Subgroups — #30664687

Order 16768.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553369	C₂	1	2	yes	maximal
2	#22553061	C₂	1	4	yes	maximal, complemented
3	C₂×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	—
5	C₂²⋊(C₁₃₁⋊C₈)	—	2	2	no	—
6	#16862157	—	2	2	no	—
7	C₂²×Q₈×C₁₃₁	C₄	1	2	yes	—
8	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
9	C₄⋊Q₈×C₁₃₁	C₄	1	2	yes	—
10	C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
12	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₃₁⋊C₈	—	2	4	no	—
14	C₂×Q₈×C₁₃₁	—	2	2	no	—
15	C₂×Q₈×C₁₃₁	—	4	4	no	—
16	C₂×Q₈×C₁₃₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₃₁	—	2	2	no	—
18	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃₁	D₄	1	2	yes	—
20	C₄⋊C₄×C₁₃₁	—	2	4	no	—
21	C₄⋊C₄×C₁₃₁	—	2	4	no	—
22	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₃₁	D₄	1	2	yes	—
24	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₁₃₁⋊C₈	—	4	4	no	—
26	C₂³×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₁₃₁	—	4	4	no	—
28	Q₈×C₁₃₁	—	4	4	no	—
29	C₂×C₄×C₁₃₁	—	2	2	no	—
30	C₂×C₄×C₁₃₁	—	2	4	no	—
31	C₂×C₄×C₁₃₁	—	2	4	no	—
32	C₂×C₄×C₁₃₁	—	4	4	no	—
33	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
37	C₂²×C₁₃₁	—	2	2	no	—
38	C₂²×C₁₃₁	#100	1	2	yes	—
39	C₂²×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₁₃₁	—	2	1	no	—
41	C₄×C₁₃₁	—	2	1	no	—
42	C₄×C₁₃₁	—	2	2	no	—
43	C₄×C₁₃₁	—	4	4	no	—
44	C₄×C₁₃₁	#137	1	4	yes	—
45	C₂×C₁₃₁	—	2	2	no	—
46	C₂×C₁₃₁	C₂×#100	1	1	yes	characteristic
47	C₂×C₁₃₁	#419	1	2	yes	—
48	C₁₃₁	#1558	1	1	yes	characteristic, complemented, Sylow
49	#1558	—	131	1	no	maximal, Sylow
50	C₂×C₄⋊Q₈	D₁₃₁	1	1	yes	characteristic
51	#432	—	131	2	no	—
52	#333	—	131	4	no	—
53	C₂²×Q₈	Dic₁₃₁	1	2	yes	—
54	C₄⋊Q₈	—	2	2	no	—
55	C₄⋊Q₈	Dic₁₃₁	1	2	yes	—
56	C₄⋊Q₈	C₂×D₁₃₁	1	2	yes	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
59	C₄⋊C₈	—	131	4	no	—
60	C₂²⋊C₈	—	262	2	no	—
61	#97	—	262	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	4	4	no	—
64	C₂×Q₈	C₂×Dic₁₃₁	1	4	yes	—
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
67	C₂²×C₄	C₂²⋊D₁₃₁	1	2	yes	—
68	C₂×C₈	—	262	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
72	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
73	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	Q₈	—	4	4	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	C₂²⋊Dic₁₃₁	1	2	yes	—
82	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
83	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
84	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
85	C₈	—	524	4	no	—
86	C₂²	—	2	2	no	—
87	C₂²	#16862163	1	2	yes	—
88	C₂²	C₂×C₂²⋊Dic₁₃₁	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	2	1	no	—
91	C₄	—	2	2	no	—
92	C₄	—	4	4	no	—
93	C₄	#16862277	1	4	yes	—
94	C₂	—	2	2	no	—
95	C₂	C₂×#16862163	1	1	yes	characteristic
96	C₂	#22553321	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₁
Fitting	C₂×C₄⋊Q₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃₁	C₂²⋊C₄
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃₁
hypocenter	C₁₃₁	#1558
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #22553061	C₂	semidirect
48. C₁₃₁	#1558	semidirect

Maximal subgroups

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

subgroup	index
#22553369	2
#22553061	2
C₂×C₄⋊Q₈×C₁₃₁	2
#1558	131