Subgroups — #36375016

Order 22336.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013718	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13063703	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063703	—	2	1	no	—
5	C₄⋊#13063703	C₄	1	2	yes	complemented
6	C₂²⋊#13063703	—	2	1	no	—
7	C₄×#13063703	—	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	#19150217	C₄	1	1	yes	characteristic, complemented
12	#19150216	—	2	1	no	—
13	C₈×D₃₄₉	—	2	1	no	—
14	M₁₆×C₃₄₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#13063703	—	2	4	no	—
16	C₂×#13063703	—	2	2	no	—
17	C₂×#13063703	—	2	2	no	—
18	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
19	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
20	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
21	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₄₉	D₄	1	1	yes	characteristic
23	C₄×D₃₄₉	D₄	1	1	yes	characteristic
24	#14584095	—	2	1	no	—
25	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₃₄₉	—	2	1	no	—
27	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₃₄₉	—	2	2	no	—
29	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
31	#13063703	—	2	4	no	—
32	#13063703	—	4	2	no	—
33	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
34	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
35	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
36	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₃₄₉	—	2	1	no	—
38	C₂×C₃₄₉	#95	1	1	yes	characteristic
39	D₃₄₉	—	2	1	no	—
40	D₃₄₉	#104	1	2	yes	—
41	C₃₄₉	#339	1	1	yes	characteristic, complemented, Sylow
42	#339	—	349	1	no	maximal, Sylow
43	C₂×M₁₆	—	349	1	no	—
44	#117	—	349	1	no	—
45	C₂×C₄²	—	349	1	no	—
46	C₂²×C₄	—	349	1	no	—
47	C₂²×C₄	—	698	1	no	—
48	M₁₆	—	349	1	no	—
49	M₁₆	—	698	1	no	—
50	M₁₆	#13063703	1	1	yes	characteristic, complemented
51	C₂×C₈	—	698	1	no	—
52	C₄⋊C₄	—	349	2	no	—
53	C₂²⋊C₄	—	698	1	no	—
54	C₄²	—	349	2	no	—
55	C₄²	—	698	1	no	—
56	C₄²	—	698	1	no	—
57	C₂³	—	349	1	no	—
58	C₂×C₄	—	349	1	no	—
59	C₂×C₄	—	349	1	no	—
60	C₂×C₄	—	349	1	no	—
61	C₂×C₄	—	349	1	no	—
62	C₂×C₄	—	349	1	no	—
63	C₂×C₄	—	698	2	no	—
64	C₂×C₄	—	698	2	no	—
65	C₂×C₄	—	698	4	no	—
66	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	698	1	no	—
69	C₂²	—	349	1	no	—
70	C₂²	—	349	1	no	—
71	C₂²	—	698	2	no	—
72	C₂²	C₄⋊#13063703	1	1	yes	characteristic
73	C₄	—	349	1	no	—
74	C₄	—	349	1	no	—
75	C₄	—	1396	2	no	—
76	C₄	—	698	4	no	—
77	C₄	C₄×#13063703	1	1	yes	characteristic
78	C₄	C₂²⋊#13063703	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	349	2	no	—
81	C₂	—	698	1	no	—
82	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13063703
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂×C₄	C₂×#13063703
Fitting	M₁₆×C₃₄₉	C₄
socle	C₂×C₃₄₉	#95
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×D₃₄₉	C₂²
hypercenter	M₁₆	#13063703
hypocenter	C₃₄₉	#339
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊#13063703	C₄	semidirect
9. C₄×#13063703	C₄	semidirect
11. #19150217	C₄	semidirect
14. M₁₆×C₃₄₉	C₄	semidirect
41. C₃₄₉	#339	semidirect
50. M₁₆	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013718	2
C₂×C₄×#13063703	2
M₁₆×D₃₄₉	2
#339	349