Subgroups — #30637027

Order 16256.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549929	C₂	1	2	yes	maximal
2	#22549598	C₂	1	2	yes	maximal, complemented
3	#22549597	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
7	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
8	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×M₁₆×C₁₂₇	C₄	1	2	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#97×C₁₂₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	2	4	no	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₁₂₇⋊C₈	—	2	4	no	—
26	C₁₂₇⋊C₈	—	2	4	no	—
27	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	—	2	2	no	—
29	C₂×C₄×C₁₂₇	—	2	4	no	—
30	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
32	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	—	2	2	no	—
35	C₈×C₁₂₇	M₁₆	1	4	yes	—
36	C₂²×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	—	2	2	no	—
38	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
42	C₄×C₁₂₇	#131	1	4	yes	—
43	C₂×C₁₂₇	—	4	1	no	—
44	C₂×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
45	C₂×C₁₂₇	#405	1	2	yes	—
46	#1453	—	127	1	no	maximal, Sylow
47	C₁₂₇	#1453	1	1	yes	characteristic, complemented, Sylow
48	#431	—	127	2	no	—
49	C₄×M₁₆	D₁₂₇	1	1	yes	characteristic
50	#322	—	127	1	no	—
51	#322	—	127	2	no	—
52	C₈²	—	127	1	no	—
53	C₂×M₁₆	Dic₁₂₇	1	2	yes	—
54	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
55	C₄⋊C₈	—	127	4	no	—
56	C₂²⋊C₈	—	127	4	no	—
57	#97	Dic₁₂₇	1	2	yes	—
58	C₄×C₈	—	127	2	no	—
59	C₄×C₈	—	127	2	no	—
60	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
62	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
63	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	127	4	no	—
66	C₂×C₈	—	127	4	no	—
67	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
68	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
69	C₄²	C₄×D₁₂₇	1	2	yes	—
70	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
71	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
72	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
76	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
78	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	254	4	no	—
81	C₈	—	254	4	no	—
82	C₈	#14523232	1	4	yes	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	2	no	—
88	C₄	C₂×#14523232	1	2	yes	—
89	C₄	#16861217	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₄×#14523232	1	1	yes	characteristic
92	C₂	#22549834	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₄×M₁₆×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	C₄×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#1453
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22549598	C₂	semidirect
3. #22549597	C₂	semidirect
4. C₈×C₁₂₇⋊C₈	C₂	semidirect
47. C₁₂₇	#1453	semidirect

Maximal subgroups

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

subgroup	index
#22549929	2
#22549598	2
#22549597	2
C₈×C₁₂₇⋊C₈	2
C₄×M₁₆×C₁₂₇	2
#1453	127