Subgroups — #30638801

Order 16256.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	#22549631	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
5	C₂²⋊(C₁₂₇⋊C₈)	—	2	4	no	—
6	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	#117×C₁₂₇	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₁₂₇⋊C₈	—	2	4	no	—
12	C₂×C₁₂₇⋊C₈	—	2	8	no	—
13	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	—	2	4	no	—
15	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₂₇	—	2	4	no	—
20	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
21	C₄²×C₁₂₇	—	2	4	no	—
22	C₁₂₇⋊C₈	—	4	4	no	—
23	C₂³×C₁₂₇	—	2	2	no	—
24	C₂³×C₁₂₇	—	2	4	no	—
25	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	—	2	8	no	—
29	C₂×C₄×C₁₂₇	—	2	4	no	—
30	C₂×C₄×C₁₂₇	—	2	8	no	—
31	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	#41	1	4	yes	—
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	—	2	4	no	—
36	C₂²×C₁₂₇	—	4	4	no	—
37	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₂₇	#117	1	2	yes	—
39	C₂²×C₁₂₇	#131	1	4	yes	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	—	4	4	no	—
42	C₄×C₁₂₇	#123	1	4	yes	—
43	C₂×C₁₂₇	—	4	2	no	—
44	C₂×C₁₂₇	#381	1	1	yes	characteristic
45	C₂×C₁₂₇	#408	1	2	yes	—
46	C₂×C₁₂₇	#433	1	4	yes	—
47	#1818	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1818	1	1	yes	characteristic, complemented, Sylow
49	C₂×#117	D₁₂₇	1	1	yes	characteristic
50	C₂×C₂²⋊C₈	—	127	2	no	—
51	#336	—	127	4	no	—
52	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂²×C₈	—	127	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₁₂₇	1	2	yes	—
57	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
58	C₂²⋊C₈	—	254	4	no	—
59	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
62	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₁₂₇	1	4	yes	—
64	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₂×C₈	—	254	4	no	—
66	C₂×C₈	—	254	8	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	4	no	—
71	C₂³	—	2	2	no	—
72	C₂³	C₂²⋊Dic₁₂₇	1	2	yes	—
73	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
80	C₂×C₄	#14523248	1	4	yes	—
81	C₈	—	508	4	no	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
86	C₂²	#16861168	1	2	yes	—
87	C₂²	#16861216	1	4	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#16861192	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#22549739	1	1	yes	characteristic
93	C₂	#22549843	1	2	yes	—
94	C₂	#22549937	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator, socle	C₂³×C₁₂₇	C₂²×C₄
Fitting	C₂×#117×C₁₂₇	C₂
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#1818
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #22549631	C₂	semidirect
48. C₁₂₇	#1818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
#22549631	2
C₂×#117×C₁₂₇	2
#1818	127