Subgroups — #28384191

Order 14272.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297299	C₂	1	4	yes	maximal
2	#20297296	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297294	C₂	1	4	yes	maximal, complemented
4	#20297280	C₂	1	2	yes	maximal
5	Q₈×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
6	#20297261	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₄⋊Dic₂₂₃	C₂²	1	4	yes	—
11	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
12	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
13	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
15	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₂₃	C₂²	1	4	yes	—
18	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
20	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₂₃	C₂³	1	2	yes	—
23	C₂×Dic₂₂₃	C₂³	1	4	yes	—
24	C₂×Dic₂₂₃	C₂³	1	1	yes	characteristic
25	C₄×D₂₂₃	—	2	2	no	—
26	Q₈×C₂₂₃	—	2	4	no	—
27	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₂₃	—	2	1	no	—
32	C₂×D₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	—	2	4	no	—
34	C₄×C₂₂₃	—	2	2	no	—
35	C₄×C₂₂₃	#41	1	2	yes	—
36	Dic₂₂₃	—	2	1	no	—
37	Dic₂₂₃	—	2	2	no	—
38	Dic₂₂₃	—	2	4	no	—
39	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
40	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
41	D₂₂₃	—	4	1	no	—
42	C₂₂₃	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	223	1	no	maximal, Sylow
44	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic, complemented
45	#126	—	223	4	no	—
46	#125	—	223	4	no	—
47	#125	—	223	1	no	—
48	C₂²⋊Q₈	—	223	2	no	—
49	C₄×Q₈	—	223	2	no	—
50	#117	—	223	1	no	—
51	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
52	C₂²×C₄	—	223	1	no	—
53	C₄⋊C₄	—	223	2	no	—
54	C₄⋊C₄	—	223	4	no	—
55	C₄⋊C₄	—	223	4	no	—
56	C₄⋊C₄	—	223	2	no	—
57	C₄⋊C₄	—	223	4	no	—
58	C₄⋊C₄	C₂×D₂₂₃	1	4	yes	—
59	C₂²⋊C₄	—	223	2	no	—
60	C₂²⋊C₄	—	223	4	no	—
61	C₄²	—	223	1	no	—
62	C₄²	—	223	4	no	—
63	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
64	C₂³	—	223	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	446	2	no	—
67	C₂×C₄	—	223	2	no	—
68	C₂×C₄	—	223	4	no	—
69	C₂×C₄	—	223	1	no	—
70	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
71	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
72	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
73	C₂²	—	446	1	no	—
74	C₂²	—	446	2	no	—
75	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	446	1	no	—
79	C₄	—	446	2	no	—
80	C₄	—	446	4	no	—
81	C₄	#16813169	1	2	yes	—
82	C₂	—	892	1	no	—
83	C₂	C₂×#16813169	1	1	yes	characteristic
84	C₂	#20297352	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	C₄⋊Q₈×C₂₂₃	C₂
prime radical	C₂²×D₂₂₃	C₂³
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#541
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20297296	C₂	semidirect
3. #20297294	C₂	semidirect
5. Q₈×Dic₂₂₃	C₂	semidirect
7. C₄⋊Q₈×C₂₂₃	C₂	semidirect
42. C₂₂₃	#541	semidirect
44. C₄⋊Q₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297299	2
#20297296	2
#20297294	2
#20297280	2
Q₈×Dic₂₂₃	2
#20297261	2
C₄⋊Q₈×C₂₂₃	2
#541	223