Subgroups — #30636812

Order 16256.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861149	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861149	—	2	2	no	—
4	#16861149	C₄	1	4	yes	complemented
5	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
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	—	2	2	no	—
10	C₁₂₇⋊C₁₆	—	2	4	no	—
11	C₂²×C₄×C₁₂₇	—	2	1	no	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	—	2	1	no	—
14	C₂×C₈×C₁₂₇	—	2	1	no	—
15	C₂×C₈×C₁₂₇	C₈	1	2	yes	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
20	C₄²×C₁₂₇	—	2	1	no	—
21	C₄²×C₁₂₇	C₈	1	2	yes	—
22	C₂³×C₁₂₇	M₁₆	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	—	2	4	no	—
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	2	no	—
32	C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
33	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	#97	1	1	yes	characteristic
36	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₁₂₇	—	2	4	no	—
39	C₄×C₁₂₇	#95	1	1	yes	characteristic
40	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₁₂₇	—	2	1	no	—
44	C₂×C₁₂₇	—	2	1	no	—
45	C₂×C₁₂₇	#336	1	1	yes	characteristic
46	C₂×C₁₂₇	#364	1	2	yes	—
47	#1376	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1376	1	1	yes	characteristic, complemented, Sylow
49	C₂×M₃₂	—	127	2	no	—
50	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
51	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
54	M₃₂	—	127	4	no	—
55	M₃₂	—	254	2	no	—
56	C₂×C₁₆	—	254	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	C₁₂₇⋊C₈	1	2	yes	—
63	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
64	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
65	C₂×C₈	D₅₀₈	1	1	yes	characteristic
66	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
67	C₄²	—	2	1	no	—
68	C₄²	C₁₂₇⋊C₈	1	2	yes	—
69	C₁₆	—	254	4	no	—
70	C₂³	#14523232	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
75	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
77	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
78	C₂×C₄	#14523232	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
81	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#16861120	1	1	yes	characteristic
84	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
85	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
88	C₄	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
89	C₄	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
90	C₄	C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#22549631	1	1	yes	characteristic
94	C₂	#22549697	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₅₀₈
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄⋊C₈
prime radical	C₂³×C₁₂₇	M₁₆
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1376
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #16861149	C₄	semidirect
48. C₁₂₇	#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861149	2
C₂×C₄×C₈×C₁₂₇	2
#1376	127