Subgroups — #36374083

Order 22336.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
2	#26013546	C₂	1	2	yes	maximal
3	#26013546	C₂	1	1	yes	maximal, characteristic
4	#26013546	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
7	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
8	C₄⋊Dic₃₄₉	—	2	2	no	—
9	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₄₉	—	2	2	no	—
13	C₂×Dic₃₄₉	—	2	2	no	—
14	C₂×Dic₃₄₉	—	2	2	no	—
15	C₂×Dic₃₄₉	—	2	2	no	—
16	C₂×Dic₃₄₉	—	2	4	no	—
17	C₂×Dic₃₄₉	—	2	2	no	—
18	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₃₄₉	—	2	1	no	—
20	C₂×C₄×C₃₄₉	—	2	2	no	—
21	C₂×C₄×C₃₄₉	—	2	1	no	—
22	C₂×C₄×C₃₄₉	—	2	1	no	—
23	C₂×C₄×C₃₄₉	—	2	2	no	—
24	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
25	C₂×C₄×C₃₄₉	Q₈	1	2	yes	—
26	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₃₄₉	#41	1	2	yes	—
29	C₂²×C₃₄₉	#41	1	1	yes	characteristic
30	C₂²×C₃₄₉	#41	1	2	yes	—
31	C₄×C₃₄₉	—	2	2	no	—
32	C₄×C₃₄₉	—	2	2	no	—
33	C₄×C₃₄₉	—	4	1	no	—
34	Dic₃₄₉	—	4	2	no	—
35	Dic₃₄₉	—	4	2	no	—
36	C₂×C₃₄₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₃₄₉	#123	1	1	yes	characteristic
38	C₂×C₃₄₉	#124	1	1	yes	characteristic
39	C₂×C₃₄₉	#125	1	2	yes	—
40	C₂×C₃₄₉	#126	1	2	yes	—
41	C₃₄₉	#400	1	1	yes	characteristic, complemented, Sylow
42	#400	—	349	1	no	maximal, Sylow
43	C₂×C₄⋊C₄	—	349	1	no	—
44	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic
45	#95	—	349	1	no	—
46	#95	—	349	2	no	—
47	#95	—	349	2	no	—
48	C₂²×C₄	—	349	2	no	—
49	C₂²×C₄	—	349	2	no	—
50	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	698	2	no	—
55	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	698	2	no	—
62	C₂×C₄	—	698	2	no	—
63	C₂×C₄	—	698	2	no	—
64	C₂×C₄	—	698	4	no	—
65	C₂×C₄	—	698	2	no	—
66	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
67	C₂×C₄	Dic₆₉₈	1	2	yes	—
68	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
69	C₂²	C₂×Dic₆₉₈	1	1	yes	characteristic
70	C₂²	#19150232	1	2	yes	—
71	C₂²	#19150233	1	1	yes	characteristic
72	C₂²	#19150234	1	2	yes	—
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	1396	2	no	—
77	C₄	—	1396	2	no	—
78	C₂	#26013618	1	1	yes	characteristic
79	C₂	#26013620	1	1	yes	characteristic
80	C₂	#26013628	1	1	yes	characteristic
81	C₂	#26013630	1	2	yes	—
82	C₂	#26013634	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₄₉
commutator, socle, prime radical, cosocle	C₂³×C₃₄₉	C₂³
Fitting	C₂×C₄⋊C₄×C₃₄₉	C₂
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	#400
LCS	G ▷ C₂³×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
41. C₃₄₉	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₃₄₉	2
#26013546	2
C₂×C₄⋊C₄×C₃₄₉	2
#400	349