Subgroups — C₂×#10599867 #12252376

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64978	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	2	yes	maximal
4	C₂×#64721	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599867	C₂	1	8	yes	maximal, direct product factor
7	C₂²×#676	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×#4018	C₄	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3947	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×#672	C₂²	1	2	yes	—
15	C₂×#3935	C₄	1	1	yes	characteristic
16	#64978	C₄	1	4	yes	—
17	#64978	C₂²	1	4	yes	—
18	C₅×#346	C₂²	1	4	yes	complemented
19	#64727	C₂²	1	8	yes	—
20	#64721	C₂²	1	8	yes	complemented
21	C₄×#672	C₂²	1	4	yes	complemented
22	C₂×#2164	—	5	1	no	maximal, Sylow
23	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×#685	C₈	1	2	yes	—
28	C₂×#683	C₈	1	2	yes	—
29	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₂²×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×#676	—	2	4	no	—
33	C₂×#676	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₅	—	2	4	no	—
35	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅	C₂³	1	2	yes	—
37	#4018	C₈	1	4	yes	—
38	#4018	C₂×C₄	1	4	yes	—
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₂×C₅×C₁₆	C₂³	1	4	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂³	1	2	yes	—
43	#3947	C₂×C₄	1	4	yes	—
44	C₈×Dic₅	C₂×C₄	1	2	yes	—
45	C₈×Dic₅	C₂³	1	2	yes	—
46	C₂×#672	—	2	4	no	—
47	C₂×#672	C₂³	1	4	yes	—
48	#3935	C₂×C₄	1	4	yes	—
49	C₂×#405	—	5	1	no	—
50	#2164	—	5	8	no	—
51	C₂×C₄⋊C₁₆	—	5	2	no	—
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×#346	D₅	1	1	yes	characteristic, complemented
54	C₂×C₄×C₁₆	—	5	1	no	—
55	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₅	—	2	2	no	—
61	C₂×C₄×D₅	—	2	2	no	—
62	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
64	C₂×C₅×C₈	M₁₆	1	4	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₅×C₈	#41	1	4	yes	—
68	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
69	C₄²×C₅	C₂×C₈	1	2	yes	—
70	C₄²×C₅	C₂²×C₄	1	2	yes	—
71	#685	C₂×C₈	1	8	yes	—
72	#683	C₂×C₈	1	8	yes	—
73	C₄×Dic₅	C₂×C₈	1	2	yes	—
74	C₄×Dic₅	C₂²×C₄	1	2	yes	—
75	C₂×#168	—	2	2	no	—
76	C₂×#168	—	2	2	no	—
77	C₂×#168	C₂²×C₄	1	2	yes	—
78	C₂×#168	C₂²×C₄	1	2	yes	—
79	#676	—	2	8	no	—
80	C₈×D₅	—	2	4	no	—
81	C₈×D₅	—	2	4	no	—
82	C₅×C₁₆	—	2	4	no	—
83	#672	—	2	4	no	—
84	C₂²×M₁₆	—	5	1	no	—
85	C₂³×C₈	—	5	1	no	—
86	C₂×#117	—	5	1	no	—
87	C₂²×C₁₆	—	5	2	no	—
88	C₂²×C₁₆	C₂×D₅	1	2	yes	—
89	#405	—	5	4	no	—
90	#405	—	5	4	no	—
91	C₂×#97	—	5	1	no	—
92	C₂×#97	—	5	1	no	—
93	C₂×C₄×C₈	—	5	1	no	—
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	C₄⋊C₁₆	—	5	8	no	—
96	C₂²⋊C₁₆	—	5	8	no	—
97	#346	C₂×D₅	1	4	yes	complemented
98	C₄×C₁₆	—	5	4	no	—
99	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₅	—	2	2	no	—
101	C₂²×D₅	—	2	4	no	—
102	C₂²×D₅	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	#117	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₅	—	2	2	no	—
110	C₂×Dic₅	—	2	4	no	—
111	C₂×Dic₅	C₂²×C₈	1	4	yes	—
112	C₂×Dic₅	C₂²×C₈	1	2	yes	—
113	C₄×D₅	—	2	2	no	—
114	C₄×D₅	—	2	2	no	—
115	C₄×D₅	—	2	4	no	—
116	C₅×C₈	#117	1	2	yes	—
117	C₅×C₈	C₂×M₁₆	1	4	yes	—
118	C₅×C₈	C₂×#41	1	2	yes	—
119	#168	—	2	2	no	—
120	#168	—	2	2	no	—
121	C₂³×C₄	—	5	1	no	—
122	C₂×M₁₆	—	5	4	no	—
123	C₂×M₁₆	—	10	4	no	—
124	C₂²×C₈	—	5	1	no	—
125	C₂²×C₈	—	5	1	no	—
126	C₂²×C₈	—	5	2	no	—
127	C₂²×C₈	—	5	2	no	—
128	C₂²×C₈	—	10	4	no	—
129	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
131	#117	—	5	4	no	—
132	#117	—	5	4	no	—
133	C₂×C₄⋊C₄	—	5	2	no	—
134	C₂×C₂²⋊C₄	—	5	2	no	—
135	C₂×C₄²	—	5	1	no	—
136	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	5	4	no	—
139	C₂×C₁₆	—	10	4	no	—
140	C₂×C₁₆	C₂²×D₅	1	4	yes	—
141	#97	—	5	4	no	—
142	#97	—	5	4	no	—
143	C₄×C₈	—	5	2	no	—
144	C₄×C₈	—	5	2	no	—
145	C₄×C₈	C₄×D₅	1	2	yes	—
146	C₄×C₈	C₂²×D₅	1	2	yes	—
147	C₂²×C₅	#431	1	2	yes	—
148	C₂²×C₅	#504	1	4	yes	—
149	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₅	—	2	4	no	—
151	C₂×D₅	—	2	2	no	—
152	C₂×D₅	—	4	4	no	—
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	#431	1	2	yes	—
155	C₄×C₅	C₂×#117	1	1	yes	characteristic
156	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
157	Dic₅	—	2	4	no	—
158	Dic₅	—	2	2	no	—
159	C₂⁴	—	5	1	no	—
160	C₂²×C₄	—	5	2	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	2	no	—
163	C₂²×C₄	—	5	2	no	—
164	C₂²×C₄	—	10	2	no	—
165	C₂²×C₄	—	10	2	no	—
166	C₂²×C₄	C₈×D₅	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
168	M₁₆	—	10	8	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	10	2	no	—
172	C₂×C₈	—	10	2	no	—
173	C₂×C₈	—	10	4	no	—
174	C₂×C₈	—	10	4	no	—
175	C₂×C₈	#676	1	4	yes	—
176	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
178	C₂×C₈	#709	1	4	yes	—
179	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
180	C₄⋊C₄	—	5	8	no	—
181	C₂²⋊C₄	—	5	8	no	—
182	C₄²	—	5	2	no	—
183	C₄²	—	5	2	no	—
184	C₄²	C₈×D₅	1	2	yes	—
185	C₄²	C₂×C₄×D₅	1	2	yes	—
186	C₁₆	—	2	4	no	—
187	C₁₆	—	10	4	no	—
188	C₂×C₅	#2164	1	4	yes	complemented
189	C₂×C₅	C₂×#431	1	1	yes	characteristic
190	C₂×C₅	C₂×#504	1	2	yes	—
191	D₅	—	4	2	no	—
192	C₂³	—	5	2	no	—
193	C₂³	—	10	4	no	—
194	C₂³	—	10	2	no	—
195	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	5	4	no	—
198	C₂×C₄	—	5	2	no	—
199	C₂×C₄	—	10	4	no	—
200	C₂×C₄	—	10	2	no	—
201	C₂×C₄	—	10	2	no	—
202	C₂×C₄	—	10	2	no	—
203	C₂×C₄	—	10	4	no	—
204	C₂×C₄	#4018	1	2	yes	—
205	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
206	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
207	C₂×C₄	C₂×#676	1	2	yes	—
208	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
209	C₈	—	10	2	no	—
210	C₈	—	10	2	no	—
211	C₈	#4018	1	2	yes	—
212	C₈	C₂×#676	1	4	yes	—
213	C₈	C₂×#709	1	2	yes	—
214	C₅	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
215	C₂²	—	10	2	no	—
216	C₂²	—	10	4	no	—
217	C₂²	—	20	4	no	—
218	C₂²	#64974	1	2	yes	—
219	C₂²	#65196	1	4	yes	—
220	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	10	2	no	—
223	C₄	—	10	4	no	—
224	C₄	#64974	1	2	yes	—
225	C₄	C₂×#4018	1	1	yes	characteristic
226	C₄	C₂²×#676	1	1	yes	characteristic
227	C₂	—	20	2	no	—
228	C₂	#10599867	1	4	yes	direct product factor
229	C₂	C₂×#64974	1	1	yes	characteristic
230	C₂	C₂×#65196	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#346	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#346	D₅
hypocenter	C₅	C₂×#2164
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#346	C₂	semidirect
4. C₂×#64721	C₂	semidirect
5. C₂×C₄×#672	C₂	semidirect
6. #10599867	C₂	direct (N × H)
18. C₅×#346	C₂²	semidirect
20. #64721	C₂²	semidirect
21. C₄×#672	C₂²	semidirect
53. C₂×#346	D₅	semidirect
97. #346	C₂×D₅	semidirect
188. C₂×C₅	#2164	semidirect
214. C₅	C₂×#2164	semidirect
228. C₂	#10599867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₂×C₅×#346	2
C₂×#64727	2
C₂×#64721	2
C₂×C₄×#672	2
#10599867	2
C₂×#2164	5

Subgroups — C2×#10599867 #12252376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10599867 #12252376