Subgroups — #30651833

Order 16256.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550293	C₂	1	8	yes	maximal, complemented
2	C₂×#16861149	C₂	1	6	yes	maximal, complemented
3	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861149	C₂²	1	4	yes	—
5	#16861149	C₂²	1	12	yes	complemented
6	C₂×C₁₂₇⋊C₁₆	C₂²	1	12	yes	complemented
7	C₂×#41×C₁₂₇	C₄	1	1	yes	characteristic
8	#131×C₁₂₇	C₄	1	4	yes	—
9	#131×C₁₂₇	C₂²	1	4	yes	—
10	C₂×M₁₆×C₁₂₇	C₄	1	3	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	3	yes	—
12	C₁₂₇⋊C₁₆	C₂³	1	8	yes	—
13	#41×C₁₂₇	C₈	1	4	yes	—
14	#41×C₁₂₇	C₂×C₄	1	4	yes	—
15	C₂×Q₈×C₁₂₇	C₈	1	1	yes	characteristic
16	C₂×D₄×C₁₂₇	C₈	1	3	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
18	M₁₆×C₁₂₇	C₂×C₄	1	12	yes	—
19	C₂×C₈×C₁₂₇	C₂×C₄	1	3	yes	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	6	yes	—
21	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂³	1	6	yes	—
23	C₂³×C₁₂₇	C₂×C₈	1	3	yes	—
24	Q₈×C₁₂₇	C₂×C₈	1	4	yes	—
25	D₄×C₁₂₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₂₇	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
30	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₂₇	C₂²×C₄	1	6	yes	—
32	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₂₇	—	4	3	no	—
34	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂²×C₈	1	6	yes	—
36	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₁₂₇	C₂²×C₈	1	6	yes	—
38	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
39	C₂×C₁₂₇	—	2	1	no	—
40	C₂×C₁₂₇	—	2	6	no	—
41	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
42	#3407	—	127	1	no	maximal, Sylow
43	C₁₂₇	#3407	1	1	yes	characteristic, complemented, Sylow
44	C₂×#131	D₁₂₇	1	1	yes	characteristic
45	#504	—	127	8	no	—
46	C₂×M₃₂	—	127	6	no	—
47	C₂×#41	Dic₁₂₇	1	1	yes	characteristic
48	#131	Dic₁₂₇	1	4	yes	—
49	#131	C₂×D₁₂₇	1	4	yes	—
50	C₂×M₁₆	Dic₁₂₇	1	3	yes	—
51	C₂²×C₈	C₂×D₁₂₇	1	3	yes	—
52	M₃₂	—	127	4	no	—
53	M₃₂	—	127	12	no	—
54	C₂×C₁₆	—	127	12	no	—
55	#41	C₁₂₇⋊C₈	1	4	yes	—
56	#41	C₂×Dic₁₂₇	1	4	yes	—
57	C₂×Q₈	C₁₂₇⋊C₈	1	1	yes	characteristic
58	C₂×D₄	C₁₂₇⋊C₈	1	3	yes	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	3	yes	—
60	M₁₆	C₂×Dic₁₂₇	1	12	yes	—
61	C₂×C₈	C₂×Dic₁₂₇	1	3	yes	—
62	C₂×C₈	C₂×Dic₁₂₇	1	6	yes	—
63	C₂×C₈	C₂²×D₁₂₇	1	6	yes	—
64	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₁₆	—	127	8	no	—
66	C₂³	C₂×C₁₂₇⋊C₈	1	3	yes	—
67	Q₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
68	D₄	C₂×C₁₂₇⋊C₈	1	12	yes	—
69	C₂×C₄	C₂×C₁₂₇⋊C₈	1	3	yes	—
70	C₂×C₄	C₂×C₁₂₇⋊C₈	1	6	yes	—
71	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
73	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
74	C₈	C₂²×Dic₁₂₇	1	6	yes	—
75	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
78	C₂²	C₂²×C₁₂₇⋊C₈	1	6	yes	—
79	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
80	C₄	C₂²×C₁₂₇⋊C₈	1	6	yes	—
81	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₁₂₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂³×C₈
Fitting	C₂×#131×C₁₂₇	C₂
prime radical	C₂×#41×C₁₂₇	C₄
cosocle	C₈×C₁₂₇	C₂⁴
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	#3407
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #22550293	C₂	semidirect
2. C₂×#16861149	C₂	semidirect
5. #16861149	C₂²	semidirect
6. C₂×C₁₂₇⋊C₁₆	C₂²	semidirect
43. C₁₂₇	#3407	semidirect

Maximal subgroups

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

subgroup	index
#22550293	2
C₂×#16861149	2
C₂×#131×C₁₂₇	2
#3407	127