Subgroups — C₁₂₇×#7198 #44570871

Order 32512.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2162	C₂	1	2	yes	maximal
2	C₁₂₇×#1452	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₂₇	C₄	1	2	yes	—
4	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
6	C₂²⋊C₁₆×C₁₂₇	C₄	1	4	yes	—
7	C₄×C₁₆×C₁₂₇	—	2	2	no	—
8	#322×C₁₂₇	—	2	2	no	—
9	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₁₂₇	—	2	4	no	—
12	C₄⋊C₈×C₁₂₇	C₈	1	4	yes	—
13	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
14	C₄×C₈×C₁₂₇	—	2	2	no	—
15	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
16	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
20	C₂×C₈×C₁₂₇	—	2	4	no	—
21	C₂×C₈×C₁₂₇	—	4	2	no	—
22	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
24	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
27	C₁₆×C₁₂₇	—	4	4	no	—
28	C₂³×C₁₂₇	#97	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	4	no	—
31	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
35	C₈×C₁₂₇	—	2	2	no	—
36	C₈×C₁₂₇	—	2	2	no	—
37	C₈×C₁₂₇	—	4	4	no	—
38	C₂²×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	#336	1	1	yes	characteristic
41	C₄×C₁₂₇	—	2	1	no	—
42	C₄×C₁₂₇	—	2	2	no	—
43	C₄×C₁₂₇	#344	1	2	yes	—
44	C₄×C₁₂₇	#350	1	4	yes	—
45	#7198	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₁₂₇	—	4	1	no	—
47	C₂×C₁₂₇	#1281	1	1	yes	characteristic
48	C₂×C₁₂₇	#1379	1	2	yes	—
49	#2162	C₂×C₁₂₇	1	2	yes	—
50	#1452	C₂×C₁₂₇	1	1	yes	characteristic
51	C₁₂₇	#7198	1	1	yes	characteristic, direct product factor, Sylow
52	#431	C₄×C₁₂₇	1	2	yes	—
53	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
54	C₄⋊C₁₆	—	2	2	no	—
55	C₂²⋊C₁₆	C₄×C₁₂₇	1	4	yes	—
56	C₄×C₁₆	—	2	2	no	—
57	#322	—	2	2	no	—
58	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
59	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
60	C₂×C₁₆	—	2	4	no	—
61	C₄⋊C₈	C₈×C₁₂₇	1	4	yes	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
65	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
66	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
67	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
68	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
72	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
73	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
74	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
75	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
76	C₁₆	—	4	4	no	—
77	C₂³	#97×C₁₂₇	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
81	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
83	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#336×C₁₂₇	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	#344×C₁₂₇	1	2	yes	—
93	C₄	#350×C₁₂₇	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	C₁₂₇×#1281	1	1	yes	characteristic
96	C₂	C₁₂₇×#1379	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#336
prime radical	C₂³×C₁₂₇	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
45. #7198	C₁₂₇	direct (N × H)
51. C₁₂₇	#7198	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2162	2
C₁₂₇×#1452	2
#7198	127

Subgroups — C127×#7198 #44570871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7198 #44570871