Subgroups — #10617268

Order 640.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65958	C₂	1	1	yes	maximal, characteristic
2	#65788	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#704	C₂	1	1	yes	maximal, characteristic, complemented
5	#65748	C₂	1	1	yes	maximal, characteristic, complemented
6	#65706	C₂	1	2	yes	maximal, complemented
7	#65705	C₂	1	1	yes	maximal, characteristic, complemented
8	#65702	C₂	1	1	yes	maximal, characteristic, complemented
9	#65701	C₂	1	2	yes	maximal, complemented
10	#65695	C₂	1	2	yes	maximal, complemented
11	#65694	C₂	1	2	yes	maximal, complemented
12	C₂×#704	C₂²	1	1	yes	characteristic
13	C₂×#704	C₂²	1	1	yes	characteristic
14	C₂×#700	C₂²	1	2	yes	—
15	C₂×D₅⋊C₈	C₂²	1	2	yes	—
16	C₅×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	#4093	C₄	1	1	yes	characteristic
18	Q₈×Dic₅	C₂²	1	1	yes	characteristic
19	#4076	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
21	#4044	C₄	1	2	yes	—
22	#4039	C₂²	1	1	yes	characteristic
23	#4038	C₂²	1	2	yes	—
24	#4036	C₄	1	1	yes	characteristic
25	#4032	C₄	1	2	yes	—
26	#4027	C₂²	1	2	yes	—
27	C₂²⋊#170	C₂²	1	2	yes	—
28	#4004	C₂²	1	2	yes	—
29	#4003	C₂²	1	2	yes	—
30	#4003	C₂²	1	2	yes	—
31	#4002	C₂²	1	2	yes	—
32	#4002	C₂²	1	1	yes	characteristic
33	#4002	C₂²	1	1	yes	characteristic
34	C₄⋊#170	C₂²	1	2	yes	—
35	C₄⋊#170	C₂²	1	1	yes	characteristic
36	C₄⋊#170	C₂²	1	1	yes	characteristic
37	C₄×#170	C₂²	1	2	yes	—
38	C₄×#170	C₂²	1	2	yes	—
39	C₄×#170	C₂²	1	1	yes	characteristic
40	C₄×#170	C₂²	1	1	yes	characteristic
41	#2969	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂³	1	2	yes	—
49	#704	—	2	2	no	—
50	#704	—	2	2	no	—
51	C₂×#170	C₂³	1	2	yes	—
52	C₂×#170	C₂³	1	2	yes	—
53	C₂×#170	C₂³	1	1	yes	characteristic
54	C₂×#170	C₂³	1	1	yes	characteristic
55	C₂×#170	C₂³	1	1	yes	characteristic
56	C₂×#170	C₂³	1	1	yes	characteristic
57	#700	—	2	4	no	—
58	D₅⋊C₈	—	2	4	no	—
59	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	#685	C₂×C₄	1	2	yes	—
63	#685	C₂×C₄	1	2	yes	—
64	#685	C₂×C₄	1	2	yes	—
65	#685	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
68	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
69	#683	C₂×C₄	1	2	yes	—
70	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₅	C₂³	1	1	yes	characteristic
73	C₄×Dic₅	C₂³	1	1	yes	characteristic
74	C₄×Dic₅	C₂³	1	2	yes	—
75	#529	—	5	1	no	—
76	#446	—	5	1	no	—
77	C₈×Q₈	—	5	1	no	—
78	#436	—	5	2	no	—
79	#436	—	5	1	no	—
80	#435	—	5	2	no	—
81	C₈×D₄	—	5	1	no	—
82	#433	—	5	2	no	—
83	#431	—	5	1	no	—
84	#405	—	5	2	no	—
85	C₄×M₁₆	—	5	1	no	—
86	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₅	C₂²×C₄	1	2	yes	—
88	C₅×Q₈	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂²⋊D₅	—	4	2	no	—
95	C₂×Dic₅	—	2	1	no	—
96	C₂×Dic₅	—	2	1	no	—
97	C₂×Dic₅	—	4	1	no	—
98	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₅	C₂²×C₄	1	2	yes	—
102	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
103	D₂₀	—	2	2	no	—
104	C₄×D₅	—	2	2	no	—
105	C₄×D₅	—	2	2	no	—
106	#170	—	2	2	no	—
107	#170	—	2	2	no	—
108	#170	—	2	1	no	—
109	#170	—	2	1	no	—
110	#170	#41	1	2	yes	—
111	#170	#41	1	2	yes	—
112	C₂×M₁₆	—	5	2	no	—
113	C₂×M₁₆	—	5	1	no	—
114	C₂×M₁₆	—	5	1	no	—
115	C₂²×C₈	—	5	2	no	—
116	#126	—	5	2	no	—
117	#125	—	5	1	no	—
118	#124	—	5	1	no	—
119	#123	—	5	2	no	—
120	C₂²⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
121	#121	—	5	1	no	—
122	C₄×Q₈	—	5	1	no	—
123	C₄×D₄	—	5	1	no	—
124	C₄×D₄	—	5	2	no	—
125	#117	—	5	2	no	—
126	C₂×C₄²	—	5	1	no	—
127	C₄⋊C₈	—	5	1	no	—
128	C₄⋊C₈	—	5	1	no	—
129	C₄⋊C₈	—	5	2	no	—
130	C₄⋊C₈	—	5	2	no	—
131	C₂²⋊C₈	—	5	2	no	—
132	C₂²⋊C₈	—	5	2	no	—
133	C₂²⋊C₈	—	5	2	no	—
134	#97	—	5	1	no	—
135	#97	—	5	1	no	—
136	#97	—	5	2	no	—
137	C₄×C₈	—	5	1	no	—
138	C₄×C₈	—	5	1	no	—
139	C₄×C₈	—	5	2	no	—
140	C₄×C₈	—	5	2	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	—	2	2	no	—
147	C₂×D₅	—	2	2	no	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	#131	1	2	yes	—
152	Dic₅	—	2	1	no	—
153	Dic₅	—	2	1	no	—
154	Dic₅	—	2	1	no	—
155	Dic₅	—	2	2	no	—
156	Dic₅	C₂×#41	1	1	yes	characteristic
157	Dic₅	C₂×#41	1	1	yes	characteristic
158	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×D₄	—	5	2	no	—
160	C₂×D₄	—	5	1	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	5	2	no	—
164	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
165	M₁₆	—	10	4	no	—
166	M₁₆	—	10	2	no	—
167	M₁₆	—	10	2	no	—
168	C₂×C₈	—	5	1	no	—
169	C₂×C₈	—	5	1	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	5	1	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	2	no	—
174	C₂×C₈	—	10	4	no	—
175	C₄⋊C₄	—	5	2	no	—
176	C₄⋊C₄	—	5	2	no	—
177	C₄⋊C₄	—	5	1	no	—
178	C₄⋊C₄	—	5	2	no	—
179	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
180	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₂²⋊C₄	—	5	2	no	—
182	C₂²⋊C₄	—	5	2	no	—
183	C₂²⋊C₄	—	5	2	no	—
184	C₂²⋊C₄	—	5	2	no	—
185	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
186	C₄²	—	5	1	no	—
187	C₄²	—	5	1	no	—
188	C₄²	—	5	2	no	—
189	C₄²	—	5	1	no	—
190	C₄²	—	5	1	no	—
191	C₂×C₅	—	4	1	no	—
192	C₂×C₅	C₄×#41	1	1	yes	characteristic
193	C₂×C₅	C₂×#131	1	1	yes	characteristic
194	C₂×C₅	#568	1	1	yes	characteristic
195	D₅	—	4	2	no	—
196	C₂³	—	5	2	no	—
197	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	D₄	—	10	2	no	—
200	D₄	—	20	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	5	1	no	—
203	C₂×C₄	—	5	1	no	—
204	C₂×C₄	—	5	1	no	—
205	C₂×C₄	—	5	1	no	—
206	C₂×C₄	—	5	2	no	—
207	C₂×C₄	—	10	2	no	—
208	C₂×C₄	—	10	2	no	—
209	C₂×C₄	—	10	1	no	—
210	C₂×C₄	—	10	1	no	—
211	C₂×C₄	—	20	1	no	—
212	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
214	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
215	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₈	—	5	2	no	—
217	C₈	—	5	2	no	—
218	C₈	—	10	2	no	—
219	C₈	—	10	2	no	—
220	C₈	—	10	1	no	—
221	C₈	—	10	1	no	—
222	C₅	#2969	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	10	2	no	—
227	C₂²	—	10	2	no	—
228	C₂²	—	10	2	no	—
229	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	5	1	no	—
234	C₄	—	5	1	no	—
235	C₄	—	10	1	no	—
236	C₄	—	10	1	no	—
237	C₄	—	10	2	no	—
238	C₄	—	10	1	no	—
239	C₄	#4131	1	2	yes	—
240	C₂	—	4	1	no	—
241	C₂	—	20	2	no	—
242	C₂	C₂×#4131	1	1	yes	characteristic
243	C₂	#66260	1	1	yes	characteristic
244	C₂	#41×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₂²⋊Q₈	C₄
prime radical	#4076	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2969
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. Q₈×#170	C₂	semidirect
4. C₄×#704	C₂	semidirect
5. #65748	C₂	semidirect
6. #65706	C₂	semidirect
7. #65705	C₂	semidirect
8. #65702	C₂	semidirect
9. #65701	C₂	semidirect
10. #65695	C₂	semidirect
11. #65694	C₂	semidirect
222. C₅	#2969	semidirect

Maximal subgroups

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

subgroup	index
#65958	2
#65788	2
Q₈×#170	2
C₄×#704	2
#65748	2
#65706	2
#65705	2
#65702	2
#65701	2
#65695	2
#65694	2
#2969	5