Subgroups — C₂×#20293104 #28349651

Order 13696.

1488 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic
2	#20293104	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₁₄	C₂	1	4	yes	maximal, complemented
4	C₂×#16812074	C₂	1	4	yes	maximal, complemented
5	C₂×#16812068	C₂	1	1	yes	maximal, characteristic
6	C₂×#16812065	C₂	1	4	yes	maximal
7	C₂×#117×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₄	C₄	1	16	yes	complemented
9	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
10	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
11	C₂²×Dic₂₁₄	C₂²	1	8	yes	—
12	C₂³×Dic₁₀₇	C₂²	1	2	yes	—
13	C₄×Dic₂₁₄	C₂²	1	32	yes	complemented
14	#16812074	C₂²	1	32	yes	complemented
15	#16812068	C₂²	1	8	yes	—
16	#16812065	C₂²	1	32	yes	—
17	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
18	C₂×Dic₁₀₇⋊C₄	C₂²	1	8	yes	—
19	C₂×C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₇	C₂²	1	8	yes	—
21	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
22	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
23	#117×C₁₀₇	C₂²	1	8	yes	—
24	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₇	C₂²	1	2	yes	—
27	C₂×Dic₂₁₄	C₂×C₄	1	16	yes	—
28	C₂×Dic₂₁₄	C₂×C₄	1	16	yes	—
29	C₂×Dic₂₁₄	C₂×C₄	1	64	yes	complemented
30	C₂×Dic₂₁₄	C₂³	1	8	yes	—
31	C₂×Dic₂₁₄	C₂³	1	8	yes	—
32	C₂²×Dic₁₀₇	C₂×C₄	1	16	yes	—
33	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
34	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
35	C₂²×Dic₁₀₇	C₂³	1	8	yes	—
36	C₄⋊Dic₁₀₇	C₂³	1	8	yes	—
37	Dic₁₀₇⋊C₄	C₂³	1	32	yes	—
38	C₂²⋊Dic₁₀₇	C₂³	1	8	yes	—
39	C₄×Dic₁₀₇	C₂³	1	32	yes	—
40	C₄×Dic₁₀₇	C₂³	1	8	yes	—
41	C₂⁴×C₁₀₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₇	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
47	C₄⋊C₄×C₁₀₇	C₂³	1	8	yes	—
48	C₂²⋊C₄×C₁₀₇	C₂³	1	8	yes	—
49	C₄²×C₁₀₇	C₂³	1	8	yes	—
50	C₂³×C₁₀₇	—	2	4	no	—
51	C₂³×C₁₀₇	C₂²×C₄	1	4	yes	—
52	C₂³×C₁₀₇	C₂⁴	1	2	yes	—
53	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₀₇	—	2	8	no	—
55	C₂×C₄×C₁₀₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₀₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₀₇	C₂²×C₄	1	16	yes	—
58	C₂×C₄×C₁₀₇	C₂⁴	1	8	yes	—
59	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
60	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
61	C₂×Dic₁₀₇	—	2	8	no	—
62	C₂×Dic₁₀₇	C₂²×C₄	1	16	yes	—
63	C₂×Dic₁₀₇	C₂²×C₄	1	32	yes	—
64	C₂×Dic₁₀₇	C₂⁴	1	8	yes	—
65	C₂×Dic₁₀₇	C₂⁴	1	8	yes	—
66	Dic₂₁₄	C₂²×C₄	1	32	yes	—
67	Dic₂₁₄	C₂²×C₄	1	32	yes	—
68	C₂²×C₁₀₇	—	2	8	no	—
69	C₂²×C₁₀₇	—	2	4	no	—
70	C₂²×C₁₀₇	C₂³×C₄	1	4	yes	—
71	C₂²×C₁₀₇	C₂³×C₄	1	2	yes	—
72	C₂²×C₁₀₇	2₋¹⁺⁴	1	4	yes	—
73	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
74	C₄×C₁₀₇	—	2	8	no	—
75	C₄×C₁₀₇	C₂³×C₄	1	4	yes	—
76	C₄×C₁₀₇	C₂³×C₄	1	4	yes	—
77	Dic₁₀₇	—	2	8	no	—
78	Dic₁₀₇	C₂³×C₄	1	16	yes	—
79	C₂×C₁₀₇	—	2	4	no	—
80	C₂×C₁₀₇	#519	1	4	yes	complemented
81	C₂×C₁₀₇	C₂⁴×C₄	1	1	yes	characteristic
82	C₂×C₁₀₇	C₂×2₋¹⁺⁴	1	2	yes	—
83	C₂×#519	—	107	1	no	maximal, Sylow
84	C₁₀₇	C₂×#519	1	1	yes	characteristic, complemented, Sylow
85	C₂³×Q₈	—	107	1	no	—
86	#519	—	107	16	no	—
87	C₂×C₄×Q₈	—	107	4	no	—
88	C₂×C₄×Q₈	—	107	4	no	—
89	C₂×#117	—	107	1	no	—
90	C₂×#117	—	107	4	no	—
91	C₂×#117	D₁₀₇	1	1	yes	characteristic
92	C₂²×Q₈	—	107	16	no	—
93	C₂²×Q₈	—	107	2	no	—
94	C₂²×Q₈	—	107	2	no	—
95	C₂²×Q₈	—	107	8	no	—
96	C₂³×C₄	—	107	2	no	—
97	C₂³×C₄	C₂×D₁₀₇	1	1	yes	characteristic
98	C₄×Q₈	—	107	32	no	—
99	C₄×Q₈	—	107	32	no	—
100	#117	—	107	8	no	—
101	#117	—	107	32	no	—
102	#117	C₂×D₁₀₇	1	8	yes	—
103	C₂×C₄⋊C₄	—	107	8	no	—
104	C₂×C₄⋊C₄	—	107	2	no	—
105	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
106	C₂×C₂²⋊C₄	—	107	2	no	—
107	C₂×C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
108	C₂×C₄²	—	107	8	no	—
109	C₂×C₄²	—	107	2	no	—
110	C₂×C₄²	C₂×D₁₀₇	1	2	yes	—
111	C₂⁴	C₂²×D₁₀₇	1	1	yes	characteristic
112	C₂×Q₈	—	107	16	no	—
113	C₂×Q₈	—	107	16	no	—
114	C₂×Q₈	—	107	64	no	—
115	C₂×Q₈	—	107	8	no	—
116	C₂×Q₈	—	107	8	no	—
117	C₂²×C₄	—	107	16	no	—
118	C₂²×C₄	—	107	8	no	—
119	C₂²×C₄	—	107	4	no	—
120	C₂²×C₄	—	107	4	no	—
121	C₂²×C₄	C₄×D₁₀₇	1	8	yes	—
122	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
123	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
124	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
126	C₄⋊C₄	—	107	32	no	—
127	C₄⋊C₄	—	107	8	no	—
128	C₄⋊C₄	C₂²×D₁₀₇	1	8	yes	—
129	C₂²⋊C₄	—	107	8	no	—
130	C₂²⋊C₄	C₂²×D₁₀₇	1	8	yes	—
131	C₄²	—	107	32	no	—
132	C₄²	—	107	8	no	—
133	C₄²	C₂²×D₁₀₇	1	8	yes	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂×C₄×D₁₀₇	1	4	yes	—
136	C₂³	C₂³×D₁₀₇	1	2	yes	—
137	C₂³	C₂³×D₁₀₇	1	1	yes	characteristic
138	Q₈	—	107	32	no	—
139	Q₈	—	107	32	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	107	32	no	—
142	C₂×C₄	—	107	16	no	—
143	C₂×C₄	—	107	8	no	—
144	C₂×C₄	—	107	8	no	—
145	C₂×C₄	—	214	8	no	—
146	C₂×C₄	C₂×C₄×D₁₀₇	1	16	yes	—
147	C₂×C₄	C₂×C₄×D₁₀₇	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₀₇	1	4	yes	—
149	C₂×C₄	C₂³×D₁₀₇	1	8	yes	—
150	C₂×C₄	C₂³×D₁₀₇	1	2	yes	—
151	C₂×C₄	C₂³×D₁₀₇	1	2	yes	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₂²×C₄×D₁₀₇	1	4	yes	—
155	C₂²	C₂²×C₄×D₁₀₇	1	2	yes	—
156	C₂²	#16812156	1	4	yes	—
157	C₂²	C₂⁴×D₁₀₇	1	1	yes	characteristic
158	C₄	—	2	8	no	—
159	C₄	—	107	16	no	—
160	C₄	—	214	8	no	—
161	C₄	C₂²×C₄×D₁₀₇	1	4	yes	—
162	C₄	C₂²×C₄×D₁₀₇	1	4	yes	—
163	C₂	—	2	4	no	—
164	C₂	#20293104	1	4	yes	direct product factor
165	C₂	C₂³×C₄×D₁₀₇	1	1	yes	characteristic
166	C₂	C₂×#16812156	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₇
commutator	C₂×C₁₀₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₇
Fitting	C₂×#117×C₁₀₇	C₂
socle	C₂³×C₁₀₇	C₂⁴
prime radical	C₂⁴×C₁₀₇	C₂³
cosocle	C₂²×C₁₀₇	C₂⁵
hypercenter	C₂×#117	D₁₀₇
hypocenter	C₁₀₇	C₂×#519
LCS	G ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #20293104	C₂	direct (N × H)
3. C₂×C₄×Dic₂₁₄	C₂	semidirect
4. C₂×#16812074	C₂	semidirect
8. C₂²×Dic₂₁₄	C₄	semidirect
13. C₄×Dic₂₁₄	C₂²	semidirect
14. #16812074	C₂²	semidirect
29. C₂×Dic₂₁₄	C₂×C₄	semidirect
80. C₂×C₁₀₇	#519	semidirect
84. C₁₀₇	C₂×#519	semidirect
164. C₂	#20293104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₁₄	2
#20293104	2
C₂×C₄×Dic₂₁₄	2
C₂×#16812074	2
C₂×#16812068	2
C₂×#16812065	2
C₂×#117×C₁₀₇	2
C₂×#519	107

Subgroups — C2×#20293104 #28349651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20293104 #28349651