Subgroups — C₂×#20297225 #28383567

Order 14272.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297225	C₂	1	4	yes	maximal, direct product factor
4	C₂²×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	complemented
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	complemented
10	C₂⁴×C₂₂₃	—	2	1	no	—
11	C₂×D₄×C₂₂₃	—	2	1	no	—
12	C₂×D₄×C₂₂₃	—	2	2	no	—
13	C₂×D₄×C₂₂₃	C₄	1	4	yes	complemented
14	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	—	4	2	no	—
18	C₂₂₃⋊C₈	—	2	2	no	—
19	C₂³×C₂₂₃	—	2	4	no	—
20	C₂³×C₂₂₃	—	2	2	no	—
21	C₂³×C₂₂₃	—	4	2	no	—
22	C₂³×C₂₂₃	—	4	2	no	—
23	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
24	D₄×C₂₂₃	—	2	4	no	—
25	D₄×C₂₂₃	—	2	2	no	—
26	D₄×C₂₂₃	C₂×C₄	1	4	yes	complemented
27	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₂₂₃	—	2	4	no	—
32	C₂²×C₂₂₃	—	4	8	no	—
33	C₂²×C₂₂₃	—	4	2	no	—
34	C₂²×C₂₂₃	—	4	2	no	—
35	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₂₂₃	D₈	1	2	yes	—
37	C₂²×C₂₂₃	SD₁₆	1	2	yes	—
38	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
39	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
40	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
42	Dic₂₂₃	—	4	2	no	—
43	C₂×C₂₂₃	—	4	4	no	—
44	C₂×C₂₂₃	#102	1	4	yes	complemented
45	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
48	C₂₂₃	C₂×#102	1	1	yes	characteristic, complemented, Sylow
49	C₂×#102	—	223	1	no	maximal, Sylow
50	C₂²×D₄	D₂₂₃	1	1	yes	characteristic
51	C₂²×C₈	—	223	1	no	—
52	C₂×C₄⋊C₄	—	223	1	no	—
53	#102	—	223	4	no	—
54	C₂⁴	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	Dic₂₂₃	1	4	yes	complemented
58	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
59	C₂²×C₄	—	446	1	no	—
60	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
61	C₂×C₈	—	446	2	no	—
62	C₂×C₈	—	223	2	no	—
63	C₄⋊C₄	—	446	1	no	—
64	C₄⋊C₄	—	223	2	no	—
65	C₂³	—	2	4	no	—
66	C₂³	—	2	2	no	—
67	C₂³	—	4	2	no	—
68	C₂³	—	4	2	no	—
69	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
70	D₄	—	2	4	no	—
71	D₄	—	2	2	no	—
72	D₄	C₂×Dic₂₂₃	1	4	yes	complemented
73	C₂×C₄	—	892	2	no	—
74	C₂×C₄	—	446	2	no	—
75	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
76	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
78	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
79	C₈	—	446	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	8	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
85	C₂²	#16813155	1	2	yes	—
86	C₂²	#16813158	1	2	yes	—
87	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
88	C₄	—	892	2	no	—
89	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
90	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
91	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
92	C₂	—	4	4	no	—
93	C₂	#20297225	1	4	yes	direct product factor
94	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
95	C₂	C₂×#16813155	1	1	yes	characteristic
96	C₂	C₂×#16813158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	C₂²×D₄×C₂₂₃	C₂
socle	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²×D₄	D₂₂₃
hypocenter	C₂₂₃	C₂×#102
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₂₃⋊C₈	C₂	semidirect
2. C₂×C₄⋊Dic₂₂₃	C₂	semidirect
3. #20297225	C₂	direct (N × H)
7. C₂×C₂₂₃⋊C₈	C₂²	semidirect
9. C₄⋊Dic₂₂₃	C₂²	semidirect
13. C₂×D₄×C₂₂₃	C₄	semidirect
26. D₄×C₂₂₃	C₂×C₄	semidirect
44. C₂×C₂₂₃	#102	semidirect
48. C₂₂₃	C₂×#102	semidirect
57. C₂×D₄	Dic₂₂₃	semidirect
72. D₄	C₂×Dic₂₂₃	semidirect
93. C₂	#20297225	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₂×C₄⋊Dic₂₂₃	2
#20297225	2
C₂²×D₄×C₂₂₃	2
C₂×#102	223

Subgroups — C2×#20297225 #28383567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297225 #28383567