Subgroups — #36375014

Order 22336.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150243	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13063703	C₂	1	1	yes	maximal, characteristic
3	#26013597	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063703	—	2	1	no	—
5	#19150243	—	2	1	no	—
6	#19150243	C₄	1	2	yes	complemented
7	C₂×#14584102	—	2	1	no	—
8	C₄×#13063703	—	2	1	no	—
9	C₄×#13063703	C₄	1	2	yes	complemented
10	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic, complemented
12	#19150210	—	2	1	no	—
13	C₄×Dic₃₄₉	—	2	1	no	—
14	C₄⋊C₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#13063703	—	2	2	no	—
16	C₂×#13063703	—	2	2	no	—
17	C₂×#13063703	—	2	2	no	—
18	#14584102	—	2	2	no	—
19	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
20	C₂×Dic₃₄₉	—	2	1	no	—
21	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
22	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₄₉	D₄	1	1	yes	characteristic
25	C₄×D₃₄₉	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₃₄₉	—	2	1	no	—
27	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₃₄₉	—	2	1	no	—
30	C₂×D₃₄₉	—	2	1	no	—
31	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₃₄₉	C₄⋊C₄	1	1	yes	characteristic
33	#13063703	—	2	4	no	—
34	C₄×C₃₄₉	—	4	1	no	—
35	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
36	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
37	Dic₃₄₉	—	4	1	no	—
38	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₃₄₉	#95	1	1	yes	characteristic
41	C₂×C₃₄₉	#104	1	1	yes	characteristic
42	C₂×C₃₄₉	#104	1	1	yes	characteristic
43	D₃₄₉	—	2	1	no	—
44	D₃₄₉	—	2	1	no	—
45	C₃₄₉	#339	1	1	yes	characteristic, complemented, Sylow
46	#339	—	349	1	no	maximal, Sylow
47	C₂×M₁₆	—	349	1	no	—
48	#117	—	349	1	no	—
49	C₂×C₄²	—	349	1	no	—
50	C₂²×C₄	—	349	1	no	—
51	C₂²×C₄	—	698	1	no	—
52	M₁₆	—	349	2	no	—
53	M₁₆	—	698	1	no	—
54	C₂×C₈	—	698	1	no	—
55	C₄⋊C₄	—	349	1	no	—
56	C₄⋊C₄	#13063703	1	1	yes	characteristic, complemented
57	C₂²⋊C₄	—	698	1	no	—
58	C₄²	—	349	2	no	—
59	C₄²	—	698	1	no	—
60	C₄²	—	698	1	no	—
61	C₂³	—	349	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	349	1	no	—
64	C₂×C₄	—	349	1	no	—
65	C₂×C₄	—	349	1	no	—
66	C₂×C₄	—	349	1	no	—
67	C₂×C₄	—	349	1	no	—
68	C₂×C₄	—	698	1	no	—
69	C₂×C₄	—	698	2	no	—
70	C₂×C₄	—	698	2	no	—
71	C₂×C₄	—	698	2	no	—
72	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
73	C₈	—	698	2	no	—
74	C₂²	—	349	1	no	—
75	C₂²	—	349	1	no	—
76	C₂²	—	698	1	no	—
77	C₂²	—	698	1	no	—
78	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	349	1	no	—
81	C₄	—	349	1	no	—
82	C₄	—	1396	1	no	—
83	C₄	—	698	4	no	—
84	C₄	C₄×#13063703	1	1	yes	characteristic
85	C₄	C₄⋊#13063703	1	1	yes	characteristic
86	C₂	—	698	1	no	—
87	C₂	—	698	1	no	—
88	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic
89	C₂	#26013703	1	1	yes	characteristic
90	C₂	#26013704	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13063703
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂×C₄	C₂×#13063703
Fitting	C₄⋊C₄×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²⋊C₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×D₃₄₉	C₂²
hypercenter	C₄⋊C₄	#13063703
hypocenter	C₃₄₉	#339
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #19150243	C₄	semidirect
9. C₄×#13063703	C₄	semidirect
11. C₄⋊Dic₃₄₉	C₄	semidirect
14. C₄⋊C₄×C₃₄₉	C₄	semidirect
45. C₃₄₉	#339	semidirect
56. C₄⋊C₄	#13063703	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150243	2
C₂×C₄×#13063703	2
#26013597	2
#339	349