Subgroups — C₂×#10600069 #12252374

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	1	yes	maximal, characteristic
4	C₂×#64727	C₂	1	1	yes	maximal, characteristic
5	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600069	C₂	1	8	yes	maximal, direct product factor
9	C₂²×#676	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₂×#4018	C₄	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3947	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	C₂²×#672	C₂²	1	1	yes	characteristic
19	C₂×#3935	C₄	1	1	yes	characteristic
20	#64978	C₄	1	4	yes	—
21	#64978	C₂²	1	4	yes	—
22	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	#64727	C₂²	1	4	yes	—
24	#64727	C₂²	1	4	yes	—
25	#64722	C₂²	1	4	yes	complemented
26	C₁₆×Dic₅	C₂²	1	4	yes	complemented
27	C₄⋊#672	C₂²	1	4	yes	complemented
28	C₂×#2164	—	5	1	no	maximal, Sylow
29	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#685	C₈	1	1	yes	characteristic
34	C₂×#685	C₈	1	1	yes	characteristic
35	C₂×#683	C₈	1	1	yes	characteristic
36	C₂×#683	C₈	1	1	yes	characteristic
37	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₂²×#168	C₂×C₄	1	1	yes	characteristic
39	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₂×#676	—	2	4	no	—
41	C₂×#676	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₅	—	2	2	no	—
43	C₂×C₈×D₅	—	2	2	no	—
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂³	1	2	yes	—
46	#4018	C₈	1	4	yes	—
47	#4018	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₂×C₅×C₁₆	—	2	2	no	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	#3947	C₂×C₄	1	4	yes	—
55	C₈×Dic₅	C₂×C₄	1	2	yes	—
56	C₈×Dic₅	C₂³	1	2	yes	—
57	C₂×#672	—	2	2	no	—
58	C₂×#672	—	2	2	no	—
59	C₂×#672	C₂³	1	2	yes	—
60	C₂×#672	C₂³	1	2	yes	—
61	#3935	C₂×C₄	1	4	yes	—
62	C₂×#405	—	5	1	no	—
63	#2164	—	5	8	no	—
64	C₂×C₄⋊C₁₆	—	5	1	no	—
65	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
66	C₂×C₂²⋊C₁₆	—	5	1	no	—
67	C₂×C₂²⋊C₁₆	—	5	1	no	—
68	C₂×#346	—	5	1	no	—
69	C₂×C₄×C₁₆	—	5	1	no	—
70	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
75	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
76	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅	—	2	2	no	—
78	C₂×C₄×D₅	—	2	2	no	—
79	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₄²×C₅	C₂×C₈	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	2	yes	—
89	#685	C₂×C₈	1	4	yes	—
90	#685	C₂×C₈	1	4	yes	—
91	#683	C₂×C₈	1	4	yes	—
92	#683	C₂×C₈	1	4	yes	—
93	C₄×Dic₅	C₂×C₈	1	2	yes	—
94	C₄×Dic₅	C₂²×C₄	1	2	yes	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	M₁₆	1	4	yes	—
97	C₂×#168	C₂²×C₄	1	2	yes	—
98	C₂×#168	C₂²×C₄	1	2	yes	—
99	#676	—	2	8	no	—
100	C₈×D₅	—	2	4	no	—
101	C₈×D₅	—	2	2	no	—
102	C₈×D₅	—	2	2	no	—
103	C₅×C₁₆	—	2	2	no	—
104	C₅×C₁₆	—	2	2	no	—
105	#672	—	2	2	no	—
106	#672	—	2	2	no	—
107	C₂²×M₁₆	—	5	1	no	—
108	C₂³×C₈	—	5	1	no	—
109	C₂×#117	—	5	1	no	—
110	C₂²×C₁₆	—	5	1	no	—
111	C₂²×C₁₆	—	5	1	no	—
112	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
113	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
114	#405	—	5	4	no	—
115	#405	—	5	4	no	—
116	C₂×#97	—	5	1	no	—
117	C₂×#97	—	5	1	no	—
118	C₂×C₄×C₈	—	5	1	no	—
119	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
120	C₄⋊C₁₆	—	5	4	no	—
121	C₄⋊C₁₆	C₂×D₅	1	4	yes	complemented
122	C₂²⋊C₁₆	—	5	4	no	—
123	C₂²⋊C₁₆	—	5	4	no	—
124	#346	—	5	4	no	—
125	C₄×C₁₆	—	5	4	no	—
126	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₅	—	2	2	no	—
128	C₂²×D₅	—	2	2	no	—
129	C₂²×D₅	—	2	2	no	—
130	C₂²×D₅	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	#117	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	2	no	—
141	C₂×Dic₅	—	2	2	no	—
142	C₂×Dic₅	C₂²×C₈	1	2	yes	—
143	C₂×Dic₅	C₂²×C₈	1	2	yes	—
144	C₂×Dic₅	C₂²×C₈	1	2	yes	—
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	4	no	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	#117	1	2	yes	—
150	C₅×C₈	C₂×#41	1	1	yes	characteristic
151	C₅×C₈	C₂×#41	1	1	yes	characteristic
152	#168	—	2	2	no	—
153	#168	C₂×M₁₆	1	4	yes	—
154	C₂³×C₄	—	5	1	no	—
155	C₂×M₁₆	—	5	4	no	—
156	C₂×M₁₆	—	10	4	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₂²×C₈	—	5	2	no	—
160	C₂²×C₈	—	5	2	no	—
161	C₂²×C₈	—	10	2	no	—
162	C₂²×C₈	—	10	2	no	—
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
165	#117	—	5	4	no	—
166	#117	—	5	4	no	—
167	C₂×C₄⋊C₄	—	5	1	no	—
168	C₂×C₄⋊C₄	—	5	1	no	—
169	C₂×C₂²⋊C₄	—	5	1	no	—
170	C₂×C₂²⋊C₄	—	5	1	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	5	2	no	—
176	C₂×C₁₆	—	5	2	no	—
177	C₂×C₁₆	—	10	2	no	—
178	C₂×C₁₆	—	10	2	no	—
179	C₂×C₁₆	C₂²×D₅	1	2	yes	—
180	C₂×C₁₆	C₂²×D₅	1	2	yes	—
181	#97	—	5	4	no	—
182	#97	—	5	4	no	—
183	C₄×C₈	—	5	2	no	—
184	C₄×C₈	—	5	2	no	—
185	C₄×C₈	C₄×D₅	1	2	yes	—
186	C₄×C₈	C₂²×D₅	1	2	yes	—
187	C₂²×C₅	#431	1	2	yes	—
188	C₂²×C₅	#504	1	2	yes	—
189	C₂²×C₅	#504	1	2	yes	—
190	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	4	4	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	#431	1	2	yes	—
198	C₄×C₅	C₂×#117	1	1	yes	characteristic
199	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
200	Dic₅	—	2	2	no	—
201	Dic₅	—	2	2	no	—
202	Dic₅	—	2	2	no	—
203	C₂⁴	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	—	10	2	no	—
210	C₂²×C₄	—	10	2	no	—
211	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
212	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	M₁₆	—	10	8	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	5	2	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	10	2	no	—
220	C₂×C₈	—	10	2	no	—
221	C₂×C₈	—	10	2	no	—
222	C₂×C₈	—	10	4	no	—
223	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
224	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₈	#711	1	2	yes	—
226	C₂×C₈	#713	1	2	yes	—
227	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
228	C₄⋊C₄	—	5	4	no	—
229	C₄⋊C₄	—	5	4	no	—
230	C₂²⋊C₄	—	5	4	no	—
231	C₂²⋊C₄	—	5	4	no	—
232	C₄²	—	5	2	no	—
233	C₄²	—	5	2	no	—
234	C₄²	C₈×D₅	1	2	yes	—
235	C₄²	C₂×C₄×D₅	1	2	yes	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	10	2	no	—
239	C₁₆	—	10	2	no	—
240	C₂×C₅	#2164	1	4	yes	complemented
241	C₂×C₅	C₂×#431	1	1	yes	characteristic
242	C₂×C₅	C₂×#504	1	1	yes	characteristic
243	C₂×C₅	C₂×#504	1	1	yes	characteristic
244	D₅	—	4	2	no	—
245	C₂³	—	5	2	no	—
246	C₂³	—	10	2	no	—
247	C₂³	—	10	2	no	—
248	C₂³	—	10	2	no	—
249	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	10	4	no	—
261	C₂×C₄	#4038	1	2	yes	—
262	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
263	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
264	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
265	C₂×C₄	D₅×M₁₆	1	2	yes	—
266	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	5	4	no	—
269	C₈	—	10	2	no	—
270	C₈	#4038	1	2	yes	—
271	C₈	C₂×#711	1	1	yes	characteristic
272	C₈	C₂×#713	1	1	yes	characteristic
273	C₅	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	10	2	no	—
275	C₂²	—	10	2	no	—
276	C₂²	—	10	2	no	—
277	C₂²	—	20	4	no	—
278	C₂²	#65122	1	2	yes	—
279	C₂²	#65190	1	2	yes	—
280	C₂²	#65196	1	2	yes	—
281	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	10	2	no	—
285	C₄	—	10	2	no	—
286	C₄	—	10	2	no	—
287	C₄	#65122	1	2	yes	—
288	C₄	C₂×#4038	1	1	yes	characteristic
289	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
290	C₂	—	20	2	no	—
291	C₂	#10600069	1	4	yes	direct product factor
292	C₂	C₂×#65122	1	1	yes	characteristic
293	C₂	C₂×#65190	1	1	yes	characteristic
294	C₂	C₂×#65196	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	C₂×#2164
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₂×#64722	C₂	semidirect
6. C₂×C₁₆×Dic₅	C₂	semidirect
7. C₂×C₄⋊#672	C₂	semidirect
8. #10600069	C₂	direct (N × H)
22. C₅×C₄⋊C₁₆	C₂²	semidirect
25. #64722	C₂²	semidirect
26. C₁₆×Dic₅	C₂²	semidirect
27. C₄⋊#672	C₂²	semidirect
65. C₂×C₄⋊C₁₆	D₅	semidirect
121. C₄⋊C₁₆	C₂×D₅	semidirect
240. C₂×C₅	#2164	semidirect
273. C₅	C₂×#2164	semidirect
291. C₂	#10600069	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×C₄⋊C₁₆	2
C₂×#64727	2
C₂×#64722	2
C₂×C₁₆×Dic₅	2
C₂×C₄⋊#672	2
#10600069	2
C₂×#2164	5

Subgroups — C2×#10600069 #12252374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600069 #12252374