Subgroups — #30743708

Order 17152.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₇⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₇₂	—	2	1	no	—
5	D₂₁₄₄	—	2	1	no	—
6	D₂₁₄₄	C₄	1	2	yes	complemented
7	C₆₇⋊C₆₄	—	2	1	no	—
8	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
9	#366×C₆₇	—	2	1	no	—
10	C₂×D₅₃₆	—	4	1	no	—
11	D₁₀₇₂	—	2	2	no	—
12	D₁₀₇₂	—	4	1	no	—
13	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
14	#107×C₆₇	—	4	1	no	—
15	C₃₂×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₆₇	D₄	1	1	yes	characteristic
17	C₂×D₂₆₈	—	8	1	no	—
18	D₅₃₆	—	4	2	no	—
19	D₅₃₆	—	8	1	no	—
20	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₆₇	—	8	1	no	—
22	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
24	C₂²×D₆₇	—	16	1	no	—
25	C₂×C₄×C₆₇	—	16	1	no	—
26	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
27	D₂₆₈	—	8	2	no	—
28	D₂₆₈	—	16	1	no	—
29	C₈×C₆₇	#102	1	1	yes	characteristic
30	C₈×C₆₇	SD₃₂	1	1	yes	characteristic
31	C₂²×C₆₇	D₃₂	1	1	yes	characteristic
32	C₂×D₆₇	—	16	2	no	—
33	C₂×D₆₇	—	32	1	no	—
34	C₂×D₆₇	—	32	1	no	—
35	C₄×C₆₇	—	32	1	no	—
36	C₄×C₆₇	#357	1	1	yes	characteristic
37	C₄×C₆₇	SD₆₄	1	1	yes	characteristic
38	#7537	—	67	1	no	maximal, Sylow
39	C₂×C₆₇	#1415	1	1	yes	characteristic
40	C₂×C₆₇	D₆₄	1	1	yes	characteristic
41	C₂×C₆₇	SD₁₂₈	1	1	yes	characteristic
42	D₆₇	—	32	2	no	—
43	C₂×D₃₂	—	67	1	no	—
44	C₂×C₆₄	—	67	1	no	—
45	#1423	D₆₇	1	1	yes	characteristic, complemented
46	C₆₇	#7537	1	1	yes	characteristic, complemented, Sylow
47	C₂×D₁₆	—	134	1	no	—
48	D₃₂	—	67	2	no	—
49	D₃₂	—	134	1	no	—
50	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
51	#366	—	2	1	no	—
52	C₆₄	—	134	1	no	—
53	C₂×D₈	—	268	1	no	—
54	D₁₆	—	268	1	no	—
55	D₁₆	—	134	2	no	—
56	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
57	#107	—	4	1	no	—
58	C₃₂	C₄×D₆₇	1	1	yes	characteristic
59	C₃₂	D₂₆₈	1	1	yes	characteristic
60	C₂×D₄	—	536	1	no	—
61	D₈	—	268	2	no	—
62	D₈	—	536	1	no	—
63	C₂×C₈	#11761613	1	1	yes	characteristic
64	C₄⋊C₄	—	8	1	no	—
65	C₁₆	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
66	C₁₆	#11761615	1	1	yes	characteristic
67	C₂³	—	1072	1	no	—
68	D₄	—	536	2	no	—
69	D₄	—	1072	1	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	#14523984	1	1	yes	characteristic
72	C₈	#14523963	1	1	yes	characteristic
73	C₈	#14523986	1	1	yes	characteristic
74	C₂²	—	1072	2	no	—
75	C₂²	—	2144	1	no	—
76	C₂²	—	2144	1	no	—
77	C₂²	#16863231	1	1	yes	characteristic
78	C₄	—	32	1	no	—
79	C₄	#16863196	1	1	yes	characteristic
80	C₄	#16863233	1	1	yes	characteristic
81	C₂	—	2144	2	no	—
82	C₂	#22558386	1	1	yes	characteristic
83	C₂	#22558421	1	1	yes	characteristic
84	C₂	#22558423	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863231
commutator	C₃₂×C₆₇	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₇
Fitting	C₆₇×#1423	C₂
socle	C₂²×C₆₇	D₃₂
prime radical	C₂×D₂₁₄₄	C₂
cosocle	C₂×C₃₂×C₆₇	C₂²
hypercenter	#1423	D₆₇
hypocenter	C₆₇	#7537
LCS	G ▷ C₃₂×C₆₇ ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₇⋊C₆₄	C₂	semidirect
3. C₆₇×#1423	C₂	semidirect
6. D₂₁₄₄	C₄	semidirect
45. #1423	D₆₇	semidirect
46. C₆₇	#7537	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁₄₄	2
C₂×C₆₇⋊C₆₄	2
C₆₇×#1423	2
#7537	67