Subgroups — #41×#89137 #20296092

Order 6976.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#89137	C₂	1	2	yes	maximal, complemented
2	D₄×#89137	C₂	1	6	yes	maximal, complemented
3	#16812778	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#89137	C₂	1	3	yes	maximal, complemented
5	#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#89137	C₂²	1	6	yes	—
7	C₄⋊#89137	C₂²	1	6	yes	—
8	C₂²⋊#89137	C₂²	1	6	yes	—
9	C₄×#89137	C₂²	1	1	yes	characteristic
10	C₄×#89137	C₂²	1	3	yes	—
11	C₄×#89137	C₂²	1	6	yes	—
12	#13182030	C₄	1	1	yes	characteristic, complemented
13	#13182029	C₄	1	3	yes	complemented
14	#13182028	C₄	1	3	yes	complemented
15	Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
16	D₄×D₁₀₉	C₂²	1	3	yes	—
17	C₂×C₄×D₁₀₉	C₂²	1	3	yes	—
18	#41×C₁₀₉	C₄	1	1	yes	characteristic, complemented
19	C₂²×D₁₀₉	C₂³	1	3	yes	—
20	C₂×#89137	—	2	12	no	—
21	C₂×#89137	C₂³	1	6	yes	—
22	C₂×#89137	C₂³	1	2	yes	—
23	Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
24	D₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
25	C₂×C₄×C₁₀₉	C₂×C₄	1	3	yes	complemented
26	C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	complemented
27	C₂×Dic₁₀₉	C₂×C₄	1	3	yes	complemented
28	D₄₃₆	C₂×C₄	1	3	yes	complemented
29	C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
30	C₄×D₁₀₉	C₂×C₄	1	3	yes	complemented
31	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
32	C₄×D₁₀₉	C₂³	1	3	yes	—
33	Dic₂₁₈	C₂×C₄	1	3	yes	complemented
34	C₂²×C₁₀₉	C₂²×C₄	1	3	yes	—
35	C₂×D₁₀₉	—	2	6	no	—
36	C₂×D₁₀₉	C₂²×C₄	1	3	yes	—
37	C₂×D₁₀₉	C₂⁴	1	1	yes	characteristic
38	#89137	—	2	6	no	—
39	#89137	#41	1	4	yes	direct product factor
40	C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
41	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	Dic₁₀₉	C₂²×C₄	1	3	yes	—
43	Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₁₀₉	—	2	3	no	—
45	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
46	D₁₀₉	—	2	3	no	—
47	D₁₀₉	C₂×#41	1	2	yes	—
48	C₁₀₉	C₄×#41	1	1	yes	characteristic, complemented, Sylow
49	C₄×#41	—	109	1	no	maximal, Sylow
50	C₂×#41	—	109	1	no	—
51	C₄×Q₈	—	109	2	no	—
52	C₄×D₄	—	109	6	no	—
53	#117	—	109	3	no	—
54	C₂×C₄²	—	109	3	no	—
55	#41	—	109	1	no	—
56	#41	—	109	3	no	—
57	#41	—	109	3	no	—
58	#41	#89137	1	1	yes	characteristic, direct product factor
59	C₂×Q₈	—	109	1	no	—
60	C₂×D₄	—	109	3	no	—
61	C₂²×C₄	—	109	3	no	—
62	C₂²×C₄	—	109	6	no	—
63	C₄⋊C₄	—	109	6	no	—
64	C₂²⋊C₄	—	109	6	no	—
65	C₄²	—	109	1	no	—
66	C₄²	—	109	3	no	—
67	C₄²	—	109	6	no	—
68	C₂³	—	109	3	no	—
69	Q₈	—	109	3	no	—
70	Q₈	C₂×#89137	1	1	yes	characteristic, complemented
71	D₄	—	109	6	no	—
72	D₄	—	109	3	no	—
73	D₄	C₂×#89137	1	3	yes	complemented
74	C₂×C₄	—	109	2	no	—
75	C₂×C₄	—	109	3	no	—
76	C₂×C₄	—	109	3	no	—
77	C₂×C₄	—	109	3	no	—
78	C₂×C₄	—	109	1	no	—
79	C₂×C₄	—	109	3	no	—
80	C₂×C₄	—	109	6	no	—
81	C₂×C₄	—	218	12	no	—
82	C₂×C₄	C₂×#89137	1	3	yes	complemented
83	C₂²	—	109	1	no	—
84	C₂²	—	109	3	no	—
85	C₂²	—	218	6	no	—
86	C₂²	C₂²×#89137	1	3	yes	—
87	C₄	—	109	4	no	—
88	C₄	—	109	1	no	—
89	C₄	—	109	3	no	—
90	C₄	—	218	6	no	—
91	C₄	C₂²×#89137	1	3	yes	—
92	C₄	C₂²×#89137	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	—	109	2	no	—
95	C₂	—	218	3	no	—
96	C₂	C₂³×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#89137
commutator, socle	C₂×C₁₀₉	C₂³×C₄
Frattini	C₂	C₂³×#89137
Fitting	#41×C₁₀₉	C₄
hypercenter	#41	#89137
hypocenter	C₁₀₉	C₄×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. Q₈×#89137	C₂	semidirect
2. D₄×#89137	C₂	semidirect
3. #16812778	C₂	semidirect
4. C₂×C₄×#89137	C₂	semidirect
12. #13182030	C₄	semidirect
13. #13182029	C₄	semidirect
14. #13182028	C₄	semidirect
18. #41×C₁₀₉	C₄	semidirect
23. Q₈×C₁₀₉	C₂×C₄	semidirect
24. D₄×C₁₀₉	C₂×C₄	semidirect
25. C₂×C₄×C₁₀₉	C₂×C₄	semidirect
26. C₂²⋊D₁₀₉	C₂×C₄	semidirect
27. C₂×Dic₁₀₉	C₂×C₄	semidirect
28. D₄₃₆	C₂×C₄	semidirect
29. C₄×D₁₀₉	C₂×C₄	semidirect
30. C₄×D₁₀₉	C₂×C₄	semidirect
33. Dic₂₁₈	C₂×C₄	semidirect
39. #89137	#41	direct (N × H)
48. C₁₀₉	C₄×#41	semidirect
58. #41	#89137	direct (N × H)
70. Q₈	C₂×#89137	semidirect
73. D₄	C₂×#89137	semidirect
82. C₂×C₄	C₂×#89137	semidirect