Subgroups — C₂²×#16812055 #28349573

Order 13696.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812055	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₄	—	2	1	no	—
6	C₂×Dic₄₂₈	—	2	3	no	—
7	C₂×Dic₄₂₈	C₂²	1	6	yes	—
8	#16812055	C₂²	1	16	yes	direct product factor
9	C₂×C₁₀₇⋊C₁₆	C₂²	1	6	yes	—
10	C₂²×Q₈×C₁₀₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₀₇	—	2	3	no	—
12	C₂×Q₁₆×C₁₀₇	C₂²	1	6	yes	—
13	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₄	—	2	6	no	—
15	C₂×Dic₂₁₄	—	4	3	no	—
16	C₂²×Dic₁₀₇	—	4	1	no	—
17	Dic₄₂₈	—	2	6	no	—
18	Dic₄₂₈	C₂³	1	4	yes	—
19	C₁₀₇⋊C₁₆	C₂³	1	4	yes	—
20	C₂×Q₈×C₁₀₇	—	2	6	no	—
21	C₂×Q₈×C₁₀₇	—	4	3	no	—
22	C₂²×C₄×C₁₀₇	—	4	1	no	—
23	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
24	Q₁₆×C₁₀₇	—	2	6	no	—
25	Q₁₆×C₁₀₇	C₂³	1	4	yes	—
26	C₂×C₈×C₁₀₇	D₄	1	3	yes	—
27	C₂×C₈×C₁₀₇	C₂³	1	3	yes	—
28	C₂³×C₁₀₇	D₈	1	1	yes	characteristic
29	Q₈×C₁₀₇	—	2	4	no	—
30	Q₈×C₁₀₇	—	4	6	no	—
31	C₂×C₄×C₁₀₇	—	4	6	no	—
32	C₂×C₄×C₁₀₇	D₈	1	3	yes	—
33	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
34	C₂×Dic₁₀₇	—	4	6	no	—
35	Dic₂₁₄	—	2	4	no	—
36	Dic₂₁₄	—	4	6	no	—
37	C₈×C₁₀₇	C₂×D₄	1	3	yes	—
38	C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
39	C₂²×C₁₀₇	Q₃₂	1	4	yes	complemented
40	C₂²×C₁₀₇	C₂×D₈	1	3	yes	—
41	C₄×C₁₀₇	—	4	4	no	—
42	C₄×C₁₀₇	C₂×D₈	1	3	yes	—
43	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
44	Dic₁₀₇	—	4	4	no	—
45	C₂×C₁₀₇	C₂×Q₃₂	1	6	yes	complemented
46	C₂×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
47	C₂²×Q₃₂	—	107	1	no	maximal, Sylow
48	C₁₀₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
49	C₂²×Q₁₆	—	107	1	no	—
50	C₂²×Q₁₆	D₁₀₇	1	1	yes	characteristic
51	C₂×Q₃₂	—	107	12	no	—
52	C₂²×C₁₆	—	107	1	no	—
53	C₂²×Q₈	—	2	1	no	—
54	C₂²×Q₈	—	214	1	no	—
55	C₂×Q₁₆	—	2	3	no	—
56	C₂×Q₁₆	—	107	6	no	—
57	C₂×Q₁₆	—	214	3	no	—
58	C₂×Q₁₆	C₂×D₁₀₇	1	6	yes	—
59	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
60	Q₃₂	—	107	16	no	—
61	C₂×C₁₆	—	107	6	no	—
62	C₂×Q₈	—	2	6	no	—
63	C₂×Q₈	—	4	3	no	—
64	C₂×Q₈	—	428	3	no	—
65	C₂×Q₈	—	214	6	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	428	1	no	—
68	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
69	Q₁₆	—	2	6	no	—
70	Q₁₆	—	107	4	no	—
71	Q₁₆	—	214	6	no	—
72	Q₁₆	C₂²×D₁₀₇	1	4	yes	—
73	C₂×C₈	C₂²⋊D₁₀₇	1	3	yes	—
74	C₂×C₈	C₂²×D₁₀₇	1	3	yes	—
75	C₁₆	—	107	4	no	—
76	C₂³	#13133803	1	1	yes	characteristic
77	Q₈	—	2	4	no	—
78	Q₈	—	4	6	no	—
79	Q₈	—	428	6	no	—
80	Q₈	—	214	4	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	428	6	no	—
83	C₂×C₄	#13133803	1	3	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
85	C₈	C₂×C₂²⋊D₁₀₇	1	3	yes	—
86	C₈	C₂³×D₁₀₇	1	1	yes	characteristic
87	C₂²	#16812055	1	4	yes	direct product factor
88	C₂²	C₂×#13133803	1	3	yes	—
89	C₄	—	4	4	no	—
90	C₄	—	428	4	no	—
91	C₄	C₂×#13133803	1	3	yes	—
92	C₄	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
93	C₂	C₂×#16812055	1	6	yes	direct product factor
94	C₂	C₂²×#13133803	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133803
commutator, cosocle	C₈×C₁₀₇	C₂⁴
Frattini	C₈	C₂³×D₁₀₇
Fitting	C₂²×Q₁₆×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	D₈
hypercenter	C₂²×Q₁₆	D₁₀₇
hypocenter	C₁₀₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16812055	C₂	direct (N × H)
8. #16812055	C₂²	direct (N × H)
39. C₂²×C₁₀₇	Q₃₂	semidirect
45. C₂×C₁₀₇	C₂×Q₃₂	semidirect
48. C₁₀₇	C₂²×Q₃₂	semidirect
87. C₂²	#16812055	direct (N × H)
93. C₂	C₂×#16812055	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₈	2
C₂×#16812055	2
C₂²×C₁₀₇⋊C₁₆	2
C₂²×Q₁₆×C₁₀₇	2
C₂²×Q₃₂	107

Subgroups — C22×#16812055 #28349573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16812055 #28349573