Subgroups — #28383638

Order 14272.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297263	C₂	1	1	yes	maximal, characteristic
2	Dic₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#20297214	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	C₄	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	C₄	1	1	yes	characteristic
12	C₂²⋊Dic₂₂₃	C₄	1	1	yes	characteristic
13	C₂²⋊Dic₂₂₃	C₄	1	1	yes	characteristic
14	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₂₃⋊C₈	—	2	1	no	—
24	C₂₂₃⋊C₈	—	2	1	no	—
25	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	—	2	1	no	—
27	C₂×C₄×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₈×C₂₂₃	—	2	1	no	—
31	C₈×C₂₂₃	—	2	1	no	—
32	C₂²×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	—	2	1	no	—
34	C₂²×C₂₂₃	—	2	1	no	—
35	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	#41	1	1	yes	characteristic
38	C₄×C₂₂₃	#41	1	1	yes	characteristic
39	Dic₂₂₃	—	2	1	no	—
40	Dic₂₂₃	—	2	1	no	—
41	Dic₂₂₃	—	2	1	no	—
42	Dic₂₂₃	—	2	1	no	—
43	C₂×C₂₂₃	—	4	1	no	—
44	C₂×C₂₂₃	#117	1	1	yes	characteristic
45	C₂×C₂₂₃	#131	1	1	yes	characteristic
46	C₂×C₂₂₃	#131	1	1	yes	characteristic
47	C₂₂₃	#433	1	1	yes	characteristic, complemented, Sylow
48	#433	—	223	1	no	maximal, Sylow
49	#117	—	223	1	no	—
50	C₄⋊C₈	—	223	1	no	—
51	C₄⋊C₈	—	223	1	no	—
52	C₂²⋊C₈	—	223	1	no	—
53	C₂²⋊C₈	D₂₂₃	1	1	yes	characteristic
54	#97	—	223	1	no	—
55	C₄×C₈	—	223	1	no	—
56	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂×C₈	—	223	1	no	—
58	C₂×C₈	—	223	1	no	—
59	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	223	1	no	—
62	C₄⋊C₄	—	223	1	no	—
63	C₂²⋊C₄	—	223	1	no	—
64	C₂²⋊C₄	—	223	1	no	—
65	C₄²	—	223	1	no	—
66	C₄²	—	223	1	no	—
67	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	223	1	no	—
71	C₂×C₄	—	223	1	no	—
72	C₂×C₄	—	223	1	no	—
73	C₂×C₄	—	223	1	no	—
74	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	446	1	no	—
79	C₈	—	446	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	446	1	no	—
86	C₄	—	446	1	no	—
87	C₄	—	446	1	no	—
88	C₄	—	446	1	no	—
89	C₄	#16813169	1	1	yes	characteristic
90	C₄	#16813169	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#20297260	1	1	yes	characteristic
93	C₂	#20297310	1	1	yes	characteristic
94	C₂	#20297312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂²⋊C₈×C₂₂₃	C₂
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#433
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. Dic₂₂₃⋊C₈	C₂	semidirect
3. Dic₂₂₃⋊C₈	C₂	semidirect
5. #20297214	C₂	semidirect
6. C₈×Dic₂₂₃	C₂	semidirect
47. C₂₂₃	#433	semidirect

Maximal subgroups

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

subgroup	index
#20297263	2
Dic₂₂₃⋊C₈	2
C₂²⋊(C₂₂₃⋊C₈)	2
#20297214	2
C₈×Dic₂₂₃	2
C₂²⋊C₈×C₂₂₃	2
#433	223