Subgroups — #30749610

Order 17152.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560337	C₂	1	6	yes	maximal
2	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863266	C₂²	1	4	yes	—
4	#16863159	—	2	12	no	—
5	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
6	C₈²×C₆₇	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₆₇	—	2	12	no	—
8	C₂²×C₈×C₆₇	D₄	1	6	yes	—
9	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₆₇	Q₈	1	6	yes	—
12	C₂²×Dic₆₇	—	4	4	no	—
13	C₄⋊Dic₆₇	—	4	24	no	—
14	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₆₇	—	2	12	no	—
16	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
17	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
19	C₂³×C₆₇	#127	1	1	yes	characteristic
20	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
21	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
22	C₂×C₄×C₆₇	#135	1	12	yes	—
23	C₂×Dic₆₇	—	8	12	no	—
24	C₈×C₆₇	—	2	12	no	—
25	C₂²×C₆₇	#384	1	1	yes	characteristic
26	C₂²×C₆₇	#495	1	6	yes	—
27	C₄×C₆₇	#427	1	6	yes	—
28	C₄×C₆₇	#499	1	6	yes	—
29	Dic₆₇	—	16	4	no	—
30	#8817	—	67	1	no	maximal, Sylow
31	C₂×C₆₇	#1711	1	4	yes	—
32	C₂×C₆₇	#1846	1	3	yes	—
33	#1834	—	67	6	no	—
34	C₂×C₈²	D₆₇	1	1	yes	characteristic
35	C₆₇	#8817	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
37	#384	—	67	4	no	—
38	#340	—	134	12	no	—
39	C₈²	Dic₆₇	1	4	yes	complemented
40	C₂²×C₈	D₂₆₈	1	6	yes	—
41	C₂×C₄⋊C₄	—	134	12	no	—
42	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
43	C₄×C₈	Dic₁₃₄	1	6	yes	—
44	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
45	C₂²×C₄	—	268	4	no	—
46	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊Dic₆₇	1	12	yes	—
49	C₄⋊C₄	—	268	24	no	—
50	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
51	C₄²	C₂×Dic₁₃₄	1	3	yes	—
52	C₂³	#14524040	1	1	yes	characteristic
53	C₂×C₄	—	536	12	no	—
54	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
55	C₂×C₄	#14524043	1	3	yes	—
56	C₂×C₄	#14524065	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#16863266	1	1	yes	characteristic
59	C₂²	#16863780	1	6	yes	—
60	C₄	—	1072	4	no	—
61	C₄	#16863434	1	6	yes	—
62	C₄	#16863798	1	6	yes	—
63	C₂	#22559821	1	4	yes	—
64	C₂	#22560373	1	3	yes	—

	subgroup	G/H
center	C₂³	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#8817
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₆₇	C₄	semidirect
35. C₆₇	#8817	semidirect
39. C₈²	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560337	2
C₂×C₈²×C₆₇	2
#8817	67