Subgroups — #12069832

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic
2	#10615059	C₂	1	2	yes	maximal
3	#10602619	C₂	1	2	yes	maximal
4	#10601698	C₂	1	4	yes	maximal, complemented
5	Q₁₆×#168	C₂	1	4	yes	maximal, complemented
6	#10600524	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	2	yes	—
9	C₅×#500	C₄	1	1	yes	characteristic
10	C₅×#494	C₄	1	1	yes	characteristic
11	C₅×#487	C₄	1	2	yes	—
12	C₅×#467	C₄	1	4	yes	—
13	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65313	—	2	2	no	—
16	#65313	C₂²	1	4	yes	—
17	Q₈×#168	—	2	2	no	—
18	Q₈×#168	C₂²	1	4	yes	—
19	#65288	—	2	2	no	—
20	#65286	—	2	2	no	—
21	#65213	C₂²	1	2	yes	—
22	C₂×C₄×#168	C₂²	1	1	yes	characteristic
23	C₂×C₄×#168	C₂²	1	1	yes	characteristic
24	Q₈⋊#168	C₂²	1	8	yes	—
25	#64678	C₂²	1	4	yes	—
26	C₈×#168	C₂²	1	2	yes	—
27	C₈×#168	C₂²	1	2	yes	—
28	#17958	—	5	1	no	maximal, Sylow
29	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₅×#125	—	2	2	no	—
34	C₅×C₂²⋊Q₈	—	2	2	no	—
35	C₅×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₄×C₅×Q₈	C₂³	1	4	yes	—
38	C₅×#117	—	2	2	no	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×#681	—	2	2	no	—
41	C₂²×#168	—	2	1	no	—
42	C₂²×#168	D₄	1	1	yes	characteristic
43	C₂²×#168	D₄	1	1	yes	characteristic
44	C₅×#107	C₂×C₄	1	4	yes	—
45	C₅×#103	C₂×C₄	1	8	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂³	1	2	yes	—
49	C₄⋊#168	—	2	2	no	—
50	C₄⋊#168	—	2	4	no	—
51	C₄⋊#168	—	2	4	no	—
52	C₄⋊#168	C₂³	1	4	yes	—
53	#3935	—	2	4	no	—
54	C₄×#168	—	2	1	no	—
55	C₄×#168	—	2	4	no	—
56	C₄×#168	D₄	1	1	yes	characteristic
57	C₄×#168	D₄	1	1	yes	characteristic
58	C₄×#168	C₂³	1	2	yes	—
59	C₄×#168	C₂³	1	1	yes	characteristic
60	C₄×#168	C₂³	1	1	yes	characteristic
61	#3170	D₅	1	1	yes	characteristic
62	#2960	—	5	2	no	—
63	#1591	—	5	4	no	—
64	C₈×Q₁₆	—	5	4	no	—
65	#1567	—	5	1	no	—
66	#1493	—	5	2	no	—
67	C₂×C₈²	—	5	1	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	8	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	4	no	—
82	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₅×C₂²⋊C₄	—	2	4	no	—
84	C₄²×C₅	—	2	4	no	—
85	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	#41	1	1	yes	characteristic
88	C₄²×C₅	C₂⁴	1	1	yes	characteristic
89	#681	—	4	4	no	—
90	C₂×#168	—	2	4	no	—
91	C₂×#168	—	2	2	no	—
92	C₂×#168	—	2	2	no	—
93	C₂×#168	—	2	4	no	—
94	C₂×#168	Q₁₆	1	4	yes	complemented
95	C₂×#168	C₂×D₄	1	1	yes	characteristic
96	C₂×#168	C₂×D₄	1	1	yes	characteristic
97	C₂×#168	C₂×D₄	1	1	yes	characteristic
98	C₂×#168	C₂×D₄	1	1	yes	characteristic
99	#533	C₂×D₅	1	2	yes	—
100	#500	Dic₅	1	1	yes	characteristic
101	#494	Dic₅	1	1	yes	characteristic
102	#487	Dic₅	1	2	yes	—
103	#467	Dic₅	1	4	yes	—
104	#446	—	5	4	no	—
105	#446	—	10	2	no	—
106	C₈×Q₈	—	5	4	no	—
107	C₈×Q₈	—	10	2	no	—
108	C₄×Q₁₆	C₂×D₅	1	4	yes	—
109	#424	—	10	2	no	—
110	#423	—	5	2	no	—
111	#405	—	10	2	no	—
112	C₂×C₄×C₈	—	5	1	no	—
113	C₂×C₄×C₈	—	5	1	no	—
114	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	#335	—	5	4	no	—
116	Q₈⋊C₈	—	5	8	no	—
117	C₈²	—	5	2	no	—
118	C₈²	—	5	2	no	—
119	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
120	C₅×Q₈	—	2	8	no	—
121	C₅×Q₈	—	4	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₅	#135	1	2	yes	—
131	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
134	C₅×C₈	—	2	2	no	—
135	C₅×C₈	#131	1	4	yes	—
136	#168	—	2	4	no	—
137	#168	—	2	1	no	—
138	#168	—	2	1	no	—
139	#168	—	4	4	no	—
140	#168	C₂×Q₁₆	1	2	yes	—
141	#168	C₂×Q₁₆	1	2	yes	—
142	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
143	C₂×M₁₆	—	10	2	no	—
144	C₂²×C₈	—	5	1	no	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	10	1	no	—
147	C₂²×C₈	C₂×Dic₅	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
149	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
150	#125	—	2	2	no	—
151	C₂²⋊Q₈	—	2	2	no	—
152	C₂²⋊Q₈	C₂×Dic₅	1	4	yes	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	C₂²×D₅	1	4	yes	—
155	#117	—	2	2	no	—
156	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
157	#107	C₂×Dic₅	1	4	yes	—
158	C₄⋊C₈	—	5	4	no	—
159	C₄⋊C₈	—	10	4	no	—
160	C₄⋊C₈	—	10	4	no	—
161	C₄⋊C₈	—	10	2	no	—
162	#103	C₂×Dic₅	1	8	yes	—
163	#97	—	10	4	no	—
164	C₄×C₈	—	5	1	no	—
165	C₄×C₈	—	5	1	no	—
166	C₄×C₈	—	5	1	no	—
167	C₄×C₈	—	5	1	no	—
168	C₄×C₈	—	5	2	no	—
169	C₄×C₈	—	10	4	no	—
170	C₄×C₈	—	10	1	no	—
171	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
172	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₅	1	2	yes	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
177	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	4	4	no	—
181	C₄×C₅	—	4	4	no	—
182	C₄×C₅	C₄×Q₁₆	1	2	yes	—
183	C₄×C₅	#443	1	2	yes	—
184	C₄×C₅	C₂×#131	1	2	yes	—
185	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
186	C₄×C₅	C₂×#135	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	C₂²×Dic₅	1	4	yes	—
189	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
190	C₂²×C₄	#711	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
192	Q₁₆	—	2	8	no	—
193	M₁₆	—	20	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	5	1	no	—
196	C₂×C₈	—	5	1	no	—
197	C₂×C₈	—	5	4	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	10	4	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	10	2	no	—
203	C₂×C₈	—	10	4	no	—
204	C₂×C₈	C₂²×Dic₅	1	2	yes	—
205	C₂×C₈	C₂²×Dic₅	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	D₄×D₅	1	1	yes	characteristic
215	C₄²	#711	1	1	yes	characteristic
216	C₄²	C₂²×Dic₅	1	1	yes	characteristic
217	C₄²	C₂³×D₅	1	1	yes	characteristic
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	#2931	1	1	yes	characteristic
220	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
221	C₂×C₅	C₂×#443	1	1	yes	characteristic
222	C₂³	D₄×Dic₅	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	Q₈	—	4	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	D₅×Q₁₆	1	2	yes	—
230	C₂×C₄	#4065	1	2	yes	—
231	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
233	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
235	C₂×C₄	C₂×#711	1	1	yes	characteristic
236	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	5	2	no	—
239	C₈	—	5	2	no	—
240	C₈	—	10	4	no	—
241	C₈	—	10	1	no	—
242	C₈	—	10	1	no	—
243	C₈	—	20	4	no	—
244	C₈	#4094	1	4	yes	—
245	C₅	#17958	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	Q₁₆×Dic₅	1	2	yes	—
249	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	Q₁₆×Dic₅	1	2	yes	—
255	C₄	#65485	1	2	yes	—
256	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
257	C₄	C₂×#4065	1	1	yes	characteristic
258	C₄	C₂×#4094	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	#10615006	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×Dic₅	1	1	yes	characteristic
262	C₂	C₂×#65485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#3170	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#3170	D₅
hypocenter	C₅	#17958
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #10601698	C₂	semidirect
5. Q₁₆×#168	C₂	semidirect
94. C₂×#168	Q₁₆	semidirect
245. C₅	#17958	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3170	2
#10615059	2
#10602619	2
#10601698	2
Q₁₆×#168	2
#10600524	2
C₂×C₈×#168	2
#17958	5