Subgroups — C₄×#19150738 #36378257

Order 22592.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150738	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₄₁₂	C₂	1	3	yes	maximal, complemented
4	#26014954	C₂	1	3	yes	maximal, complemented
5	#26014950	C₂	1	3	yes	maximal, complemented
6	C₄²×D₃₅₃	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150738	C₄	1	8	yes	direct product factor
9	C₂×D₁₄₁₂	C₂²	1	3	yes	—
10	C₂×C₄×D₃₅₃	C₂²	1	3	yes	—
11	C₂×C₄×D₃₅₃	C₂²	1	3	yes	—
12	C₂×C₄×D₃₅₃	C₂²	1	3	yes	—
13	C₄⋊Dic₃₅₃	C₂²	1	3	yes	—
14	#19150714	C₂²	1	6	yes	—
15	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₃₅₃	C₂²	1	3	yes	—
17	C₄×Dic₃₅₃	C₂²	1	3	yes	—
18	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₅₃	C₂²	1	3	yes	—
20	C₄²×C₃₅₃	C₂²	1	3	yes	—
21	C₂²×D₃₅₃	C₂³	1	3	yes	—
22	D₁₄₁₂	C₂×C₄	1	12	yes	complemented
23	C₂×Dic₃₅₃	C₂³	1	3	yes	—
24	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
26	C₄×D₃₅₃	—	2	6	no	—
27	C₄×D₃₅₃	—	2	6	no	—
28	C₄×D₃₅₃	C₂×C₄	1	12	yes	complemented
29	Q₈×C₃₅₃	C₂×C₄	1	4	yes	complemented
30	C₂×C₄×C₃₅₃	C₂³	1	3	yes	—
31	C₂×C₄×C₃₅₃	C₂³	1	3	yes	—
32	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
34	C₂×D₃₅₃	—	2	3	no	—
35	C₂×D₃₅₃	—	2	3	no	—
36	C₂×D₃₅₃	C₂²×C₄	1	6	yes	—
37	C₄×C₃₅₃	—	2	3	no	—
38	C₄×C₃₅₃	C₂²×C₄	1	6	yes	—
39	C₄×C₃₅₃	#41	1	2	yes	complemented
40	Dic₃₅₃	—	2	3	no	—
41	Dic₃₅₃	C₂²×C₄	1	2	yes	—
42	Dic₃₅₃	#41	1	2	yes	complemented
43	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₃₅₃	C₂×#41	1	1	yes	characteristic
45	C₂×C₃₅₃	C₂×#41	1	1	yes	characteristic
46	D₃₅₃	—	2	6	no	—
47	C₃₅₃	C₄×#41	1	1	yes	characteristic, complemented, Sylow
48	C₄×#41	—	353	1	no	maximal, Sylow
49	C₂×#41	—	353	1	no	—
50	C₄×Q₈	—	353	1	no	—
51	C₄×Q₈	D₃₅₃	1	1	yes	characteristic, complemented
52	C₄×D₄	—	353	3	no	—
53	C₄×D₄	—	353	3	no	—
54	#117	—	353	3	no	—
55	C₂×C₄²	—	353	3	no	—
56	#41	—	353	8	no	—
57	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
58	C₂×D₄	—	353	3	no	—
59	C₂²×C₄	—	353	3	no	—
60	C₂²×C₄	—	353	3	no	—
61	C₂²×C₄	—	353	3	no	—
62	C₄⋊C₄	—	353	3	no	—
63	C₄⋊C₄	C₂×D₃₅₃	1	3	yes	—
64	C₂²⋊C₄	—	353	6	no	—
65	C₄²	—	353	3	no	—
66	C₄²	—	353	3	no	—
67	C₄²	—	353	1	no	—
68	C₄²	C₂×D₃₅₃	1	3	yes	—
69	C₂³	—	353	3	no	—
70	Q₈	C₄×D₃₅₃	1	4	yes	complemented
71	D₄	—	353	12	no	—
72	C₂×C₄	—	353	1	no	—
73	C₂×C₄	—	353	1	no	—
74	C₂×C₄	—	353	12	no	—
75	C₂×C₄	—	353	3	no	—
76	C₂×C₄	—	706	6	no	—
77	C₂×C₄	—	706	6	no	—
78	C₂×C₄	C₂²×D₃₅₃	1	3	yes	—
79	C₂×C₄	C₂²×D₃₅₃	1	3	yes	—
80	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
81	C₂²	—	353	6	no	—
82	C₂²	—	706	3	no	—
83	C₂²	—	706	3	no	—
84	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
85	C₄	—	2	3	no	—
86	C₄	—	353	2	no	—
87	C₄	—	353	2	no	—
88	C₄	—	706	3	no	—
89	C₄	C₂×C₄×D₃₅₃	1	6	yes	—
90	C₄	#19150738	1	2	yes	direct product factor
91	C₂	—	706	6	no	—
92	C₂	C₂²×C₄×D₃₅₃	1	1	yes	characteristic
93	C₂	#41×D₃₅₃	1	1	yes	characteristic
94	C₂	C₂×#19150738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₃
commutator	C₂×C₃₅₃	C₂³×C₄
Frattini	C₂²	C₂³×D₃₅₃
Fitting	C₄×Q₈×C₃₅₃	C₂
socle, cosocle	C₂²×C₃₅₃	C₂⁴
prime radical	C₂×#19150738	C₂
hypercenter	C₄×Q₈	D₃₅₃
hypocenter	C₃₅₃	C₄×#41
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₃₅₃	C₂	semidirect
3. C₄×D₁₄₁₂	C₂	semidirect
4. #26014954	C₂	semidirect
5. #26014950	C₂	semidirect
6. C₄²×D₃₅₃	C₂	semidirect
7. C₄×Q₈×C₃₅₃	C₂	semidirect
8. #19150738	C₄	direct (N × H)
22. D₁₄₁₂	C₂×C₄	semidirect
28. C₄×D₃₅₃	C₂×C₄	semidirect
29. Q₈×C₃₅₃	C₂×C₄	semidirect
39. C₄×C₃₅₃	#41	semidirect
42. Dic₃₅₃	#41	semidirect
47. C₃₅₃	C₄×#41	semidirect
51. C₄×Q₈	D₃₅₃	semidirect
70. Q₈	C₄×D₃₅₃	semidirect
90. C₄	#19150738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19150738	2
Q₈×Dic₃₅₃	2
C₄×D₁₄₁₂	2
#26014954	2
#26014950	2
C₄²×D₃₅₃	2
C₄×Q₈×C₃₅₃	2
C₄×#41	353

Subgroups — C4×#19150738 #36378257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19150738 #36378257