Subgroups — #30636676

Order 16256.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549602	C₂	1	2	yes	maximal
2	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
5	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₈	—	4	2	no	—
9	C₂²×C₄×C₁₂₇	—	2	1	no	—
10	C₂²×C₄×C₁₂₇	—	2	1	no	—
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
13	C₄⋊C₄×C₁₂₇	—	4	1	no	—
14	C₄⋊C₄×C₁₂₇	—	4	1	no	—
15	C₄⋊C₄×C₁₂₇	—	4	4	no	—
16	C₄²×C₁₂₇	—	2	1	no	—
17	C₄²×C₁₂₇	C₈	1	2	yes	complemented
18	C₁₂₇⋊C₈	—	8	2	no	—
19	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	2	2	no	—
21	C₂×C₄×C₁₂₇	—	2	2	no	—
22	C₂×C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	—	2	1	no	—
25	C₂×C₄×C₁₂₇	—	2	2	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	—	4	2	no	—
28	C₂×C₄×C₁₂₇	—	4	1	no	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
31	C₂²×C₁₂₇	—	2	1	no	—
32	C₂²×C₁₂₇	—	2	1	no	—
33	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂²×C₁₂₇	#99	1	1	yes	characteristic
35	C₂²×C₁₂₇	#101	1	1	yes	characteristic
36	C₄×C₁₂₇	—	2	4	no	—
37	C₄×C₁₂₇	—	2	1	no	—
38	C₄×C₁₂₇	—	2	1	no	—
39	C₄×C₁₂₇	—	4	2	no	—
40	C₄×C₁₂₇	—	8	1	no	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	—	2	1	no	—
43	C₂×C₁₂₇	#324	1	1	yes	characteristic
44	C₂×C₁₂₇	#353	1	1	yes	characteristic
45	C₂×C₁₂₇	#356	1	1	yes	characteristic
46	#1325	—	127	1	no	maximal, Sylow
47	C₁₂₇	#1325	1	1	yes	characteristic, complemented, Sylow
48	#384	D₁₂₇	1	1	yes	characteristic
49	#324	—	127	2	no	—
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	Dic₁₂₇	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
54	C₂²⋊C₈	—	254	2	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
59	C₂×C₈	—	508	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄²	—	2	1	no	—
64	C₄²	C₁₂₇⋊C₈	1	2	yes	complemented
65	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#14523232	1	1	yes	characteristic
77	C₈	—	1016	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
81	C₂²	#16861124	1	1	yes	characteristic
82	C₂²	#16861128	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	8	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#22549602	1	1	yes	characteristic
91	C₂	#22549669	1	1	yes	characteristic
92	C₂	#22549675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861124
commutator	C₂×C₄×C₁₂₇	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₁₂₇
Fitting	#384×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#99
prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₁₂₇	C₂²
hypercenter	#384	D₁₂₇
hypocenter	C₁₂₇	#1325
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #384

Complements

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

N	H ≅ G/N	type
17. C₄²×C₁₂₇	C₈	semidirect
47. C₁₂₇	#1325	semidirect
64. C₄²	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549602	2
#384×C₁₂₇	2
#1325	127