Subgroups — #36375109

Order 22336.

130 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×D₃₄₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#26013694	C₂	1	2	yes	maximal, complemented
4	C₄×#14584102	C₂	1	2	yes	maximal, complemented
5	#26013597	C₂	1	1	yes	maximal, characteristic
6	#19150243	C₄	1	4	yes	complemented
7	D₃₄₉⋊C₈	C₄	1	4	yes	complemented
8	C₂×#14584102	C₂²	1	1	yes	characteristic
9	C₂×#14584102	C₂²	1	1	yes	characteristic
10	C₂×#14584102	C₂²	1	2	yes	—
11	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic
13	#19150210	C₄	1	2	yes	—
14	C₄×Dic₃₄₉	C₂²	1	2	yes	—
15	C₄⋊C₄×C₃₄₉	C₄	1	1	yes	characteristic
16	#14584102	C₂×C₄	1	2	yes	—
17	#14584102	C₂×C₄	1	2	yes	—
18	#14584102	C₂×C₄	1	4	yes	—
19	C₂²×D₃₄₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
22	C₄×D₃₄₉	C₂×C₄	1	2	yes	—
23	C₄×D₃₄₉	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₄₉	—	2	1	no	—
28	C₂×D₃₄₉	—	2	1	no	—
29	C₂×D₃₄₉	C₄²	1	2	yes	—
30	C₄×C₃₄₉	—	2	2	no	—
31	C₄×C₃₄₉	C₄²	1	2	yes	—
32	Dic₃₄₉	—	2	2	no	—
33	Dic₃₄₉	C₂²×C₄	1	1	yes	characteristic
34	Dic₃₄₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₃₄₉	#131	1	1	yes	characteristic
37	C₂×C₃₄₉	#131	1	1	yes	characteristic
38	D₃₄₉	—	2	2	no	—
39	C₃₄₉	#405	1	1	yes	characteristic, complemented, Sylow
40	#405	—	349	1	no	maximal, Sylow
41	C₂×M₁₆	—	349	1	no	—
42	C₂²×C₈	—	349	1	no	—
43	#117	—	349	1	no	—
44	#97	—	349	2	no	—
45	C₄×C₈	—	349	2	no	—
46	C₂²×C₄	—	349	1	no	—
47	M₁₆	—	349	4	no	—
48	C₂×C₈	—	349	1	no	—
49	C₂×C₈	—	349	2	no	—
50	C₂×C₈	—	349	1	no	—
51	C₂×C₈	—	349	4	no	—
52	C₄⋊C₄	—	349	1	no	—
53	C₄⋊C₄	#13063703	1	1	yes	characteristic
54	C₂²⋊C₄	—	349	2	no	—
55	C₄²	—	349	2	no	—
56	C₂³	—	349	1	no	—
57	C₂×C₄	—	349	1	no	—
58	C₂×C₄	—	349	2	no	—
59	C₂×C₄	—	349	2	no	—
60	C₂×C₄	—	349	2	no	—
61	C₂×C₄	C₂×#13063703	1	2	yes	—
62	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
63	C₈	—	349	2	no	—
64	C₈	—	349	2	no	—
65	C₈	—	349	4	no	—
66	C₂²	—	349	2	no	—
67	C₂²	—	698	1	no	—
68	C₂²	—	698	1	no	—
69	C₂²	C₂²×#13063703	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	349	1	no	—
72	C₄	—	349	1	no	—
73	C₄	—	698	2	no	—
74	C₄	C₄×#13063703	1	2	yes	—
75	C₂	—	698	2	no	—
76	C₂	C₂×C₄×#13063703	1	1	yes	characteristic
77	C₂	#26013726	1	1	yes	characteristic
78	C₂	#26013727	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13063703
commutator	C₂×C₃₄₉	C₂×C₄²
Fitting	C₄⋊C₄×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²×C₄
prime radical	C₂²×D₃₄₉	C₂×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₄⋊C₄	#13063703
hypocenter	C₃₄₉	#405
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #26013694	C₂	semidirect
4. C₄×#14584102	C₂	semidirect
6. #19150243	C₄	semidirect
7. D₃₄₉⋊C₈	C₄	semidirect
39. C₃₄₉	#405	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×D₃₄₉⋊C₈	2
#26013694	2
C₄×#14584102	2
#26013597	2
#405	349