Subgroups — C₂×#10601707 #12253544

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 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, complemented
2	C₂²×#3960	C₂	1	2	yes	maximal, complemented
3	C₂×#64984	C₂	1	1	yes	maximal, characteristic
4	#10601707	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64711	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Q₈	—	2	1	no	—
9	C₂²×D₄₀	—	2	1	no	—
10	C₂²×D₄₀	C₂²	1	2	yes	—
11	C₂×#4020	—	2	1	no	—
12	C₄×C₅×Q₁₆	—	2	2	no	—
13	C₄×C₅×Q₁₆	C₂²	1	4	yes	complemented
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×#3960	—	2	8	no	—
18	C₂×#3960	C₂²	1	8	yes	complemented
19	C₂²×#672	C₂²	1	2	yes	—
20	#64984	—	2	2	no	—
21	#64984	C₂²	1	4	yes	—
22	#64711	C₂²	1	8	yes	complemented
23	C₄⋊#672	C₂²	1	4	yes	complemented
24	C₂×#2207	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×D₂₀	—	2	2	no	—
27	C₂²×D₂₀	—	4	1	no	—
28	C₂²×D₂₀	—	4	1	no	—
29	C₂×C₅×Q₁₆	—	2	1	no	—
30	C₂×C₅×Q₁₆	—	2	2	no	—
31	C₂×C₅×Q₁₆	D₄	1	4	yes	complemented
32	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₅×Q₈	—	2	4	no	—
36	C₄×C₅×Q₈	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄²×C₅	—	4	1	no	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×D₄₀	—	2	4	no	—
42	C₂×D₄₀	—	2	4	no	—
43	C₂×D₄₀	—	2	4	no	—
44	C₂×D₄₀	—	2	2	no	—
45	C₂×D₄₀	—	4	4	no	—
46	C₂×D₄₀	C₂³	1	4	yes	—
47	#4020	—	2	4	no	—
48	#4020	—	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	#3960	—	2	16	no	—
55	C₂×#672	—	2	4	no	—
56	C₂×#672	C₂³	1	4	yes	—
57	C₂²×SD₃₂	—	5	2	no	—
58	C₂×#493	—	5	1	no	—
59	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic, complemented
60	#2207	—	5	8	no	—
61	C₂×C₄⋊C₁₆	—	5	1	no	—
62	C₂×#357	—	5	2	no	—
63	C₂³×D₅	—	4	2	no	—
64	C₂×C₅×Q₈	—	2	4	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×Q₈	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×D₂₀	—	2	4	no	—
74	C₂×D₂₀	—	4	4	no	—
75	C₂×D₂₀	—	4	4	no	—
76	C₂×D₂₀	—	4	2	no	—
77	C₂×D₂₀	—	4	4	no	—
78	C₂×D₂₀	—	4	4	no	—
79	C₂×D₂₀	—	8	4	no	—
80	C₂×D₂₀	—	8	4	no	—
81	C₅×Q₁₆	—	2	4	no	—
82	C₅×Q₁₆	—	2	2	no	—
83	C₅×Q₁₆	C₂×D₄	1	4	yes	complemented
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₄²×C₅	—	4	4	no	—
94	C₄²×C₅	D₈	1	2	yes	—
95	C₄²×C₅	C₂×D₄	1	2	yes	—
96	D₄₀	—	2	4	no	—
97	D₄₀	—	2	4	no	—
98	D₄₀	—	4	8	no	—
99	D₄₀	—	4	4	no	—
100	#672	—	2	4	no	—
101	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
102	C₂²×D₈	—	5	2	no	—
103	C₂²×D₈	—	10	1	no	—
104	C₂×#127	—	10	1	no	—
105	C₂×C₄×Q₈	—	2	1	no	—
106	C₂×SD₃₂	—	5	8	no	—
107	C₂×SD₃₂	—	10	8	no	—
108	C₂²×C₁₆	—	5	2	no	—
109	#493	—	5	4	no	—
110	#493	—	10	2	no	—
111	C₄×Q₁₆	—	2	2	no	—
112	C₄×Q₁₆	C₂×D₅	1	4	yes	complemented
113	C₂×#107	C₂×D₅	1	1	yes	characteristic
114	C₂×#103	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	C₄⋊C₁₆	—	5	4	no	—
117	#357	—	5	8	no	—
118	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
119	C₂²×D₅	—	4	4	no	—
120	C₂²×D₅	—	8	4	no	—
121	C₂²×D₅	—	8	4	no	—
122	C₂²×D₅	—	8	4	no	—
123	C₅×Q₈	—	2	4	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	4	4	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	8	2	no	—
131	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
132	C₂×C₄×C₅	#121	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	D₂₀	—	4	4	no	—
139	D₂₀	—	4	4	no	—
140	D₂₀	—	8	8	no	—
141	D₂₀	—	8	4	no	—
142	D₂₀	—	8	8	no	—
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	#121	1	2	yes	—
145	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₅×C₈	C₂×#41	1	1	yes	characteristic
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×D₄	—	10	2	no	—
149	C₂²×D₄	—	20	1	no	—
150	C₂²×D₄	—	20	1	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	D₂₀	1	4	yes	complemented
154	C₂×Q₁₆	C₂²×D₅	1	2	yes	—
155	C₂×D₈	—	5	4	no	—
156	C₂×D₈	—	10	4	no	—
157	C₂×D₈	—	10	4	no	—
158	C₂×D₈	—	10	2	no	—
159	C₂×D₈	—	10	4	no	—
160	C₂×D₈	—	20	4	no	—
161	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	#127	—	10	4	no	—
164	#127	—	20	2	no	—
165	C₄×Q₈	—	2	4	no	—
166	C₄×Q₈	—	4	2	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	4	1	no	—
169	C₂×C₄²	—	4	1	no	—
170	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	SD₃₂	—	10	16	no	—
172	C₂×C₁₆	—	5	4	no	—
173	C₂×C₁₆	—	10	4	no	—
174	#107	—	2	1	no	—
175	#107	C₂²×D₅	1	2	yes	—
176	#103	—	2	4	no	—
177	C₄×C₈	C₂²⋊D₅	1	2	yes	—
178	C₄×C₈	C₂²×D₅	1	2	yes	—
179	C₂²×C₅	#459	1	2	yes	—
180	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
181	C₂²×C₅	#509	1	2	yes	—
182	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₅	—	8	4	no	—
184	C₂×D₅	—	8	4	no	—
185	C₂×D₅	—	8	4	no	—
186	C₂×D₅	—	16	8	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	4	4	no	—
189	C₄×C₅	—	8	2	no	—
190	C₄×C₅	#459	1	2	yes	—
191	C₄×C₅	C₂×SD₃₂	1	4	yes	complemented
192	C₄×C₅	C₂×#121	1	1	yes	characteristic
193	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂⁴	—	20	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×D₄	—	10	4	no	—
200	C₂×D₄	—	20	4	no	—
201	C₂×D₄	—	20	4	no	—
202	C₂×D₄	—	20	4	no	—
203	C₂×D₄	—	20	2	no	—
204	C₂×D₄	—	20	4	no	—
205	C₂×D₄	—	40	4	no	—
206	C₂×D₄	—	40	4	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	#686	1	1	yes	characteristic
210	C₂²×C₄	#686	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	Q₁₆	—	2	2	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	C₂×D₂₀	1	4	yes	complemented
215	D₈	—	10	4	no	—
216	D₈	—	10	4	no	—
217	D₈	—	20	8	no	—
218	D₈	—	20	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×D₂₀	1	2	yes	—
221	C₂×C₈	#709	1	2	yes	—
222	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄²	—	4	4	no	—
229	C₄²	#686	1	2	yes	—
230	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
231	C₁₆	—	10	4	no	—
232	C₂×C₅	#2207	1	4	yes	complemented
233	C₂×C₅	C₂×#459	1	1	yes	characteristic
234	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
235	C₂×C₅	C₂×#509	1	1	yes	characteristic
236	D₅	—	16	4	no	—
237	C₂³	—	20	4	no	—
238	C₂³	—	40	4	no	—
239	C₂³	—	40	4	no	—
240	C₂³	—	40	4	no	—
241	C₂³	C₂×#686	1	1	yes	characteristic
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	2	no	—
245	D₄	—	20	4	no	—
246	D₄	—	20	4	no	—
247	D₄	—	40	8	no	—
248	D₄	—	40	4	no	—
249	D₄	—	40	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	#3960	1	4	yes	complemented
256	C₂×C₄	#4076	1	2	yes	—
257	C₂×C₄	C₂×#686	1	2	yes	—
258	C₂×C₄	C₂×#686	1	1	yes	characteristic
259	C₂×C₄	C₂×#686	1	2	yes	—
260	C₂×C₄	#4095	1	2	yes	—
261	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	#4076	1	2	yes	—
264	C₈	C₂²×D₂₀	1	1	yes	characteristic
265	C₈	C₂×#709	1	1	yes	characteristic
266	C₅	C₂×#2207	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	40	4	no	—
268	C₂²	—	40	4	no	—
269	C₂²	—	40	4	no	—
270	C₂²	—	80	8	no	—
271	C₂²	#65304	1	2	yes	—
272	C₂²	C₂×#3960	1	2	yes	—
273	C₂²	#65482	1	2	yes	—
274	C₂²	C₂²×#686	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	#65304	1	2	yes	—
279	C₄	C₂×#3960	1	4	yes	complemented
280	C₄	C₂×#4076	1	1	yes	characteristic
281	C₄	C₂×#4095	1	1	yes	characteristic
282	C₂	—	80	4	no	—
283	C₂	#10601707	1	4	yes	direct product factor
284	C₂	C₂×#65304	1	1	yes	characteristic
285	C₂	C₂²×#3960	1	1	yes	characteristic
286	C₂	C₂×#65482	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	C₂³×C₅	C₂×D₈
prime radical	C₂×#64984	C₂
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	C₂×#2207
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
1. C₂×C₄×C₅×Q₁₆	C₂	semidirect
2. C₂²×#3960	C₂	semidirect
4. #10601707	C₂	direct (N × H)
5. C₂×#64711	C₂	semidirect
6. C₂×C₄⋊#672	C₂	semidirect
13. C₄×C₅×Q₁₆	C₂²	semidirect
18. C₂×#3960	C₂²	semidirect
22. #64711	C₂²	semidirect
23. C₄⋊#672	C₂²	semidirect
31. C₂×C₅×Q₁₆	D₄	semidirect
59. C₂×C₄×Q₁₆	D₅	semidirect
83. C₅×Q₁₆	C₂×D₄	semidirect
112. C₄×Q₁₆	C₂×D₅	semidirect
131. C₂×C₄×C₅	SD₃₂	semidirect
153. C₂×Q₁₆	D₂₀	semidirect
191. C₄×C₅	C₂×SD₃₂	semidirect
214. Q₁₆	C₂×D₂₀	semidirect
232. C₂×C₅	#2207	semidirect
255. C₂×C₄	#3960	semidirect
266. C₅	C₂×#2207	semidirect
279. C₄	C₂×#3960	semidirect
283. C₂	#10601707	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×#3960	2
C₂×#64984	2
#10601707	2
C₂×#64711	2
C₂×C₄⋊#672	2
C₂×#2207	5

Subgroups — C2×#10601707 #12253544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601707 #12253544