Subgroups — #30745196

Order 17152.

526 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559955	C₂	1	2	yes	maximal
2	C₂×#16863118	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
5	#16863151	C₄	1	8	yes	—
6	#16863118	C₄	1	16	yes	complemented
7	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
9	C₄³×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
11	C₄×C₆₇⋊C₈	C₂×C₄	1	16	yes	—
12	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₇	—	2	8	no	—
17	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
18	C₂×C₆₇⋊C₈	—	2	16	no	—
19	C₂×C₆₇⋊C₈	C₄²	1	16	yes	—
20	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
21	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
22	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	—	2	8	no	—
24	C₂×C₈×C₆₇	C₄²	1	8	yes	—
25	C₄²×C₆₇	—	2	8	no	—
26	C₄²×C₆₇	M₁₆	1	8	yes	—
27	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
28	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	8	no	—
30	C₂×C₄×C₆₇	#97	1	8	yes	—
31	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
32	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₆₇	#131	1	8	yes	—
35	C₈×C₆₇	—	2	8	no	—
36	C₆₇⋊C₈	—	2	16	no	—
37	C₂²×C₆₇	C₄³	1	1	yes	characteristic
38	C₂²×C₆₇	C₂×#97	1	2	yes	—
39	C₂²×C₆₇	#405	1	4	yes	—
40	C₄×C₆₇	—	2	8	no	—
41	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
42	C₄×C₆₇	#405	1	8	yes	—
43	#7843	—	67	1	no	maximal, Sylow
44	C₂×C₆₇	#1452	1	4	yes	—
45	C₂×C₆₇	C₄×#97	1	1	yes	characteristic
46	C₂×C₆₇	#1726	1	2	yes	—
47	#1744	—	67	2	no	—
48	C₄²×C₈	D₆₇	1	1	yes	characteristic
49	C₂×#322	—	67	4	no	—
50	C₆₇	#7843	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄×C₈	—	67	4	no	—
52	C₂×C₄×C₈	Dic₆₇	1	4	yes	—
53	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
54	C₄³	C₂×D₆₇	1	1	yes	characteristic
55	#336	—	67	8	no	—
56	#322	—	67	16	no	—
57	C₂²×C₈	—	67	8	no	—
58	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
59	C₂×C₄²	C₄×D₆₇	1	4	yes	—
60	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
61	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
62	C₄×C₈	—	2	8	no	—
63	C₄×C₈	—	67	16	no	—
64	C₄×C₈	C₄×D₆₇	1	8	yes	—
65	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
66	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
67	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
68	C₂×C₈	—	2	8	no	—
69	C₂×C₈	—	67	16	no	—
70	C₂×C₈	—	134	16	no	—
71	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
72	C₄²	—	2	8	no	—
73	C₄²	#11761612	1	8	yes	—
74	C₄²	C₂×C₄×D₆₇	1	4	yes	—
75	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	#14523954	1	8	yes	—
78	C₂×C₄	C₄²×D₆₇	1	4	yes	—
79	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
80	C₂×C₄	C₂×#11761612	1	4	yes	—
81	C₂×C₄	#14524049	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	134	16	no	—
84	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
85	C₂²	C₂×#14523954	1	2	yes	—
86	C₂²	#16863351	1	4	yes	—
87	C₄	—	2	8	no	—
88	C₄	C₄×#11761612	1	4	yes	—
89	C₄	#16863350	1	8	yes	—
90	C₂	#22558501	1	4	yes	—
91	C₂	C₄×#14523954	1	1	yes	characteristic
92	C₂	#22559895	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₄³
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#7843
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #16863118	C₄	semidirect
50. C₆₇	#7843	semidirect

Maximal subgroups

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

subgroup	index
#22559955	2
C₂×#16863118	2
C₄²×C₈×C₆₇	2
#7843	67