Subgroups — #28383440

Order 14272.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊Dic₂₂₃	C₂	1	2	yes	maximal
3	C₂₂₃⋊#95	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₂²	1	4	yes	—
6	C₄⋊Dic₂₂₃	—	2	4	no	—
7	C₂²⋊Dic₂₂₃	—	2	4	no	—
8	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₂₃	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	C₄	1	4	yes	complemented
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₂₃	—	2	4	no	—
14	C₂×Dic₂₂₃	—	2	4	no	—
15	C₂×Dic₂₂₃	—	2	4	no	—
16	C₂³×C₂₂₃	—	2	2	no	—
17	C₂³×C₂₂₃	—	2	2	no	—
18	C₂³×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂³×C₂₂₃	D₄	1	2	yes	—
20	C₂³×C₂₂₃	Q₈	1	2	yes	—
21	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	—	2	8	no	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	D₄	1	4	yes	—
27	C₂²×C₂₂₃	—	2	2	no	—
28	C₂²×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	—	2	4	no	—
30	C₂²×C₂₂₃	—	2	4	no	—
31	C₂²×C₂₂₃	C₄⋊C₄	1	4	yes	complemented
32	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₂₂₃	C₂×D₄	1	2	yes	—
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₂₂₃	#41	1	2	yes	—
37	C₄×C₂₂₃	—	2	2	no	—
38	C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	complemented
39	Dic₂₂₃	—	4	4	no	—
40	C₂×C₂₂₃	—	2	4	no	—
41	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₂₂₃	#117	1	1	yes	characteristic
44	C₂×C₂₂₃	#121	1	2	yes	—
45	C₂×C₂₂₃	C₂²⋊Q₈	1	2	yes	—
46	C₂₂₃	#380	1	1	yes	characteristic, complemented, Sylow
47	#380	—	223	1	no	maximal, Sylow
48	C₂³×C₄	D₂₂₃	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	223	2	no	—
50	C₂×C₂²⋊C₄	—	223	2	no	—
51	#95	—	223	2	no	—
52	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	—	223	4	no	—
55	C₂²×C₄	Dic₂₂₃	1	4	yes	complemented
56	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
58	C₄⋊C₄	—	446	4	no	—
59	C₂²⋊C₄	—	446	4	no	—
60	C₂³	—	2	2	no	—
61	C₂³	—	2	2	no	—
62	C₂³	C₂×Dic₂₂₃	1	2	yes	—
63	C₂³	D₈₉₂	1	2	yes	—
64	C₂³	Dic₄₄₆	1	2	yes	—
65	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	446	4	no	—
69	C₂×C₄	—	446	4	no	—
70	C₂×C₄	—	446	4	no	—
71	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
72	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
73	C₂×C₄	C₂²⋊D₂₂₃	1	4	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	C₄⋊Dic₂₂₃	1	4	yes	complemented
79	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
80	C₂²	C₂×C₂²⋊D₂₂₃	1	2	yes	—
81	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
82	C₂²	C₂×Dic₄₄₆	1	1	yes	characteristic
83	C₂²	#16813170	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	892	4	no	—
86	C₄	C₂²⋊Dic₂₂₃	1	4	yes	complemented
87	C₂	—	2	4	no	—
88	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
89	C₂	C₂×C₄⋊Dic₂₂₃	1	1	yes	characteristic
90	C₂	#20297263	1	1	yes	characteristic
91	C₂	#20297278	1	2	yes	—
92	C₂	#20297283	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂³×C₄×C₂₂₃	C₂
socle, cosocle	C₂³×C₂₂₃	C₂³
prime radical	C₂⁴×C₂₂₃	C₂²
hypercenter	C₂³×C₄	D₂₂₃
hypocenter	C₂₂₃	#380
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₂₂₃	C₂	semidirect
3. C₂₂₃⋊#95	C₂	semidirect
10. C₂²×C₄×C₂₂₃	C₄	semidirect
31. C₂²×C₂₂₃	C₄⋊C₄	semidirect
38. C₄×C₂₂₃	C₂²⋊C₄	semidirect
46. C₂₂₃	#380	semidirect
55. C₂²×C₄	Dic₂₂₃	semidirect
78. C₂²	C₄⋊Dic₂₂₃	semidirect
86. C₄	C₂²⋊Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₂₃	2
C₂×C₂²⋊Dic₂₂₃	2
C₂₂₃⋊#95	2
C₂³×C₄×C₂₂₃	2
#380	223