Subgroups — #30636611

Order 16256.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861120	C₂	1	2	yes	maximal
2	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
4	#16861120	—	2	2	no	—
5	#16861120	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	#95×C₁₂₇	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	8	no	—
10	C₂×C₁₂₇⋊C₈	—	2	4	no	—
11	C₂²×C₄×C₁₂₇	—	2	2	no	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
14	C₄⋊C₄×C₁₂₇	—	4	2	no	—
15	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
18	C₁₂₇⋊C₈	—	2	8	no	—
19	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄×C₁₂₇	—	2	1	no	—
22	C₂×C₄×C₁₂₇	—	2	4	no	—
23	C₂×C₄×C₁₂₇	—	4	2	no	—
24	C₂×C₄×C₁₂₇	—	4	4	no	—
25	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₂₇	#95	1	1	yes	characteristic
30	C₂²×C₁₂₇	#100	1	1	yes	characteristic
31	C₂²×C₁₂₇	#101	1	1	yes	characteristic
32	C₂²×C₁₂₇	#104	1	4	yes	—
33	C₄×C₁₂₇	—	2	2	no	—
34	C₄×C₁₂₇	—	2	2	no	—
35	C₄×C₁₂₇	—	2	1	no	—
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	—	8	2	no	—
38	C₂×C₁₂₇	#329	1	2	yes	—
39	C₂×C₁₂₇	#330	1	2	yes	—
40	C₂×C₁₂₇	#339	1	2	yes	—
41	C₂×C₁₂₇	#343	1	1	yes	characteristic
42	#1295	—	127	1	no	maximal, Sylow
43	C₁₂₇	#1295	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	127	2	no	—
45	#382	D₁₂₇	1	1	yes	characteristic
46	C₂²×C₈	—	254	2	no	—
47	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
48	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
49	#97	—	127	4	no	—
50	#97	—	254	2	no	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
55	C₂×C₈	—	254	8	no	—
56	C₂×C₈	—	254	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	C₄×D₁₂₇	1	2	yes	—
59	C₄²	D₅₀₈	1	1	yes	characteristic
60	C₄²	Dic₂₅₄	1	1	yes	characteristic
61	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
68	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
69	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
71	C₈	—	254	8	no	—
72	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
73	C₂²	#16861126	1	1	yes	characteristic
74	C₂²	#16861128	1	1	yes	characteristic
75	C₂²	#16861136	1	4	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	8	2	no	—
81	C₂	#22549616	1	2	yes	—
82	C₂	#22549618	1	2	yes	—
83	C₂	#22549635	1	2	yes	—
84	C₂	#22549645	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	#382×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	#382	D₁₂₇
hypocenter	C₁₂₇	#1295
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
5. #16861120	C₄	semidirect
43. C₁₂₇	#1295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861120	2
#382×C₁₂₇	2
#1295	127