Subgroups — C₂×#10598307 #12014201

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3949	C₂	1	1	yes	maximal, characteristic
4	#10598307	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₅	—	2	1	no	—
7	C₂²×#681	—	2	1	no	—
8	C₂×C₅×M₃₂	—	2	2	no	—
9	C₂×C₅×M₃₂	—	2	2	no	—
10	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
13	C₂×#3951	—	2	1	no	—
14	C₂×#3951	—	2	2	no	—
15	C₂×#3951	C₄	1	4	yes	complemented
16	C₂×#3951	C₂²	1	2	yes	—
17	C₂×#3949	—	2	1	no	—
18	C₂×#3949	—	2	2	no	—
19	C₂×#3949	C₄	1	4	yes	—
20	C₂×#3949	C₂²	1	2	yes	—
21	C₂×#1383	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×Dic₅	—	4	1	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
32	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅	—	2	4	no	—
35	C₂×C₄⋊Dic₅	—	2	2	no	—
36	C₂×C₄⋊Dic₅	—	4	2	no	—
37	C₂×C₄⋊Dic₅	—	4	1	no	—
38	C₂×#681	—	2	4	no	—
39	C₂×#681	—	2	2	no	—
40	C₂×#681	—	4	2	no	—
41	C₂×#681	—	4	1	no	—
42	C₂²×#168	—	4	1	no	—
43	C₅×M₃₂	—	2	4	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	2	no	—
46	C₂×C₅×C₁₆	—	2	4	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
48	#3951	—	2	4	no	—
49	#3951	—	2	2	no	—
50	#3951	C₂×C₄	1	4	yes	complemented
51	#3949	—	2	4	no	—
52	#3949	—	2	2	no	—
53	#3949	C₂×C₄	1	4	yes	—
54	C₂²×M₃₂	D₅	1	1	yes	characteristic
55	C₂²×#108	—	5	1	no	—
56	C₂²×#106	—	5	1	no	—
57	#1383	—	5	4	no	—
58	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	D₈	1	2	yes	—
64	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅	—	4	4	no	—
70	C₂²×Dic₅	—	4	2	no	—
71	C₂²×Dic₅	—	8	4	no	—
72	C₂×C₅×C₈	C₄²	1	2	yes	—
73	C₂×C₅×C₈	C₄²	1	2	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊Dic₅	—	2	4	no	—
91	C₄⋊Dic₅	—	4	4	no	—
92	C₄⋊Dic₅	—	4	2	no	—
93	#681	—	2	4	no	—
94	#681	—	4	4	no	—
95	#681	—	4	2	no	—
96	C₂×#168	—	4	4	no	—
97	C₂×#168	—	4	2	no	—
98	C₅×C₁₆	—	2	4	no	—
99	C₂²×M₁₆	—	10	1	no	—
100	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₄⋊C₄	—	10	1	no	—
102	C₂×M₃₂	—	2	2	no	—
103	C₂×M₃₂	—	2	2	no	—
104	C₂×M₃₂	Dic₅	1	2	yes	complemented
105	C₂×M₃₂	C₂×D₅	1	2	yes	—
106	C₂²×C₁₆	Dic₅	1	2	yes	complemented
107	C₂×#108	—	5	4	no	—
108	C₂×#108	—	5	2	no	—
109	C₂×#108	—	10	1	no	—
110	C₂×#108	—	10	2	no	—
111	C₂×#106	—	5	4	no	—
112	C₂×#106	—	5	2	no	—
113	C₂×#106	—	10	1	no	—
114	C₂×#106	—	10	2	no	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	#102	1	2	yes	—
117	C₂³×C₅	#103	1	2	yes	—
118	C₂³×C₅	#106	1	1	yes	characteristic
119	C₂³×C₅	#106	1	1	yes	characteristic
120	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	#95	1	2	yes	—
122	C₂×C₄×C₅	#95	1	2	yes	—
123	C₂×C₄×C₅	#102	1	2	yes	—
124	C₂×C₄×C₅	#102	1	2	yes	—
125	C₂×C₄×C₅	#102	1	2	yes	—
126	C₂×C₄×C₅	#103	1	2	yes	—
127	C₂×C₄×C₅	#103	1	2	yes	—
128	C₂×C₄×C₅	#103	1	2	yes	—
129	C₂×C₄×C₅	#106	1	1	yes	characteristic
130	C₂×C₄×C₅	#106	1	1	yes	characteristic
131	C₂×C₄×C₅	#107	1	2	yes	—
132	C₂×C₄×C₅	#107	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×Dic₅	—	4	4	no	—
140	C₂×Dic₅	—	8	8	no	—
141	C₂×Dic₅	—	8	4	no	—
142	C₅×C₈	#95	1	2	yes	—
143	C₅×C₈	#95	1	2	yes	—
144	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
145	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
148	#168	—	4	4	no	—
149	C₂³×C₄	—	20	1	no	—
150	C₂³×C₄	D₂₀	1	1	yes	characteristic
151	C₂×M₁₆	—	10	4	no	—
152	C₂×M₁₆	—	10	2	no	—
153	C₂×M₁₆	—	20	1	no	—
154	C₂×M₁₆	—	20	2	no	—
155	C₂²×C₈	—	20	1	no	—
156	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
157	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₅	1	2	yes	—
159	C₂²×C₈	C₄×D₅	1	2	yes	—
160	C₂²×C₈	D₂₀	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
165	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	10	4	no	—
167	C₂×C₄⋊C₄	—	10	2	no	—
168	C₂×C₄⋊C₄	—	20	1	no	—
169	C₂×C₄⋊C₄	—	20	2	no	—
170	M₃₂	—	2	4	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	2	no	—
173	C₂×C₁₆	—	2	4	no	—
174	C₂×C₁₆	C₂×Dic₅	1	4	yes	complemented
175	#108	—	5	4	no	—
176	#108	—	10	4	no	—
177	#108	—	10	2	no	—
178	#106	—	5	4	no	—
179	#106	—	10	4	no	—
180	#106	—	10	2	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	4	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	#340	1	2	yes	—
186	C₂²×C₅	#340	1	2	yes	—
187	C₂²×C₅	#365	1	4	yes	—
188	C₂²×C₅	C₂×#102	1	1	yes	characteristic
189	C₂²×C₅	C₂×#103	1	1	yes	characteristic
190	C₂²×C₅	C₂×#106	1	1	yes	characteristic
191	C₄×C₅	#340	1	2	yes	—
192	C₄×C₅	#340	1	2	yes	—
193	C₄×C₅	C₂×#95	1	1	yes	characteristic
194	C₄×C₅	C₂×#102	1	1	yes	characteristic
195	C₄×C₅	C₂×#103	1	1	yes	characteristic
196	C₄×C₅	C₂×#107	1	1	yes	characteristic
197	Dic₅	—	8	4	no	—
198	C₂⁴	#677	1	1	yes	characteristic
199	C₂²×C₄	—	20	4	no	—
200	C₂²×C₄	—	20	2	no	—
201	C₂²×C₄	—	40	4	no	—
202	C₂²×C₄	#677	1	1	yes	characteristic
203	C₂²×C₄	#677	1	1	yes	characteristic
204	C₂²×C₄	#677	1	1	yes	characteristic
205	C₂²×C₄	D₄₀	1	2	yes	—
206	C₂²×C₄	Dic₂₀	1	2	yes	—
207	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
209	C₂²×C₄	#685	1	2	yes	—
210	C₂²×C₄	#685	1	2	yes	—
211	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
212	M₁₆	—	10	4	no	—
213	M₁₆	—	20	4	no	—
214	M₁₆	—	20	2	no	—
215	C₂×C₈	—	20	4	no	—
216	C₂×C₈	—	20	2	no	—
217	C₂×C₈	C₄×Dic₅	1	2	yes	—
218	C₂×C₈	C₄×Dic₅	1	2	yes	—
219	C₂×C₈	#683	1	2	yes	—
220	C₂×C₈	#683	1	2	yes	—
221	C₂×C₈	#683	1	2	yes	—
222	C₂×C₈	#683	1	2	yes	—
223	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
224	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
225	C₂×C₈	#685	1	2	yes	—
226	C₂×C₈	#685	1	2	yes	—
227	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
229	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	10	4	no	—
236	C₄⋊C₄	—	20	4	no	—
237	C₄⋊C₄	—	20	2	no	—
238	C₁₆	—	2	4	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	#1383	1	4	yes	complemented
243	C₂×C₅	C₂×#340	1	1	yes	characteristic
244	C₂×C₅	C₂×#365	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	#3948	1	2	yes	—
247	C₂³	#3949	1	1	yes	characteristic
248	C₂³	#3949	1	1	yes	characteristic
249	C₂³	#3953	1	2	yes	—
250	C₂³	C₂×#677	1	1	yes	characteristic
251	C₂×C₄	—	20	4	no	—
252	C₂×C₄	—	40	8	no	—
253	C₂×C₄	—	40	4	no	—
254	C₂×C₄	#3948	1	2	yes	—
255	C₂×C₄	#3948	1	2	yes	—
256	C₂×C₄	#3948	1	2	yes	—
257	C₂×C₄	#3949	1	1	yes	characteristic
258	C₂×C₄	#3949	1	1	yes	characteristic
259	C₂×C₄	#3950	1	2	yes	—
260	C₂×C₄	#3950	1	2	yes	—
261	C₂×C₄	#3953	1	2	yes	—
262	C₂×C₄	#3953	1	2	yes	—
263	C₂×C₄	#3953	1	2	yes	—
264	C₂×C₄	#3963	1	2	yes	—
265	C₂×C₄	#3963	1	2	yes	—
266	C₂×C₄	C₂×#677	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
268	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
270	C₂×C₄	C₂×#685	1	1	yes	characteristic
271	C₂×C₄	C₂×#685	1	1	yes	characteristic
272	C₈	—	20	4	no	—
273	C₈	#3963	1	2	yes	—
274	C₈	#3963	1	2	yes	—
275	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
276	C₈	C₂×#683	1	1	yes	characteristic
277	C₈	C₂×#683	1	1	yes	characteristic
278	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
279	C₅	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#64733	1	4	yes	—
285	C₂²	#64771	1	2	yes	—
286	C₂²	#64771	1	2	yes	—
287	C₂²	C₂×#3948	1	1	yes	characteristic
288	C₂²	C₂×#3949	1	1	yes	characteristic
289	C₂²	C₂×#3953	1	1	yes	characteristic
290	C₄	—	40	4	no	—
291	C₄	#64771	1	2	yes	—
292	C₄	#64771	1	2	yes	—
293	C₄	C₂×#3948	1	1	yes	characteristic
294	C₄	C₂×#3950	1	1	yes	characteristic
295	C₄	C₂×#3953	1	1	yes	characteristic
296	C₄	C₂×#3963	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	#10598307	1	4	yes	direct product factor
301	C₂	C₂×#64733	1	2	yes	—
302	C₂	C₂×#64771	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3949
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂³×C₅	#106
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	C₂×#1383
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598307	C₂	direct (N × H)
10. C₂×C₅×M₃₂	C₄	semidirect
12. C₂²×C₅×C₁₆	C₄	semidirect
15. C₂×#3951	C₄	semidirect
47. C₂×C₅×C₁₆	C₂×C₄	semidirect
50. #3951	C₂×C₄	semidirect
104. C₂×M₃₂	Dic₅	semidirect
106. C₂²×C₁₆	Dic₅	semidirect
174. C₂×C₁₆	C₂×Dic₅	semidirect
242. C₂×C₅	#1383	semidirect
279. C₅	C₂×#1383	semidirect
300. C₂	#10598307	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×#3951	2
C₂²×#3949	2
#10598307	2
C₂×#1383	5

Subgroups — C2×#10598307 #12014201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598307 #12014201