Subgroups — #30647108

Order 16256.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861209	C₂	1	2	yes	maximal
2	#22549782	C₂	1	8	yes	maximal
3	#22549742	C₂	1	4	yes	maximal
4	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
6	#16861209	—	2	8	no	—
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
9	C₁₂₇⋊#95	C₂²	1	16	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
13	C₂×Dic₂₅₄	—	2	8	no	—
14	C₂×Dic₂₅₄	—	2	16	no	—
15	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
16	C₄⋊Dic₁₂₇	—	2	8	no	—
17	C₄⋊Dic₁₂₇	—	2	8	no	—
18	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
21	C₄²×C₁₂₇	—	2	8	no	—
22	C₄²×C₁₂₇	D₄	1	4	yes	—
23	C₄²×C₁₂₇	Q₈	1	8	yes	complemented
24	Dic₂₅₄	—	4	16	no	—
25	C₂×Dic₁₂₇	—	2	8	no	—
26	C₂×Dic₁₂₇	—	2	16	no	—
27	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	—	2	8	no	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂×Q₈	1	8	yes	—
32	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₁₂₇	#41	1	8	yes	—
34	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
36	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
37	C₄×C₁₂₇	—	2	8	no	—
38	C₄×C₁₂₇	#124	1	8	yes	—
39	C₄×C₁₂₇	C₄⋊Q₈	1	4	yes	—
40	Dic₁₂₇	—	4	8	no	—
41	C₂×C₁₂₇	C₂×#124	1	2	yes	—
42	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₁₂₇	#533	1	4	yes	—
44	#2862	—	127	1	no	maximal, Sylow
45	C₁₂₇	#2862	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄⋊Q₈	—	127	2	no	—
47	#391	—	127	8	no	—
48	#382	—	127	4	no	—
49	C₄³	D₁₂₇	1	1	yes	characteristic
50	C₂²×Q₈	—	127	4	no	—
51	C₄⋊Q₈	—	254	8	no	—
52	C₂×C₄⋊C₄	—	127	4	no	—
53	C₂×C₄⋊C₄	—	127	4	no	—
54	C₂×C₄²	C₂×D₁₂₇	1	4	yes	—
55	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
57	#95	—	127	16	no	—
58	C₂×Q₈	—	254	8	no	—
59	C₂×Q₈	—	254	16	no	—
60	C₂²×C₄	—	127	8	no	—
61	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
62	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
64	C₄⋊C₄	—	254	8	no	—
65	C₄⋊C₄	—	254	8	no	—
66	C₄²	—	2	8	no	—
67	C₄²	D₅₀₈	1	4	yes	—
68	C₄²	Dic₂₅₄	1	8	yes	complemented
69	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
70	Q₈	—	508	16	no	—
71	C₂×C₄	—	2	8	no	—
72	C₂×C₄	—	254	8	no	—
73	C₂×C₄	—	254	16	no	—
74	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
75	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
76	C₂×C₄	C₂×Dic₂₅₄	1	8	yes	—
77	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
78	C₂×C₄	#14523248	1	8	yes	—
79	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
80	C₂²	C₂²×Dic₂₅₄	1	2	yes	—
81	C₂²	C₂×#14523248	1	4	yes	—
82	C₄	—	2	8	no	—
83	C₄	—	508	8	no	—
84	C₄	#16861197	1	8	yes	—
85	C₄	#16861209	1	4	yes	—
86	C₂	C₂×#16861197	1	2	yes	—
87	C₂	C₂×#16861209	1	1	yes	characteristic
88	C₂	#22550433	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator, socle, prime radical, cosocle	C₂³×C₁₂₇	C₂⁴
Fitting	C₄³×C₁₂₇	C₂
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#2862
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
23. C₄²×C₁₂₇	Q₈	semidirect
45. C₁₂₇	#2862	semidirect
68. C₄²	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861209	2
#22549782	2
#22549742	2
C₄³×C₁₂₇	2
#2862	127