Subgroups — C₂×#10601709 #12253550

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3961	C₂	1	2	yes	maximal
3	C₂×#64988	C₂	1	1	yes	maximal, characteristic
4	#10601709	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64712	C₂	1	2	yes	maximal
6	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Q₈	—	2	1	no	—
9	C₂²×Dic₂₀	—	2	1	no	—
10	C₂²×Dic₂₀	C₂²	1	2	yes	—
11	C₂×#4015	—	2	1	no	—
12	C₄×C₅×Q₁₆	—	2	2	no	—
13	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₅×#107	C₂²	1	1	yes	characteristic
15	C₂×C₅×#103	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×#3961	—	2	8	no	—
18	C₂×#3961	C₂²	1	8	yes	—
19	C₂²×#672	C₂²	1	2	yes	—
20	#64988	—	2	2	no	—
21	#64988	C₂²	1	4	yes	—
22	#64712	C₂²	1	8	yes	—
23	C₄⋊#672	C₂²	1	4	yes	—
24	C₂×#2209	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×Dic₁₀	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	D₄	1	4	yes	—
30	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×Q₈	—	2	4	no	—
34	C₄×C₅×Q₈	—	4	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₅	—	4	1	no	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	—	4	1	no	—
40	C₂×C₄⋊Dic₅	—	4	1	no	—
41	C₂×Dic₂₀	—	2	4	no	—
42	C₂×Dic₂₀	—	2	4	no	—
43	C₂×Dic₂₀	—	2	4	no	—
44	C₂×Dic₂₀	—	2	2	no	—
45	C₂×Dic₂₀	—	4	4	no	—
46	C₂×Dic₂₀	C₂³	1	4	yes	—
47	#4015	—	2	4	no	—
48	#4015	—	4	2	no	—
49	C₅×#107	—	2	1	no	—
50	C₅×#107	C₂³	1	2	yes	—
51	C₅×#103	—	2	4	no	—
52	C₄×C₅×C₈	D₄	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	#3961	—	2	16	no	—
55	C₂×#672	—	2	4	no	—
56	C₂×#672	C₂³	1	4	yes	—
57	C₂²×Q₃₂	—	5	2	no	—
58	C₂×#494	—	5	1	no	—
59	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
60	#2209	—	5	8	no	—
61	C₂×C₄⋊C₁₆	—	5	1	no	—
62	C₂×#358	—	5	2	no	—
63	C₂×C₅×Q₈	—	2	4	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂×C₅×Q₈	—	4	1	no	—
66	C₂×C₅×Q₈	—	4	2	no	—
67	C₂²×C₄×C₅	—	4	1	no	—
68	C₂²×C₄×C₅	—	4	1	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	2	no	—
73	C₂×Dic₁₀	—	2	4	no	—
74	C₂×Dic₁₀	—	4	4	no	—
75	C₂×Dic₁₀	—	4	4	no	—
76	C₂×Dic₁₀	—	4	2	no	—
77	C₅×Q₁₆	—	2	4	no	—
78	C₅×Q₁₆	—	2	2	no	—
79	C₅×Q₁₆	C₂×D₄	1	4	yes	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	2	yes	—
85	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	1	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₄²×C₅	—	4	4	no	—
90	C₄²×C₅	D₈	1	2	yes	—
91	C₄²×C₅	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₅	—	4	4	no	—
93	C₄⋊Dic₅	—	4	4	no	—
94	Dic₂₀	—	2	4	no	—
95	Dic₂₀	—	2	4	no	—
96	Dic₂₀	—	4	8	no	—
97	Dic₂₀	—	4	4	no	—
98	#672	—	2	4	no	—
99	C₂²×Q₁₆	—	5	2	no	—
100	C₂²×Q₁₆	—	10	1	no	—
101	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	—	10	1	no	—
103	C₂×C₄×Q₈	—	2	1	no	—
104	C₂×Q₃₂	—	5	8	no	—
105	C₂×Q₃₂	—	10	8	no	—
106	C₂²×C₁₆	—	5	2	no	—
107	#494	—	5	4	no	—
108	#494	—	10	2	no	—
109	C₄×Q₁₆	—	2	2	no	—
110	C₄×Q₁₆	C₂×D₅	1	4	yes	—
111	C₂×#107	C₂×D₅	1	1	yes	characteristic
112	C₂×#103	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₄⋊C₁₆	—	5	4	no	—
115	#358	—	5	8	no	—
116	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×Q₈	—	4	4	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	—	8	2	no	—
125	C₂×C₄×C₅	Q₃₂	1	4	yes	complemented
126	C₂×C₄×C₅	#121	1	2	yes	—
127	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₅	—	4	4	no	—
133	C₂×Dic₅	—	8	4	no	—
134	Dic₁₀	—	4	4	no	—
135	Dic₁₀	—	4	4	no	—
136	Dic₁₀	—	8	4	no	—
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	#121	1	2	yes	—
139	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	—	10	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	5	4	no	—
146	C₂×Q₁₆	—	10	4	no	—
147	C₂×Q₁₆	—	10	4	no	—
148	C₂×Q₁₆	—	10	2	no	—
149	C₂×Q₁₆	—	10	4	no	—
150	C₂×Q₁₆	—	20	4	no	—
151	C₂×Q₁₆	D₂₀	1	4	yes	—
152	C₂×Q₁₆	C₂²×D₅	1	2	yes	—
153	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
155	C₄⋊Q₈	—	10	4	no	—
156	C₄⋊Q₈	—	20	2	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	20	1	no	—
162	C₂×C₄⋊C₄	—	20	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	Q₃₂	—	10	16	no	—
166	C₂×C₁₆	—	5	4	no	—
167	C₂×C₁₆	—	10	4	no	—
168	#107	—	2	1	no	—
169	#107	C₂²×D₅	1	2	yes	—
170	#103	—	2	4	no	—
171	C₄×C₈	C₂²⋊D₅	1	2	yes	—
172	C₄×C₈	C₂²×D₅	1	2	yes	—
173	C₂²×C₅	#459	1	2	yes	—
174	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
175	C₂²×C₅	#510	1	2	yes	—
176	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	4	4	no	—
179	C₄×C₅	—	8	2	no	—
180	C₄×C₅	#459	1	2	yes	—
181	C₄×C₅	C₂×Q₃₂	1	4	yes	complemented
182	C₄×C₅	C₂×#121	1	1	yes	characteristic
183	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	Dic₅	—	8	4	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	10	4	no	—
190	C₂×Q₈	—	20	2	no	—
191	C₂×Q₈	—	20	4	no	—
192	C₂×Q₈	—	20	4	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	20	2	no	—
196	C₂²×C₄	#686	1	1	yes	characteristic
197	C₂²×C₄	#686	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	10	4	no	—
202	Q₁₆	—	10	4	no	—
203	Q₁₆	—	20	8	no	—
204	Q₁₆	—	20	4	no	—
205	Q₁₆	C₂×D₂₀	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₂₀	1	2	yes	—
208	C₂×C₈	#709	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	20	4	no	—
216	C₄⋊C₄	—	20	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	#686	1	2	yes	—
219	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
220	C₁₆	—	10	4	no	—
221	C₂×C₅	#2209	1	4	yes	complemented
222	C₂×C₅	C₂×#459	1	1	yes	characteristic
223	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
224	C₂×C₅	C₂×#510	1	1	yes	characteristic
225	C₂³	C₂×#686	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	20	4	no	—
230	Q₈	—	20	4	no	—
231	Q₈	—	40	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	20	4	no	—
238	C₂×C₄	—	40	4	no	—
239	C₂×C₄	#3961	1	4	yes	complemented
240	C₂×C₄	#4076	1	2	yes	—
241	C₂×C₄	C₂×#686	1	2	yes	—
242	C₂×C₄	C₂×#686	1	1	yes	characteristic
243	C₂×C₄	C₂×#686	1	2	yes	—
244	C₂×C₄	#4095	1	2	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	#4076	1	2	yes	—
248	C₈	C₂²×D₂₀	1	1	yes	characteristic
249	C₈	C₂×#709	1	1	yes	characteristic
250	C₅	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
251	C₂²	#65304	1	2	yes	—
252	C₂²	C₂×#3961	1	2	yes	—
253	C₂²	#65484	1	2	yes	—
254	C₂²	C₂²×#686	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	—	40	4	no	—
259	C₄	#65304	1	2	yes	—
260	C₄	C₂×#3961	1	4	yes	complemented
261	C₄	C₂×#4076	1	1	yes	characteristic
262	C₄	C₂×#4095	1	1	yes	characteristic
263	C₂	#10601709	1	4	yes	direct product factor
264	C₂	C₂×#65304	1	1	yes	characteristic
265	C₂	C₂²×#3961	1	1	yes	characteristic
266	C₂	C₂×#65484	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	C₂×#2209
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #10601709	C₂	direct (N × H)
125. C₂×C₄×C₅	Q₃₂	semidirect
181. C₄×C₅	C₂×Q₃₂	semidirect
221. C₂×C₅	#2209	semidirect
239. C₂×C₄	#3961	semidirect
250. C₅	C₂×#2209	semidirect
260. C₄	C₂×#3961	semidirect
263. C₂	#10601709	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×C₄×C₅×Q₁₆	2
C₂²×#3961	2
C₂×#64988	2
#10601709	2
C₂×#64712	2
C₂×C₄⋊#672	2
C₂×#2209	5

Subgroups — C2×#10601709 #12253550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601709 #12253550