Subgroups — C₂²×#16813719 #28407823

Order 14464.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813719	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14268659	C₂	1	2	yes	maximal, complemented
3	C₂²×#14268655	C₂	1	1	yes	maximal, characteristic
4	#16813719	C₂²	1	16	yes	direct product factor
5	C₂×#14268659	C₂²	1	12	yes	complemented
6	C₂×#14268655	C₄	1	6	yes	—
7	C₂×#14268655	C₂²	1	6	yes	—
8	C₂²×#11744753	C₄	1	1	yes	characteristic
9	C₂²×#11744753	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₁₃	C₄	1	1	yes	characteristic
11	#14268659	C₂³	1	8	yes	complemented
12	#14268655	C₂×C₄	1	12	yes	—
13	#14268655	C₂³	1	4	yes	—
14	C₂×#11744753	C₂×C₄	1	6	yes	—
15	C₂×#11744753	C₂×C₄	1	3	yes	—
16	C₂×#11744753	C₂³	1	3	yes	—
17	C₂³×D₁₁₃	C₈	1	1	yes	characteristic
18	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₁₃	C₈	1	6	yes	—
20	C₂×C₄×D₁₁₃	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₁₁₃	C₈	1	1	yes	characteristic
22	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
23	C₂²×D₁₁₃	—	2	4	no	—
24	C₂²×D₁₁₃	C₂×C₈	1	6	yes	—
25	C₂×C₄×C₁₁₃	C₂×C₈	1	6	yes	—
26	C₂×Dic₁₁₃	C₂×C₈	1	3	yes	—
27	C₂×Dic₁₁₃	C₂²×C₄	1	3	yes	—
28	C₄×D₁₁₃	C₂×C₈	1	12	yes	—
29	C₄×D₁₁₃	C₂²×C₄	1	4	yes	—
30	#11744753	C₂²×C₄	1	4	yes	—
31	#11744753	C₂²×C₄	1	3	yes	—
32	#11744753	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₁₃	M₃₂	1	4	yes	complemented
34	C₂²×C₁₁₃	C₂²×C₈	1	3	yes	—
35	C₂×D₁₁₃	—	2	12	no	—
36	C₂×D₁₁₃	C₂²×C₈	1	4	yes	—
37	C₄×C₁₁₃	C₂²×C₈	1	4	yes	—
38	Dic₁₁₃	C₂²×C₈	1	3	yes	—
39	Dic₁₁₃	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₁₃	C₂×M₃₂	1	6	yes	complemented
41	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
42	D₁₁₃	—	2	4	no	—
43	C₂²×M₃₂	—	113	1	no	maximal, Sylow
44	C₁₁₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₈	—	113	1	no	—
46	C₂×M₃₂	—	113	12	no	—
47	C₂²×C₁₆	—	113	2	no	—
48	C₂³×C₄	—	113	1	no	—
49	C₂²×C₈	—	113	1	no	—
50	C₂²×C₈	—	113	1	no	—
51	C₂²×C₈	—	113	6	no	—
52	C₂²×C₈	—	113	6	no	—
53	M₃₂	—	113	16	no	—
54	C₂×C₁₆	—	113	12	no	—
55	C₂⁴	—	113	1	no	—
56	C₂²×C₄	—	113	1	no	—
57	C₂²×C₄	—	113	6	no	—
58	C₂²×C₄	—	113	6	no	—
59	C₂²×C₄	#11744762	1	1	yes	characteristic
60	C₂×C₈	—	113	3	no	—
61	C₂×C₈	—	113	4	no	—
62	C₂×C₈	—	113	6	no	—
63	C₂×C₈	—	113	3	no	—
64	C₂×C₈	—	113	12	no	—
65	C₁₆	—	113	8	no	—
66	C₂³	—	113	6	no	—
67	C₂³	—	226	4	no	—
68	C₂³	C₂×#11744762	1	1	yes	characteristic
69	C₂×C₄	—	113	3	no	—
70	C₂×C₄	—	113	3	no	—
71	C₂×C₄	—	113	4	no	—
72	C₂×C₄	—	113	12	no	—
73	C₂×C₄	C₂×#11744762	1	6	yes	—
74	C₈	—	113	1	no	—
75	C₈	—	113	3	no	—
76	C₈	—	113	4	no	—
77	C₂²	—	113	4	no	—
78	C₂²	—	226	12	no	—
79	C₂²	#16813719	1	4	yes	direct product factor
80	C₂²	C₂²×#11744762	1	3	yes	—
81	C₄	—	113	1	no	—
82	C₄	—	113	3	no	—
83	C₄	C₂²×#11744762	1	4	yes	—
84	C₂	—	226	4	no	—
85	C₂	C₂×#16813719	1	6	yes	direct product factor
86	C₂	C₂³×#11744762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11744762
commutator	C₂×C₁₁₃	C₂³×C₈
Frattini	C₂	C₂³×#11744762
Fitting	C₂²×C₄×C₁₁₃	C₈
socle	C₂³×C₁₁₃	C₂×C₈
prime radical	C₂³×D₁₁₃	C₈
cosocle	#11744753	C₂⁴
hypercenter	C₂²×C₄	#11744762
hypocenter	C₁₁₃	C₂²×M₃₂
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16813719	C₂	direct (N × H)
2. C₂²×#14268659	C₂	semidirect
4. #16813719	C₂²	direct (N × H)
5. C₂×#14268659	C₂²	semidirect
11. #14268659	C₂³	semidirect
33. C₂²×C₁₁₃	M₃₂	semidirect
40. C₂×C₁₁₃	C₂×M₃₂	semidirect
44. C₁₁₃	C₂²×M₃₂	semidirect
79. C₂²	#16813719	direct (N × H)
85. C₂	C₂×#16813719	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×#16813719	2
C₂²×#14268659	2
C₂²×#14268655	2
C₂²×M₃₂	113

Subgroups — C22×#16813719 #28407823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16813719 #28407823