Subgroups — #30642011

Order 16256.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₀₈	C₂	1	6	yes	maximal, complemented
2	#22549813	C₂	1	4	yes	maximal, complemented
3	#22549786	C₂	1	4	yes	maximal, complemented
4	#376×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₀₈	C₂²	1	6	yes	—
6	C₂²×C₄×D₁₂₇	C₂²	1	4	yes	—
7	C₄×D₅₀₈	—	2	24	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	6	yes	—
9	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	12	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
11	#95×C₁₂₇	C₂²	1	4	yes	—
12	C₂³×D₁₂₇	C₂³	1	4	yes	—
13	C₂×D₅₀₈	—	2	24	no	—
14	C₂×D₅₀₈	—	2	12	no	—
15	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
16	C₂×C₄×D₁₂₇	—	2	24	no	—
17	C₄⋊Dic₁₂₇	—	2	12	no	—
18	(C₂×D₁₂₇)⋊C₄	—	2	24	no	—
19	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
20	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₂₇	D₄	1	12	yes	complemented
22	C₂²×D₁₂₇	—	2	12	no	—
23	C₂²×D₁₂₇	—	2	12	no	—
24	C₂²×D₁₂₇	#41	1	8	yes	—
25	D₅₀₈	—	2	48	no	—
26	C₂×Dic₁₂₇	—	2	12	no	—
27	C₄×D₁₂₇	—	4	16	no	—
28	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	12	no	—
30	C₂×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
32	C₂²×C₁₂₇	C₂²×D₄	1	3	yes	—
33	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
34	C₂×D₁₂₇	—	2	24	no	—
35	C₂×D₁₂₇	—	4	12	no	—
36	C₂×D₁₂₇	—	4	4	no	—
37	C₄×C₁₂₇	—	2	12	no	—
38	C₄×C₁₂₇	#127	1	4	yes	—
39	Dic₁₂₇	—	4	4	no	—
40	C₂×C₁₂₇	#525	1	6	yes	—
41	C₂×C₁₂₇	C₂×#127	1	1	yes	characteristic
42	D₁₂₇	—	4	8	no	—
43	#2397	—	127	1	no	maximal, Sylow
44	C₁₂₇	#2397	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×D₄	—	127	6	no	—
46	#399	—	127	4	no	—
47	#392	—	127	4	no	—
48	#376	D₁₂₇	1	1	yes	characteristic, complemented
49	C₂²×D₄	—	127	6	no	—
50	C₂³×C₄	—	127	4	no	—
51	C₄×D₄	—	254	24	no	—
52	C₂×C₄⋊C₄	—	127	6	no	—
53	C₂×C₂²⋊C₄	—	127	12	no	—
54	C₂×C₄²	C₂×D₁₂₇	1	3	yes	—
55	#95	C₂×D₁₂₇	1	4	yes	—
56	C₂⁴	—	127	4	no	—
57	C₂×D₄	—	254	24	no	—
58	C₂×D₄	—	254	12	no	—
59	C₂²×C₄	—	127	4	no	—
60	C₂²×C₄	—	254	24	no	—
61	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
62	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₄⋊C₄	—	254	12	no	—
64	C₂²⋊C₄	—	254	24	no	—
65	C₄²	D₅₀₈	1	12	yes	complemented
66	C₂³	—	127	8	no	—
67	C₂³	—	254	12	no	—
68	C₂³	—	254	12	no	—
69	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
70	D₄	—	254	48	no	—
71	C₂×C₄	—	2	12	no	—
72	C₂×C₄	—	508	16	no	—
73	C₂×C₄	—	254	12	no	—
74	C₂×C₄	C₂×D₅₀₈	1	12	yes	—
75	C₂×C₄	C₂×D₅₀₈	1	6	yes	—
76	C₂²	—	508	12	no	—
77	C₂²	—	508	4	no	—
78	C₂²	—	254	24	no	—
79	C₂²	C₂²×D₅₀₈	1	3	yes	—
80	C₂²	#41×D₁₂₇	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	508	4	no	—
83	C₄	#16861206	1	4	yes	—
84	C₂	—	508	8	no	—
85	C₂	#22550388	1	6	yes	—
86	C₂	C₂×#16861206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator, socle, cosocle	C₂³×C₁₂₇	C₂⁴
Frattini	C₂³	C₂³×D₁₂₇
Fitting	#376×C₁₂₇	C₂
hypercenter	#376	D₁₂₇
hypocenter	C₁₂₇	#2397
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₅₀₈	C₂	semidirect
2. #22549813	C₂	semidirect
3. #22549786	C₂	semidirect
4. #376×C₁₂₇	C₂	semidirect
21. C₄²×C₁₂₇	D₄	semidirect
44. C₁₂₇	#2397	semidirect
48. #376	D₁₂₇	semidirect
65. C₄²	D₅₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₀₈	2
#22549813	2
#22549786	2
#376×C₁₂₇	2
#2397	127