Subgroups — #12014183

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 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	C₂×#65395	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
4	#10598297	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#4072	—	2	1	no	—
7	C₂²×C₄⋊Dic₅	—	2	1	no	—
8	C₂×C₅×M₃₂	—	2	2	no	—
9	C₂×C₅×M₃₂	—	2	2	no	—
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	#65395	—	2	2	no	—
13	#65395	C₄	1	4	yes	complemented
14	C₂×#3950	—	2	1	no	—
15	C₂×#3950	—	2	2	no	—
16	C₂×#3950	C₄	1	4	yes	complemented
17	C₂×#3950	C₂²	1	2	yes	—
18	C₂×#3950	C₂²	1	2	yes	—
19	C₂×#3949	—	2	1	no	—
20	#12557	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂³×Dic₅	—	4	1	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₅	—	4	1	no	—
32	#4072	—	2	4	no	—
33	#4072	—	4	2	no	—
34	C₂×C₄⋊Dic₅	—	2	4	no	—
35	C₂×C₄⋊Dic₅	—	2	2	no	—
36	C₂×C₄⋊Dic₅	—	2	2	no	—
37	C₂×C₄⋊Dic₅	—	4	1	no	—
38	C₂×C₄⋊Dic₅	—	4	2	no	—
39	C₂×C₄×Dic₅	—	4	1	no	—
40	C₅×M₃₂	—	2	8	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
44	#3950	—	2	4	no	—
45	#3950	—	2	2	no	—
46	#3950	—	2	2	no	—
47	#3950	C₂×C₄	1	4	yes	—
48	#3950	C₂×C₄	1	4	yes	complemented
49	#3949	—	2	4	no	—
50	C₂²×M₃₂	D₅	1	1	yes	characteristic
51	C₂×#427	—	5	1	no	—
52	C₂²×#107	—	5	1	no	—
53	#1380	—	5	4	no	—
54	C₂⁴×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₅	—	4	4	no	—
64	C₂²×Dic₅	—	4	2	no	—
65	C₂²×Dic₅	—	4	2	no	—
66	C₂²×Dic₅	—	8	4	no	—
67	C₂×C₅×C₈	C₄²	1	4	yes	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂²⋊Dic₅	—	4	4	no	—
81	C₄⋊Dic₅	—	2	4	no	—
82	C₄⋊Dic₅	—	2	4	no	—
83	C₄⋊Dic₅	—	4	4	no	—
84	C₄⋊Dic₅	—	4	2	no	—
85	C₄⋊Dic₅	—	4	2	no	—
86	C₄×Dic₅	—	4	4	no	—
87	C₅×C₁₆	—	2	4	no	—
88	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
89	C₂×#117	—	10	1	no	—
90	C₂²×C₄⋊C₄	—	10	1	no	—
91	C₂×M₃₂	—	2	2	no	—
92	C₂×M₃₂	—	2	2	no	—
93	C₂×M₃₂	Dic₅	1	4	yes	complemented
94	C₂²×C₁₆	C₂×D₅	1	2	yes	—
95	#427	—	5	4	no	—
96	#427	—	10	2	no	—
97	C₂×#107	—	5	4	no	—
98	C₂×#107	—	5	2	no	—
99	C₂×#107	—	5	2	no	—
100	C₂×#107	—	10	1	no	—
101	C₂×#107	—	10	2	no	—
102	C₂×#106	—	10	1	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	#102	1	2	yes	—
106	C₂³×C₅	#102	1	2	yes	—
107	C₂³×C₅	#107	1	2	yes	—
108	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅	#95	1	4	yes	—
110	C₂×C₄×C₅	#102	1	2	yes	—
111	C₂×C₄×C₅	#102	1	2	yes	—
112	C₂×C₄×C₅	#103	1	4	yes	—
113	C₂×C₄×C₅	#103	1	4	yes	—
114	C₂×C₄×C₅	#106	1	4	yes	—
115	C₂×C₄×C₅	#107	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×Dic₅	—	4	4	no	—
123	C₂×Dic₅	—	4	4	no	—
124	C₂×Dic₅	—	8	8	no	—
125	C₂×Dic₅	—	8	2	no	—
126	C₂×Dic₅	—	8	4	no	—
127	C₂×Dic₅	—	8	2	no	—
128	C₅×C₈	#95	1	4	yes	—
129	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
130	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂³×C₄	—	20	1	no	—
134	C₂³×C₄	D₂₀	1	1	yes	characteristic
135	C₂²×C₈	Dic₁₀	1	2	yes	—
136	C₂²×C₈	C₄×D₅	1	2	yes	—
137	C₂²×C₈	C₄×D₅	1	2	yes	—
138	C₂²×C₈	D₂₀	1	1	yes	characteristic
139	C₂²×C₈	C₂×Dic₅	1	2	yes	—
140	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
142	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
143	#117	—	10	4	no	—
144	#117	—	20	2	no	—
145	C₂×C₄⋊C₄	—	10	4	no	—
146	C₂×C₄⋊C₄	—	10	2	no	—
147	C₂×C₄⋊C₄	—	10	2	no	—
148	C₂×C₄⋊C₄	—	20	2	no	—
149	C₂×C₄⋊C₄	—	20	1	no	—
150	C₂×C₂²⋊C₄	—	20	1	no	—
151	C₂×C₄²	—	20	1	no	—
152	M₃₂	—	2	8	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	C₂×Dic₅	1	4	yes	complemented
156	#107	—	5	4	no	—
157	#107	—	5	4	no	—
158	#107	—	10	4	no	—
159	#107	—	10	2	no	—
160	#107	—	10	2	no	—
161	#106	—	10	4	no	—
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	#340	1	4	yes	—
167	C₂²×C₅	C₂×#102	1	1	yes	characteristic
168	C₂²×C₅	C₂×#102	1	1	yes	characteristic
169	C₂²×C₅	C₂×#107	1	1	yes	characteristic
170	C₂²×C₅	#509	1	2	yes	—
171	C₂²×C₅	#510	1	2	yes	—
172	C₄×C₅	#340	1	4	yes	—
173	C₄×C₅	C₂×#95	1	1	yes	characteristic
174	C₄×C₅	C₂×#103	1	1	yes	characteristic
175	C₄×C₅	C₂×#103	1	1	yes	characteristic
176	C₄×C₅	C₂×#106	1	1	yes	characteristic
177	Dic₅	—	8	4	no	—
178	Dic₅	—	8	4	no	—
179	C₂⁴	D₄₀	1	1	yes	characteristic
180	C₂²×C₄	—	20	4	no	—
181	C₂²×C₄	—	20	2	no	—
182	C₂²×C₄	—	20	2	no	—
183	C₂²×C₄	—	40	4	no	—
184	C₂²×C₄	#677	1	2	yes	—
185	C₂²×C₄	#677	1	2	yes	—
186	C₂²×C₄	D₄₀	1	1	yes	characteristic
187	C₂²×C₄	Dic₂₀	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
189	C₂²×C₄	#685	1	2	yes	—
190	C₂²×C₄	#685	1	2	yes	—
191	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
192	C₂×C₈	C₄×Dic₅	1	4	yes	—
193	C₂×C₈	#683	1	4	yes	—
194	C₂×C₈	#683	1	4	yes	—
195	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
196	C₂×C₈	#685	1	2	yes	—
197	C₂×C₈	#685	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
199	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
205	C₄⋊C₄	—	10	4	no	—
206	C₄⋊C₄	—	10	4	no	—
207	C₄⋊C₄	—	20	4	no	—
208	C₄⋊C₄	—	20	2	no	—
209	C₄⋊C₄	—	20	2	no	—
210	C₂²⋊C₄	—	20	4	no	—
211	C₄²	—	20	4	no	—
212	C₁₆	—	2	4	no	—
213	C₂×C₅	—	2	4	no	—
214	C₂×C₅	C₂×#340	1	1	yes	characteristic
215	C₂×C₅	#2139	1	2	yes	—
216	C₂×C₅	#2140	1	1	yes	characteristic
217	C₂×C₅	#2141	1	1	yes	characteristic
218	C₂×C₅	#2159	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#3950	1	2	yes	—
222	C₂³	#3953	1	2	yes	—
223	C₂³	#3953	1	2	yes	—
224	C₂³	C₂×D₄₀	1	1	yes	characteristic
225	C₂×C₄	—	20	4	no	—
226	C₂×C₄	—	20	4	no	—
227	C₂×C₄	—	40	8	no	—
228	C₂×C₄	—	40	4	no	—
229	C₂×C₄	—	40	2	no	—
230	C₂×C₄	—	40	2	no	—
231	C₂×C₄	#3948	1	4	yes	—
232	C₂×C₄	#3948	1	4	yes	—
233	C₂×C₄	#3949	1	4	yes	—
234	C₂×C₄	#3950	1	2	yes	—
235	C₂×C₄	#3953	1	2	yes	—
236	C₂×C₄	#3953	1	2	yes	—
237	C₂×C₄	#3963	1	4	yes	—
238	C₂×C₄	C₂×#677	1	1	yes	characteristic
239	C₂×C₄	C₂×#677	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
242	C₂×C₄	C₂×#685	1	1	yes	characteristic
243	C₂×C₄	C₂×#685	1	1	yes	characteristic
244	C₈	#3963	1	4	yes	—
245	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
246	C₈	C₂×#683	1	1	yes	characteristic
247	C₈	C₂×#683	1	1	yes	characteristic
248	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
249	C₅	#12557	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	#64771	1	4	yes	—
255	C₂²	#65197	1	2	yes	—
256	C₂²	#65198	1	2	yes	—
257	C₂²	C₂×#3950	1	1	yes	characteristic
258	C₂²	C₂×#3953	1	1	yes	characteristic
259	C₂²	C₂×#3953	1	1	yes	characteristic
260	C₄	—	40	4	no	—
261	C₄	—	40	4	no	—
262	C₄	#64771	1	4	yes	—
263	C₄	C₂×#3948	1	1	yes	characteristic
264	C₄	C₂×#3948	1	1	yes	characteristic
265	C₄	C₂×#3949	1	1	yes	characteristic
266	C₄	C₂×#3963	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	#10602221	1	2	yes	—
269	C₂	#10602232	1	2	yes	—
270	C₂	#10602233	1	1	yes	characteristic
271	C₂	#10602238	1	1	yes	characteristic
272	C₂	C₂×#64771	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂⁴×C₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12557
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. #10598297	C₂	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
13. #65395	C₄	semidirect
16. C₂×#3950	C₄	semidirect
43. C₂×C₅×C₁₆	C₂×C₄	semidirect
48. #3950	C₂×C₄	semidirect
93. C₂×M₃₂	Dic₅	semidirect
155. C₂×C₁₆	C₂×Dic₅	semidirect
249. C₅	#12557	semidirect

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
C₂×#65395	2
C₂²×#3950	2
#10598297	2
#12557	5