Finite Group Databasebeta

← C₂×#20266092 #27187445

Subgroups — C₂×#20266092 #27187445

Order 12928.

224 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 66 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×D₁₀₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692326	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266092	C₂	1	4	yes	maximal, direct product factor
4	C₂×#15692278	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	—	2	1	no	—
6	D₁₀₁⋊C₁₆	—	2	1	no	—
7	D₁₀₁⋊C₁₆	C₂²	1	2	yes	complemented
8	C₂×#13108077	—	2	1	no	—
9	#15692326	—	2	1	no	—
10	#15692326	C₂²	1	2	yes	complemented
11	C₂×#13108075	—	2	1	no	—
12	#15692278	—	2	1	no	—
13	#15692278	—	2	1	no	—
14	#15692278	C₄	1	2	yes	—
15	#15692278	C₂²	1	2	yes	—
16	C₂×#13108061	—	2	1	no	—
17	C₂×D₈₀₈	C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₀₁	C₄	1	1	yes	characteristic
20	#13108084	—	2	2	no	—
21	#13108084	—	4	1	no	—
22	C₂×#11717581	—	4	1	no	—
23	#13108077	—	2	2	no	—
24	#13108075	—	2	2	no	—
25	#13108075	—	2	2	no	—
26	#13108075	—	4	1	no	—
27	#13108075	—	4	1	no	—
28	C₂×D₄₀₄	—	2	1	no	—
29	C₂×D₄₀₄	—	4	1	no	—
30	C₂×C₄×D₁₀₁	—	4	1	no	—
31	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
32	#13108061	—	2	2	no	—
33	#13108061	—	2	2	no	—
34	D₈₀₈	—	2	1	no	—
35	D₈₀₈	C₂×C₄	1	2	yes	—
36	C₂×#11717579	D₄	1	1	yes	characteristic
37	C₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₀₁	D₄	1	1	yes	characteristic
39	C₈×D₁₀₁	D₄	1	1	yes	characteristic
40	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₁₀₁	—	2	1	no	—
42	Q₁₆×C₁₀₁	—	2	1	no	—
43	Q₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₁₀₁	—	4	1	no	—
46	C₂²×D₁₀₁	SD₁₆	1	1	yes	characteristic
47	Q₈×C₁₀₁	—	2	2	no	—
48	Q₈×C₁₀₁	—	4	1	no	—
49	C₂×C₄×C₁₀₁	—	4	1	no	—
50	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₁	D₈	1	1	yes	characteristic
52	D₄₀₄	—	2	2	no	—
53	D₄₀₄	—	4	2	no	—
54	D₄₀₄	—	4	2	no	—
55	D₄₀₄	—	4	1	no	—
56	C₄×D₁₀₁	—	4	2	no	—
57	C₄×D₁₀₁	—	4	2	no	—
58	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₁₀₁	D₈	1	1	yes	characteristic
60	C₄×D₁₀₁	SD₁₆	1	1	yes	characteristic
61	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
62	#11717581	—	4	2	no	—
63	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
65	#11717579	C₂²⋊C₄	1	1	yes	characteristic
66	#11717579	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₁₀₁	#102	1	1	yes	characteristic
68	C₂×D₁₀₁	—	2	2	no	—
69	C₂×D₁₀₁	—	4	2	no	—
70	C₂×D₁₀₁	—	8	2	no	—
71	C₂×D₁₀₁	#102	1	1	yes	characteristic
72	C₂×D₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₁₀₁	—	4	2	no	—
74	C₄×C₁₀₁	#102	1	1	yes	characteristic
75	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₁₀₁	#102	1	1	yes	characteristic
77	Dic₁₀₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₁₀₁	#359	1	2	yes	complemented
79	C₂×C₁₀₁	C₂×#102	1	1	yes	characteristic
80	D₁₀₁	—	2	1	no	—
81	D₁₀₁	—	2	1	no	—
82	D₁₀₁	—	8	2	no	—
83	C₂×#359	—	101	1	no	maximal, Sylow
84	C₁₀₁	C₂×#359	1	1	yes	characteristic, complemented, Sylow
85	C₂×#135	—	101	1	no	—
86	C₂²×C₁₆	—	101	1	no	—
87	C₂×#108	—	101	1	no	—
88	#359	—	101	4	no	—
89	C₂×#41	—	202	1	no	—
90	#135	—	101	2	no	—
91	#135	—	101	2	no	—
92	#135	—	202	1	no	—
93	#135	—	202	1	no	—
94	C₂×Q₁₆	#87938	1	1	yes	characteristic
95	C₂×SD₁₆	—	202	1	no	—
96	C₂×D₈	—	101	1	no	—
97	C₂×M₁₆	—	202	1	no	—
98	C₂²×C₈	—	101	1	no	—
99	C₂×C₁₆	—	101	2	no	—
100	C₂×C₁₆	—	202	1	no	—
101	C₂×C₁₆	—	202	1	no	—
102	#108	—	101	2	no	—
103	#108	—	202	1	no	—
104	#41	—	404	1	no	—
105	#41	—	404	1	no	—
106	#41	—	202	2	no	—
107	#41	—	202	2	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	404	1	no	—
110	C₂×D₄	—	202	1	no	—
111	C₂²×C₄	—	101	1	no	—
112	C₂²×C₄	—	404	1	no	—
113	Q₁₆	—	2	1	no	—
114	Q₁₆	C₂×#87938	1	2	yes	—
115	SD₁₆	—	202	2	no	—
116	SD₁₆	—	202	2	no	—
117	D₈	—	101	2	no	—
118	D₈	—	202	1	no	—
119	M₁₆	—	404	1	no	—
120	M₁₆	—	202	2	no	—
121	C₂×C₈	—	101	1	no	—
122	C₂×C₈	—	101	1	no	—
123	C₂×C₈	—	101	1	no	—
124	C₂×C₈	—	101	1	no	—
125	C₂×C₈	—	101	1	no	—
126	C₂×C₈	—	404	1	no	—
127	C₂×C₈	C₂×#87938	1	1	yes	characteristic
128	C₁₆	—	202	2	no	—
129	C₂³	—	101	1	no	—
130	C₂³	—	404	1	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	404	2	no	—
134	D₄	—	404	2	no	—
135	D₄	—	404	1	no	—
136	D₄	—	202	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	101	1	no	—
139	C₂×C₄	—	101	1	no	—
140	C₂×C₄	—	101	1	no	—
141	C₂×C₄	—	101	1	no	—
142	C₂×C₄	—	101	1	no	—
143	C₂×C₄	—	404	2	no	—
144	C₂×C₄	—	404	2	no	—
145	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
146	C₈	—	101	1	no	—
147	C₈	—	101	1	no	—
148	C₈	—	404	2	no	—
149	C₈	C₂²⋊#87938	1	1	yes	characteristic
150	C₈	C₂²×#87938	1	1	yes	characteristic
151	C₂²	—	808	2	no	—
152	C₂²	—	101	1	no	—
153	C₂²	—	101	1	no	—
154	C₂²	—	404	2	no	—
155	C₂²	—	202	2	no	—
156	C₂²	#15692317	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	101	1	no	—
159	C₄	—	101	1	no	—
160	C₄	#15692317	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊#87938	1	1	yes	characteristic
162	C₂	—	808	2	no	—
163	C₂	—	202	1	no	—
164	C₂	—	202	1	no	—
165	C₂	#20266092	1	2	yes	direct product factor
166	C₂	C₂×#15692317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692317
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₈	C₂²×#87938
Fitting	C₂×Q₁₆×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#102
prime radical	C₂×#15692278	C₂
cosocle	C₈×D₁₀₁	C₂³
hypercenter	C₂×Q₁₆	#87938
hypocenter	C₁₀₁	C₂×#359
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₀₁⋊C₁₆	C₂	semidirect
2. C₂×#15692326	C₂	semidirect
3. #20266092	C₂	direct (N × H)
7. D₁₀₁⋊C₁₆	C₂²	semidirect
10. #15692326	C₂²	semidirect
78. C₂×C₁₀₁	#359	semidirect
84. C₁₀₁	C₂×#359	semidirect
165. C₂	#20266092	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×D₁₀₁⋊C₁₆	2
C₂×#15692326	2
#20266092	2
C₂×#15692278	2
C₂×#359	101