Subgroups — #101×D₃₄₉ #36374122

Order 22336.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₃₄₉	C₂	1	2	yes	maximal
3	#26013558	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013557	C₂	1	2	yes	maximal, complemented
5	#101×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₉₈	C₄	1	2	yes	—
7	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
8	Q₈×D₃₄₉	—	2	2	no	—
9	Q₈×D₃₄₉	—	2	2	no	—
10	C₂×C₄×D₃₄₉	C₄	1	2	yes	—
11	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
12	#19150217	C₂²	1	2	yes	—
13	#19150216	—	2	2	no	—
14	C₈×D₃₄₉	—	2	2	no	—
15	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
16	M₁₆×C₃₄₉	C₂²	1	2	yes	—
17	C₂²×D₃₄₉	C₂×C₄	1	1	yes	characteristic
18	Dic₆₉₈	—	2	2	no	—
19	Dic₆₉₈	—	2	4	no	—
20	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₄₉	—	2	2	no	—
23	C₄×D₃₄₉	—	2	2	no	—
24	C₄×D₃₄₉	D₄	1	2	yes	—
25	C₄×D₃₄₉	D₄	1	2	yes	—
26	#14584095	—	2	2	no	—
27	Q₈×C₃₄₉	—	2	2	no	—
28	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
30	C₈×C₃₄₉	—	2	2	no	—
31	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×D₃₄₉	—	2	2	no	—
33	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₃₄₉	—	2	2	no	—
36	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
37	Dic₃₄₉	—	2	2	no	—
38	Dic₃₄₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₃₄₉	—	2	1	no	—
40	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	D₃₄₉	—	2	1	no	—
42	D₃₄₉	#101	1	2	yes	direct product factor
43	C₃₄₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
44	C₂×#101	—	349	1	no	maximal, Sylow
45	C₂²×Q₈	—	349	1	no	—
46	C₂×M₁₆	—	349	2	no	—
47	#101	—	349	1	no	—
48	#101	—	349	2	no	—
49	#101	D₃₄₉	1	1	yes	characteristic, direct product factor
50	C₂×Q₈	—	349	2	no	—
51	C₂×Q₈	—	349	1	no	—
52	C₂×Q₈	—	698	2	no	—
53	C₂×Q₈	—	698	2	no	—
54	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
55	C₂²×C₄	—	349	1	no	—
56	C₂²×C₄	—	349	2	no	—
57	M₁₆	—	349	2	no	—
58	M₁₆	—	698	2	no	—
59	M₁₆	C₂×D₃₄₉	1	2	yes	—
60	C₂×C₈	—	698	2	no	—
61	C₂³	—	349	1	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	698	4	no	—
64	Q₈	—	698	2	no	—
65	C₂×C₄	—	349	1	no	—
66	C₂×C₄	—	349	2	no	—
67	C₂×C₄	—	349	2	no	—
68	C₂×C₄	—	349	2	no	—
69	C₂×C₄	—	698	2	no	—
70	C₂×C₄	—	698	2	no	—
71	C₂×C₄	C₄×D₃₄₉	1	2	yes	—
72	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	698	2	no	—
75	C₂²	—	349	1	no	—
76	C₂²	—	349	1	no	—
77	C₂²	—	698	2	no	—
78	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	349	2	no	—
81	C₄	—	698	2	no	—
82	C₄	D₄×D₃₄₉	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	349	2	no	—
85	C₂	—	698	1	no	—
86	C₂	C₂²⋊C₄×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₃₄₉
commutator	C₂²×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#101×C₃₄₉	C₂
socle	C₂×C₃₄₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₄₉	C₂×C₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#101	D₃₄₉
hypocenter	C₃₄₉	C₂×#101
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
3. #26013558	C₂	semidirect
4. #26013557	C₂	semidirect
5. #101×C₃₄₉	C₂	semidirect
42. D₃₄₉	#101	direct (N × H)
43. C₃₄₉	C₂×#101	semidirect
49. #101	D₃₄₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₃₄₉	2
M₁₆×D₃₄₉	2
#26013558	2
#26013557	2
#101×C₃₄₉	2
C₂×#101	349

Subgroups — #101×D349 #36374122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×D₃₄₉ #36374122