Subgroups — C₂×#10602108 #12253527

Order 1280.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#3960	C₂	1	2	yes	maximal
3	C₂×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
4	C₂×#64784	C₂	1	2	yes	maximal, complemented
5	#10602108	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₂²×D₄₀	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₀	—	2	1	no	—
12	C₅×#494	—	2	2	no	—
13	C₅×#494	C₂²	1	4	yes	complemented
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3960	—	2	8	no	—
16	C₂×#3960	C₂²	1	8	yes	—
17	C₂×#3953	—	2	1	no	—
18	C₂×#3950	C₂²	1	1	yes	characteristic
19	C₂²×#672	C₂²	1	2	yes	—
20	C₄×D₄₀	—	2	2	no	—
21	C₄×D₄₀	C₂²	1	4	yes	—
22	#64784	C₂²	1	8	yes	complemented
23	C₄⋊#672	C₂²	1	4	yes	complemented
24	C₂×#2208	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	—	2	2	no	—
26	C₂²×D₂₀	—	2	1	no	—
27	C₂²×C₄×D₅	—	4	1	no	—
28	C₂×C₅×Q₁₆	—	2	4	no	—
29	C₂×C₅×Q₁₆	—	2	2	no	—
30	C₂×C₅×Q₁₆	—	2	4	no	—
31	C₂×C₅×Q₁₆	—	2	4	no	—
32	C₂×C₅×Q₁₆	—	4	4	no	—
33	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	—	2	4	no	—
37	C₅×C₄⋊Q₈	—	4	2	no	—
38	C₂×C₅×C₄⋊C₄	—	4	1	no	—
39	C₂×C₅×C₄⋊C₄	—	4	1	no	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×#685	—	4	1	no	—
42	C₂×C₄⋊Dic₅	—	2	1	no	—
43	C₂×D₄₀	—	2	1	no	—
44	C₂×D₄₀	—	2	2	no	—
45	C₂×D₄₀	D₄	1	4	yes	—
46	C₂×D₄₀	C₂³	1	2	yes	—
47	C₄×D₂₀	—	2	4	no	—
48	C₄×D₂₀	—	4	2	no	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	#3960	—	2	16	no	—
52	#3953	—	2	4	no	—
53	#3950	—	2	1	no	—
54	#3950	C₂³	1	2	yes	—
55	C₂×#672	—	2	4	no	—
56	C₂×#672	C₂³	1	4	yes	—
57	C₂²×SD₃₂	—	5	2	no	—
58	C₂×#494	D₅	1	1	yes	characteristic, complemented
59	C₂×C₄×D₈	—	5	1	no	—
60	#2208	—	5	8	no	—
61	C₂×C₄⋊C₁₆	—	5	1	no	—
62	C₂×#358	—	5	2	no	—
63	C₂³×D₅	—	4	1	no	—
64	C₂×C₅×Q₈	—	2	4	no	—
65	C₂×C₅×Q₈	—	4	2	no	—
66	C₂×C₅×Q₈	—	4	4	no	—
67	C₂×C₅×Q₈	—	4	4	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₅	—	4	1	no	—
73	C₂×D₂₀	—	2	4	no	—
74	C₂×D₂₀	—	2	2	no	—
75	C₂×D₂₀	—	4	1	no	—
76	C₂×D₂₀	—	4	2	no	—
77	C₂×C₄×D₅	—	4	4	no	—
78	C₂×C₄×D₅	—	8	4	no	—
79	C₅×Q₁₆	—	2	4	no	—
80	C₅×Q₁₆	—	2	4	no	—
81	C₅×Q₁₆	—	4	8	no	—
82	C₅×Q₁₆	—	4	4	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	2	yes	—
88	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₄²×C₅	D₈	1	2	yes	—
92	C₄²×C₅	C₂×D₄	1	2	yes	—
93	#685	—	4	4	no	—
94	C₄⋊Dic₅	—	2	2	no	—
95	C₄⋊Dic₅	—	4	1	no	—
96	D₄₀	—	2	4	no	—
97	D₄₀	—	2	2	no	—
98	D₄₀	C₂×D₄	1	4	yes	—
99	#672	—	2	4	no	—
100	C₂²×Q₁₆	—	2	1	no	—
101	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
102	C₂²×D₈	—	5	1	no	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×C₄×D₄	—	10	1	no	—
105	C₂×SD₃₂	—	5	8	no	—
106	C₂×SD₃₂	—	10	8	no	—
107	C₂²×C₁₆	—	5	2	no	—
108	#494	—	2	2	no	—
109	#494	C₂×D₅	1	4	yes	complemented
110	C₄×D₈	—	5	4	no	—
111	C₄×D₈	—	10	2	no	—
112	C₂×#107	—	5	1	no	—
113	C₂×#102	—	10	1	no	—
114	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	C₄⋊C₁₆	—	5	4	no	—
116	#358	—	5	8	no	—
117	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
118	C₂²×D₅	—	4	2	no	—
119	C₂²×D₅	—	4	4	no	—
120	C₂²×D₅	—	8	2	no	—
121	C₂²×D₅	—	8	2	no	—
122	C₅×Q₈	—	4	4	no	—
123	C₅×Q₈	—	4	4	no	—
124	C₅×Q₈	—	8	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	8	4	no	—
128	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
129	C₂×C₄×C₅	#121	1	2	yes	—
130	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₅	—	4	2	no	—
136	C₂×Dic₅	—	8	2	no	—
137	D₂₀	—	2	4	no	—
138	D₂₀	—	4	2	no	—
139	D₂₀	—	4	4	no	—
140	C₄×D₅	—	8	8	no	—
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	#121	1	2	yes	—
143	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₅×C₈	C₂×#41	1	1	yes	characteristic
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×D₄	—	10	1	no	—
147	C₂³×C₄	—	20	1	no	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	4	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	4	4	no	—
153	C₂×Q₁₆	C₂²×D₅	1	4	yes	—
154	C₂×D₈	—	5	4	no	—
155	C₂×D₈	—	5	2	no	—
156	C₂×D₈	—	10	2	no	—
157	C₂×D₈	—	10	1	no	—
158	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	4	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	C₄×D₄	—	10	4	no	—
163	C₄×D₄	—	20	2	no	—
164	C₂×C₄⋊C₄	—	4	1	no	—
165	C₂×C₄⋊C₄	—	4	1	no	—
166	C₂×C₄⋊C₄	—	10	1	no	—
167	C₂×C₂²⋊C₄	—	20	1	no	—
168	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	SD₃₂	—	10	16	no	—
170	C₂×C₁₆	—	5	4	no	—
171	C₂×C₁₆	—	10	4	no	—
172	#107	—	5	2	no	—
173	#107	—	10	1	no	—
174	#102	—	10	4	no	—
175	C₄×C₈	C₂²⋊D₅	1	2	yes	—
176	C₄×C₈	C₂²×D₅	1	2	yes	—
177	C₂²×C₅	#459	1	2	yes	—
178	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
179	C₂²×C₅	#510	1	2	yes	—
180	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₅	—	4	4	no	—
182	C₂×D₅	—	8	2	no	—
183	C₂×D₅	—	8	2	no	—
184	C₂×D₅	—	8	8	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	8	4	no	—
187	C₄×C₅	#459	1	2	yes	—
188	C₄×C₅	C₂×SD₃₂	1	4	yes	complemented
189	C₄×C₅	C₂×#121	1	1	yes	characteristic
190	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	Dic₅	—	8	2	no	—
192	C₂⁴	—	20	1	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×D₄	—	10	2	no	—
198	C₂×D₄	—	10	4	no	—
199	C₂×D₄	—	20	1	no	—
200	C₂×D₄	—	20	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	20	1	no	—
203	C₂²×C₄	—	20	4	no	—
204	C₂²×C₄	—	40	4	no	—
205	C₂²×C₄	#686	1	1	yes	characteristic
206	C₂²×C₄	#686	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	Q₁₆	—	2	4	no	—
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	8	no	—
211	Q₁₆	—	4	4	no	—
212	D₈	—	5	4	no	—
213	D₈	—	10	2	no	—
214	D₈	—	10	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	D₄×D₅	1	2	yes	—
217	C₂×C₈	#711	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
219	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	10	2	no	—
224	C₄⋊C₄	—	20	1	no	—
225	C₂²⋊C₄	—	20	4	no	—
226	C₄²	#686	1	2	yes	—
227	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
228	C₁₆	—	10	4	no	—
229	C₂×C₅	#2208	1	4	yes	complemented
230	C₂×C₅	C₂×#459	1	1	yes	characteristic
231	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
232	C₂×C₅	C₂×#510	1	1	yes	characteristic
233	D₅	—	8	4	no	—
234	C₂³	—	20	2	no	—
235	C₂³	—	20	4	no	—
236	C₂³	—	40	2	no	—
237	C₂³	—	40	2	no	—
238	C₂³	C₂×#686	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	4	no	—
241	Q₈	—	8	4	no	—
242	D₄	—	10	4	no	—
243	D₄	—	20	2	no	—
244	D₄	—	20	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	20	2	no	—
249	C₂×C₄	—	40	2	no	—
250	C₂×C₄	—	40	8	no	—
251	C₂×C₄	#3960	1	4	yes	complemented
252	C₂×C₄	C₂×#686	1	2	yes	—
253	C₂×C₄	C₂×#686	1	1	yes	characteristic
254	C₂×C₄	C₂×#686	1	2	yes	—
255	C₂×C₄	#4078	1	2	yes	—
256	C₂×C₄	#4084	1	2	yes	—
257	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	#4084	1	2	yes	—
260	C₈	C₂×D₄×D₅	1	1	yes	characteristic
261	C₈	C₂×#711	1	1	yes	characteristic
262	C₅	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
263	C₂²	—	20	4	no	—
264	C₂²	—	40	2	no	—
265	C₂²	—	40	2	no	—
266	C₂²	—	40	8	no	—
267	C₂²	#65361	1	2	yes	—
268	C₂²	C₂×#3960	1	2	yes	—
269	C₂²	#65468	1	2	yes	—
270	C₂²	C₂²×#686	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	40	2	no	—
274	C₄	#65361	1	2	yes	—
275	C₄	C₂×#3960	1	4	yes	complemented
276	C₄	C₂×#4078	1	1	yes	characteristic
277	C₄	C₂×#4084	1	1	yes	characteristic
278	C₂	—	40	4	no	—
279	C₂	#10602108	1	4	yes	direct product factor
280	C₂	C₂×#65361	1	1	yes	characteristic
281	C₂	C₂²×#3960	1	1	yes	characteristic
282	C₂	C₂×#65468	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂²×D₄₀	C₂²
hypercenter	C₂×#494	D₅
hypocenter	C₅	C₂×#2208
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
4. C₂×#64784	C₂	semidirect
5. #10602108	C₂	direct (N × H)
6. C₂×C₄⋊#672	C₂	semidirect
13. C₅×#494	C₂²	semidirect
22. #64784	C₂²	semidirect
23. C₄⋊#672	C₂²	semidirect
58. C₂×#494	D₅	semidirect
109. #494	C₂×D₅	semidirect
128. C₂×C₄×C₅	SD₃₂	semidirect
188. C₄×C₅	C₂×SD₃₂	semidirect
229. C₂×C₅	#2208	semidirect
251. C₂×C₄	#3960	semidirect
262. C₅	C₂×#2208	semidirect
275. C₄	C₂×#3960	semidirect
279. C₂	#10602108	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×C₅×#494	2
C₂²×#3960	2
C₂×C₄×D₄₀	2
C₂×#64784	2
#10602108	2
C₂×C₄⋊#672	2
C₂×#2208	5

Subgroups — C2×#10602108 #12253527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602108 #12253527