Subgroups — #12018369

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	C₅×#1969	C₂	1	1	yes	maximal, characteristic
3	C₂×#64784	C₂	1	2	yes	maximal
4	#10598213	C₂	1	2	yes	maximal
5	#10598148	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×#4015	—	2	1	no	—
8	C₂×C₅×#107	C₂²	1	1	yes	characteristic
9	C₂×C₅×#103	—	2	1	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#391	—	2	1	no	—
12	C₂×#3950	—	2	1	no	—
13	C₂×#3950	C₂²	1	2	yes	—
14	C₂²×#672	C₂²	1	2	yes	—
15	#64971	—	2	4	no	—
16	#64784	—	2	4	no	—
17	#13364	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	1	no	—
19	C₂²×Dic₁₀	—	4	1	no	—
20	C₂×C₅×Q₁₆	—	2	2	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	1	no	—
23	C₂×C₅×Q₁₆	Q₈	1	2	yes	complemented
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	4	1	no	—
30	#4015	—	2	4	no	—
31	#4015	—	4	2	no	—
32	C₅×#107	—	2	2	no	—
33	C₅×#103	—	4	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₅×#95	—	4	1	no	—
36	C₅×#95	—	4	1	no	—
37	#3950	—	2	4	no	—
38	#3950	—	2	2	no	—
39	#3950	D₄	1	4	yes	—
40	C₂×#672	—	2	2	no	—
41	C₂×#672	—	2	2	no	—
42	C₂×#672	—	2	2	no	—
43	C₂×#500	—	5	1	no	—
44	C₂×#358	—	5	2	no	—
45	#1969	D₅	1	1	yes	characteristic
46	#1380	—	5	2	no	—
47	#1374	—	5	1	no	—
48	C₂×C₅×Q₈	—	2	2	no	—
49	C₂×C₅×Q₈	—	4	1	no	—
50	C₂×C₅×Q₈	—	4	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂²×C₄×C₅	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅	—	4	2	no	—
60	C₂×Dic₁₀	—	4	4	no	—
61	C₂×Dic₁₀	—	8	2	no	—
62	C₂×Dic₁₀	—	8	2	no	—
63	C₅×Q₁₆	—	2	4	no	—
64	C₅×Q₁₆	—	4	2	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	SD₁₆	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	2	yes	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄⋊Dic₅	—	2	4	no	—
77	C₄⋊Dic₅	—	4	4	no	—
78	C₄⋊Dic₅	—	4	2	no	—
79	#672	—	4	2	no	—
80	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	—	10	1	no	—
82	C₂²×C₁₆	—	5	2	no	—
83	#500	—	10	4	no	—
84	C₂×#107	—	5	2	no	—
85	C₂×#107	—	10	1	no	—
86	C₂×#107	C₂×D₅	1	1	yes	characteristic
87	C₂×#103	—	2	1	no	—
88	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
89	#391	—	2	1	no	—
90	#358	—	10	4	no	—
91	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
92	C₅×Q₈	—	4	2	no	—
93	C₅×Q₈	—	4	2	no	—
94	C₅×Q₈	—	8	1	no	—
95	C₅×Q₈	—	8	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	8	1	no	—
102	C₂×C₄×C₅	—	8	1	no	—
103	C₂×C₄×C₅	—	8	1	no	—
104	C₂×C₄×C₅	—	8	1	no	—
105	C₂×C₄×C₅	—	8	1	no	—
106	C₂×C₄×C₅	SD₃₂	1	2	yes	—
107	C₂×C₄×C₅	Q₃₂	1	2	yes	—
108	C₂×C₄×C₅	#120	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#124	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₅	#137	1	1	yes	characteristic
114	C₂×Dic₅	—	4	4	no	—
115	C₂×Dic₅	—	8	2	no	—
116	C₂×Dic₅	—	8	2	no	—
117	Dic₁₀	—	8	8	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
120	C₅×C₈	#124	1	2	yes	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×Q₈	—	20	1	no	—
123	C₂×Q₁₆	—	2	1	no	—
124	C₂×Q₁₆	—	2	2	no	—
125	C₂×Q₁₆	—	2	2	no	—
126	C₂×Q₁₆	Dic₁₀	1	2	yes	complemented
127	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
129	C₄⋊Q₈	—	10	4	no	—
130	C₄⋊Q₈	—	20	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	10	2	no	—
133	C₂×C₄⋊C₄	—	20	1	no	—
134	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
135	C₂×C₁₆	—	10	2	no	—
136	C₂×C₁₆	—	10	2	no	—
137	C₂×C₁₆	—	10	2	no	—
138	#107	—	2	2	no	—
139	#107	—	5	4	no	—
140	#107	—	10	4	no	—
141	#107	—	10	2	no	—
142	#103	—	4	2	no	—
143	C₄×C₈	—	2	2	no	—
144	#95	—	4	1	no	—
145	#95	—	4	1	no	—
146	C₂²×C₅	#447	1	1	yes	characteristic
147	C₂²×C₅	#474	1	1	yes	characteristic
148	C₂²×C₅	#486	1	1	yes	characteristic
149	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
150	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
151	C₂²×C₅	#509	1	1	yes	characteristic
152	C₂²×C₅	#510	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	—	16	1	no	—
157	C₄×C₅	#393	1	1	yes	characteristic
158	C₄×C₅	#448	1	1	yes	characteristic
159	C₄×C₅	#476	1	1	yes	characteristic
160	C₄×C₅	#489	1	1	yes	characteristic
161	Dic₅	—	8	4	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	20	4	no	—
169	C₂×Q₈	—	40	2	no	—
170	C₂×Q₈	—	40	2	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	20	2	no	—
174	C₂²×C₄	#686	1	1	yes	characteristic
175	C₂²×C₄	#686	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	Q₁₆	—	2	4	no	—
178	Q₁₆	—	4	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	#687	1	2	yes	—
182	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
183	C₂×C₈	#709	1	1	yes	characteristic
184	C₂×C₈	D₄×D₅	1	2	yes	—
185	C₂×C₈	#711	1	2	yes	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	10	4	no	—
189	C₄⋊C₄	—	20	4	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₁₆	—	20	2	no	—
194	C₂×C₅	#2023	1	1	yes	characteristic
195	C₂×C₅	#2187	1	1	yes	characteristic
196	C₂×C₅	#2188	1	1	yes	characteristic
197	C₂×C₅	#2225	1	1	yes	characteristic
198	C₂×C₅	#2227	1	1	yes	characteristic
199	C₂×C₅	#2241	1	1	yes	characteristic
200	C₂×C₅	#2243	1	1	yes	characteristic
201	C₂³	C₂×#686	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	8	1	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	40	8	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	20	4	no	—
218	C₂×C₄	—	40	2	no	—
219	C₂×C₄	—	40	2	no	—
220	C₂×C₄	#3960	1	2	yes	—
221	C₂×C₄	#3961	1	2	yes	—
222	C₂×C₄	#4070	1	1	yes	characteristic
223	C₂×C₄	C₂×#687	1	1	yes	characteristic
224	C₂×C₄	#4082	1	1	yes	characteristic
225	C₂×C₄	#4083	1	1	yes	characteristic
226	C₂×C₄	#4095	1	1	yes	characteristic
227	C₂×C₄	#4097	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	#4024	1	2	yes	—
230	C₈	#4031	1	2	yes	—
231	C₅	#13364	1	1	yes	characteristic, complemented, Sylow
232	C₂²	#65296	1	1	yes	characteristic
233	C₂²	#65325	1	1	yes	characteristic
234	C₂²	#65359	1	1	yes	characteristic
235	C₂²	C₂×#3960	1	1	yes	characteristic
236	C₂²	C₂×#3961	1	1	yes	characteristic
237	C₂²	#65482	1	1	yes	characteristic
238	C₂²	#65484	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	40	4	no	—
244	C₄	#65236	1	1	yes	characteristic
245	C₄	#65297	1	1	yes	characteristic
246	C₄	#65329	1	1	yes	characteristic
247	C₄	#65344	1	1	yes	characteristic
248	C₂	#10600431	1	1	yes	characteristic
249	C₂	#10601703	1	1	yes	characteristic
250	C₂	#10601704	1	1	yes	characteristic
251	C₂	#10601878	1	1	yes	characteristic
252	C₂	#10601880	1	1	yes	characteristic
253	C₂	#10602103	1	1	yes	characteristic
254	C₂	#10602112	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator, cosocle	C₂²×C₅×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1969	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
hypercenter	#1969	D₅
hypocenter	C₅	#13364
LCS	G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
23. C₂×C₅×Q₁₆	Q₈	semidirect
126. C₂×Q₁₆	Dic₁₀	semidirect
231. C₅	#13364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64971	2
C₅×#1969	2
C₂×#64784	2
#10598213	2
#10598148	2
#13364	5