Subgroups — #30743641

Order 17152.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863229	C₂	1	1	yes	maximal, characteristic
2	#22558420	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₆₇	C₄	1	1	yes	characteristic
5	#16863229	—	2	1	no	—
6	#16863229	—	2	1	no	—
7	C₂×C₆₇⋊C₃₂	C₄	1	1	yes	characteristic
8	C₆₇⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
10	C₆₄×C₆₇	—	2	1	no	—
11	C₂×C₈×D₆₇	C₈	1	1	yes	characteristic
12	C₂×C₆₇⋊C₁₆	C₈	1	1	yes	characteristic
13	C₁₆×D₆₇	—	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	C₂×C₄×D₆₇	C₁₆	1	1	yes	characteristic
20	C₂×C₆₇⋊C₈	C₁₆	1	1	yes	characteristic
21	C₈×D₆₇	—	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	C₂²×D₆₇	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₆₇	C₃₂	1	1	yes	characteristic
30	C₄×D₆₇	—	2	1	no	—
31	C₄×D₆₇	—	2	1	no	—
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	C₂×C₃₂	1	1	yes	characteristic
36	C₂×D₆₇	—	2	1	no	—
37	C₂×D₆₇	—	4	1	no	—
38	C₄×C₆₇	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₆₇	M₆₄	1	1	yes	characteristic
40	Dic₆₇	—	2	1	no	—
41	#7513	—	67	1	no	maximal, Sylow
42	C₂×C₆₇	—	2	1	no	—
43	C₂×C₆₇	C₂²⋊C₃₂	1	1	yes	characteristic
44	D₆₇	—	4	1	no	—
45	C₂×M₆₄	—	67	1	no	—
46	M₁₂₈	—	67	1	no	—
47	M₁₂₈	D₆₇	1	1	yes	characteristic, complemented
48	C₆₇	#7513	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	67	1	no	—
50	M₆₄	—	134	1	no	—
51	M₆₄	—	134	1	no	—
52	C₂×C₃₂	—	67	1	no	—
53	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
54	C₆₄	—	2	1	no	—
55	C₆₄	—	134	1	no	—
56	C₂²×C₈	—	67	1	no	—
57	C₂×C₁₆	—	67	1	no	—
58	C₂×C₁₆	—	134	1	no	—
59	C₂×C₁₆	—	134	1	no	—
60	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
61	C₃₂	—	134	1	no	—
62	C₃₂	D₂₆₈	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₆₇	1	1	yes	characteristic
64	C₂²×C₄	—	67	1	no	—
65	C₂×C₈	—	67	1	no	—
66	C₂×C₈	—	134	1	no	—
67	C₂×C₈	—	134	1	no	—
68	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
69	C₁₆	—	134	1	no	—
70	C₁₆	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
71	C₁₆	#11761611	1	1	yes	characteristic
72	C₂³	—	67	1	no	—
73	C₂×C₄	—	67	1	no	—
74	C₂×C₄	—	134	1	no	—
75	C₂×C₄	—	134	1	no	—
76	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
77	C₈	—	134	1	no	—
78	C₈	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
79	C₈	#14523982	1	1	yes	characteristic
80	C₂²	—	268	1	no	—
81	C₂²	—	134	1	no	—
82	C₂²	C₃₂×D₆₇	1	1	yes	characteristic
83	C₄	—	134	1	no	—
84	C₄	(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
85	C₄	#16863229	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	268	1	no	—
88	C₂	(C₂×D₆₇)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₆₇
Fitting	C₆₇×M₁₂₈	C₂
socle	C₂×C₆₇	C₂²⋊C₃₂
prime radical	C₂²×D₆₇	C₃₂
cosocle	C₂×C₃₂×C₆₇	C₂²
hypercenter	M₁₂₈	D₆₇
hypocenter	C₆₇	#7513
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #22558420	C₂	semidirect
3. C₆₇×M₁₂₈	C₂	semidirect
47. M₁₂₈	D₆₇	semidirect
48. C₆₇	#7513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863229	2
#22558420	2
C₆₇×M₁₂₈	2
#7513	67