Subgroups — C₂×#10600270 #12097244

Order 1280.

1310 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×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3935	C₂	1	2	yes	maximal
4	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
5	C₂×#64744	C₂	1	2	yes	maximal
6	#10600270	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×C₄⋊C₄	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂³×#168	C₂²	1	1	yes	characteristic
11	C₂³×#168	C₂²	1	2	yes	—
12	C₂³×#168	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	—
14	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₅×#95	C₄	1	2	yes	—
16	C₂×C₄⋊#168	C₄	1	8	yes	complemented
17	C₂×C₄⋊#168	C₂²	1	4	yes	—
18	C₂×#3935	—	2	8	no	—
19	C₂×#3935	C₂²	1	8	yes	—
20	C₂×C₄×#168	C₄	1	8	yes	complemented
21	C₂×C₄×#168	C₂²	1	4	yes	—
22	#64744	C₂²	1	8	yes	—
23	C₂×#1770	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	—	2	8	no	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₅	—	2	4	no	—
31	C₂×C₄²×C₅	—	2	4	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	C₂³	1	4	yes	—
37	C₂×C₄²×C₅	C₂³	1	2	yes	—
38	C₂²×#168	—	2	4	no	—
39	C₂²×#168	—	2	2	no	—
40	C₂²×#168	—	2	2	no	—
41	C₂²×#168	—	2	4	no	—
42	C₂²×#168	C₂×C₄	1	4	yes	—
43	C₂²×#168	C₂×C₄	1	8	yes	—
44	C₂²×#168	C₂×C₄	1	4	yes	—
45	C₂²×#168	D₄	1	4	yes	—
46	C₂²×#168	Q₈	1	4	yes	—
47	C₂²×#168	C₂³	1	2	yes	—
48	C₂²×#168	C₂³	1	4	yes	—
49	C₂²×#168	C₂³	1	2	yes	—
50	C₅×#95	C₂×C₄	1	8	yes	—
51	C₄⋊#168	C₂×C₄	1	16	yes	complemented
52	#3935	—	2	16	no	—
53	C₄×#168	C₂×C₄	1	16	yes	complemented
54	C₂²×C₄⋊C₈	—	5	1	no	—
55	C₂²×#97	—	5	2	no	—
56	C₂²×C₄×C₈	—	5	1	no	—
57	C₂×C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
58	#1770	—	5	8	no	—
59	C₂×#336	—	5	2	no	—
60	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	8	no	—
63	C₂²×C₄×C₅	—	2	8	no	—
64	C₂²×C₄×C₅	—	2	8	no	—
65	C₂²×C₄×C₅	C₄²	1	4	yes	—
66	C₂²×C₄×C₅	C₄²	1	4	yes	—
67	C₂²×C₄×C₅	M₁₆	1	8	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	2	yes	—
77	C₂²×C₄×C₅	#41	1	2	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	16	no	—
80	C₄²×C₅	—	2	8	no	—
81	C₄²×C₅	—	2	4	no	—
82	C₄²×C₅	—	2	4	no	—
83	C₄²×C₅	C₄²	1	8	yes	—
84	C₄²×C₅	C₂²×C₄	1	4	yes	—
85	C₄²×C₅	C₂²×C₄	1	4	yes	—
86	C₂×#168	—	2	8	no	—
87	C₂×#168	—	2	16	no	—
88	C₂×#168	—	2	4	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	—	2	2	no	—
91	C₂×#168	—	2	4	no	—
92	C₂×#168	C₄⋊C₄	1	16	yes	complemented
93	C₂×#168	C₂²×C₄	1	4	yes	—
94	C₂×#168	C₂²×C₄	1	8	yes	—
95	C₂×#168	C₂²×C₄	1	4	yes	—
96	C₂×#168	C₂×D₄	1	4	yes	—
97	C₂×#168	C₂×Q₈	1	4	yes	—
98	C₂³×C₈	—	5	1	no	—
99	C₂³×C₈	—	5	1	no	—
100	C₂³×C₈	—	5	2	no	—
101	C₂²×C₄⋊C₄	Dic₅	1	2	yes	—
102	C₂²×C₄²	C₂×D₅	1	2	yes	—
103	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	5	8	no	—
105	C₂×C₄⋊C₈	—	5	4	no	—
106	C₂×#97	—	5	8	no	—
107	C₂×#97	—	10	8	no	—
108	C₂×C₄×C₈	—	5	8	no	—
109	C₂×C₄×C₈	—	5	4	no	—
110	C₄×C₄⋊C₄	Dic₅	1	4	yes	—
111	C₄×C₄⋊C₄	C₂×D₅	1	4	yes	—
112	C₂×#95	Dic₅	1	2	yes	—
113	#336	—	5	8	no	—
114	C₂³×C₅	C₂×C₄²	1	2	yes	—
115	C₂³×C₅	C₄×D₄	1	2	yes	—
116	C₂³×C₅	C₄×Q₈	1	2	yes	—
117	C₂³×C₅	C₂×M₁₆	1	4	yes	—
118	C₂³×C₅	#131	1	4	yes	—
119	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	8	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	8	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	4	16	no	—
126	C₂×C₄×C₅	#97	1	16	yes	complemented
127	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₅	#117	1	4	yes	—
132	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
133	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
135	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	#168	—	2	4	no	—
144	#168	—	2	8	no	—
145	#168	C₂×C₄⋊C₄	1	8	yes	complemented
146	C₂³×C₄	C₂×Dic₅	1	4	yes	—
147	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
148	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂²×C₈	—	5	4	no	—
151	C₂²×C₈	—	5	4	no	—
152	C₂²×C₈	—	5	4	no	—
153	C₂²×C₈	—	5	2	no	—
154	C₂²×C₈	—	5	4	no	—
155	C₂²×C₈	—	5	8	no	—
156	C₂²×C₈	—	5	2	no	—
157	C₂²×C₈	—	5	4	no	—
158	C₂²×C₈	—	10	4	no	—
159	C₂²×C₈	—	10	2	no	—
160	C₂²×C₈	—	10	2	no	—
161	C₂²×C₈	—	10	4	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	C₂×Dic₅	1	8	yes	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	—	2	4	no	—
166	C₂×C₄²	C₄×D₅	1	4	yes	—
167	C₂×C₄²	C₄×D₅	1	4	yes	—
168	C₂×C₄²	C₂×Dic₅	1	4	yes	—
169	C₂×C₄²	C₂×Dic₅	1	2	yes	—
170	C₂×C₄²	C₂²×D₅	1	4	yes	—
171	C₂×C₄²	C₂²×D₅	1	2	yes	—
172	C₄⋊C₈	—	5	16	no	—
173	#97	—	10	16	no	—
174	C₄×C₈	—	5	16	no	—
175	#95	C₂×Dic₅	1	8	yes	—
176	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
177	C₂²×C₅	C₂×#97	1	4	yes	—
178	C₂²×C₅	#405	1	4	yes	—
179	C₂²×C₅	#436	1	8	yes	—
180	C₂²×C₅	#446	1	8	yes	—
181	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
182	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
184	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
185	C₂²×C₅	C₂×#131	1	2	yes	—
186	C₄×C₅	—	2	4	no	—
187	C₄×C₅	—	4	8	no	—
188	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
189	C₄×C₅	C₂×#97	1	8	yes	complemented
190	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₄×C₅	C₂×#117	1	1	yes	characteristic
192	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
194	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	#681	1	8	yes	—
200	C₂²×C₄	C₄×Dic₅	1	4	yes	—
201	C₂²×C₄	C₄×Dic₅	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
204	C₂²×C₄	D₄×D₅	1	2	yes	—
205	C₂²×C₄	#711	1	2	yes	—
206	C₂²×C₄	Q₈×D₅	1	2	yes	—
207	C₂²×C₄	#713	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
209	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
213	C₂×C₈	—	5	4	no	—
214	C₂×C₈	—	5	4	no	—
215	C₂×C₈	—	5	4	no	—
216	C₂×C₈	—	5	16	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	5	8	no	—
219	C₂×C₈	—	10	4	no	—
220	C₂×C₈	—	10	2	no	—
221	C₂×C₈	—	10	2	no	—
222	C₂×C₈	—	10	4	no	—
223	C₂×C₈	—	10	8	no	—
224	C₂×C₈	—	10	16	no	—
225	C₄⋊C₄	—	2	16	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	8	no	—
229	C₄²	C₄×Dic₅	1	8	yes	—
230	C₄²	C₂×C₄×D₅	1	4	yes	—
231	C₄²	C₂×C₄×D₅	1	4	yes	—
232	C₂×C₅	#1770	1	8	yes	complemented
233	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₅	C₂²×#97	1	1	yes	characteristic
235	C₂×C₅	C₂×#405	1	1	yes	characteristic
236	C₂×C₅	C₂×#436	1	2	yes	—
237	C₂×C₅	C₂×#446	1	2	yes	—
238	C₂³	C₂×#681	1	4	yes	—
239	C₂³	C₂×C₄×Dic₅	1	2	yes	—
240	C₂³	D₄×Dic₅	1	2	yes	—
241	C₂³	Q₈×Dic₅	1	2	yes	—
242	C₂³	#4094	1	4	yes	—
243	C₂³	C₂³×Dic₅	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	#3935	1	16	yes	complemented
251	C₂×C₄	#4033	1	4	yes	—
252	C₂×C₄	D₅×C₄⋊C₄	1	4	yes	—
253	C₂×C₄	#4038	1	4	yes	—
254	C₂×C₄	#4039	1	4	yes	—
255	C₂×C₄	C₂×#681	1	8	yes	—
256	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
257	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
258	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
259	C₂×C₄	D₄×Dic₅	1	2	yes	—
260	C₂×C₄	Q₈×Dic₅	1	2	yes	—
261	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
264	C₂×C₄	C₂×#711	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
266	C₂×C₄	C₂×#713	1	1	yes	characteristic
267	C₈	—	5	8	no	—
268	C₈	—	10	4	no	—
269	C₈	—	10	8	no	—
270	C₅	C₂×#1770	1	1	yes	characteristic, complemented, Sylow
271	C₂²	C₂×#3935	1	4	yes	—
272	C₂²	C₄⋊C₄×Dic₅	1	4	yes	—
273	C₂²	#65286	1	4	yes	—
274	C₂²	#65301	1	8	yes	—
275	C₂²	#65313	1	8	yes	—
276	C₂²	C₂²×#681	1	2	yes	—
277	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
278	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
279	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
280	C₂²	C₂×#4094	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	4	8	no	—
283	C₄	C₂×#3935	1	8	yes	complemented
284	C₄	C₄⋊C₄×Dic₅	1	4	yes	—
285	C₄	C₂×#4033	1	1	yes	characteristic
286	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#4038	1	1	yes	characteristic
288	C₄	C₂×#4039	1	1	yes	characteristic
289	C₂	#10600270	1	8	yes	direct product factor
290	C₂	C₂²×#3935	1	1	yes	characteristic
291	C₂	C₂×C₄⋊C₄×Dic₅	1	1	yes	characteristic
292	C₂	C₂×#65286	1	1	yes	characteristic
293	C₂	C₂×#65301	1	2	yes	—
294	C₂	C₂×#65313	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₄×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₅
hypocenter	C₅	C₂×#1770
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #10600270	C₂	direct (N × H)
16. C₂×C₄⋊#168	C₄	semidirect
20. C₂×C₄×#168	C₄	semidirect
51. C₄⋊#168	C₂×C₄	semidirect
53. C₄×#168	C₂×C₄	semidirect
92. C₂×#168	C₄⋊C₄	semidirect
126. C₂×C₄×C₅	#97	semidirect
145. #168	C₂×C₄⋊C₄	semidirect
189. C₄×C₅	C₂×#97	semidirect
232. C₂×C₅	#1770	semidirect
250. C₂×C₄	#3935	semidirect
270. C₅	C₂×#1770	semidirect
283. C₄	C₂×#3935	semidirect
289. C₂	#10600270	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×C₄⋊C₄	2
C₂²×C₄⋊#168	2
C₂²×#3935	2
C₂²×C₄×#168	2
C₂×#64744	2
#10600270	2
C₂×#1770	5

Subgroups — C2×#10600270 #12097244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600270 #12097244