Subgroups — #13023722

Order 1280.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65485	C₂	1	1	yes	maximal, characteristic
2	C₂×#64889	C₂	1	1	yes	maximal, characteristic, complemented
3	#10604687	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598815	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#135	C₄	1	1	yes	characteristic
6	C₂×#4094	—	2	1	no	—
7	#65748	—	2	1	no	—
8	C₂×#3998	—	2	1	no	—
9	C₂×#3969	—	2	1	no	—
10	#65485	—	2	1	no	—
11	#65485	—	2	1	no	—
12	#65485	C₄	1	2	yes	—
13	#65485	C₂²	1	2	yes	—
14	C₂×#3951	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	#64889	—	2	1	no	—
17	#64889	C₂²	1	2	yes	complemented
18	#64879	—	2	1	no	—
19	C₈×#170	C₂²	1	2	yes	complemented
20	#11602	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	—	2	1	no	—
22	C₅×#135	—	2	1	no	—
23	C₅×#135	—	2	1	no	—
24	C₅×#135	C₈	1	2	yes	complemented
25	C₅×#135	C₂×C₄	1	2	yes	—
26	C₂×C₅×Q₁₆	C₈	1	1	yes	characteristic, complemented
27	C₂×C₅×SD₁₆	—	2	1	no	—
28	C₂×C₅×D₈	C₈	1	1	yes	characteristic, complemented
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	#4094	—	2	2	no	—
31	#4094	—	2	2	no	—
32	#4094	—	4	1	no	—
33	#4094	—	4	1	no	—
34	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
35	C₂×#681	—	2	1	no	—
36	C₂×#681	—	4	1	no	—
37	C₂²×#168	—	4	1	no	—
38	C₂²×#168	D₄	1	1	yes	characteristic
39	C₂²⋊#170	—	2	2	no	—
40	C₄⋊#170	—	2	2	no	—
41	C₄×#170	—	2	2	no	—
42	#3998	—	2	2	no	—
43	#3998	—	4	1	no	—
44	C₂×#674	—	4	1	no	—
45	#3969	—	2	2	no	—
46	#3969	—	2	2	no	—
47	#3951	—	2	1	no	—
48	#3951	C₂×C₄	1	2	yes	—
49	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₈×Dic₅	D₄	1	1	yes	characteristic
51	C₈×Dic₅	D₄	1	1	yes	characteristic
52	C₈×Dic₅	C₂³	1	1	yes	characteristic
53	C₂×#443	—	5	1	no	—
54	C₂×#364	—	5	1	no	—
55	#1450	—	5	1	no	—
56	#1335	—	5	4	no	—
57	C₅×#41	—	2	2	no	—
58	C₅×#41	—	2	2	no	—
59	C₅×#41	—	4	1	no	—
60	C₅×#41	—	4	1	no	—
61	C₂×C₅×Q₈	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₅	—	2	1	no	—
67	C₂×#170	—	2	4	no	—
68	C₅×Q₁₆	—	2	1	no	—
69	C₅×Q₁₆	C₂×C₈	1	2	yes	—
70	C₅×SD₁₆	—	2	2	no	—
71	C₅×SD₁₆	—	2	2	no	—
72	C₅×D₈	—	2	1	no	—
73	C₅×D₈	C₂×C₈	1	2	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
77	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
78	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₄×Dic₅	—	2	1	no	—
81	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
83	#681	—	2	2	no	—
84	#681	—	4	1	no	—
85	#681	—	4	2	no	—
86	#681	—	4	2	no	—
87	C₂×#168	—	4	1	no	—
88	C₂×#168	—	4	2	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#168	D₈	1	1	yes	characteristic
92	C₂×#168	D₈	1	1	yes	characteristic
93	C₂×#168	SD₁₆	1	1	yes	characteristic
94	C₂×#168	SD₁₆	1	1	yes	characteristic
95	C₂×#168	C₂×D₄	1	1	yes	characteristic
96	#674	—	4	2	no	—
97	C₂×#135	AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×#131	—	10	1	no	—
99	C₂×M₃₂	—	10	1	no	—
100	#443	—	5	2	no	—
101	#443	—	5	2	no	—
102	#443	—	10	1	no	—
103	#443	—	10	1	no	—
104	#431	—	10	1	no	—
105	C₂×#108	—	5	1	no	—
106	C₂×#104	—	10	1	no	—
107	C₂×C₄×C₈	—	5	1	no	—
108	#364	—	5	2	no	—
109	#364	—	10	1	no	—
110	#322	—	10	1	no	—
111	C₈²	—	5	2	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	#104	1	1	yes	characteristic
114	C₅×Q₈	—	2	2	no	—
115	C₅×Q₈	—	4	1	no	—
116	C₅×D₄	—	2	2	no	—
117	C₅×D₄	—	4	2	no	—
118	C₅×D₄	—	4	2	no	—
119	C₅×D₄	—	4	1	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₄×C₅	#102	1	1	yes	characteristic
126	C₂×C₄×C₅	#102	1	1	yes	characteristic
127	C₂×C₄×C₅	#104	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Dic₅	—	2	1	no	—
130	C₂×Dic₅	—	2	1	no	—
131	C₂×Dic₅	—	4	2	no	—
132	#170	—	4	4	no	—
133	C₅×C₈	C₂²⋊C₈	1	1	yes	characteristic
134	C₅×C₈	C₂²⋊C₈	1	1	yes	characteristic
135	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
136	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
137	#168	—	4	2	no	—
138	#168	#102	1	1	yes	characteristic
139	#168	#102	1	1	yes	characteristic
140	#168	C₂×D₈	1	1	yes	characteristic
141	#168	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×#41	—	2	1	no	—
143	#135	—	2	1	no	—
144	#135	—	2	1	no	—
145	#135	#170	1	2	yes	complemented
146	#135	C₂×AGL(1,𝔽₅)	1	2	yes	—
147	C₂×Q₁₆	#170	1	1	yes	characteristic, complemented
148	C₂×SD₁₆	—	2	1	no	—
149	C₂×D₈	#170	1	1	yes	characteristic, complemented
150	#131	—	10	2	no	—
151	#131	—	10	2	no	—
152	#131	—	20	1	no	—
153	#131	—	20	1	no	—
154	C₂×M₁₆	—	10	1	no	—
155	C₂×M₁₆	—	20	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	20	1	no	—
158	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₄²	—	5	1	no	—
160	M₃₂	—	10	2	no	—
161	M₃₂	—	20	1	no	—
162	C₂×C₁₆	—	20	1	no	—
163	#108	—	5	2	no	—
164	#108	—	10	1	no	—
165	C₄⋊C₈	—	10	2	no	—
166	#104	—	10	2	no	—
167	#104	—	10	2	no	—
168	C₂²⋊C₈	—	10	2	no	—
169	C₄×C₈	—	5	1	no	—
170	C₄×C₈	—	5	1	no	—
171	C₄×C₈	—	5	1	no	—
172	C₄×C₈	—	5	1	no	—
173	C₄×C₈	—	10	2	no	—
174	C₂²×C₅	—	4	1	no	—
175	C₂²×C₅	—	4	2	no	—
176	C₂²×C₅	—	8	2	no	—
177	C₂²×C₅	D₄⋊C₈	1	1	yes	characteristic
178	C₂²×C₅	D₄⋊C₈	1	1	yes	characteristic
179	C₂²×C₅	C₂×#104	1	1	yes	characteristic
180	C₄×C₅	—	4	2	no	—
181	C₄×C₅	D₄⋊C₈	1	1	yes	characteristic
182	C₄×C₅	D₄⋊C₈	1	1	yes	characteristic
183	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₄×C₅	C₂×#102	1	1	yes	characteristic
185	Dic₅	—	2	2	no	—
186	Dic₅	—	4	1	no	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	—	4	1	no	—
190	#41	—	4	1	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	10	1	no	—
196	C₂²×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
197	Q₁₆	—	2	1	no	—
198	Q₁₆	C₂×#170	1	2	yes	—
199	SD₁₆	—	2	2	no	—
200	SD₁₆	—	2	2	no	—
201	D₈	—	2	1	no	—
202	D₈	C₂×#170	1	2	yes	—
203	M₁₆	—	10	2	no	—
204	M₁₆	—	20	2	no	—
205	M₁₆	—	20	2	no	—
206	M₁₆	—	20	1	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	—	5	1	no	—
211	C₂×C₈	—	5	1	no	—
212	C₂×C₈	—	5	1	no	—
213	C₂×C₈	—	10	4	no	—
214	C₂×C₈	—	20	2	no	—
215	C₂×C₈	—	20	2	no	—
216	C₂×C₈	—	20	1	no	—
217	C₂×C₈	C₂×#170	1	1	yes	characteristic
218	C₂×C₈	C₂×#170	1	1	yes	characteristic
219	C₂×C₈	#704	1	1	yes	characteristic
220	C₂×C₈	#704	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₄²	—	5	1	no	—
224	C₄²	—	5	1	no	—
225	C₄²	—	10	1	no	—
226	C₁₆	—	20	2	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	2	1	no	—
229	C₂×C₅	—	2	1	no	—
230	C₂×C₅	—	8	2	no	—
231	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
232	C₂³	—	4	1	no	—
233	C₂³	#4008	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	10	1	no	—
244	C₂×C₄	—	10	1	no	—
245	C₂×C₄	—	20	2	no	—
246	C₂×C₄	#4007	1	1	yes	characteristic
247	C₂×C₄	#4007	1	1	yes	characteristic
248	C₂×C₄	#4008	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊#170	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#170	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
252	C₈	—	5	1	no	—
253	C₈	—	5	1	no	—
254	C₈	—	5	1	no	—
255	C₈	—	5	1	no	—
256	C₈	—	20	4	no	—
257	C₈	—	20	2	no	—
258	C₈	C₂²⋊#170	1	1	yes	characteristic
259	C₈	C₂²⋊#170	1	1	yes	characteristic
260	C₈	C₂²×#170	1	1	yes	characteristic
261	C₈	C₂×#704	1	1	yes	characteristic
262	C₅	#11602	1	1	yes	characteristic, complemented, Sylow
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊#170	1	1	yes	characteristic
267	C₂²	D₄⋊#170	1	1	yes	characteristic
268	C₂²	C₂×#4008	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	10	2	no	—
271	C₄	—	20	1	no	—
272	C₄	D₄⋊#170	1	1	yes	characteristic
273	C₄	D₄⋊#170	1	1	yes	characteristic
274	C₄	C₂×#4007	1	1	yes	characteristic
275	C₄	C₂×C₂²⋊#170	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊#170	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4007
commutator	C₅×C₈	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting, prime radical	C₂×C₅×#135	C₄
socle	C₂×C₅	C₂×D₄⋊C₈
cosocle	C₈×Dic₅	C₂³
hypercenter	C₂×#135	AGL(1,𝔽₅)
hypocenter	C₅	#11602
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#64889	C₂	semidirect
3. #10604687	C₂	semidirect
4. #10598815	C₂	semidirect
17. #64889	C₂²	semidirect
19. C₈×#170	C₂²	semidirect
24. C₅×#135	C₈	semidirect
26. C₂×C₅×Q₁₆	C₈	semidirect
28. C₂×C₅×D₈	C₈	semidirect
145. #135	#170	semidirect
147. C₂×Q₁₆	#170	semidirect
149. C₂×D₈	#170	semidirect
262. C₅	#11602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65485	2
C₂×#64889	2
#10604687	2
#10598815	2
#11602	5