Finite Group Databasebeta

← C₂×Q₃₂×D₁₀₁ #27184243

Subgroups — C₂×Q₃₂×D₁₀₁ #27184243

Order 12928.

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×D₁₀₁	C₂	1	2	yes	maximal
2	C₂×Dic₈₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692201	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₁₀₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₁₀₁	—	2	2	no	—
8	C₂×Dic₄₀₄	C₂²	1	2	yes	—
9	C₂×#13108064	—	2	2	no	—
10	Q₁₆×D₁₀₁	—	2	4	no	—
11	Q₁₆×D₁₀₁	C₂²	1	8	yes	—
12	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
13	Dic₈₀₈	C₂²	1	4	yes	complemented
14	#15692201	C₂²	1	8	yes	complemented
15	C₂×#13108049	C₂²	1	1	yes	characteristic
16	C₁₆×D₁₀₁	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₁₀₁	C₂²	1	2	yes	—
18	Q₃₂×C₁₀₁	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	—	4	2	no	—
22	Q₈×D₁₀₁	—	2	8	no	—
23	Q₈×D₁₀₁	—	4	4	no	—
24	C₂×C₄×D₁₀₁	—	4	2	no	—
25	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
26	Dic₄₀₄	—	2	2	no	—
27	Dic₄₀₄	C₂³	1	4	yes	—
28	#13108064	—	2	8	no	—
29	C₂×#11717579	D₄	1	1	yes	characteristic
30	C₈×D₁₀₁	D₄	1	2	yes	—
31	C₈×D₁₀₁	C₂³	1	2	yes	—
32	#13108049	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₀₁	—	2	2	no	—
34	Q₁₆×C₁₀₁	—	2	2	no	—
35	Q₁₆×C₁₀₁	C₂³	1	4	yes	—
36	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₁₆×C₁₀₁	C₂³	1	2	yes	—
38	C₂²×D₁₀₁	D₈	1	1	yes	characteristic
39	Q₈×C₁₀₁	—	2	4	no	—
40	Q₈×C₁₀₁	—	4	2	no	—
41	C₂×C₄×C₁₀₁	—	4	2	no	—
42	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₀₁	—	4	2	no	—
44	C₂×Dic₁₀₁	D₈	1	1	yes	characteristic
45	C₄×D₁₀₁	—	4	8	no	—
46	C₄×D₁₀₁	D₈	1	2	yes	—
47	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
48	Dic₂₀₂	—	2	4	no	—
49	Dic₂₀₂	—	4	2	no	—
50	Dic₂₀₂	—	4	8	no	—
51	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₀₁	C₂⁴	1	1	yes	characteristic
53	#11717579	C₂×D₄	1	2	yes	—
54	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
55	C₂×D₁₀₁	Q₃₂	1	4	yes	direct product factor
56	C₂×D₁₀₁	C₂×D₈	1	2	yes	—
57	C₄×C₁₀₁	—	4	4	no	—
58	C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
59	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
60	Dic₁₀₁	—	4	4	no	—
61	Dic₁₀₁	C₂×D₈	1	2	yes	—
62	C₂×C₁₀₁	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₁₀₁	C₂²×D₈	1	1	yes	characteristic
64	D₁₀₁	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	101	1	no	maximal, Sylow
66	C₁₀₁	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	101	2	no	—
68	C₂×Q₃₂	—	101	2	no	—
69	C₂×Q₃₂	—	101	8	no	—
70	C₂×Q₃₂	—	101	1	no	—
71	C₂×Q₃₂	D₁₀₁	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	101	1	no	—
73	C₂²×Q₈	—	202	2	no	—
74	C₂×Q₁₆	—	101	8	no	—
75	C₂×Q₁₆	—	101	2	no	—
76	C₂×Q₁₆	—	202	4	no	—
77	C₂×Q₁₆	—	202	2	no	—
78	C₂×Q₁₆	C₂×D₁₀₁	1	2	yes	—
79	C₂²×C₈	—	101	1	no	—
80	Q₃₂	—	101	8	no	—
81	Q₃₂	—	101	4	no	—
82	Q₃₂	C₂×D₁₀₁	1	4	yes	direct product factor
83	C₂×C₁₆	—	101	1	no	—
84	C₂×C₁₆	—	101	4	no	—
85	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	404	2	no	—
88	C₂×Q₈	—	404	4	no	—
89	C₂×Q₈	—	202	8	no	—
90	C₂×Q₈	—	202	2	no	—
91	C₂²×C₄	—	101	1	no	—
92	C₂²×C₄	—	404	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	101	4	no	—
95	Q₁₆	—	202	8	no	—
96	Q₁₆	—	202	2	no	—
97	Q₁₆	C₂²×D₁₀₁	1	4	yes	—
98	C₂×C₈	—	101	1	no	—
99	C₂×C₈	—	101	2	no	—
100	C₂×C₈	—	101	2	no	—
101	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
102	C₁₆	—	101	2	no	—
103	C₁₆	C₂²×D₁₀₁	1	2	yes	—
104	C₂³	—	101	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	404	8	no	—
108	Q₈	—	404	2	no	—
109	Q₈	—	202	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	101	1	no	—
112	C₂×C₄	—	101	2	no	—
113	C₂×C₄	—	101	2	no	—
114	C₂×C₄	—	404	8	no	—
115	C₂×C₄	—	404	2	no	—
116	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
117	C₈	—	101	2	no	—
118	C₈	D₄×D₁₀₁	1	1	yes	characteristic
119	C₈	C₂³×D₁₀₁	1	1	yes	characteristic
120	C₂²	—	101	4	no	—
121	C₂²	—	101	2	no	—
122	C₂²	D₈×D₁₀₁	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	101	2	no	—
125	C₄	—	404	4	no	—
126	C₄	D₈×D₁₀₁	1	1	yes	characteristic
127	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
128	C₂	—	101	4	no	—
129	C₂	Q₃₂×D₁₀₁	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₁
commutator, cosocle	C₈×C₁₀₁	C₂⁴
Frattini	C₈	C₂³×D₁₀₁
Fitting	C₂×Q₃₂×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×D₈
prime radical	C₂²×D₁₀₁	D₈
hypercenter	C₂×Q₃₂	D₁₀₁
hypocenter	C₁₀₁	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
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×Dic₈₀₈	C₂	semidirect
3. C₂×#15692201	C₂	semidirect
4. Q₃₂×D₁₀₁	C₂	direct (N × H)
6. C₂×Q₃₂×C₁₀₁	C₂	semidirect
13. Dic₈₀₈	C₂²	semidirect
14. #15692201	C₂²	semidirect
18. Q₃₂×C₁₀₁	C₂²	semidirect
55. C₂×D₁₀₁	Q₃₂	direct (N × H)
62. C₂×C₁₀₁	C₂×Q₃₂	semidirect
64. D₁₀₁	C₂×Q₃₂	direct (N × H)
66. C₁₀₁	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₁₀₁	direct (N × H)
82. Q₃₂	C₂×D₁₀₁	direct (N × H)
129. C₂	Q₃₂×D₁₀₁	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×D₁₀₁	2
C₂×Dic₈₀₈	2
C₂×#15692201	2
Q₃₂×D₁₀₁	2
C₂×C₁₆×D₁₀₁	2
C₂×Q₃₂×C₁₀₁	2
C₂²×Q₃₂	101