Subgroups — #36374326

Order 22336.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013687	C₂	1	1	yes	maximal, characteristic
2	#26013670	C₂	1	2	yes	maximal, complemented
3	#26013567	C₂	1	2	yes	maximal, complemented
4	#26013558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150234	—	2	2	no	—
7	#19150233	—	2	2	no	—
8	#19150233	C₂²	1	2	yes	—
9	Q₈×D₃₄₉	—	2	2	no	—
10	#19150223	—	2	2	no	—
11	#19150220	—	2	2	no	—
12	#19150217	C₂²	1	2	yes	—
13	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₄₉	—	2	2	no	—
16	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
17	Dic₆₉₈	—	2	2	no	—
18	Dic₆₉₈	D₄	1	2	yes	—
19	D₁₃₉₆	—	2	2	no	—
20	D₁₃₉₆	D₄	1	2	yes	—
21	C₂²⋊D₃₄₉	—	2	2	no	—
22	C₄×D₃₄₉	—	2	4	no	—
23	C₄×D₃₄₉	—	2	2	no	—
24	#14584095	—	2	2	no	—
25	Q₈×C₃₄₉	—	2	2	no	—
26	Q₈×C₃₄₉	—	2	2	no	—
27	C₂×C₄×C₃₄₉	—	2	1	no	—
28	C₂×C₄×C₃₄₉	—	2	1	no	—
29	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
30	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
31	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₄₉	—	2	2	no	—
33	C₄×C₃₄₉	—	2	2	no	—
34	C₄×C₃₄₉	—	2	2	no	—
35	C₄×C₃₄₉	—	4	1	no	—
36	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
37	Dic₃₄₉	—	2	2	no	—
38	C₂×C₃₄₉	—	2	1	no	—
39	C₂×C₃₄₉	#120	1	1	yes	characteristic
40	D₃₄₉	—	4	2	no	—
41	C₃₄₉	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	349	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	349	1	no	—
44	#137	—	349	2	no	—
45	C₄⋊Q₈	D₃₄₉	1	1	yes	characteristic, complemented
46	#104	—	349	2	no	—
47	#101	—	349	1	no	—
48	#41	—	349	2	no	—
49	#41	—	698	2	no	—
50	#41	—	698	2	no	—
51	C₂×Q₈	—	698	2	no	—
52	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
54	Q₁₆	—	698	2	no	—
55	SD₁₆	—	698	2	no	—
56	M₁₆	—	349	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	349	2	no	—
62	Q₈	—	698	2	no	—
63	D₄	—	349	2	no	—
64	D₄	—	698	2	no	—
65	D₄	—	698	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	698	4	no	—
69	C₂×C₄	—	698	2	no	—
70	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
71	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
72	C₈	—	698	2	no	—
73	C₂²	—	698	2	no	—
74	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	698	2	no	—
79	C₄	D₄×D₃₄₉	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	1396	2	no	—
82	C₂	#26013607	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013607
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₄⋊Q₈×C₃₄₉	C₂
socle	C₂×C₃₄₉	#120
prime radical	#26013687	C₂
hypercenter	C₄⋊Q₈	D₃₄₉
hypocenter	C₃₄₉	#456
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #26013670	C₂	semidirect
3. #26013567	C₂	semidirect
4. #26013558	C₂	semidirect
5. C₄⋊Q₈×C₃₄₉	C₂	semidirect
41. C₃₄₉	#456	semidirect
45. C₄⋊Q₈	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013687	2
#26013670	2
#26013567	2
#26013558	2
C₄⋊Q₈×C₃₄₉	2
#456	349