Subgroups — #30668218

Order 16768.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862189	C₂	1	4	yes	maximal, complemented
2	#22553689	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₄₈	—	2	1	no	—
6	C₂×D₁₀₄₈	C₂²	1	2	yes	—
7	#16862243	—	2	1	no	—
8	#16862189	—	2	8	no	—
9	C₂×C₁₃₁⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
11	C₂×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
13	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×D₅₂₄	—	2	2	no	—
15	C₂×D₅₂₄	—	4	2	no	—
16	D₁₀₄₈	—	2	4	no	—
17	D₁₀₄₈	—	4	2	no	—
18	C₁₃₁⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×Q₈×C₁₃₁	—	2	2	no	—
20	Q₁₆×C₁₃₁	—	2	4	no	—
21	Q₁₆×C₁₃₁	—	4	2	no	—
22	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₃₁	—	4	2	no	—
25	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂²×D₁₃₁	—	4	2	no	—
27	D₅₂₄	—	4	2	no	—
28	D₅₂₄	—	4	2	no	—
29	D₅₂₄	—	8	4	no	—
30	Q₈×C₁₃₁	—	4	2	no	—
31	Q₈×C₁₃₁	—	4	2	no	—
32	C₂×C₄×C₁₃₁	—	4	2	no	—
33	C₂×C₄×C₁₃₁	D₈	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₁	D₈	1	2	yes	—
36	C₈×C₁₃₁	C₂×D₄	1	2	yes	—
37	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
38	C₂×D₁₃₁	—	8	2	no	—
39	C₂×D₁₃₁	—	8	4	no	—
40	C₄×C₁₃₁	—	8	2	no	—
41	C₄×C₁₃₁	SD₃₂	1	4	yes	complemented
42	C₄×C₁₃₁	#127	1	1	yes	characteristic
43	C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
44	C₂×C₁₃₁	#493	1	1	yes	characteristic
45	C₂×C₁₃₁	C₂×SD₃₂	1	2	yes	—
46	D₁₃₁	—	16	2	no	—
47	C₁₃₁	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	131	1	no	maximal, Sylow
49	C₂×SD₃₂	—	131	4	no	—
50	#494	D₁₃₁	1	1	yes	characteristic, complemented
51	#493	—	131	1	no	—
52	C₄×C₁₆	—	131	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₁₃₁	1	2	yes	—
55	C₂×D₈	—	262	1	no	—
56	C₂×D₈	—	131	2	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	262	1	no	—
59	SD₃₂	—	262	8	no	—
60	C₂×C₁₆	—	131	2	no	—
61	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	262	2	no	—
64	C₂×D₄	—	524	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	262	4	no	—
68	D₈	—	524	2	no	—
69	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
73	C₁₆	—	131	4	no	—
74	C₂³	—	524	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	524	2	no	—
78	D₄	—	524	2	no	—
79	D₄	—	1048	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#14523498	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
83	C₈	#14523498	1	2	yes	—
84	C₈	D₄×D₁₃₁	1	2	yes	—
85	C₂²	—	1048	2	no	—
86	C₂²	—	1048	4	no	—
87	C₂²	C₂×#14523498	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#16862189	1	4	yes	complemented
90	C₄	#16862242	1	1	yes	characteristic
91	C₄	C₂×#14523498	1	1	yes	characteristic
92	C₂	—	2096	2	no	—
93	C₂	#22553688	1	1	yes	characteristic
94	C₂	C₂×#16862189	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523498
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#494×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₈
prime radical	#22553689	C₂
hypercenter	#494	D₁₃₁
hypocenter	C₁₃₁	#2248
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#16862189	C₂	semidirect
3. C₄×C₁₃₁⋊C₁₆	C₂	semidirect
4. #494×C₁₃₁	C₂	semidirect
18. C₁₃₁⋊C₁₆	D₄	semidirect
41. C₄×C₁₃₁	SD₃₂	semidirect
47. C₁₃₁	#2248	semidirect
50. #494	D₁₃₁	semidirect
89. C₄	#16862189	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862189	2
#22553689	2
C₄×C₁₃₁⋊C₁₆	2
#494×C₁₃₁	2
#2248	131