Subgroups — #28383576

Order 14272.

158 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	#20297263	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297228	C₂	1	2	yes	maximal, complemented
4	#20297225	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₂₃⋊C₈	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	C₂²⋊Dic₂₂₃	—	2	1	no	—
10	C₄×Dic₂₂₃	—	2	1	no	—
11	#41×C₂₂₃	—	2	2	no	—
12	#41×C₂₂₃	C₄	1	4	yes	complemented
13	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₂₃	—	2	1	no	—
15	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₂₂₃	—	2	1	no	—
17	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₂₃	—	2	2	no	—
19	C₂₂₃⋊C₈	—	2	2	no	—
20	C₂³×C₂₂₃	—	2	1	no	—
21	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
22	Q₈×C₂₂₃	—	2	1	no	—
23	Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
24	D₄×C₂₂₃	—	2	4	no	—
25	D₄×C₂₂₃	—	2	1	no	—
26	D₄×C₂₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	—	2	4	no	—
29	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
31	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₂₂₃	—	2	2	no	—
34	C₂²×C₂₂₃	—	2	2	no	—
35	C₂²×C₂₂₃	—	4	2	no	—
36	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₂₃	—	2	2	no	—
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	—	2	2	no	—
44	C₂×C₂₂₃	—	4	2	no	—
45	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₂₂₃	#135	1	2	yes	—
47	C₂₂₃	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	223	1	no	maximal, Sylow
49	C₂×#41	D₂₂₃	1	1	yes	characteristic
50	C₂²×C₈	—	223	1	no	—
51	#117	—	223	1	no	—
52	#103	—	223	2	no	—
53	#102	—	223	2	no	—
54	#41	—	2	2	no	—
55	#41	Dic₂₂₃	1	4	yes	complemented
56	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
59	C₂²×C₄	—	2	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	—	223	2	no	—
64	C₂²⋊C₄	—	446	1	no	—
65	C₄²	—	446	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	C₂×Dic₂₂₃	1	2	yes	—
70	D₄	—	2	4	no	—
71	D₄	—	2	1	no	—
72	D₄	C₂×Dic₂₂₃	1	2	yes	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	446	2	no	—
76	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
77	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
78	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
80	C₈	—	446	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	892	2	no	—
88	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
89	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
94	C₂	#20297324	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	C₂×#41×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	#416
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₂₃⋊C₈	C₂	semidirect
2. #20297263	C₂	semidirect
3. #20297228	C₂	semidirect
4. #20297225	C₂	semidirect
12. #41×C₂₂₃	C₄	semidirect
47. C₂₂₃	#416	semidirect
55. #41	Dic₂₂₃	semidirect

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
#20297263	2
#20297228	2
#20297225	2
C₂×#41×C₂₂₃	2
#416	223