Finite Group Databasebeta

← C₂×Q₃₂×D₁₂₇ #30651841

Subgroups — C₂×Q₃₂×D₁₂₇ #30651841

Order 16256.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₁₂₇	C₂	1	2	yes	maximal
2	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16861155	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₁₂₇	—	2	2	no	—
8	C₂×Dic₅₀₈	C₂²	1	2	yes	—
9	C₂×#14523238	—	2	2	no	—
10	Q₁₆×D₁₂₇	—	2	4	no	—
11	Q₁₆×D₁₂₇	C₂²	1	8	yes	—
12	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
13	Dic₁₀₁₆	C₂²	1	4	yes	complemented
14	#16861155	C₂²	1	8	yes	complemented
15	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₁₂₇	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
18	Q₃₂×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	—	4	2	no	—
22	Q₈×D₁₂₇	—	2	8	no	—
23	Q₈×D₁₂₇	—	4	4	no	—
24	C₂×C₄×D₁₂₇	—	4	2	no	—
25	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
26	Dic₅₀₈	—	2	2	no	—
27	Dic₅₀₈	C₂³	1	4	yes	—
28	#14523238	—	2	8	no	—
29	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₁₂₇	D₄	1	2	yes	—
31	C₈×D₁₂₇	C₂³	1	2	yes	—
32	C₁₂₇⋊C₁₆	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₂₇	—	2	2	no	—
34	Q₁₆×C₁₂₇	—	2	2	no	—
35	Q₁₆×C₁₂₇	C₂³	1	4	yes	—
36	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₁₆×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×D₁₂₇	D₈	1	1	yes	characteristic
39	Dic₂₅₄	—	2	4	no	—
40	Dic₂₅₄	—	4	2	no	—
41	Dic₂₅₄	—	4	8	no	—
42	C₂×Dic₁₂₇	—	4	2	no	—
43	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
44	C₄×D₁₂₇	—	4	8	no	—
45	C₄×D₁₂₇	D₈	1	2	yes	—
46	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
47	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
48	Q₈×C₁₂₇	—	2	4	no	—
49	Q₈×C₁₂₇	—	4	2	no	—
50	C₂×C₄×C₁₂₇	—	4	2	no	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
55	C₂×D₁₂₇	Q₃₂	1	4	yes	direct product factor
56	C₂×D₁₂₇	C₂×D₈	1	2	yes	—
57	C₄×C₁₂₇	—	4	4	no	—
58	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
59	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
60	Dic₁₂₇	—	4	4	no	—
61	Dic₁₂₇	C₂×D₈	1	2	yes	—
62	C₂×C₁₂₇	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
64	D₁₂₇	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	127	1	no	maximal, Sylow
66	C₁₂₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	127	2	no	—
68	C₂×Q₃₂	—	127	2	no	—
69	C₂×Q₃₂	—	127	8	no	—
70	C₂×Q₃₂	—	127	1	no	—
71	C₂×Q₃₂	D₁₂₇	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	127	1	no	—
73	C₂²×Q₈	—	254	2	no	—
74	C₂×Q₁₆	—	127	8	no	—
75	C₂×Q₁₆	—	127	2	no	—
76	C₂×Q₁₆	—	254	4	no	—
77	C₂×Q₁₆	—	254	2	no	—
78	C₂×Q₁₆	C₂×D₁₂₇	1	2	yes	—
79	C₂²×C₈	—	127	1	no	—
80	Q₃₂	—	127	8	no	—
81	Q₃₂	—	127	4	no	—
82	Q₃₂	C₂×D₁₂₇	1	4	yes	direct product factor
83	C₂×C₁₆	—	127	1	no	—
84	C₂×C₁₆	—	127	4	no	—
85	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	508	2	no	—
88	C₂×Q₈	—	508	4	no	—
89	C₂×Q₈	—	254	8	no	—
90	C₂×Q₈	—	254	2	no	—
91	C₂²×C₄	—	127	1	no	—
92	C₂²×C₄	—	508	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	127	4	no	—
95	Q₁₆	—	254	8	no	—
96	Q₁₆	—	254	2	no	—
97	Q₁₆	C₂²×D₁₂₇	1	4	yes	—
98	C₂×C₈	—	127	1	no	—
99	C₂×C₈	—	127	2	no	—
100	C₂×C₈	—	127	2	no	—
101	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
102	C₁₆	—	127	2	no	—
103	C₁₆	C₂²×D₁₂₇	1	2	yes	—
104	C₂³	—	127	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	508	8	no	—
108	Q₈	—	508	2	no	—
109	Q₈	—	254	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	127	1	no	—
112	C₂×C₄	—	127	2	no	—
113	C₂×C₄	—	127	2	no	—
114	C₂×C₄	—	508	8	no	—
115	C₂×C₄	—	508	2	no	—
116	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
117	C₈	—	127	2	no	—
118	C₈	D₄×D₁₂₇	1	1	yes	characteristic
119	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
120	C₂²	—	127	4	no	—
121	C₂²	—	127	2	no	—
122	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	127	2	no	—
125	C₄	—	508	4	no	—
126	C₄	D₈×D₁₂₇	1	1	yes	characteristic
127	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
128	C₂	—	127	4	no	—
129	C₂	Q₃₂×D₁₂₇	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₈×C₁₂₇	C₂⁴
Frattini	C₈	C₂³×D₁₂₇
Fitting	C₂×Q₃₂×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂²×D₁₂₇	D₈
hypercenter	C₂×Q₃₂	D₁₂₇
hypocenter	C₁₂₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₀₁₆	C₂	semidirect
3. C₂×#16861155	C₂	semidirect
4. Q₃₂×D₁₂₇	C₂	direct (N × H)
6. C₂×Q₃₂×C₁₂₇	C₂	semidirect
13. Dic₁₀₁₆	C₂²	semidirect
14. #16861155	C₂²	semidirect
18. Q₃₂×C₁₂₇	C₂²	semidirect
55. C₂×D₁₂₇	Q₃₂	direct (N × H)
62. C₂×C₁₂₇	C₂×Q₃₂	semidirect
64. D₁₂₇	C₂×Q₃₂	direct (N × H)
66. C₁₂₇	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₁₂₇	direct (N × H)
82. Q₃₂	C₂×D₁₂₇	direct (N × H)
129. C₂	Q₃₂×D₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₁₂₇	2
C₂×Dic₁₀₁₆	2
C₂×#16861155	2
Q₃₂×D₁₂₇	2
C₂×C₁₆×D₁₂₇	2
C₂×Q₃₂×C₁₂₇	2
C₂²×Q₃₂	127