Subgroups — C₂×#20292676 #28345068

Order 13696.

460 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133802	C₂	1	2	yes	maximal
2	#20292676	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812045	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133802	—	2	2	no	—
6	C₂×#13133802	—	2	4	no	—
7	C₂×#13133802	C₂²	1	4	yes	—
8	C₂×#13133802	C₂²	1	8	yes	—
9	C₂²×C₁₀₇⋊C₈	—	2	2	no	—
10	#16812045	C₂²	1	16	yes	complemented
11	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₀₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₀₇	C₄	1	4	yes	—
14	C₂×M₁₆×C₁₀₇	C₂²	1	4	yes	—
15	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
16	#13133802	—	2	8	no	—
17	#13133802	—	2	4	no	—
18	#13133802	C₂³	1	8	yes	—
19	C₂×C₁₀₇⋊C₈	—	2	8	no	—
20	C₂×C₁₀₇⋊C₈	—	2	4	no	—
21	C₂⁴×C₁₀₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₇	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
30	M₁₆×C₁₀₇	C₂×C₄	1	16	yes	—
31	C₂×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₁₀₇	C₂³	1	4	yes	—
34	C₂³×C₁₀₇	—	4	2	no	—
35	C₂³×C₁₀₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₁₀₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₁₀₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₀₇	C₂²×C₄	1	8	yes	—
53	C₁₀₇⋊C₈	—	2	8	no	—
54	C₂²×C₁₀₇	—	2	4	no	—
55	C₂²×C₁₀₇	—	2	2	no	—
56	C₂²×C₁₀₇	—	4	2	no	—
57	C₂²×C₁₀₇	—	4	2	no	—
58	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₁₀₇	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₁₀₇	—	2	4	no	—
68	C₂×C₁₀₇	—	2	1	no	—
69	C₂×C₁₀₇	—	2	1	no	—
70	C₂×C₁₀₇	#430	1	2	yes	complemented
71	C₂×C₁₀₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂×#430	—	107	1	no	maximal, Sylow
73	C₁₀₇	C₂×#430	1	1	yes	characteristic, complemented, Sylow
74	C₂²×M₁₆	—	107	2	no	—
75	C₂²×M₁₆	D₁₀₇	1	1	yes	characteristic
76	#430	—	107	8	no	—
77	C₂×#108	—	107	4	no	—
78	C₂³×C₄	C₂×D₁₀₇	1	1	yes	characteristic
79	C₂×M₁₆	—	107	4	no	—
80	C₂×M₁₆	—	107	8	no	—
81	C₂×M₁₆	—	214	4	no	—
82	C₂×M₁₆	—	214	2	no	—
83	C₂×M₁₆	Dic₁₀₇	1	4	yes	—
84	C₂×M₁₆	Dic₁₀₇	1	4	yes	—
85	C₂×M₁₆	C₂×D₁₀₇	1	4	yes	—
86	C₂²×C₈	—	214	2	no	—
87	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
88	#108	—	107	16	no	—
89	C₂⁴	Dic₂₁₄	1	1	yes	characteristic
90	C₂²×C₄	Dic₂₁₄	1	1	yes	characteristic
91	C₂²×C₄	Dic₂₁₄	1	2	yes	—
92	C₂²×C₄	D₄₂₈	1	2	yes	—
93	C₂²×C₄	D₄₂₈	1	2	yes	—
94	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
95	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
98	M₁₆	—	107	8	no	—
99	M₁₆	—	214	8	no	—
100	M₁₆	—	214	4	no	—
101	M₁₆	C₂×Dic₁₀₇	1	16	yes	—
102	C₂×C₈	—	214	8	no	—
103	C₂×C₈	—	214	4	no	—
104	C₂×C₈	C₂×Dic₁₀₇	1	4	yes	—
105	C₂×C₈	C₂×Dic₁₀₇	1	4	yes	—
106	C₂×C₈	C₂²×D₁₀₇	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₁₀₇	1	2	yes	—
109	C₂³	C₄⋊Dic₁₀₇	1	2	yes	—
110	C₂³	C₂×Dic₂₁₄	1	1	yes	characteristic
111	C₂³	C₂×Dic₂₁₄	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₁₀₇	1	4	yes	—
113	C₂×C₄	C₄⋊Dic₁₀₇	1	2	yes	—
114	C₂×C₄	C₄⋊Dic₁₀₇	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₁₀₇	1	4	yes	—
116	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×Dic₁₀₇	1	4	yes	—
119	C₂×C₄	C₂×D₄₂₈	1	4	yes	—
120	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
122	C₂×C₄	C₂×Dic₂₁₄	1	2	yes	—
123	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
125	C₈	—	214	8	no	—
126	C₈	C₂²×Dic₁₀₇	1	8	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₁₀₇	1	4	yes	—
133	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
134	C₂²	C₂²×Dic₂₁₄	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
136	C₄	C₂×C₄⋊Dic₁₀₇	1	4	yes	—
137	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
138	C₄	C₂³×Dic₁₀₇	1	1	yes	characteristic
139	C₄	C₂²×D₄₂₈	1	1	yes	characteristic
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#20292676	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₈
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂²×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₇	Q₈
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₇
hypocenter	C₁₀₇	C₂×#430
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20292676	C₂	direct (N × H)
3. C₂×#16812045	C₂	semidirect
10. #16812045	C₂²	semidirect
70. C₂×C₁₀₇	#430	semidirect
73. C₁₀₇	C₂×#430	semidirect
143. C₂	#20292676	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×#13133802	2
#20292676	2
C₂×#16812045	2
C₂²×M₁₆×C₁₀₇	2
C₂×#430	107

Subgroups — C2×#20292676 #28345068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20292676 #28345068