Subgroups — C₂²×#235 #5954

Order 192.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#67	C₂	1	2	yes	maximal, complemented
3	C₂×#235	C₂	1	24	yes	maximal, direct product factor
4	C₂²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₃	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41	—	3	1	no	maximal, Sylow
8	C₂⁴×C₃	C₂²	1	1	yes	characteristic
9	C₂³×D₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂²	1	12	yes	complemented
13	C₂×#67	C₂²	1	24	yes	complemented
14	C₂²×Dic₃	C₂²	1	2	yes	—
15	#235	C₂²	1	64	yes	direct product factor
16	C₂×D₁₂	C₂²	1	12	yes	complemented
17	C₂×C₄×D₃	C₂²	1	24	yes	complemented
18	C₂×Dic₆	C₂²	1	12	yes	complemented
19	C₂×#41	—	3	24	no	—
20	C₂²×Q₈	—	3	1	no	—
21	C₂²×D₄	—	3	2	no	—
22	C₂²×D₄	—	3	1	no	—
23	C₂³×C₄	—	3	2	no	—
24	C₂³×C₄	D₃	1	1	yes	characteristic, complemented
25	C₂³×C₃	—	2	4	no	—
26	C₂³×C₃	C₂³	1	6	yes	—
27	C₂³×C₃	C₂³	1	1	yes	characteristic
28	C₂²×D₃	—	2	8	no	—
29	C₂²×D₃	C₂³	1	12	yes	—
30	C₂×C₃×C₄	C₂³	1	6	yes	—
31	C₂×C₃×C₄	C₂³	1	6	yes	—
32	C₂×C₃×C₄	C₂³	1	16	yes	complemented
33	#67	C₂³	1	32	yes	complemented
34	C₂×Dic₃	C₂³	1	12	yes	—
35	D₁₂	C₂³	1	16	yes	complemented
36	C₄×D₃	C₂³	1	32	yes	complemented
37	Dic₆	C₂³	1	16	yes	complemented
38	C₂⁴	—	3	2	no	—
39	C₂⁴	C₂×D₃	1	1	yes	characteristic
40	#41	—	3	64	no	—
41	C₂×Q₈	—	3	12	no	—
42	C₂×D₄	—	3	24	no	—
43	C₂×D₄	—	3	12	no	—
44	C₂²×C₄	—	3	2	no	—
45	C₂²×C₄	—	3	24	no	—
46	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₃	1	12	yes	complemented
49	C₂²×C₃	—	2	12	no	—
50	C₂²×C₃	#41	1	4	yes	complemented
51	C₂²×C₃	C₂⁴	1	4	yes	—
52	C₂²×C₃	C₂⁴	1	3	yes	—
53	C₂×D₃	—	2	24	no	—
54	C₂×D₃	C₂⁴	1	8	yes	—
55	C₃×C₄	C₂⁴	1	4	yes	—
56	C₃×C₄	C₂⁴	1	4	yes	—
57	Dic₃	C₂⁴	1	8	yes	—
58	C₂³	—	2	4	no	—
59	C₂³	—	3	12	no	—
60	C₂³	—	6	8	no	—
61	C₂³	C₂²×D₃	1	6	yes	—
62	C₂³	C₂²×D₃	1	1	yes	characteristic
63	Q₈	—	3	16	no	—
64	D₄	—	3	32	no	—
65	D₄	—	3	16	no	—
66	C₂×C₄	—	3	12	no	—
67	C₂×C₄	—	3	32	no	—
68	C₂×C₄	C₂²×D₃	1	6	yes	—
69	C₂×C₄	C₂²×D₃	1	6	yes	—
70	C₂×C₄	C₂²×D₃	1	16	yes	complemented
71	C₂×C₃	—	2	4	no	—
72	C₂×C₃	C₂×#41	1	6	yes	complemented
73	C₂×C₃	C₂⁵	1	1	yes	characteristic
74	D₃	—	2	8	no	—
75	C₂²	—	2	12	no	—
76	C₂²	—	3	8	no	—
77	C₂²	—	6	24	no	—
78	C₂²	#235	1	4	yes	direct product factor
79	C₂²	C₂³×D₃	1	4	yes	—
80	C₂²	C₂³×D₃	1	3	yes	—
81	C₄	—	3	8	no	—
82	C₄	C₂³×D₃	1	4	yes	—
83	C₄	C₂³×D₃	1	4	yes	—
84	C₃	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
85	C₂	—	2	4	no	—
86	C₂	—	6	8	no	—
87	C₂	C₂×#235	1	6	yes	direct product factor
88	C₂	C₂⁴×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁵
Frattini	C₂	C₂⁴×D₃
Fitting	C₂³×C₃×C₄	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂³×C₄	D₃
hypocenter	C₃	C₂²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₄	C₂	semidirect
2. C₂²×#67	C₂	semidirect
3. C₂×#235	C₂	direct (N × H)
4. C₂²×D₁₂	C₂	semidirect
5. C₂²×C₄×D₃	C₂	semidirect
6. C₂²×Dic₆	C₂	semidirect
12. C₂²×C₃×C₄	C₂²	semidirect
13. C₂×#67	C₂²	semidirect
15. #235	C₂²	direct (N × H)
16. C₂×D₁₂	C₂²	semidirect
17. C₂×C₄×D₃	C₂²	semidirect
18. C₂×Dic₆	C₂²	semidirect
24. C₂³×C₄	D₃	semidirect
32. C₂×C₃×C₄	C₂³	semidirect
33. #67	C₂³	semidirect
35. D₁₂	C₂³	semidirect
36. C₄×D₃	C₂³	semidirect
37. Dic₆	C₂³	semidirect
48. C₂²×C₄	C₂×D₃	semidirect
50. C₂²×C₃	#41	semidirect
70. C₂×C₄	C₂²×D₃	semidirect
72. C₂×C₃	C₂×#41	semidirect
78. C₂²	#235	direct (N × H)
84. C₃	C₂²×#41	semidirect
87. C₂	C₂×#235	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂²×#67	2
C₂×#235	2
C₂²×D₁₂	2
C₂²×C₄×D₃	2
C₂²×Dic₆	2
C₂²×#41	3

Subgroups — C22×#235 #5954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#235 #5954