Subgroups — #30749598

Order 17152.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560366	C₂	1	6	yes	maximal
2	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523975	—	2	12	no	—
4	#16863266	C₂²	1	4	yes	—
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	#14523975	—	4	24	no	—
9	C₂²×C₈×C₆₇	D₄	1	6	yes	—
10	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₆₇	Q₈	1	6	yes	—
13	C₂²×Dic₆₇	—	4	4	no	—
14	C₄⋊Dic₆₇	—	4	24	no	—
15	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
16	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
17	C₂×C₈×C₆₇	SD₁₆	1	24	yes	—
18	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
19	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
20	C₂³×C₆₇	#127	1	1	yes	characteristic
21	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
22	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
23	C₂×C₄×C₆₇	C₂×SD₁₆	1	12	yes	—
24	C₂×Dic₆₇	—	8	12	no	—
25	C₈×C₆₇	#106	1	24	yes	complemented
26	C₂²×C₆₇	#384	1	1	yes	characteristic
27	C₂²×C₆₇	#492	1	6	yes	—
28	C₄×C₆₇	C₂×#106	1	6	yes	—
29	C₄×C₆₇	#498	1	6	yes	—
30	Dic₆₇	—	16	4	no	—
31	#8813	—	67	1	no	maximal, Sylow
32	C₂×C₆₇	#1705	1	4	yes	—
33	C₂×C₆₇	#1844	1	3	yes	—
34	#1844	—	67	6	no	—
35	C₂×C₈²	D₆₇	1	1	yes	characteristic
36	C₆₇	#8813	1	1	yes	characteristic, complemented, Sylow
37	C₂×#106	—	134	12	no	—
38	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
39	#384	—	67	4	no	—
40	C₈²	Dic₆₇	1	4	yes	complemented
41	C₂²×C₈	D₂₆₈	1	6	yes	—
42	C₂×C₄⋊C₄	—	134	12	no	—
43	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
44	#106	—	268	24	no	—
45	C₄×C₈	Dic₁₃₄	1	6	yes	—
46	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
47	C₂²×C₄	—	268	4	no	—
48	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
49	C₂×C₈	C₄⋊Dic₆₇	1	12	yes	—
50	C₂×C₈	#11761617	1	24	yes	—
51	C₄⋊C₄	—	268	24	no	—
52	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
53	C₄²	C₂×Dic₁₃₄	1	3	yes	—
54	C₂³	#14524040	1	1	yes	characteristic
55	C₂×C₄	—	536	12	no	—
56	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
57	C₂×C₄	#14524043	1	3	yes	—
58	C₂×C₄	C₂×#11761617	1	12	yes	—
59	C₈	#14523975	1	24	yes	complemented
60	C₂²	#16863266	1	1	yes	characteristic
61	C₂²	#16863769	1	6	yes	—
62	C₄	—	1072	4	no	—
63	C₄	C₂×#14523975	1	6	yes	—
64	C₄	#16863794	1	6	yes	—
65	C₂	#22559796	1	4	yes	—
66	C₂	#22560366	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	#8813
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
25. C₈×C₆₇	#106	semidirect
36. C₆₇	#8813	semidirect
40. C₈²	Dic₆₇	semidirect
59. C₈	#14523975	semidirect

Maximal subgroups

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

subgroup	index
#22560366	2
C₂×C₈²×C₆₇	2
#8813	67