Subgroups — #28383441

Order 14272.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₂₃⋊#95	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	—	2	2	no	—
6	C₂²×Dic₂₂₃	—	2	2	no	—
7	C₂²×Dic₂₂₃	C₄	1	4	yes	complemented
8	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
9	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
10	C₂²⋊Dic₂₂₃	—	2	2	no	—
11	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
14	C₂×Dic₂₂₃	—	2	8	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	—	2	2	no	—
18	C₂×Dic₂₂₃	—	2	4	no	—
19	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
20	C₂³×C₂₂₃	—	2	4	no	—
21	C₂³×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂³×C₂₂₃	D₄	1	2	yes	—
23	C₂³×C₂₂₃	D₄	1	2	yes	—
24	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	—	2	2	no	—
26	C₂×C₄×C₂₂₃	—	2	4	no	—
27	C₂²×C₂₂₃	—	2	4	no	—
28	C₂²×C₂₂₃	—	2	4	no	—
29	C₂²×C₂₂₃	—	2	4	no	—
30	C₂²×C₂₂₃	C₂²⋊C₄	1	4	yes	—
31	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₂₂₃	#41	1	4	yes	—
35	C₄×C₂₂₃	—	4	2	no	—
36	Dic₂₂₃	—	2	4	no	—
37	Dic₂₂₃	—	4	2	no	—
38	C₂×C₂₂₃	—	2	4	no	—
39	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₂₂₃	#117	1	2	yes	—
41	C₂×C₂₂₃	#123	1	2	yes	—
42	C₂×C₂₂₃	#123	1	2	yes	—
43	C₂₂₃	#381	1	1	yes	characteristic, complemented, Sylow
44	#381	—	223	1	no	maximal, Sylow
45	C₂³×C₄	—	223	1	no	—
46	C₂×C₂²⋊C₄	—	223	1	no	—
47	C₂×C₂²⋊C₄	D₂₂₃	1	1	yes	characteristic
48	#95	—	223	4	no	—
49	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic
50	C₂²×C₄	—	446	2	no	—
51	C₂²×C₄	—	446	2	no	—
52	C₂²×C₄	—	223	2	no	—
53	C₂²×C₄	—	223	4	no	—
54	C₂²×C₄	—	223	2	no	—
55	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	—	446	2	no	—
58	C₂³	—	2	4	no	—
59	C₂³	C₄×D₂₂₃	1	2	yes	—
60	C₂³	C₂²⋊D₂₂₃	1	2	yes	—
61	C₂³	D₈₉₂	1	2	yes	—
62	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	446	2	no	—
66	C₂×C₄	—	446	4	no	—
67	C₂×C₄	—	446	8	no	—
68	C₂×C₄	—	446	2	no	—
69	C₂×C₄	—	446	2	no	—
70	C₂×C₄	—	223	4	no	—
71	C₂²	—	2	4	no	—
72	C₂²	—	2	4	no	—
73	C₂²	—	2	4	no	—
74	C₂²	(C₂×D₂₂₃)⋊C₄	1	4	yes	—
75	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
76	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
77	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
78	C₂²	#16813169	1	4	yes	—
79	C₄	—	4	2	no	—
80	C₄	—	892	2	no	—
81	C₄	—	446	4	no	—
82	C₂	—	2	4	no	—
83	C₂	C₂×(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
84	C₂	#20297260	1	2	yes	—
85	C₂	#20297285	1	2	yes	—
86	C₂	#20297288	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂×C₂²⋊C₄×C₂₂₃	C₂
socle, cosocle	C₂³×C₂₂₃	C₂³
prime radical	C₂⁴×C₂₂₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#381
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂₂₃⋊#95	C₂	semidirect
7. C₂²×Dic₂₂₃	C₄	semidirect
43. C₂₂₃	#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₂₃	2
C₂×C₂²⋊Dic₂₂₃	2
C₂₂₃⋊#95	2
C₂×C₂²⋊C₄×C₂₂₃	2
#381	223