₄	—	6	2	no	—
75	D₄	—	78	8	no	—
76	D₄	—	78	2	no	—
77	C₂×C₄	—	13	2	no	—
78	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
79	C₂×C₃	D₄×D₁₃	1	2	yes	—
80	C₂×C₃	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
81	D₃	—	4	2	no	—
82	D₃	—	52	2	no	—
83	C₂²	—	6	4	no	—
84	C₂²	—	6	2	no	—
85	C₂²	—	78	4	no	—
86	C₂²	—	78	2	no	—
87	C₂²	C₂×D₃×D₁₃	1	1	yes	characteristic
88	C₄	—	13	4	no	—
89	C₄	#64604	1	2	yes	complemented
90	C₃	#88194	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	12	2	no	—
92	C₂	—	156	2	no	—
93	C₂	D₁₂×D₁₃	1	2	yes	—
94	C₂	C₂×#64604	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₁₃
commutator, socle, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₂×C₄	D₃×D₁₃
hypocenter	C₃×C₁₃	#127
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₆	C₂	semidirect
2. C₂×C₁₃×D₁₂	C₂	semidirect
3. C₂×#64604	C₂	semidirect
4. C₃×C₄×Dic₁₃	C₂	semidirect
16. C₄×Dic₁₃	D₃	semidirect
22. C₃×C₄×C₁₃	D₄	semidirect
23. C₃×Dic₁₃	D₄	semidirect
41. C₄×C₁₃	D₁₂	semidirect
42. Dic₁₃	D₁₂	semidirect
45. C₂×D₁₂	D₁₃	semidirect
47. C₃×C₁₃	#127	semidirect
64. C₁₃	#874	semidirect
71. C₃×C₄	C₂²⋊D₁₃	semidirect
89. C₄	#64604	semidirect
90. C₃	#88194	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₆	2
C₂×C₁₃×D₁₂	2
C₂×#64604	2
C₃×C₄×Dic₁₃	2
#88194	3
#874	13