Subgroups — #30745206

Order 17152.

582 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523953	C₂	1	3	yes	maximal
2	C₂×#16863117	C₂	1	3	yes	maximal
3	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523953	C₄	1	12	yes	—
5	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
6	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
7	#16863117	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₈²×C₆₇	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
12	#14523953	—	2	24	no	—
13	C₄×C₆₇⋊C₈	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
17	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
18	C₂²×Dic₆₇	C₄²	1	4	yes	—
19	C₂×C₆₇⋊C₈	—	2	12	no	—
20	C₂×C₆₇⋊C₈	C₄²	1	12	yes	—
21	C₄×Dic₆₇	—	2	12	no	—
22	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
23	C₂×C₈×C₆₇	C₄²	1	12	yes	—
24	C₂×C₈×C₆₇	M₁₆	1	24	yes	—
25	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
26	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
31	C₂×Dic₆₇	—	2	12	no	—
32	C₈×C₆₇	#97	1	24	yes	complemented
33	C₆₇⋊C₈	—	2	12	no	—
34	C₂²×C₆₇	C₄³	1	1	yes	characteristic
35	C₂²×C₆₇	C₄×M₁₆	1	6	yes	—
36	C₄×C₆₇	C₂×#97	1	6	yes	—
37	C₄×C₆₇	C₄×M₁₆	1	6	yes	—
38	Dic₆₇	—	4	4	no	—
39	#7846	—	67	1	no	maximal, Sylow
40	C₂×C₆₇	#1455	1	4	yes	—
41	C₂×C₆₇	C₄×#97	1	3	yes	—
42	C₄×#97	—	67	3	no	—
43	C₂×#322	—	67	3	no	—
44	C₂×C₈²	D₆₇	1	1	yes	characteristic
45	C₆₇	#7846	1	1	yes	characteristic, complemented, Sylow
46	C₂×#97	—	67	12	no	—
47	C₂×C₄×C₈	—	67	3	no	—
48	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
49	C₄³	—	67	1	no	—
50	#322	—	67	12	no	—
51	C₈²	Dic₆₇	1	4	yes	complemented
52	C₂²×C₈	—	67	6	no	—
53	C₂²×C₈	C₄×D₆₇	1	6	yes	—
54	C₂×C₄²	—	67	6	no	—
55	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
56	#97	—	134	24	no	—
57	C₄×C₈	—	67	12	no	—
58	C₄×C₈	C₄×D₆₇	1	6	yes	—
59	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
60	C₂²×C₄	—	67	4	no	—
61	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
62	C₂×C₈	—	67	12	no	—
63	C₂×C₈	—	134	12	no	—
64	C₂×C₈	C₄×Dic₆₇	1	12	yes	—
65	C₂×C₈	#11761611	1	24	yes	—
66	C₄²	—	134	12	no	—
67	C₄²	C₂×C₄×D₆₇	1	3	yes	—
68	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
69	C₂³	C₄²×D₆₇	1	1	yes	characteristic
70	C₂×C₄	—	134	12	no	—
71	C₂×C₄	C₄²×D₆₇	1	3	yes	—
72	C₂×C₄	C₂×C₄×Dic₆₇	1	3	yes	—
73	C₂×C₄	C₂×#11761611	1	12	yes	—
74	C₈	—	134	12	no	—
75	C₈	#14523953	1	24	yes	complemented
76	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
77	C₂²	C₄×#11761611	1	6	yes	—
78	C₄	—	268	4	no	—
79	C₄	C₂×#14523953	1	6	yes	—
80	C₄	C₄×#11761611	1	6	yes	—
81	C₂	#22558511	1	4	yes	—
82	C₂	C₄×#14523953	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₄³
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#7846
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. #16863117	C₄	semidirect
9. C₈²×C₆₇	C₄	semidirect
32. C₈×C₆₇	#97	semidirect
45. C₆₇	#7846	semidirect
51. C₈²	Dic₆₇	semidirect
75. C₈	#14523953	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523953	2
C₂×#16863117	2
C₂×C₈²×C₆₇	2
#7846	67