Subgroups — #30772233

Order 17152.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558287	C₂	1	6	yes	maximal
2	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
4	C₂×#97×C₆₇	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
6	#376×C₆₇	C₄	1	1	yes	characteristic
7	C₂×C₆₇⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
11	#97×C₆₇	—	2	6	no	—
12	C₄×C₈×C₆₇	—	2	6	no	—
13	#95×C₆₇	C₈	1	4	yes	—
14	C₆₇⋊C₁₆	—	4	4	no	—
15	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₈×C₆₇	—	2	3	no	—
18	C₂×C₈×C₆₇	—	2	3	no	—
19	C₂×C₈×C₆₇	M₁₆	1	6	yes	—
20	C₂×C₈×C₆₇	#41	1	6	yes	—
21	C₄²×C₆₇	—	2	3	no	—
22	C₄²×C₆₇	—	2	3	no	—
23	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₆₇	—	2	6	no	—
25	C₂×C₄×C₆₇	—	2	12	no	—
26	C₂×C₄×C₆₇	#117	1	3	yes	—
27	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
28	C₈×C₆₇	—	2	6	no	—
29	C₈×C₆₇	#126	1	4	yes	—
30	C₂²×C₆₇	#431	1	3	yes	—
31	C₂²×C₆₇	#504	1	4	yes	—
32	C₄×C₆₇	—	4	6	no	—
33	C₄×C₆₇	#383	1	1	yes	characteristic
34	C₄×C₆₇	#432	1	3	yes	—
35	#12895	—	67	1	no	maximal, Sylow
36	C₂×C₆₇	#1839	1	1	yes	characteristic
37	C₂×C₆₇	#2164	1	6	yes	—
38	#1726	D₆₇	1	1	yes	characteristic
39	#1374	—	67	6	no	—
40	C₆₇	#12895	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₁₆	—	67	4	no	—
42	C₂×#97	Dic₆₇	1	3	yes	—
43	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
44	#376	Dic₆₇	1	1	yes	characteristic
45	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
46	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
48	C₂×C₁₆	—	134	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	C₆₇⋊C₈	1	4	yes	—
52	C₂²×C₄	C₂×C₆₇⋊C₈	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11761612	1	6	yes	—
57	C₂×C₈	#11761628	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	268	4	no	—
61	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	#14524002	1	3	yes	—
65	C₂×C₄	C₂×#11761612	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#14524037	1	4	yes	—
68	C₂²	#16863449	1	3	yes	—
69	C₂²	#16863813	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#16863264	1	1	yes	characteristic
72	C₄	#16863453	1	3	yes	—
73	C₂	#22560352	1	1	yes	characteristic
74	C₂	#22562016	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₆₇×#1726	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#12895
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
40. C₆₇	#12895	semidirect

Maximal subgroups

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

subgroup	index
#22558287	2
C₆₇×#1726	2
#12895	67