Subgroups — #10617273

Order 640.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65956	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	#65759	C₂	1	1	yes	maximal, characteristic
5	#65748	C₂	1	1	yes	maximal, characteristic
6	#65709	C₂	1	2	yes	maximal, complemented
7	#65709	C₂	1	1	yes	maximal, characteristic, complemented
8	#65707	C₂	1	2	yes	maximal, complemented
9	#65703	C₂	1	1	yes	maximal, characteristic, complemented
10	#65697	C₂	1	2	yes	maximal
11	#65697	C₂	1	2	yes	maximal
12	C₅×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	Q₈×Dic₅	C₂²	1	1	yes	characteristic
14	#4090	C₄	1	1	yes	characteristic
15	#4070	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
17	#4035	C₄	1	2	yes	—
18	#4034	C₄	1	1	yes	characteristic
19	#4033	C₂²	1	1	yes	characteristic
20	#4033	C₂²	1	2	yes	—
21	#4024	C₄	1	2	yes	—
22	#4023	C₂²	1	2	yes	—
23	C₂²⋊#170	C₂²	1	1	yes	characteristic
24	C₂²⋊#170	C₂²	1	1	yes	characteristic
25	C₂²⋊#170	C₂²	1	2	yes	—
26	#4004	C₂²	1	2	yes	—
27	#4004	C₂²	1	2	yes	—
28	#4002	C₂²	1	2	yes	—
29	#4002	C₂²	1	2	yes	—
30	#4002	C₂²	1	2	yes	—
31	C₄⋊#170	C₂²	1	2	yes	—
32	C₄⋊#170	C₂²	1	2	yes	—
33	C₄⋊#170	C₂²	1	1	yes	characteristic
34	C₄⋊#170	C₂²	1	1	yes	characteristic
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	#2985	—	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	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
48	C₂×#170	C₂³	1	1	yes	characteristic
49	C₂×#170	C₂³	1	1	yes	characteristic
50	C₂×#170	C₂³	1	2	yes	—
51	C₂×#170	C₂³	1	1	yes	characteristic
52	C₂×#170	C₂³	1	2	yes	—
53	C₂×#170	C₂³	1	1	yes	characteristic
54	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
60	#683	C₂×C₄	1	2	yes	—
61	#683	C₂×C₄	1	2	yes	—
62	#683	C₂×C₄	1	2	yes	—
63	#683	C₂×C₄	1	2	yes	—
64	#683	C₂×C₄	1	2	yes	—
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂³	1	1	yes	characteristic
68	C₄×Dic₅	C₂³	1	2	yes	—
69	C₄×Dic₅	C₂³	1	2	yes	—
70	C₄×Dic₅	C₂³	1	1	yes	characteristic
71	#533	—	5	1	no	—
72	#446	—	5	2	no	—
73	#446	—	5	1	no	—
74	#446	—	5	2	no	—
75	#446	—	5	1	no	—
76	C₈×Q₈	—	5	1	no	—
77	C₈×Q₈	—	5	1	no	—
78	#433	—	5	2	no	—
79	#433	—	5	2	no	—
80	#432	—	5	1	no	—
81	#431	—	5	1	no	—
82	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₅×Q₈	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅	—	2	1	no	—
90	C₂×Dic₅	—	2	1	no	—
91	C₂×Dic₅	—	4	1	no	—
92	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₅	C₂²×C₄	1	2	yes	—
96	C₂×Dic₅	C₂²×C₄	1	2	yes	—
97	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
98	Dic₁₀	—	2	2	no	—
99	Dic₁₀	—	4	2	no	—
100	#170	—	2	1	no	—
101	#170	—	2	1	no	—
102	#170	—	2	2	no	—
103	#170	—	2	1	no	—
104	#170	—	2	2	no	—
105	#170	—	2	1	no	—
106	C₄⋊Q₈	—	5	1	no	—
107	C₄⋊Q₈	—	5	1	no	—
108	#125	—	5	2	no	—
109	C₂²⋊Q₈	—	5	1	no	—
110	C₂²⋊Q₈	—	5	2	no	—
111	C₂²⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₄×Q₈	—	5	1	no	—
113	C₄×Q₈	—	5	1	no	—
114	C₄×Q₈	—	5	2	no	—
115	#117	—	5	2	no	—
116	C₂×C₄²	—	5	1	no	—
117	C₄⋊C₈	—	5	1	no	—
118	C₄⋊C₈	—	5	1	no	—
119	C₄⋊C₈	—	5	1	no	—
120	C₄⋊C₈	—	5	1	no	—
121	C₄⋊C₈	—	5	2	no	—
122	C₄⋊C₈	—	5	2	no	—
123	C₄⋊C₈	—	5	2	no	—
124	C₄⋊C₈	—	5	2	no	—
125	C₂²⋊C₈	—	5	1	no	—
126	C₂²⋊C₈	—	5	1	no	—
127	C₂²⋊C₈	—	5	2	no	—
128	#97	—	5	2	no	—
129	#97	—	5	2	no	—
130	#97	—	5	2	no	—
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	—	5	2	no	—
134	C₄×C₈	—	5	2	no	—
135	C₂²×C₅	—	2	1	no	—
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	#131	1	2	yes	—
143	Dic₅	—	2	1	no	—
144	Dic₅	—	2	2	no	—
145	Dic₅	—	2	1	no	—
146	Dic₅	—	2	2	no	—
147	Dic₅	—	2	1	no	—
148	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×Q₈	—	5	1	no	—
151	C₂×Q₈	—	5	2	no	—
152	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	5	1	no	—
155	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₂×C₈	—	5	1	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	2	no	—
161	C₂×C₈	—	5	2	no	—
162	C₄⋊C₄	—	5	2	no	—
163	C₄⋊C₄	—	5	2	no	—
164	C₄⋊C₄	—	5	2	no	—
165	C₄⋊C₄	—	5	2	no	—
166	C₄⋊C₄	—	5	2	no	—
167	C₄⋊C₄	—	5	2	no	—
168	C₄⋊C₄	—	5	1	no	—
169	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
170	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂²⋊C₄	—	5	2	no	—
172	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
173	C₄²	—	5	1	no	—
174	C₄²	—	5	2	no	—
175	C₄²	—	5	2	no	—
176	C₄²	—	5	1	no	—
177	C₄²	—	5	1	no	—
178	C₄²	—	5	1	no	—
179	C₂×C₅	—	4	1	no	—
180	C₂×C₅	#519	1	1	yes	characteristic
181	C₂×C₅	C₂×#131	1	1	yes	characteristic
182	C₂×C₅	#568	1	1	yes	characteristic
183	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	Q₈	—	2	2	no	—
185	Q₈	—	10	2	no	—
186	Q₈	—	20	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	—	5	1	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	2	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	10	1	no	—
195	C₂×C₄	—	10	1	no	—
196	C₂×C₄	—	20	1	no	—
197	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
199	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
200	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₈	—	10	1	no	—
202	C₈	—	10	1	no	—
203	C₈	—	10	1	no	—
204	C₈	—	10	1	no	—
205	C₈	—	10	2	no	—
206	C₈	—	10	2	no	—
207	C₅	#2985	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	5	1	no	—
216	C₄	—	5	1	no	—
217	C₄	—	10	2	no	—
218	C₄	—	10	1	no	—
219	C₄	—	10	1	no	—
220	C₄	—	10	1	no	—
221	C₄	—	10	2	no	—
222	C₄	#4131	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	C₂×#4131	1	1	yes	characteristic
225	C₂	#66262	1	1	yes	characteristic
226	C₂	#66264	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	C₂³×C₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2985
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
6. #65709	C₂	semidirect
7. #65709	C₂	semidirect
8. #65707	C₂	semidirect
9. #65703	C₂	semidirect
207. C₅	#2985	semidirect

Maximal subgroups

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

subgroup	index
#65956	2
#65788	2
Q₈×#170	2
#65759	2
#65748	2
#65709	2
#65707	2
#65703	2
#65697	2
#2985	5