Subgroups — C₂×#13099522 #72269210

Order 3200.

928 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11717224	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11716679	C₂	1	1	yes	maximal, characteristic, complemented
4	#13099522	C₂	1	4	yes	maximal, direct product factor
5	C₂×#87900	—	2	1	no	—
6	C₂²×C₅²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅²×C₄⋊C₄	—	2	1	no	—
8	C₂×#87802	C₄	1	1	yes	characteristic, complemented
9	#11717224	—	2	1	no	—
10	#11717224	—	2	1	no	—
11	#11717224	C₄	1	2	yes	complemented
12	#11717224	C₂²	1	2	yes	—
13	C₂×#87801	C₄	1	1	yes	characteristic, complemented
14	C₂×#87800	—	2	1	no	—
15	C₅²×#106	—	2	1	no	—
16	C₅²×#106	C₂²	1	2	yes	complemented
17	#11716679	—	2	1	no	—
18	#11716679	C₂²	1	2	yes	complemented
19	C₂×#87756	—	2	1	no	—
20	C₂×#64708	—	5	6	no	maximal
21	C₂²×C₄×C₅²	—	4	1	no	—
22	C₂²×C₄×C₅²	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊(C₅⋊D₅)	—	4	1	no	—
24	#87900	—	2	2	no	—
25	#87900	—	2	2	no	—
26	#87900	—	4	1	no	—
27	#87900	—	4	1	no	—
28	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
29	C₂×C₄×C₅⋊D₅	—	4	1	no	—
30	C₂×#6045	—	2	1	no	—
31	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅²×C₈	D₄	1	1	yes	characteristic
34	C₂×C₅²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₅²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₅²×C₈	C₂³	1	1	yes	characteristic
37	C₅²×C₄⋊C₄	—	2	2	no	—
38	C₅²×C₄⋊C₄	—	4	1	no	—
39	#87802	—	2	1	no	—
40	#87802	C₂×C₄	1	2	yes	complemented
41	#87801	—	2	1	no	—
42	#87801	C₂×C₄	1	2	yes	complemented
43	#87800	—	2	2	no	—
44	#87800	—	2	2	no	—
45	#87756	—	2	2	no	—
46	C₂×#4050	—	5	6	no	—
47	C₂×C₅×#106	D₅	1	6	yes	complemented
48	C₂×#3944	—	5	6	no	—
49	#64708	—	5	24	no	—
50	C₂³×C₅²	D₈	1	1	yes	characteristic
51	C₂²×C₅⋊D₅	—	4	1	no	—
52	C₂×C₄×C₅²	—	4	2	no	—
53	C₂×C₄×C₅²	—	8	2	no	—
54	C₂×C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅²	D₈	1	1	yes	characteristic
57	C₂×C₄×C₅²	SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅²	SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊(C₅⋊D₅)	—	4	2	no	—
61	C₂²⋊(C₅⋊D₅)	—	4	2	no	—
62	C₂×#1039	—	4	1	no	—
63	C₄⋊(C₅⋊D₅)	—	2	2	no	—
64	C₄⋊(C₅⋊D₅)	—	4	1	no	—
65	C₄×C₅⋊D₅	—	4	2	no	—
66	C₄×C₅⋊D₅	—	4	2	no	—
67	#6045	—	2	2	no	—
68	#6045	—	4	1	no	—
69	C₅²×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₅²×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₅²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₅²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×#709	—	10	6	no	—
74	C₂²×C₅×C₈	C₂×D₅	1	6	yes	—
75	C₂×C₅×C₄⋊C₄	—	2	6	no	—
76	C₂×Dic₂₀	—	5	6	no	—
77	#4050	—	5	12	no	—
78	#4050	—	5	12	no	—
79	#4050	—	10	6	no	—
80	#4050	—	10	6	no	—
81	C₂×D₄₀	—	5	6	no	—
82	C₂×#677	—	10	6	no	—
83	C₅×#106	—	2	6	no	—
84	C₅×#106	C₂×D₅	1	12	yes	complemented
85	#3944	—	5	12	no	—
86	#3944	—	10	6	no	—
87	C₂×#672	—	10	6	no	—
88	C₂×#360	—	25	1	no	Sylow
89	C₂²×C₅²	—	2	2	no	—
90	C₂²×C₅²	#102	1	1	yes	characteristic
91	C₂²×C₅²	#102	1	1	yes	characteristic
92	C₂²×C₅²	C₂×D₈	1	1	yes	characteristic
93	C₂×C₅⋊D₅	—	4	2	no	—
94	C₂×C₅⋊D₅	—	8	2	no	—
95	C₄×C₅²	—	8	2	no	—
96	C₄×C₅²	#102	1	1	yes	characteristic
97	C₄×C₅²	#102	1	1	yes	characteristic
98	C₄×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₅²	C₂×SD₁₆	1	1	yes	characteristic
100	#1039	—	4	2	no	—
101	C₂²×C₄×C₅	—	4	6	no	—
102	C₂²×C₄×C₅	C₂²⋊D₅	1	6	yes	—
103	C₂×C₂²⋊D₅	—	20	6	no	—
104	#709	—	10	12	no	—
105	#709	—	10	12	no	—
106	#709	—	20	6	no	—
107	#709	—	20	6	no	—
108	C₂×D₂₀	—	10	6	no	—
109	C₂×C₄×D₅	—	20	6	no	—
110	C₂×Dic₁₀	—	10	6	no	—
111	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
112	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
113	C₂×C₅×C₈	D₂₀	1	6	yes	—
114	C₂×C₅×C₈	D₂₀	1	6	yes	—
115	C₂×C₅×C₈	C₂²⋊D₅	1	6	yes	—
116	C₂×C₅×C₈	C₂²×D₅	1	6	yes	—
117	C₅×C₄⋊C₄	—	2	12	no	—
118	C₅×C₄⋊C₄	—	4	6	no	—
119	Dic₂₀	—	5	12	no	—
120	Dic₂₀	—	10	6	no	—
121	D₄₀	—	5	12	no	—
122	D₄₀	—	10	6	no	—
123	#677	—	10	12	no	—
124	#677	—	10	12	no	—
125	#672	—	10	12	no	—
126	C₂×#135	—	25	1	no	—
127	C₂×M₃₂	—	25	1	no	—
128	C₂×#106	C₅⋊D₅	1	1	yes	characteristic, complemented
129	#360	—	25	4	no	—
130	C₂×C₅²	—	2	1	no	—
131	C₂×C₅²	—	2	1	no	—
132	C₂×C₅²	#360	1	2	yes	complemented
133	C₂×C₅²	C₂×#102	1	1	yes	characteristic
134	C₅⋊D₅	—	8	2	no	—
135	C₂³×C₅	#686	1	6	yes	—
136	C₂²×D₅	—	20	6	no	—
137	C₂×C₄×C₅	—	4	12	no	—
138	C₂×C₄×C₅	—	8	12	no	—
139	C₂×C₄×C₅	#685	1	6	yes	—
140	C₂×C₄×C₅	#685	1	6	yes	—
141	C₂×C₄×C₅	#686	1	6	yes	—
142	C₂×C₄×C₅	#688	1	6	yes	—
143	C₂×C₄×C₅	#688	1	6	yes	—
144	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
145	C₂²⋊D₅	—	20	12	no	—
146	C₂²⋊D₅	—	20	12	no	—
147	C₂×Dic₅	—	20	6	no	—
148	D₂₀	—	10	12	no	—
149	D₂₀	—	20	6	no	—
150	C₄×D₅	—	20	12	no	—
151	C₄×D₅	—	20	12	no	—
152	Dic₁₀	—	10	12	no	—
153	Dic₁₀	—	20	6	no	—
154	C₅×C₈	#685	1	6	yes	—
155	C₅×C₈	#685	1	6	yes	—
156	C₅×C₈	C₂×C₄×D₅	1	6	yes	—
157	C₅×C₈	C₂×D₂₀	1	6	yes	—
158	C₂×#41	—	50	1	no	—
159	#135	—	25	2	no	—
160	#135	—	25	2	no	—
161	#135	—	50	1	no	—
162	#135	—	50	1	no	—
163	C₂×Q₁₆	—	25	1	no	—
164	C₂×SD₁₆	—	50	1	no	—
165	C₂×D₈	—	25	1	no	—
166	C₂²×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	1	no	—
168	M₃₂	—	25	2	no	—
169	M₃₂	—	50	1	no	—
170	C₂×C₁₆	—	50	1	no	—
171	#106	—	2	1	no	—
172	#106	C₂×C₅⋊D₅	1	2	yes	complemented
173	C₅²	C₂×#360	1	1	yes	characteristic, complemented, Sylow
174	C₂²×C₅	—	2	12	no	—
175	C₂²×C₅	#3941	1	6	yes	—
176	C₂²×C₅	#3941	1	6	yes	—
177	C₂²×C₅	C₂×#686	1	6	yes	—
178	C₂×D₅	—	20	12	no	—
179	C₂×D₅	—	40	12	no	—
180	C₄×C₅	—	8	12	no	—
181	C₄×C₅	#3941	1	6	yes	—
182	C₄×C₅	#3941	1	6	yes	—
183	C₄×C₅	C₂×#685	1	6	yes	—
184	C₄×C₅	C₂×#688	1	6	yes	—
185	Dic₅	—	20	12	no	—
186	#41	—	50	2	no	—
187	#41	—	50	2	no	—
188	#41	—	100	1	no	—
189	#41	—	100	1	no	—
190	C₂×Q₈	—	50	1	no	—
191	C₂×D₄	—	50	1	no	—
192	C₂×D₄	—	100	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	100	1	no	—
195	C₂²×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
196	Q₁₆	—	25	2	no	—
197	Q₁₆	—	50	1	no	—
198	SD₁₆	—	50	2	no	—
199	SD₁₆	—	50	2	no	—
200	D₈	—	25	2	no	—
201	D₈	—	50	1	no	—
202	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
203	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
204	C₂×C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
205	C₂×C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₁₆	—	50	2	no	—
211	C₂×C₅	—	2	6	no	—
212	C₂×C₅	—	2	6	no	—
213	C₂×C₅	#64708	1	12	yes	complemented
214	C₂×C₅	C₂×#3941	1	6	yes	—
215	D₅	—	40	12	no	—
216	C₂³	—	100	1	no	—
217	C₂³	#87809	1	1	yes	characteristic
218	Q₈	—	50	2	no	—
219	Q₈	—	100	1	no	—
220	D₄	—	50	2	no	—
221	D₄	—	100	2	no	—
222	D₄	—	100	2	no	—
223	D₄	—	100	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	100	1	no	—
227	C₂×C₄	—	100	2	no	—
228	C₂×C₄	—	100	2	no	—
229	C₂×C₄	#87808	1	1	yes	characteristic
230	C₂×C₄	#87808	1	1	yes	characteristic
231	C₂×C₄	#87809	1	1	yes	characteristic
232	C₂×C₄	#87811	1	1	yes	characteristic
233	C₂×C₄	#87811	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
235	C₈	#87808	1	1	yes	characteristic
236	C₈	#87808	1	1	yes	characteristic
237	C₈	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
238	C₈	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
239	C₅	C₂×#64708	1	6	yes	complemented
240	C₂²	—	2	2	no	—
241	C₂²	—	100	2	no	—
242	C₂²	—	200	2	no	—
243	C₂²	#11716676	1	1	yes	characteristic
244	C₂²	#11716676	1	1	yes	characteristic
245	C₂²	C₂×#87809	1	1	yes	characteristic
246	C₄	—	8	2	no	—
247	C₄	—	100	2	no	—
248	C₄	#11716676	1	1	yes	characteristic
249	C₄	#11716676	1	1	yes	characteristic
250	C₄	C₂×#87808	1	1	yes	characteristic
251	C₄	C₂×#87811	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	200	2	no	—
255	C₂	#13099522	1	2	yes	direct product factor
256	C₂	C₂×#11716676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11716676
commutator	C₅²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×#106	C₂
socle	C₂²×C₅²	#102
prime radical	C₂×#11717224	C₂
cosocle	C₂×C₅²×C₈	C₂³
hypercenter	C₂×#106	C₅⋊D₅
hypocenter	C₅²	C₂×#360
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×#106	C₂	semidirect
3. C₂×#11716679	C₂	semidirect
4. #13099522	C₂	direct (N × H)
8. C₂×#87802	C₄	semidirect
11. #11717224	C₄	semidirect
13. C₂×#87801	C₄	semidirect
16. C₅²×#106	C₂²	semidirect
18. #11716679	C₂²	semidirect
40. #87802	C₂×C₄	semidirect
42. #87801	C₂×C₄	semidirect
47. C₂×C₅×#106	D₅	semidirect
84. C₅×#106	C₂×D₅	semidirect
128. C₂×#106	C₅⋊D₅	semidirect
132. C₂×C₅²	#360	semidirect
172. #106	C₂×C₅⋊D₅	semidirect
173. C₅²	C₂×#360	semidirect
213. C₂×C₅	#64708	semidirect
239. C₅	C₂×#64708	semidirect
255. C₂	#13099522	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×#11717224	2
C₂×C₅²×#106	2
C₂×#11716679	2
#13099522	2
C₂×#64708	5

Subgroups — C2×#13099522 #72269210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13099522 #72269210