Subgroups — #13051421

Order 1344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#10620658	C₂	1	1	yes	maximal, characteristic
4	#89312	—	3	1	no	maximal, Hall
5	C₂×Dic₄₂	—	2	1	no	—
6	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
7	#66627	—	2	1	no	—
8	Dic₈₄	—	2	1	no	—
9	Dic₈₄	C₄	1	2	yes	—
10	C₃×C₇×C₁₆	—	2	1	no	—
11	C₂×Dic₂₈	—	3	1	no	—
12	C₇×M₃₂	D₃	1	1	yes	characteristic
13	#6393	—	3	1	no	—
14	#4517	—	7	1	no	maximal, Hall
15	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×Dic₂₁	—	4	1	no	—
17	Dic₄₂	—	2	2	no	—
18	Dic₄₂	—	4	1	no	—
19	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₇×C₈	D₄	1	1	yes	characteristic
21	C₃⋊(C₇⋊C₈)	—	4	1	no	—
22	C₂×Dic₁₄	—	6	1	no	—
23	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
24	#1136	—	6	1	no	—
25	Dic₂₈	—	3	2	no	—
26	Dic₂₈	—	6	1	no	—
27	C₇×C₁₆	—	2	1	no	—
28	C₂×Dic₁₂	—	7	1	no	—
29	C₃×M₃₂	D₇	1	1	yes	characteristic
30	#819	—	7	1	no	—
31	C₂²×C₃×C₇	SD₁₆	1	1	yes	characteristic
32	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₄×C₇	D₈	1	1	yes	characteristic
34	Dic₂₁	—	4	2	no	—
35	#362	—	21	1	no	Sylow
36	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
37	C₂×Dic₇	—	12	1	no	—
38	Dic₁₄	—	6	2	no	—
39	Dic₁₄	—	12	1	no	—
40	C₇×C₈	C₄×D₃	1	1	yes	characteristic
41	C₇×C₈	#67	1	1	yes	characteristic
42	C₇⋊C₈	—	12	1	no	—
43	C₂×Dic₆	—	14	1	no	—
44	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
45	#208	—	14	1	no	—
46	Dic₁₂	—	7	2	no	—
47	Dic₁₂	—	14	1	no	—
48	C₃×C₁₆	—	2	1	no	—
49	C₂×C₃×C₇	—	2	1	no	—
50	C₂×C₃×C₇	#102	1	1	yes	characteristic
51	C₂×Q₁₆	—	21	1	no	—
52	M₃₂	D₂₁	1	1	yes	characteristic
53	#108	—	21	1	no	—
54	C₂²×C₇	#204	1	1	yes	characteristic
55	C₄×C₇	D₂₄	1	1	yes	characteristic
56	C₄×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
57	Dic₇	—	12	2	no	—
58	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
59	C₂×Dic₃	—	28	1	no	—
60	Dic₆	—	14	2	no	—
61	Dic₆	—	28	1	no	—
62	C₃×C₈	C₄×D₇	1	1	yes	characteristic
63	C₃×C₈	C₂²⋊D₇	1	1	yes	characteristic
64	C₃⋊C₈	—	28	1	no	—
65	C₃×C₇	#362	1	1	yes	characteristic, complemented, Hall
66	C₂×Q₈	—	42	1	no	—
67	Q₁₆	—	21	2	no	—
68	Q₁₆	—	42	1	no	—
69	M₁₆	—	42	1	no	—
70	C₂×C₈	C₂×D₂₁	1	1	yes	characteristic
71	C₁₆	—	2	1	no	—
72	C₂×C₇	—	2	1	no	—
73	C₂×C₇	#821	1	1	yes	characteristic
74	C₂²×C₃	#1132	1	1	yes	characteristic
75	C₃×C₄	D₅₆	1	1	yes	characteristic
76	C₃×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
77	Dic₃	—	28	2	no	—
78	Q₈	—	42	2	no	—
79	Q₈	—	84	1	no	—
80	C₂×C₄	—	84	1	no	—
81	C₂×C₄	D₈₄	1	1	yes	characteristic
82	C₈	—	84	1	no	—
83	C₈	C₄×D₂₁	1	1	yes	characteristic
84	C₈	#4268	1	1	yes	characteristic
85	C₇	#4517	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	2	1	no	—
87	C₂×C₃	#6395	1	1	yes	characteristic
88	C₂²	#66623	1	1	yes	characteristic
89	C₄	—	84	2	no	—
90	C₄	D₁₆₈	1	1	yes	characteristic
91	C₄	#66631	1	1	yes	characteristic
92	C₃	#89312	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	2	1	no	—
94	C₂	#10620660	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10620660
commutator	C₃×C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	#102
prime radical	C₂²×C₃×C₇	SD₁₆
cosocle	C₂×C₃×C₇×C₈	C₂²
hypercenter	M₃₂	D₂₁
hypocenter	C₃×C₇	#362
LCS	G ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
65. C₃×C₇	#362	semidirect
85. C₇	#4517	semidirect
92. C₃	#89312	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄	2
C₃×C₇×M₃₂	2
#10620658	2
#89312	3
#4517	7