Finite Group Databasebeta

← C₄×#15692200 #27172986

Subgroups — C₄×#15692200 #27172986

Order 12928.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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×#15692200	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈₀₈	C₂	1	1	yes	maximal, characteristic
4	#20265005	C₂	1	1	yes	maximal, characteristic, complemented
5	#20264982	C₂	1	1	yes	maximal, characteristic, complemented
6	#20264979	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₀₄	C₂²	1	1	yes	characteristic
9	C₂×D₈₀₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₀₂	—	2	1	no	—
11	C₄×D₄₀₄	—	2	1	no	—
12	#15692200	C₄	1	4	yes	direct product factor
13	#15692189	C₂²	1	1	yes	characteristic
14	#15692189	C₂²	1	1	yes	characteristic
15	#15692180	—	2	1	no	—
16	#15692177	—	2	1	no	—
17	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₀₂	—	2	1	no	—
21	C₂×D₄₀₄	—	2	1	no	—
22	C₂×C₄×D₁₀₁	—	4	1	no	—
23	Dic₄₀₄	—	2	1	no	—
24	Dic₄₀₄	C₂×C₄	1	2	yes	complemented
25	D₈₀₈	—	2	1	no	—
26	D₈₀₈	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₁₀₁	—	2	1	no	—
28	C₄⋊Dic₁₀₁	—	2	1	no	—
29	#13108053	—	4	1	no	—
30	#13108051	—	4	1	no	—
31	C₄×Dic₁₀₁	—	4	1	no	—
32	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
35	C₁₆×C₁₀₁	—	2	1	no	—
36	C₁₆×C₁₀₁	C₂×C₄	1	2	yes	complemented
37	C₂²×D₁₀₁	—	4	1	no	—
38	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₁	—	4	1	no	—
42	C₂×Dic₁₀₁	—	4	1	no	—
43	C₂×Dic₁₀₁	—	4	1	no	—
44	D₄₀₄	—	2	2	no	—
45	D₄₀₄	—	4	1	no	—
46	C₄×D₁₀₁	—	8	2	no	—
47	Dic₂₀₂	—	2	2	no	—
48	Dic₂₀₂	—	4	1	no	—
49	C₈×C₁₀₁	—	2	1	no	—
50	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₀₁	#41	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₀₁	—	4	2	no	—
54	C₂×D₁₀₁	—	8	1	no	—
55	C₂×D₁₀₁	—	8	1	no	—
56	C₄×C₁₀₁	—	2	1	no	—
57	C₄×C₁₀₁	SD₃₂	1	2	yes	complemented
58	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₀₁	#135	1	1	yes	characteristic
60	Dic₁₀₁	—	4	2	no	—
61	Dic₁₀₁	—	8	1	no	—
62	Dic₁₀₁	—	8	1	no	—
63	C₂×C₁₀₁	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₀₁	C₂×SD₃₂	1	1	yes	characteristic
65	C₂×C₁₀₁	#508	1	1	yes	characteristic
66	D₁₀₁	—	8	2	no	—
67	C₄×SD₃₂	—	101	1	no	maximal, Sylow
68	C₁₀₁	C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₃₂	—	101	1	no	—
70	C₄×Q₁₆	—	101	1	no	—
71	C₄×D₈	—	101	1	no	—
72	#367	—	101	1	no	—
73	#358	—	101	1	no	—
74	#357	—	101	1	no	—
75	C₄×C₁₆	D₁₀₁	1	1	yes	characteristic, complemented
76	C₂×Q₁₆	—	101	1	no	—
77	C₂×D₈	—	101	1	no	—
78	C₄×Q₈	—	202	1	no	—
79	C₄×D₄	—	202	1	no	—
80	SD₃₂	—	101	4	no	—
81	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
83	#107	—	101	1	no	—
84	#107	—	101	1	no	—
85	#103	—	202	1	no	—
86	#102	—	202	1	no	—
87	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
88	C₂×Q₈	—	202	1	no	—
89	C₂×D₄	—	202	1	no	—
90	C₂²×C₄	—	404	1	no	—
91	Q₁₆	—	101	2	no	—
92	Q₁₆	—	202	1	no	—
93	D₈	—	101	2	no	—
94	D₈	—	202	1	no	—
95	C₂×C₈	D₄₀₄	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
97	C₄⋊C₄	—	404	1	no	—
98	C₄⋊C₄	—	202	1	no	—
99	C₄⋊C₄	—	202	1	no	—
100	C₂²⋊C₄	—	404	1	no	—
101	C₄²	—	404	1	no	—
102	C₄²	D₄₀₄	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	C₄×D₁₀₁	1	2	yes	complemented
105	C₂³	—	404	1	no	—
106	Q₈	—	404	1	no	—
107	Q₈	—	202	2	no	—
108	D₄	—	404	1	no	—
109	D₄	—	202	2	no	—
110	C₂×C₄	—	808	2	no	—
111	C₂×C₄	—	404	1	no	—
112	C₂×C₄	—	404	1	no	—
113	C₂×C₄	—	404	1	no	—
114	C₂×C₄	D₈₀₈	1	1	yes	characteristic
115	C₂×C₄	D₈₀₈	1	1	yes	characteristic
116	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	C₂×C₄×D₁₀₁	1	1	yes	characteristic
119	C₈	#13108074	1	1	yes	characteristic
120	C₂²	—	808	1	no	—
121	C₂²	—	808	1	no	—
122	C₂²	—	404	2	no	—
123	C₂²	C₂×D₈₀₈	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	808	1	no	—
126	C₄	—	808	1	no	—
127	C₄	—	404	2	no	—
128	C₄	#15692200	1	2	yes	direct product factor
129	C₄	C₄×D₄₀₄	1	1	yes	characteristic
130	C₄	#15692277	1	1	yes	characteristic
131	C₂	—	808	2	no	—
132	C₂	C₄×D₈₀₈	1	1	yes	characteristic
133	C₂	C₂×#15692200	1	1	yes	characteristic
134	C₂	#20265609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈₀₈
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	C₄×C₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×D₈
prime radical	C₂×D₈₀₈	C₂²
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	C₄×C₁₆	D₁₀₁
hypocenter	C₁₀₁	C₄×SD₃₂
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₄₀₄	C₂	semidirect
4. #20265005	C₂	semidirect
5. #20264982	C₂	semidirect
7. C₄×C₁₆×C₁₀₁	C₂	semidirect
12. #15692200	C₄	direct (N × H)
24. Dic₄₀₄	C₂×C₄	semidirect
36. C₁₆×C₁₀₁	C₂×C₄	semidirect
57. C₄×C₁₀₁	SD₃₂	semidirect
68. C₁₀₁	C₄×SD₃₂	semidirect
75. C₄×C₁₆	D₁₀₁	semidirect
104. C₁₆	C₄×D₁₀₁	semidirect
128. C₄	#15692200	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×#15692200	2
C₄×Dic₄₀₄	2
C₄×D₈₀₈	2
#20265005	2
#20264982	2
#20264979	2
C₄×C₁₆×C₁₀₁	2
C₄×SD₃₂	101