Subgroups — C₂×#124×C₂₂₃ #28383238

Order 14272.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#124×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₄×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₂₂₃	C₂²	1	2	yes	—
7	C₂×Q₈×C₂₂₃	—	2	4	no	—
8	C₂×Q₈×C₂₂₃	C₂²	1	4	yes	complemented
9	C₂×D₄×C₂₂₃	—	2	4	no	—
10	C₂×D₄×C₂₂₃	C₂²	1	4	yes	—
11	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₂₃	C₂²	1	16	yes	complemented
15	C₄²×C₂₂₃	C₂²	1	4	yes	complemented
16	C₂³×C₂₂₃	—	2	4	no	—
17	C₂³×C₂₂₃	—	2	8	no	—
18	C₂³×C₂₂₃	C₂³	1	4	yes	—
19	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
20	Q₈×C₂₂₃	—	2	8	no	—
21	D₄×C₂₂₃	—	2	8	no	—
22	C₂×C₄×C₂₂₃	—	2	4	no	—
23	C₂×C₄×C₂₂₃	—	2	4	no	—
24	C₂×C₄×C₂₂₃	D₄	1	4	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
26	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
27	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
28	C₂²×C₂₂₃	—	2	4	no	—
29	C₂²×C₂₂₃	—	2	8	no	—
30	C₂²×C₂₂₃	—	4	8	no	—
31	C₂²×C₂₂₃	C₂×D₄	1	2	yes	—
32	C₂²×C₂₂₃	#41	1	4	yes	—
33	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₂₂₃	—	2	4	no	—
35	C₄×C₂₂₃	—	2	4	no	—
36	C₄×C₂₂₃	C₂×D₄	1	4	yes	—
37	C₂×C₂₂₃	—	4	4	no	—
38	C₂×C₂₂₃	#124	1	4	yes	direct product factor
39	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×#41	1	2	yes	—
41	C₂₂₃	C₂×#124	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×#124	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
44	C₂²×D₄	C₂×C₂₂₃	1	1	yes	characteristic
45	#124	C₂×C₂₂₃	1	8	yes	direct product factor
46	C₂×C₂²⋊C₄	C₂×C₂₂₃	1	4	yes	complemented
47	C₂×C₄²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
48	C₂⁴	C₂²×C₂₂₃	1	2	yes	—
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	C₂²×C₂₂₃	1	4	yes	complemented
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	C₂²×C₂₂₃	1	4	yes	—
53	C₂²×C₄	C₂²×C₂₂₃	1	2	yes	—
54	C₂²×C₄	C₂²×C₂₂₃	1	2	yes	—
55	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
56	C₂²⋊C₄	C₂²×C₂₂₃	1	16	yes	complemented
57	C₄²	C₂²×C₂₂₃	1	4	yes	complemented
58	C₂³	—	2	4	no	—
59	C₂³	—	2	8	no	—
60	C₂³	C₂³×C₂₂₃	1	4	yes	—
61	C₂³	C₂³×C₂₂₃	1	1	yes	characteristic
62	Q₈	—	2	8	no	—
63	D₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	D₄×C₂₂₃	1	4	yes	—
67	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
68	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
69	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	4	8	no	—
73	C₂²	C₂×D₄×C₂₂₃	1	2	yes	—
74	C₂²	#41×C₂₂₃	1	4	yes	—
75	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	4	no	—
78	C₄	C₂×D₄×C₂₂₃	1	4	yes	—
79	C₂	—	4	4	no	—
80	C₂	#124×C₂₂₃	1	4	yes	direct product factor
81	C₂	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
82	C₂	C₂×#41×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
prime radical	C₂²×D₄×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₂₃	C₂	semidirect
3. #124×C₂₂₃	C₂	direct (N × H)
4. C₂×C₂²⋊C₄×C₂₂₃	C₂	semidirect
5. C₂×C₄²×C₂₂₃	C₂	semidirect
8. C₂×Q₈×C₂₂₃	C₂²	semidirect
14. C₂²⋊C₄×C₂₂₃	C₂²	semidirect
15. C₄²×C₂₂₃	C₂²	semidirect
38. C₂×C₂₂₃	#124	direct (N × H)
41. C₂₂₃	C₂×#124	direct (N × H)
42. C₂×#124	C₂₂₃	direct (N × H)
43. C₂²×Q₈	C₂×C₂₂₃	semidirect
45. #124	C₂×C₂₂₃	direct (N × H)
46. C₂×C₂²⋊C₄	C₂×C₂₂₃	semidirect
47. C₂×C₄²	C₂×C₂₂₃	semidirect
50. C₂×Q₈	C₂²×C₂₂₃	semidirect
56. C₂²⋊C₄	C₂²×C₂₂₃	semidirect
57. C₄²	C₂²×C₂₂₃	semidirect
80. C₂	#124×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₂₃	2
C₂²×D₄×C₂₂₃	2
#124×C₂₂₃	2
C₂×C₂²⋊C₄×C₂₂₃	2
C₂×C₄²×C₂₂₃	2
C₂×#124	223

Subgroups — C2×#124×C223 #28383238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#124×C₂₂₃ #28383238