Subgroups — #30649254

Order 16256.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550433	C₂	1	2	yes	maximal
2	#22550285	C₂	1	8	yes	maximal, complemented
3	#22549914	C₂	1	4	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861209	C₂²	1	4	yes	—
6	#16861201	C₂²	1	4	yes	—
7	#16861188	—	2	4	no	—
8	C₄×Dic₂₅₄	—	2	4	no	—
9	#16861168	C₂²	1	4	yes	—
10	#16861143	C₂²	1	8	yes	—
11	#16861141	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	#97×C₁₂₇	C₂²	1	4	yes	—
15	C₂×Dic₂₅₄	—	2	4	no	—
16	C₄⋊Dic₁₂₇	—	2	4	no	—
17	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
18	Dic₁₂₇⋊C₄	—	2	8	no	—
19	C₂²⋊Dic₁₂₇	—	2	4	no	—
20	C₄×Dic₁₂₇	—	2	4	no	—
21	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	Q₈	1	8	yes	complemented
24	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
25	C₄²×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	—	4	8	no	—
28	C₂×Dic₁₂₇	—	2	8	no	—
29	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	8	yes	—
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	4	no	—
40	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
41	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
42	Dic₁₂₇	—	4	8	no	—
43	C₂×C₁₂₇	—	2	2	no	—
44	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₂₇	#575	1	2	yes	—
46	#3162	—	127	1	no	maximal, Sylow
47	C₁₂₇	#3162	1	1	yes	characteristic, complemented, Sylow
48	#533	—	127	2	no	—
49	#501	—	127	8	no	—
50	#427	—	127	4	no	—
51	C₂×#97	D₁₂₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
53	C₄⋊Q₈	—	127	4	no	—
54	#125	—	127	4	no	—
55	C₂²⋊Q₈	—	254	4	no	—
56	C₄×Q₈	—	254	4	no	—
57	#117	—	127	4	no	—
58	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
59	#107	—	127	8	no	—
60	#106	—	127	8	no	—
61	#97	C₂×D₁₂₇	1	4	yes	—
62	C₂×Q₈	—	254	4	no	—
63	C₂²×C₄	D₅₀₈	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₂×C₈	Dic₂₅₄	1	8	yes	complemented
66	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
67	C₄⋊C₄	—	127	8	no	—
68	C₄⋊C₄	—	254	8	no	—
69	C₄⋊C₄	—	254	4	no	—
70	C₂²⋊C₄	—	254	4	no	—
71	C₄²	—	254	4	no	—
72	C₄²	D₅₀₈	1	2	yes	—
73	C₄²	C₂²×D₁₂₇	1	2	yes	—
74	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
75	Q₈	—	508	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	254	8	no	—
78	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
79	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
81	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
82	C₈	C₂×Dic₂₅₄	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#16861209	1	2	yes	—
85	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	508	8	no	—
88	C₄	#16861209	1	2	yes	—
89	C₄	C₂²×Dic₂₅₄	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×#16861209	1	1	yes	characteristic
92	C₂	#22550676	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#3162
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22550285	C₂	semidirect
23. C₂×C₈×C₁₂₇	Q₈	semidirect
47. C₁₂₇	#3162	semidirect
65. C₂×C₈	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
#22550433	2
#22550285	2
#22549914	2
C₂×#97×C₁₂₇	2
#3162	127