Subgroups — #322×D₁₀₉ #28361306

Order 13952.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×D₁₀₉	C₂	1	2	yes	maximal
3	#20294748	C₂	1	2	yes	maximal, complemented
4	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	4	yes	—
7	C₂×C₈×D₁₀₉	C₄	1	2	yes	—
8	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₄×#11724784	C₂²	1	2	yes	—
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	4	yes	—
12	C₈×Dic₁₀₉	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
17	C₂×#11724784	C₂×C₄	1	4	yes	—
18	C₂×#11724784	C₂×C₄	1	2	yes	—
19	C₈×D₁₀₉	—	2	8	no	—
20	C₈×D₁₀₉	C₈	1	8	yes	complemented
21	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
29	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₄²	1	2	yes	—
31	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
32	C₄×D₁₀₉	M₁₆	1	8	yes	—
33	C₈×C₁₀₉	—	2	4	no	—
34	C₈×C₁₀₉	C₂×C₈	1	4	yes	complemented
35	#11724784	—	2	4	no	—
36	#11724784	C₂×C₈	1	4	yes	complemented
37	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
38	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
39	C₂×D₁₀₉	#97	1	4	yes	—
40	C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
41	C₄×C₁₀₉	C₂×M₁₆	1	4	yes	—
42	Dic₁₀₉	C₄×C₈	1	2	yes	—
43	Dic₁₀₉	#97	1	4	yes	—
44	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
45	C₂×C₁₀₉	C₂×#97	1	2	yes	—
46	D₁₀₉	#322	1	4	yes	direct product factor
47	C₂×#322	—	109	1	no	maximal, Sylow
48	C₁₀₉	C₂×#322	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₄×C₈	—	109	1	no	—
50	C₂×C₄×C₈	—	109	2	no	—
51	#322	—	109	1	no	—
52	#322	—	109	2	no	—
53	#322	D₁₀₉	1	1	yes	characteristic, direct product factor
54	C₂²×C₈	—	109	2	no	—
55	C₂²×C₈	—	109	4	no	—
56	C₂×C₄²	—	109	1	no	—
57	C₄×C₈	—	109	2	no	—
58	C₄×C₈	—	109	4	no	—
59	C₄×C₈	—	109	1	no	—
60	C₄×C₈	—	109	2	no	—
61	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
62	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	109	2	no	—
65	C₂×C₈	—	109	2	no	—
66	C₂×C₈	—	109	8	no	—
67	C₂×C₈	—	109	4	no	—
68	C₂×C₈	—	218	8	no	—
69	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
70	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
71	C₄²	—	109	1	no	—
72	C₄²	—	109	2	no	—
73	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
74	C₂³	—	109	1	no	—
75	C₂×C₄	—	109	1	no	—
76	C₂×C₄	—	109	2	no	—
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	109	8	no	—
79	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	—	109	4	no	—
83	C₈	—	218	4	no	—
84	C₈	C₈×D₁₀₉	1	4	yes	complemented
85	C₂²	—	109	2	no	—
86	C₂²	—	109	4	no	—
87	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
88	C₄	—	109	2	no	—
89	C₄	—	109	4	no	—
90	C₄	C₂×C₈×D₁₀₉	1	2	yes	—
91	C₄	M₁₆×D₁₀₉	1	4	yes	—
92	C₂	—	109	4	no	—
93	C₂	C₄×C₈×D₁₀₉	1	1	yes	characteristic
94	C₂	#97×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	#322×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄²
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#322	D₁₀₉
hypocenter	C₁₀₉	C₂×#322
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #20294748	C₂	semidirect
4. #20294747	C₂	semidirect
5. #322×C₁₀₉	C₂	semidirect
20. C₈×D₁₀₉	C₈	semidirect
34. C₈×C₁₀₉	C₂×C₈	semidirect
36. #11724784	C₂×C₈	semidirect
46. D₁₀₉	#322	direct (N × H)
48. C₁₀₉	C₂×#322	semidirect
53. #322	D₁₀₉	direct (N × H)
84. C₈	C₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×D₁₀₉	2
#20294748	2
#20294747	2
#322×C₁₀₉	2
C₂×#322	109

Subgroups — #322×D109 #28361306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #322×D₁₀₉ #28361306