Subgroups — #28366313

Order 13952.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₃₆	C₂	1	6	yes	maximal, complemented
2	#20294963	C₂	1	4	yes	maximal, complemented
3	#20294936	C₂	1	4	yes	maximal, complemented
4	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₃₆	C₂²	1	6	yes	—
6	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	—
7	C₄×D₄₃₆	—	2	24	no	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	6	yes	—
9	C₂×#13182006	C₂²	1	12	yes	—
10	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
11	#95×C₁₀₉	C₂²	1	4	yes	—
12	C₂³×D₁₀₉	C₂³	1	4	yes	—
13	C₂×D₄₃₆	—	2	24	no	—
14	C₂×D₄₃₆	—	2	12	no	—
15	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
16	C₂×C₄×D₁₀₉	—	2	24	no	—
17	C₄⋊Dic₁₀₉	—	2	12	no	—
18	#13182006	—	2	24	no	—
19	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
20	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₄²×C₁₀₉	D₄	1	12	yes	complemented
22	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
23	C₂²×D₁₀₉	—	2	12	no	—
24	C₂²×D₁₀₉	—	2	12	no	—
25	C₂²×D₁₀₉	#41	1	8	yes	—
26	C₂×C₄×C₁₀₉	—	2	12	no	—
27	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
28	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
29	C₂×Dic₁₀₉	—	2	12	no	—
30	D₄₃₆	—	2	48	no	—
31	C₄×D₁₀₉	—	4	16	no	—
32	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
33	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
34	C₂×D₁₀₉	—	2	24	no	—
35	C₂×D₁₀₉	—	4	12	no	—
36	C₂×D₁₀₉	—	4	4	no	—
37	C₄×C₁₀₉	—	2	12	no	—
38	C₄×C₁₀₉	#127	1	4	yes	—
39	Dic₁₀₉	—	4	4	no	—
40	C₂×C₁₀₉	#525	1	6	yes	—
41	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
42	D₁₀₉	—	4	8	no	—
43	#2397	—	109	1	no	maximal, Sylow
44	C₁₀₉	#2397	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×D₄	—	109	6	no	—
46	#399	—	109	4	no	—
47	#392	—	109	4	no	—
48	#376	D₁₀₉	1	1	yes	characteristic, complemented
49	C₂²×D₄	—	109	6	no	—
50	C₂³×C₄	—	109	4	no	—
51	C₄×D₄	—	218	24	no	—
52	C₂×C₄⋊C₄	—	109	6	no	—
53	C₂×C₂²⋊C₄	—	109	12	no	—
54	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
55	#95	C₂×D₁₀₉	1	4	yes	—
56	C₂⁴	—	109	4	no	—
57	C₂×D₄	—	218	24	no	—
58	C₂×D₄	—	218	12	no	—
59	C₂²×C₄	—	109	4	no	—
60	C₂²×C₄	—	218	24	no	—
61	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
62	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
63	C₄⋊C₄	—	218	12	no	—
64	C₂²⋊C₄	—	218	24	no	—
65	C₄²	D₄₃₆	1	12	yes	complemented
66	C₂³	—	109	8	no	—
67	C₂³	—	218	12	no	—
68	C₂³	—	218	12	no	—
69	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
70	D₄	—	218	48	no	—
71	C₂×C₄	—	2	12	no	—
72	C₂×C₄	—	436	16	no	—
73	C₂×C₄	—	218	12	no	—
74	C₂×C₄	C₂×D₄₃₆	1	12	yes	—
75	C₂×C₄	C₂×D₄₃₆	1	6	yes	—
76	C₂²	—	436	12	no	—
77	C₂²	—	436	4	no	—
78	C₂²	—	218	24	no	—
79	C₂²	C₂²×D₄₃₆	1	3	yes	—
80	C₂²	#41×D₁₀₉	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	436	4	no	—
83	C₄	#16812696	1	4	yes	—
84	C₂	—	436	8	no	—
85	C₂	#20295538	1	6	yes	—
86	C₂	C₂×#16812696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Frattini	C₂³	C₂³×D₁₀₉
Fitting	#376×C₁₀₉	C₂
hypercenter	#376	D₁₀₉
hypocenter	C₁₀₉	#2397
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₄₃₆	C₂	semidirect
2. #20294963	C₂	semidirect
3. #20294936	C₂	semidirect
4. #376×C₁₀₉	C₂	semidirect
21. C₄²×C₁₀₉	D₄	semidirect
44. C₁₀₉	#2397	semidirect
48. #376	D₁₀₉	semidirect
65. C₄²	D₄₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₃₆	2
#20294963	2
#20294936	2
#376×C₁₀₉	2
#2397	109