Subgroups — C₂×C₄⋊#16812367 #42122923

Order 27712.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16812367	C₂	1	2	yes	maximal
2	C₄⋊#16812367	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#13181884	C₂	1	1	yes	maximal, characteristic
4	C₂×#16812367	—	2	4	no	—
5	C₂×#16812367	C₂²	1	4	yes	—
6	C₂²×#13181884	C₄	1	1	yes	characteristic
7	C₂²×#13181884	C₂²	1	1	yes	characteristic
8	C₄×#13181884	C₄	1	2	yes	—
9	C₄×#13181884	C₂²	1	2	yes	—
10	C₂×C₄×D₄₃₃	C₄	1	1	yes	characteristic
11	#16812367	—	2	4	no	—
12	C₂×#13181884	—	2	2	no	—
13	C₂×#13181884	C₂×C₄	1	2	yes	—
14	C₂×#13181884	C₂×C₄	1	1	yes	characteristic
15	C₂×#13181884	D₄	1	2	yes	—
16	C₂×#13181884	Q₈	1	2	yes	—
17	C₂×#13181884	C₂³	1	1	yes	characteristic
18	C₂²×D₄₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₃₃	C₈	1	1	yes	characteristic, complemented
20	C₄×D₄₃₃	C₈	1	2	yes	complemented
21	C₄×D₄₃₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₃₃	C₈	1	1	yes	characteristic, complemented
23	C₂²×C₄₃₃	C₂×C₈	1	1	yes	characteristic
24	C₂×D₄₃₃	C₄⋊C₄	1	2	yes	—
25	C₂×D₄₃₃	C₂×C₈	1	1	yes	characteristic
26	C₂×D₄₃₃	M₁₆	1	2	yes	—
27	C₂×D₄₃₃	C₂²×C₄	1	1	yes	characteristic
28	#13181884	—	2	2	no	—
29	#13181884	C₄⋊C₄	1	2	yes	—
30	#13181884	C₂×D₄	1	1	yes	characteristic
31	#13181884	C₂×Q₈	1	1	yes	characteristic
32	C₄×C₄₃₃	C₂×C₈	1	2	yes	complemented
33	Dic₄₃₃	C₂×C₈	1	2	yes	complemented
34	C₂×C₄₃₃	C₄⋊C₈	1	2	yes	complemented
35	C₂×C₄₃₃	C₂²×C₈	1	1	yes	characteristic
36	D₄₃₃	C₄⋊C₈	1	2	yes	complemented
37	D₄₃₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	D₄₃₃	C₂×M₁₆	1	1	yes	characteristic
39	C₄₃₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄⋊C₈	—	433	1	no	maximal, Sylow
41	C₂²×C₈	—	433	2	no	—
42	C₂×C₄²	—	433	1	no	—
43	C₄⋊C₈	—	433	4	no	—
44	C₂²×C₄	—	433	1	no	—
45	C₂²×C₄	—	433	1	no	—
46	C₂²×C₄	—	433	1	no	—
47	C₂×C₈	—	866	4	no	—
48	C₂×C₈	—	433	4	no	—
49	C₄²	—	433	2	no	—
50	C₄²	—	433	2	no	—
51	C₂³	—	433	1	no	—
52	C₂×C₄	—	866	2	no	—
53	C₂×C₄	—	433	2	no	—
54	C₂×C₄	—	433	1	no	—
55	C₂×C₄	—	433	1	no	—
56	C₂×C₄	—	433	2	no	—
57	C₂×C₄	—	433	2	no	—
58	C₂×C₄	—	433	1	no	—
59	C₂×C₄	—	433	2	no	—
60	C₂×C₄	—	433	2	no	—
61	C₂×C₄	#16812367	1	1	yes	characteristic, complemented
62	C₈	—	866	4	no	—
63	C₂²	—	433	2	no	—
64	C₂²	—	433	2	no	—
65	C₂²	—	433	1	no	—
66	C₂²	—	433	1	no	—
67	C₂²	C₂×#16812367	1	1	yes	characteristic
68	C₄	—	866	2	no	—
69	C₄	—	433	1	no	—
70	C₄	—	433	2	no	—
71	C₄	—	433	1	no	—
72	C₄	—	433	2	no	—
73	C₄	C₂×#16812367	1	2	yes	complemented
74	C₂	—	433	1	no	—
75	C₂	—	433	1	no	—
76	C₂	—	433	2	no	—
77	C₂	C₄⋊#16812367	1	2	yes	direct product factor
78	C₂	C₂²×#16812367	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#16812367
commutator	C₂×C₄₃₃	C₂²×C₈
Frattini	C₂	C₂²×#16812367
Fitting	C₂×C₄×C₄₃₃	C₈
socle	C₂²×C₄₃₃	C₂×C₈
prime radical	C₂²×D₄₃₃	C₂×C₄
cosocle	C₂×#13181884	C₂³
hypercenter	C₂×C₄	#16812367
hypocenter	C₄₃₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16812367	C₂	direct (N × H)
19. C₂×Dic₄₃₃	C₈	semidirect
20. C₄×D₄₃₃	C₈	semidirect
22. C₂×C₄×C₄₃₃	C₈	semidirect
32. C₄×C₄₃₃	C₂×C₈	semidirect
33. Dic₄₃₃	C₂×C₈	semidirect
34. C₂×C₄₃₃	C₄⋊C₈	semidirect
36. D₄₃₃	C₄⋊C₈	semidirect
39. C₄₃₃	C₂×C₄⋊C₈	semidirect
61. C₂×C₄	#16812367	semidirect
73. C₄	C₂×#16812367	semidirect
77. C₂	C₄⋊#16812367	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×#16812367	2
C₄⋊#16812367	2
C₂×C₄×#13181884	2
C₂×C₄⋊C₈	433

Subgroups — C2×C4⋊#16812367 #42122923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#16812367 #42122923