Subgroups — #29507923

Order 15424.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838759	C₂	1	2	yes	maximal
2	#21440642	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₉₆₄	C₂	1	1	yes	maximal, characteristic
4	#21440567	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₈₂	C₂²	1	2	yes	—
7	C₂×D₉₆₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₄₁	—	2	1	no	—
9	#16838759	—	2	4	no	—
10	C₄⋊Dic₂₄₁	—	2	1	no	—
11	C₄⋊Dic₂₄₁	—	2	1	no	—
12	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
13	#16838747	—	2	1	no	—
14	C₂×C₈×C₂₄₁	C₂²	1	2	yes	—
15	C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
16	C₂²×D₂₄₁	—	2	1	no	—
17	Dic₄₈₂	—	2	2	no	—
18	Dic₄₈₂	—	2	2	no	—
19	D₉₆₄	—	2	1	no	—
20	D₉₆₄	D₄	1	2	yes	—
21	C₂×Dic₂₄₁	—	2	1	no	—
22	C₂×Dic₂₄₁	—	2	2	no	—
23	C₄×D₂₄₁	—	4	2	no	—
24	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
27	C₈×C₂₄₁	—	2	2	no	—
28	C₂²×C₂₄₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₄₁	—	2	2	no	—
30	C₂×D₂₄₁	—	4	1	no	—
31	C₂×D₂₄₁	—	4	1	no	—
32	C₄×C₂₄₁	—	2	1	no	—
33	C₄×C₂₄₁	SD₁₆	1	2	yes	complemented
34	C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₄₁	#41	1	1	yes	characteristic
36	Dic₂₄₁	—	4	2	no	—
37	Dic₂₄₁	—	4	1	no	—
38	C₂×C₂₄₁	#121	1	1	yes	characteristic
39	C₂×C₂₄₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₂₄₁	#137	1	1	yes	characteristic
41	D₂₄₁	—	4	2	no	—
42	C₂₄₁	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	241	1	no	maximal, Sylow
44	C₂×SD₁₆	—	241	2	no	—
45	C₄⋊Q₈	—	241	1	no	—
46	C₄×D₄	—	241	1	no	—
47	C₄⋊C₈	D₂₄₁	1	1	yes	characteristic, complemented
48	#103	—	241	2	no	—
49	C₂×Q₈	—	241	2	no	—
50	C₂×D₄	—	241	1	no	—
51	C₂²×C₄	—	482	1	no	—
52	SD₁₆	—	482	4	no	—
53	C₂×C₈	C₂×D₂₄₁	1	2	yes	—
54	C₄⋊C₄	—	482	1	no	—
55	C₄⋊C₄	—	482	1	no	—
56	C₄⋊C₄	—	241	1	no	—
57	C₂²⋊C₄	—	482	1	no	—
58	C₄²	C₂×D₂₄₁	1	1	yes	characteristic
59	C₂³	—	482	1	no	—
60	Q₈	—	482	2	no	—
61	Q₈	—	482	2	no	—
62	D₄	—	482	1	no	—
63	D₄	—	241	2	no	—
64	C₂×C₄	—	964	2	no	—
65	C₂×C₄	—	482	1	no	—
66	C₂×C₄	—	482	2	no	—
67	C₂×C₄	D₉₆₄	1	1	yes	characteristic
68	C₂×C₄	D₉₆₄	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₂²	—	964	1	no	—
72	C₂²	—	964	1	no	—
73	C₂²	—	482	2	no	—
74	C₂²	C₂×D₉₆₄	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	964	2	no	—
77	C₄	—	964	1	no	—
78	C₄	#16838759	1	2	yes	complemented
79	C₄	D₄×D₂₄₁	1	1	yes	characteristic
80	C₄	#16838771	1	1	yes	characteristic
81	C₂	—	964	2	no	—
82	C₂	#21440617	1	1	yes	characteristic
83	C₂	C₂×#16838759	1	1	yes	characteristic
84	C₂	#21440675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₆₄
commutator, cosocle	C₂×C₄×C₂₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₄₁
Fitting	C₄⋊C₈×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂×D₄
prime radical	C₂×D₉₆₄	C₂²
hypercenter	C₄⋊C₈	D₂₄₁
hypocenter	C₂₄₁	#461
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #21440642	C₂	semidirect
4. #21440567	C₂	semidirect
5. C₄⋊C₈×C₂₄₁	C₂	semidirect
33. C₄×C₂₄₁	SD₁₆	semidirect
42. C₂₄₁	#461	semidirect
47. C₄⋊C₈	D₂₄₁	semidirect
78. C₄	#16838759	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838759	2
#21440642	2
C₄×D₉₆₄	2
#21440567	2
C₄⋊C₈×C₂₄₁	2
#461	241