Subgroups — #36375108

Order 22336.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150242	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14584102	C₂	1	1	yes	maximal, characteristic
3	#26013694	C₂	1	2	yes	maximal, complemented
4	C₄×#14584102	C₂	1	2	yes	maximal, complemented
5	#26013595	C₂	1	1	yes	maximal, characteristic
6	#19150242	C₄	1	4	yes	complemented
7	C₂×#14584102	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	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
12	#19150212	C₄	1	2	yes	—
13	C₂²⋊Dic₃₄₉	C₄	1	1	yes	characteristic
14	C₄×Dic₃₄₉	C₂²	1	2	yes	—
15	C₂²⋊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	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
22	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
23	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
24	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
26	C₂²×C₃₄₉	—	2	2	no	—
27	C₂²×C₃₄₉	C₄²	1	2	yes	—
28	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₄₉	—	2	2	no	—
30	Dic₃₄₉	—	2	2	no	—
31	Dic₃₄₉	C₄²	1	2	yes	—
32	Dic₃₄₉	C₂²×C₄	1	1	yes	characteristic
33	Dic₃₄₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₃₄₉	—	2	2	no	—
35	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₃₄₉	#131	1	2	yes	—
37	C₃₄₉	#405	1	1	yes	characteristic, complemented, Sylow
38	#405	—	349	1	no	maximal, Sylow
39	C₂×M₁₆	—	349	1	no	—
40	C₂²×C₈	—	349	1	no	—
41	#117	—	349	1	no	—
42	#97	—	349	2	no	—
43	C₄×C₈	—	349	2	no	—
44	C₂²×C₄	—	349	1	no	—
45	M₁₆	—	349	4	no	—
46	C₂×C₈	—	349	1	no	—
47	C₂×C₈	—	349	2	no	—
48	C₂×C₈	—	349	1	no	—
49	C₂×C₈	—	349	4	no	—
50	C₄⋊C₄	—	349	2	no	—
51	C₂²⋊C₄	—	349	1	no	—
52	C₂²⋊C₄	#13063703	1	1	yes	characteristic
53	C₄²	—	349	2	no	—
54	C₂³	C₂×#13063703	1	1	yes	characteristic
55	C₂×C₄	—	349	2	no	—
56	C₂×C₄	—	349	2	no	—
57	C₂×C₄	—	349	1	no	—
58	C₂×C₄	—	349	1	no	—
59	C₂×C₄	—	349	2	no	—
60	C₂×C₄	C₂×#13063703	1	2	yes	—
61	C₈	—	349	2	no	—
62	C₈	—	349	2	no	—
63	C₈	—	349	4	no	—
64	C₂²	—	2	2	no	—
65	C₂²	C₄×#13063703	1	2	yes	—
66	C₂²	C₂²×#13063703	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	349	1	no	—
69	C₄	—	349	1	no	—
70	C₄	—	349	2	no	—
71	C₄	—	698	2	no	—
72	C₂	—	2	2	no	—
73	C₂	C₂×C₄×#13063703	1	1	yes	characteristic
74	C₂	#26013726	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13063703
commutator	C₂×C₃₄₉	C₂×C₄²
Fitting	C₂²⋊C₄×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²×C₄
prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₂²⋊C₄	#13063703
hypocenter	C₃₄₉	#405
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊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. #19150242	C₄	semidirect
7. C₂×#14584102	C₄	semidirect
37. 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×#19150242	2
C₂²×#14584102	2
#26013694	2
C₄×#14584102	2
#26013595	2
#405	349