Subgroups — #2397×C₂₂₃ #43251412

Order 28544.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₂₃	C₂	1	6	yes	maximal, complemented
2	#399×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#392×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	#376×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₂₂₃	C₂²	1	6	yes	—
6	C₂³×C₄×C₂₂₃	C₂²	1	4	yes	—
7	C₄×D₄×C₂₂₃	—	2	24	no	—
8	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
9	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	12	yes	—
10	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
11	#95×C₂₂₃	C₂²	1	4	yes	—
12	C₂⁴×C₂₂₃	C₂³	1	4	yes	—
13	C₂×D₄×C₂₂₃	—	2	24	no	—
14	C₂×D₄×C₂₂₃	—	2	12	no	—
15	C₂²×C₄×C₂₂₃	—	2	24	no	—
16	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
18	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	—	2	12	no	—
20	C₂²⋊C₄×C₂₂₃	—	2	24	no	—
21	C₄²×C₂₂₃	D₄	1	12	yes	complemented
22	C₂³×C₂₂₃	—	2	12	no	—
23	C₂³×C₂₂₃	—	2	12	no	—
24	C₂³×C₂₂₃	#41	1	8	yes	—
25	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
26	D₄×C₂₂₃	—	2	48	no	—
27	C₂×C₄×C₂₂₃	—	2	12	no	—
28	C₂×C₄×C₂₂₃	—	2	12	no	—
29	C₂×C₄×C₂₂₃	—	4	16	no	—
30	C₂×C₄×C₂₂₃	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₂₂₃	C₂×D₄	1	6	yes	—
32	C₂²×C₂₂₃	—	2	24	no	—
33	C₂²×C₂₂₃	—	4	12	no	—
34	C₂²×C₂₂₃	—	4	4	no	—
35	C₂²×C₂₂₃	C₂²×D₄	1	3	yes	—
36	C₂²×C₂₂₃	C₂×#41	1	4	yes	—
37	C₄×C₂₂₃	—	2	12	no	—
38	C₄×C₂₂₃	—	4	4	no	—
39	C₄×C₂₂₃	#127	1	4	yes	—
40	C₂×C₂₂₃	—	4	8	no	—
41	C₂×C₂₂₃	#525	1	6	yes	—
42	C₂×C₂₂₃	C₂×#127	1	1	yes	characteristic
43	C₂₂₃	#2397	1	1	yes	characteristic, direct product factor, Sylow
44	#2397	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₄×D₄	C₂×C₂₂₃	1	6	yes	complemented
46	#399	C₂×C₂₂₃	1	4	yes	complemented
47	#392	C₂×C₂₂₃	1	4	yes	complemented
48	#376	C₂×C₂₂₃	1	1	yes	characteristic, complemented
49	C₂²×D₄	C₂²×C₂₂₃	1	6	yes	—
50	C₂³×C₄	C₂²×C₂₂₃	1	4	yes	—
51	C₄×D₄	—	2	24	no	—
52	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
53	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	12	yes	—
54	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
55	#95	C₂²×C₂₂₃	1	4	yes	—
56	C₂⁴	C₂³×C₂₂₃	1	4	yes	—
57	C₂×D₄	—	2	24	no	—
58	C₂×D₄	—	2	12	no	—
59	C₂²×C₄	—	2	24	no	—
60	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
61	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
62	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₄⋊C₄	—	2	12	no	—
64	C₂²⋊C₄	—	2	24	no	—
65	C₄²	D₄×C₂₂₃	1	12	yes	complemented
66	C₂³	—	2	12	no	—
67	C₂³	—	2	12	no	—
68	C₂³	#41×C₂₂₃	1	8	yes	—
69	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
70	D₄	—	2	48	no	—
71	C₂×C₄	—	2	12	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	4	16	no	—
74	C₂×C₄	C₂×D₄×C₂₂₃	1	12	yes	—
75	C₂×C₄	C₂×D₄×C₂₂₃	1	6	yes	—
76	C₂²	—	2	24	no	—
77	C₂²	—	4	12	no	—
78	C₂²	—	4	4	no	—
79	C₂²	C₂²×D₄×C₂₂₃	1	3	yes	—
80	C₂²	C₂×#41×C₂₂₃	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	4	4	no	—
83	C₄	#127×C₂₂₃	1	4	yes	—
84	C₂	—	4	8	no	—
85	C₂	#525×C₂₂₃	1	6	yes	—
86	C₂	C₂×#127×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₂₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
socle	C₂³×C₂₂₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₄×C₂₂₃	C₂	semidirect
2. #399×C₂₂₃	C₂	semidirect
3. #392×C₂₂₃	C₂	semidirect
4. #376×C₂₂₃	C₂	semidirect
21. C₄²×C₂₂₃	D₄	semidirect
43. C₂₂₃	#2397	direct (N × H)
44. #2397	C₂₂₃	direct (N × H)
45. C₂×C₄×D₄	C₂×C₂₂₃	semidirect
46. #399	C₂×C₂₂₃	semidirect
47. #392	C₂×C₂₂₃	semidirect
48. #376	C₂×C₂₂₃	semidirect
65. C₄²	D₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₂₃	2
#399×C₂₂₃	2
#392×C₂₂₃	2
#376×C₂₂₃	2
#2397	223

Subgroups — #2397×C223 #43251412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2397×C₂₂₃ #43251412