Subgroups — #36374236

Order 22336.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150223	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#26013604	C₂	1	1	yes	maximal, characteristic
4	#26013572	C₂	1	1	yes	maximal, characteristic
5	#26013564	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
9	#19150223	C₄	1	4	yes	complemented
10	C₂×#14584095	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₄₉	—	2	1	no	—
12	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
13	#19150212	—	2	1	no	—
14	C₄×Dic₃₄₉	—	2	1	no	—
15	C₄×Dic₃₄₉	—	2	1	no	—
16	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₄₉	C₂²	1	1	yes	characteristic
20	Dic₆₉₈	—	2	1	no	—
21	Dic₆₉₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₃₄₉	—	2	1	no	—
23	C₂×Dic₃₄₉	—	2	1	no	—
24	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
25	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
26	#14584095	C₂×C₄	1	2	yes	—
27	Q₈×C₃₄₉	—	2	1	no	—
28	Q₈×C₃₄₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₄₉	—	2	1	no	—
30	C₂×C₄×C₃₄₉	—	2	1	no	—
31	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
32	C₈×C₃₄₉	—	2	1	no	—
33	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₄₉	—	2	2	no	—
35	C₄×C₃₄₉	—	4	1	no	—
36	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₃₄₉	#41	1	1	yes	characteristic
38	Dic₃₄₉	—	2	2	no	—
39	Dic₃₄₉	—	2	1	no	—
40	Dic₃₄₉	—	4	1	no	—
41	Dic₃₄₉	Q₁₆	1	2	yes	complemented
42	C₂×C₃₄₉	C₄×D₄	1	1	yes	characteristic
43	C₂×C₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₃₄₉	#135	1	1	yes	characteristic
45	C₃₄₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	349	1	no	maximal, Sylow
47	C₂×Q₁₆	—	349	1	no	—
48	C₄×Q₈	—	349	1	no	—
49	C₄×Q₈	—	349	1	no	—
50	#107	—	349	1	no	—
51	#103	—	349	1	no	—
52	#103	D₃₄₉	1	1	yes	characteristic
53	C₄×C₈	—	349	1	no	—
54	C₂×Q₈	—	349	1	no	—
55	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
56	Q₁₆	—	349	4	no	—
57	C₂×C₈	—	349	1	no	—
58	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
59	C₄⋊C₄	—	349	1	no	—
60	C₄⋊C₄	—	698	1	no	—
61	C₄⋊C₄	—	698	1	no	—
62	C₄⋊C₄	C₂×D₃₄₉	1	1	yes	characteristic
63	C₄²	—	349	1	no	—
64	C₄²	—	698	1	no	—
65	C₄²	—	698	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	349	2	no	—
68	Q₈	—	698	1	no	—
69	Q₈	C₄×D₃₄₉	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	349	1	no	—
73	C₂×C₄	—	349	1	no	—
74	C₂×C₄	—	698	1	no	—
75	C₂×C₄	—	698	1	no	—
76	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	349	2	no	—
79	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	349	2	no	—
83	C₄	—	1396	1	no	—
84	C₄	—	698	2	no	—
85	C₄	—	698	1	no	—
86	C₄	C₂×C₄×D₃₄₉	1	1	yes	characteristic
87	C₄	#19150232	1	1	yes	characteristic
88	C₂	#26013599	1	1	yes	characteristic
89	C₂	Q₁₆×D₃₄₉	1	1	yes	characteristic
90	C₂	#26013660	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#103×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×D₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#103	D₃₄₉
hypocenter	C₃₄₉	C₄×Q₁₆
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #19150223	C₄	semidirect
41. Dic₃₄₉	Q₁₆	semidirect
45. C₃₄₉	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150223	2
Q₈×Dic₃₄₉	2
#26013604	2
#26013572	2
#26013564	2
C₈×Dic₃₄₉	2
#103×C₃₄₉	2
C₄×Q₁₆	349