Subgroups — C₂×#22549652 #30640320

Order 16256.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₂₇⋊C₁₆	C₂	1	2	yes	maximal
2	#22549652	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	C₄	1	8	yes	complemented
5	C₂×C₁₂₇⋊C₁₆	C₂²	1	4	yes	—
6	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₁₂₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
12	C₂²×C₄×C₁₂₇	C₈	1	2	yes	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	C₈	1	4	yes	—
15	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₈	1	2	yes	—
20	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	—	2	4	no	—
23	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
24	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₂₇	C₄²	1	2	yes	—
29	C₈×C₁₂₇	C₂×C₈	1	4	yes	—
30	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₁₂₇	C₄×C₈	1	2	yes	—
32	C₂²×C₁₂₇	M₃₂	1	4	yes	—
33	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₁₂₇	—	2	4	no	—
35	C₄×C₁₂₇	C₄×C₈	1	2	yes	—
36	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₁₂₇	#346	1	4	yes	complemented
39	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×M₃₂	1	2	yes	—
41	C₂×#346	—	127	1	no	maximal, Sylow
42	C₁₂₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₁₆	—	127	2	no	—
44	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
45	#346	—	127	4	no	—
46	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
47	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
49	C₂×C₁₆	—	127	8	no	—
50	C₂×C₁₆	—	127	4	no	—
51	C₄×C₈	Dic₁₂₇	1	2	yes	—
52	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
53	C₂²×C₄	C₁₂₇⋊C₈	1	2	yes	—
54	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
55	C₂×C₈	C₁₂₇⋊C₈	1	4	yes	—
56	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
57	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
58	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
59	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
60	C₄²	C₁₂₇⋊C₈	1	2	yes	—
61	C₄²	C₂×Dic₁₂₇	1	2	yes	—
62	C₁₆	—	127	8	no	—
63	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
66	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
67	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
68	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
69	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
70	C₈	C₄×Dic₁₂₇	1	2	yes	—
71	C₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
72	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
73	C₂²	C₄×C₁₂₇⋊C₈	1	2	yes	—
74	C₂²	#16861149	1	4	yes	—
75	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×C₁₂₇⋊C₈	1	2	yes	—
78	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
79	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
80	C₂	#22549652	1	4	yes	direct product factor
81	C₂	C₂×C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
82	C₂	C₂×#16861149	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#346
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22549652	C₂	direct (N × H)
4. C₂×C₁₂₇⋊C₁₆	C₄	semidirect
11. C₁₂₇⋊C₁₆	C₂×C₄	semidirect
38. C₂×C₁₂₇	#346	semidirect
42. C₁₂₇	C₂×#346	semidirect
80. C₂	#22549652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₂₇⋊C₁₆	2
#22549652	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#346	127

Subgroups — C2×#22549652 #30640320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549652 #30640320