Subgroups — C₂²×#65167 #12968893

Order 1280.

1966 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4036	C₂	1	2	yes	maximal
2	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂²×#3940	C₂	1	2	yes	maximal
6	C₂×#65167	C₂	1	24	yes	maximal, direct product factor
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₅	—	2	2	no	—
10	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂²×#683	—	2	2	no	—
12	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂³×#168	C₂²	1	1	yes	characteristic
14	C₂×#4036	—	2	12	no	—
15	C₂×#4036	C₂²	1	24	yes	—
16	C₂×C₅×#107	C₂²	1	12	yes	—
17	C₂×#3950	C₂²	1	12	yes	—
18	C₂×C₈×Dic₅	C₂²	1	12	yes	—
19	C₂×#3940	C₂²	1	24	yes	—
20	#65167	C₂²	1	64	yes	direct product factor
21	C₂²×#499	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	—	2	2	no	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	—	2	2	no	—
25	C₂³×Dic₅	D₄	1	1	yes	characteristic
26	C₂³×Dic₅	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	Q₈	1	8	yes	—
28	C₂²×C₅×C₈	C₂³	1	6	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	6	no	—
30	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
31	C₂×C₄⋊Dic₅	—	2	6	no	—
32	C₂×C₄⋊Dic₅	—	2	24	no	—
33	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
34	C₂×#683	—	2	24	no	—
35	C₂×C₄×Dic₅	D₄	1	6	yes	—
36	C₂×C₄×Dic₅	C₂³	1	6	yes	—
37	C₂²×#168	Q₈	1	8	yes	—
38	C₂²×#168	C₂³	1	6	yes	—
39	#4036	—	2	48	no	—
40	#4036	C₂³	1	32	yes	—
41	C₅×#107	C₂³	1	16	yes	—
42	#3950	C₂³	1	16	yes	—
43	C₈×Dic₅	C₂³	1	16	yes	—
44	#3940	C₂³	1	32	yes	—
45	C₂²×#125	—	5	2	no	—
46	C₂×#499	—	5	24	no	—
47	C₂²×#107	—	5	1	no	—
48	C₂²×#107	D₅	1	1	yes	characteristic
49	C₂²×#106	—	5	2	no	—
50	C₂²×C₄×C₈	—	5	1	no	—
51	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	12	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
58	C₂²×Dic₅	—	2	4	no	—
59	C₂²×Dic₅	—	2	4	no	—
60	C₂²×Dic₅	—	2	12	no	—
61	C₂²×Dic₅	—	4	8	no	—
62	C₂²×Dic₅	C₂×D₄	1	6	yes	—
63	C₂²×Dic₅	C₂×D₄	1	6	yes	—
64	C₂×C₅×C₈	C₂×Q₈	1	24	yes	—
65	C₂×C₅×C₈	C₂⁴	1	4	yes	—
66	C₅×C₄⋊C₄	—	2	12	no	—
67	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
68	C₄⋊Dic₅	—	2	12	no	—
69	C₄⋊Dic₅	—	2	32	no	—
70	C₄⋊Dic₅	C₂⁴	1	8	yes	—
71	#683	—	2	32	no	—
72	C₄×Dic₅	C₂×D₄	1	12	yes	—
73	C₄×Dic₅	C₂⁴	1	4	yes	—
74	C₂×#168	C₂×Q₈	1	24	yes	—
75	C₂×#168	C₂⁴	1	4	yes	—
76	C₂³×C₈	—	5	1	no	—
77	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
78	C₂×#125	—	5	24	no	—
79	C₂×#125	—	10	12	no	—
80	C₂²×C₄⋊C₄	—	5	2	no	—
81	C₂²×C₄⋊C₄	—	10	2	no	—
82	C₂²×C₄⋊C₄	—	10	2	no	—
83	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
84	C₂²×C₄²	—	5	1	no	—
85	#499	—	5	64	no	—
86	C₂×#107	—	5	12	no	—
87	C₂×#107	C₂×D₅	1	12	yes	—
88	C₂×#106	—	5	24	no	—
89	C₂×C₄×C₈	—	5	12	no	—
90	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
91	C₂³×C₅	#135	1	4	yes	—
92	C₂³×C₅	#135	1	4	yes	—
93	C₂³×C₅	C₂²×D₄	1	3	yes	—
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	—	4	24	no	—
96	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
97	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
98	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
99	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
100	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
101	C₂×Dic₅	—	2	12	no	—
102	C₂×Dic₅	—	2	12	no	—
103	C₂×Dic₅	—	2	8	no	—
104	C₂×Dic₅	—	4	24	no	—
105	C₂×Dic₅	C₂²×D₄	1	4	yes	—
106	C₂×Dic₅	C₂²×D₄	1	4	yes	—
107	C₅×C₈	C₂²×Q₈	1	8	yes	—
108	#168	C₂²×Q₈	1	8	yes	—
109	C₂³×C₄	—	2	2	no	—
110	C₂³×C₄	—	5	1	no	—
111	C₂³×C₄	—	5	1	no	—
112	C₂³×C₄	—	10	2	no	—
113	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂²×C₈	—	5	8	no	—
115	C₂²×C₈	—	5	6	no	—
116	C₂²×C₈	Dic₁₀	1	8	yes	—
117	C₂²×C₈	C₂²×D₅	1	6	yes	—
118	#125	—	5	32	no	—
119	#125	—	10	48	no	—
120	C₂×C₄⋊C₄	—	2	6	no	—
121	C₂×C₄⋊C₄	—	5	12	no	—
122	C₂×C₄⋊C₄	—	10	24	no	—
123	C₂×C₄⋊C₄	—	10	24	no	—
124	C₂×C₄⋊C₄	—	10	6	no	—
125	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
126	C₂×C₄²	—	5	6	no	—
127	C₂×C₄²	—	5	6	no	—
128	#107	—	5	16	no	—
129	#107	C₂²×D₅	1	16	yes	—
130	#106	—	5	32	no	—
131	C₄×C₈	—	5	16	no	—
132	C₂²×C₅	#499	1	16	yes	complemented
133	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
134	C₂²×C₅	C₂×#135	1	6	yes	—
135	C₂²×C₅	C₂×#135	1	6	yes	—
136	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
137	C₄×C₅	—	4	8	no	—
138	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
139	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
140	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
141	Dic₅	—	2	4	no	—
142	Dic₅	—	2	4	no	—
143	Dic₅	—	4	8	no	—
144	C₂⁴	D₄×D₅	1	1	yes	characteristic
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	5	6	no	—
148	C₂²×C₄	—	5	6	no	—
149	C₂²×C₄	—	10	4	no	—
150	C₂²×C₄	—	10	4	no	—
151	C₂²×C₄	—	10	12	no	—
152	C₂²×C₄	—	20	8	no	—
153	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
154	C₂²×C₄	D₄×D₅	1	3	yes	—
155	C₂²×C₄	Q₈×D₅	1	4	yes	—
156	C₂²×C₄	C₂³×D₅	1	3	yes	—
157	C₂×C₈	—	5	24	no	—
158	C₂×C₈	—	5	4	no	—
159	C₂×C₈	C₂×Dic₁₀	1	24	yes	—
160	C₂×C₈	C₂³×D₅	1	4	yes	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	—	5	8	no	—
163	C₄⋊C₄	—	10	32	no	—
164	C₄⋊C₄	—	10	32	no	—
165	C₄⋊C₄	—	10	12	no	—
166	C₄⋊C₄	C₂³×D₅	1	8	yes	—
167	C₄²	—	5	12	no	—
168	C₄²	—	5	4	no	—
169	C₂×C₅	C₂×#499	1	12	yes	complemented
170	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₅	C₂²×#135	1	1	yes	characteristic
172	C₂×C₅	C₂²×#135	1	1	yes	characteristic
173	C₂³	#4034	1	4	yes	—
174	C₂³	#4058	1	4	yes	—
175	C₂³	#4065	1	4	yes	—
176	C₂³	C₂×D₄×D₅	1	3	yes	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	24	no	—
179	C₂×C₄	—	5	4	no	—
180	C₂×C₄	—	5	4	no	—
181	C₂×C₄	—	10	12	no	—
182	C₂×C₄	—	10	12	no	—
183	C₂×C₄	—	10	8	no	—
184	C₂×C₄	—	20	24	no	—
185	C₂×C₄	#4034	1	12	yes	—
186	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
187	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
188	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
189	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
190	C₈	—	5	8	no	—
191	C₈	C₂²×Dic₁₀	1	8	yes	—
192	C₅	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
193	C₂²	#65167	1	16	yes	direct product factor
194	C₂²	C₂×#4034	1	6	yes	—
195	C₂²	C₂×#4058	1	6	yes	—
196	C₂²	C₂×#4065	1	6	yes	—
197	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	10	4	no	—
200	C₄	—	10	4	no	—
201	C₄	—	20	8	no	—
202	C₄	C₂×#4034	1	6	yes	—
203	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
204	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
205	C₂	C₂×#65167	1	12	yes	direct product factor
206	C₂	C₂²×#4034	1	1	yes	characteristic
207	C₂	C₂²×#4058	1	1	yes	characteristic
208	C₂	C₂²×#4065	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×#107	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×#107	D₅
hypocenter	C₅	C₂²×#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
6. C₂×#65167	C₂	direct (N × H)
20. #65167	C₂²	direct (N × H)
132. C₂²×C₅	#499	semidirect
169. C₂×C₅	C₂×#499	semidirect
192. C₅	C₂²×#499	semidirect
193. C₂²	#65167	direct (N × H)
205. C₂	C₂×#65167	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×#4036	2
C₂²×C₅×#107	2
C₂²×#3950	2
C₂²×C₈×Dic₅	2
C₂²×#3940	2
C₂×#65167	2
C₂²×#499	5

Subgroups — C22×#65167 #12968893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65167 #12968893