Subgroups — #30743643

Order 17152.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562502	C₂	1	1	yes	maximal, characteristic
2	#22558420	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863812	C₄	1	1	yes	characteristic
5	#16863230	C₄	1	1	yes	characteristic
6	#16863229	—	2	1	no	—
7	C₃₂×D₆₇	—	2	1	no	—
8	C₆₇⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
10	C₆₄×C₆₇	—	2	1	no	—
11	#14524049	C₈	1	1	yes	characteristic
12	#14523983	C₈	1	1	yes	characteristic
13	#14523982	—	2	1	no	—
14	C₁₆×D₆₇	—	2	1	no	—
15	C₆₇⋊C₃₂	—	2	1	no	—
16	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₆₇	D₄	1	1	yes	characteristic
18	C₃₂×C₆₇	D₄	1	1	yes	characteristic
19	#11761628	C₁₆	1	1	yes	characteristic
20	#11761612	C₁₆	1	1	yes	characteristic
21	#11761611	—	2	1	no	—
22	C₈×D₆₇	—	2	1	no	—
23	C₆₇⋊C₁₆	—	2	1	no	—
24	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₆₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₆₇	—	2	1	no	—
29	D₂₆₈	C₃₂	1	1	yes	characteristic
30	C₄×D₆₇	—	2	1	no	—
31	Dic₁₃₄	C₃₂	1	1	yes	characteristic
32	C₈×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₆₇	M₃₂	1	1	yes	characteristic
34	C₆₇⋊C₈	—	2	1	no	—
35	C₂²×C₆₇	M₆₄	1	1	yes	characteristic
36	C₂×D₆₇	—	2	1	no	—
37	C₄×C₆₇	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₆₇	C₂×C₃₂	1	1	yes	characteristic
39	Dic₆₇	—	2	1	no	—
40	#7514	—	67	1	no	maximal, Sylow
41	C₂×C₆₇	—	2	1	no	—
42	C₂×C₆₇	C₂²⋊C₃₂	1	1	yes	characteristic
43	D₆₇	—	4	1	no	—
44	#2258	—	67	1	no	—
45	M₁₂₈	—	67	1	no	—
46	C₂×C₆₄	D₆₇	1	1	yes	characteristic, complemented
47	C₆₇	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	—	67	1	no	—
49	M₆₄	—	67	1	no	—
50	M₆₄	—	134	1	no	—
51	C₂×C₃₂	—	134	1	no	—
52	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	134	1	no	—
55	#131	—	67	1	no	—
56	M₃₂	—	67	1	no	—
57	M₃₂	—	134	1	no	—
58	C₂×C₁₆	—	134	1	no	—
59	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
60	C₃₂	—	134	1	no	—
61	C₃₂	D₂₆₈	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₆₇	1	1	yes	characteristic
63	#41	—	67	1	no	—
64	M₁₆	—	67	1	no	—
65	M₁₆	—	134	1	no	—
66	C₂×C₈	—	134	1	no	—
67	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
68	C₁₆	—	134	1	no	—
69	C₁₆	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
70	C₁₆	#11761611	1	1	yes	characteristic
71	Q₈	—	67	1	no	—
72	D₄	—	67	1	no	—
73	D₄	—	134	1	no	—
74	C₂×C₄	—	134	1	no	—
75	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
76	C₈	—	134	1	no	—
77	C₈	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
78	C₈	#14523982	1	1	yes	characteristic
79	C₂²	—	134	1	no	—
80	C₂²	#16863229	1	1	yes	characteristic
81	C₄	—	134	1	no	—
82	C₄	(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
83	C₄	C₃₂×D₆₇	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	268	1	no	—
86	C₂	(C₂×D₆₇)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₂₆₈
commutator	C₄×C₆₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₆₇
Fitting	C₂×C₆₄×C₆₇	C₂
socle	C₂×C₆₇	C₂²⋊C₃₂
prime radical	#11761628	C₁₆
cosocle	C₂×C₃₂×C₆₇	C₂²
hypercenter	C₂×C₆₄	D₆₇
hypocenter	C₆₇	#7514
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #22558420	C₂	semidirect
3. C₂×C₆₄×C₆₇	C₂	semidirect
46. C₂×C₆₄	D₆₇	semidirect
47. C₆₇	#7514	semidirect

Maximal subgroups

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

subgroup	index
#22562502	2
#22558420	2
C₂×C₆₄×C₆₇	2
#7514	67