Subgroups — #10614862

Order 640.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#529	C₂	1	1	yes	maximal, characteristic
2	#65313	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#65301	C₂	1	1	yes	maximal, characteristic, complemented
5	#65301	C₂	1	2	yes	maximal, complemented
6	#65300	C₂	1	2	yes	maximal, complemented
7	D₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#65289	C₂	1	2	yes	maximal, complemented
9	#65288	C₂	1	2	yes	maximal, complemented
10	#65214	C₂	1	1	yes	maximal, characteristic, complemented
11	#65213	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#126	C₄	1	2	yes	—
13	C₅×#125	C₄	1	1	yes	characteristic
14	C₅×#124	C₄	1	1	yes	characteristic
15	C₅×#123	C₄	1	2	yes	—
16	C₅×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₅×#121	C₄	1	1	yes	characteristic
18	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
19	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
20	C₄×C₅×D₄	C₂²	1	2	yes	—
21	C₅×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₂×#681	C₂²	1	2	yes	—
24	C₂×#681	C₂²	1	2	yes	—
25	C₂²×#168	C₂²	1	2	yes	—
26	C₂²⋊#168	C₂²	1	2	yes	—
27	C₂²⋊#168	C₂²	1	2	yes	—
28	C₂²⋊#168	C₂²	1	1	yes	characteristic
29	C₂²⋊#168	C₂²	1	1	yes	characteristic
30	C₄⋊#168	C₂²	1	1	yes	characteristic
31	C₄⋊#168	C₂²	1	2	yes	—
32	C₄⋊#168	C₂²	1	2	yes	—
33	C₄⋊#168	C₂²	1	2	yes	—
34	C₄⋊#168	C₂²	1	1	yes	characteristic
35	C₄⋊#168	C₂²	1	1	yes	characteristic
36	C₄⋊#168	C₂²	1	1	yes	characteristic
37	#3935	C₂²	1	2	yes	—
38	C₄×#168	C₂²	1	2	yes	—
39	C₄×#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2993	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂³	1	2	yes	—
49	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₄²×C₅	C₂³	1	2	yes	—
65	#681	—	4	2	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	4	2	no	—
68	C₂×#168	C₂³	1	1	yes	characteristic
69	C₂×#168	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	1	yes	characteristic
71	C₂×#168	C₂³	1	2	yes	—
72	C₂×#168	C₂³	1	1	yes	characteristic
73	C₂×#168	C₂³	1	2	yes	—
74	#529	D₅	1	1	yes	characteristic
75	#446	—	5	1	no	—
76	C₈×Q₈	—	5	1	no	—
77	#436	—	5	1	no	—
78	#436	—	5	2	no	—
79	#435	—	5	2	no	—
80	C₈×D₄	—	5	1	no	—
81	#433	—	5	2	no	—
82	#431	—	5	1	no	—
83	#424	—	5	2	no	—
84	#423	—	5	1	no	—
85	C₂³×C₅	C₂²×C₄	1	2	yes	—
86	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₅×Q₈	—	2	2	no	—
88	C₅×D₄	—	2	2	no	—
89	C₅×D₄	—	4	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	4	1	no	—
96	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
105	#168	—	2	1	no	—
106	#168	—	2	2	no	—
107	#168	—	2	2	no	—
108	#168	—	2	1	no	—
109	#168	—	2	1	no	—
110	#168	—	2	1	no	—
111	C₂×M₁₆	—	5	2	no	—
112	C₂×M₁₆	—	5	2	no	—
113	C₂²×C₈	—	5	2	no	—
114	#126	Dic₅	1	2	yes	—
115	#125	Dic₅	1	1	yes	characteristic
116	#124	Dic₅	1	1	yes	characteristic
117	#123	Dic₅	1	2	yes	—
118	C₂²⋊Q₈	Dic₅	1	1	yes	characteristic
119	#121	Dic₅	1	1	yes	characteristic
120	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
121	C₄×D₄	C₂×D₅	1	1	yes	characteristic
122	C₄×D₄	C₂×D₅	1	2	yes	—
123	#117	C₂×D₅	1	2	yes	—
124	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	—	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	C₄⋊C₈	—	5	1	no	—
132	C₂²⋊C₈	—	5	1	no	—
133	C₂²⋊C₈	—	5	1	no	—
134	C₂²⋊C₈	—	5	2	no	—
135	C₂²⋊C₈	—	5	2	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	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	#131	1	2	yes	—
155	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
156	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×Q₈	C₂×Dic₅	1	1	yes	characteristic
158	C₂×D₄	C₂×Dic₅	1	1	yes	characteristic
159	C₂×D₄	C₂×Dic₅	1	2	yes	—
160	C₂²×C₄	C₂×Dic₅	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₅	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₅	1	2	yes	—
164	M₁₆	—	20	2	no	—
165	M₁₆	—	20	2	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	5	1	no	—
168	C₂×C₈	—	5	1	no	—
169	C₂×C₈	—	5	1	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	5	2	no	—
172	C₂×C₈	—	20	2	no	—
173	C₄⋊C₄	C₂×Dic₅	1	2	yes	—
174	C₄⋊C₄	C₂×Dic₅	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
176	C₄⋊C₄	C₂×Dic₅	1	2	yes	—
177	C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
178	C₄⋊C₄	C₂×Dic₅	1	2	yes	—
179	C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
180	C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
181	C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
182	C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
183	C₂²⋊C₄	C₂×Dic₅	1	2	yes	—
184	C₄²	C₂×Dic₅	1	1	yes	characteristic
185	C₄²	C₂×Dic₅	1	1	yes	characteristic
186	C₄²	C₂²×D₅	1	1	yes	characteristic
187	C₄²	C₂²×D₅	1	2	yes	—
188	C₄²	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₅	—	4	1	no	—
190	C₂×C₅	—	4	2	no	—
191	C₂×C₅	#520	1	1	yes	characteristic
192	C₂×C₅	C₂×#131	1	1	yes	characteristic
193	C₂×C₅	#568	1	1	yes	characteristic
194	C₂³	C₂²×Dic₅	1	2	yes	—
195	C₂³	C₂²×Dic₅	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
206	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₅	1	2	yes	—
210	C₂×C₄	C₂²×Dic₅	1	2	yes	—
211	C₂×C₄	C₂²×Dic₅	1	2	yes	—
212	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
214	C₈	—	10	1	no	—
215	C₈	—	10	2	no	—
216	C₈	—	10	1	no	—
217	C₈	—	10	2	no	—
218	C₈	—	10	1	no	—
219	C₈	—	10	1	no	—
220	C₅	#2993	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₂³×Dic₅	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	#4094	1	2	yes	—
236	C₄	#4149	1	1	yes	characteristic
237	C₄	#4150	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	4	2	no	—
240	C₂	C₂×#4094	1	1	yes	characteristic
241	C₂	#66153	1	1	yes	characteristic
242	C₂	#66161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#529	C₂
prime radical	C₅×#121	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#529	D₅
hypocenter	C₅	#2993
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
2. #65313	C₂	semidirect
3. Q₈×#168	C₂	semidirect
4. #65301	C₂	semidirect
5. #65301	C₂	semidirect
6. #65300	C₂	semidirect
7. D₄×#168	C₂	semidirect
8. #65289	C₂	semidirect
9. #65288	C₂	semidirect
10. #65214	C₂	semidirect
11. #65213	C₂	semidirect
220. C₅	#2993	semidirect

Maximal subgroups

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

subgroup	index
C₅×#529	2
#65313	2
Q₈×#168	2
#65301	2
#65300	2
D₄×#168	2
#65289	2
#65288	2
#65214	2
#65213	2
#2993	5