Subgroups — #10611676

Order 640.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#713	C₂	1	1	yes	maximal, characteristic
2	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#65313	C₂	1	1	yes	maximal, characteristic, complemented
4	#65126	C₂	1	1	yes	maximal, characteristic, complemented
5	#65125	C₂	1	2	yes	maximal, complemented
6	#65123	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	#64995	C₂	1	2	yes	maximal, complemented
9	#64994	C₂	1	2	yes	maximal, complemented
10	C₈×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#64974	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#713	C₄	1	1	yes	characteristic
13	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅	C₄	1	1	yes	characteristic
15	C₂×#676	C₂²	1	2	yes	—
16	C₂×C₈×D₅	C₂²	1	2	yes	—
17	C₂×C₈×D₅	C₂²	1	2	yes	—
18	#4039	C₄	1	2	yes	—
19	#4039	C₄	1	1	yes	characteristic
20	#4038	C₄	1	1	yes	characteristic
21	#4038	C₄	1	2	yes	—
22	C₄×D₂₀	C₂²	1	1	yes	characteristic
23	C₄×D₂₀	C₂²	1	2	yes	—
24	C₄²×D₅	C₂²	1	2	yes	—
25	C₄²×D₅	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊C₈	C₂²	1	2	yes	—
27	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₅×#97	C₂²	1	2	yes	—
29	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	#3952	C₂²	1	2	yes	—
31	#3952	C₂²	1	1	yes	characteristic
32	#3952	C₂²	1	1	yes	characteristic
33	#3952	C₂²	1	2	yes	—
34	#3946	C₂²	1	1	yes	characteristic
35	#3946	C₂²	1	2	yes	—
36	#3946	C₂²	1	1	yes	characteristic
37	C₄⋊#168	C₂²	1	2	yes	—
38	C₄⋊#168	C₂²	1	1	yes	characteristic
39	#3935	C₂²	1	2	yes	—
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2990	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	#713	—	2	4	no	—
44	C₂×D₂₀	C₂×C₄	1	2	yes	—
45	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₅	C₂³	1	2	yes	—
51	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂³	1	2	yes	—
55	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₄²×C₅	C₂³	1	2	yes	—
59	#685	C₂×C₄	1	2	yes	—
60	#685	C₂×C₄	1	2	yes	—
61	#685	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₅	C₂×C₄	1	2	yes	—
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×C₄	1	2	yes	—
67	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₅	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	#676	—	4	2	no	—
73	C₈×D₅	—	4	2	no	—
74	C₈×D₅	—	4	2	no	—
75	C₄×#41	—	5	1	no	—
76	#446	—	5	1	no	—
77	#446	D₅	1	1	yes	characteristic, complemented
78	#435	—	5	2	no	—
79	#435	—	5	2	no	—
80	C₈×D₄	—	5	1	no	—
81	C₈×D₄	—	5	1	no	—
82	#432	—	5	2	no	—
83	#431	—	5	1	no	—
84	#423	—	5	1	no	—
85	C₂×C₄⋊C₈	—	5	2	no	—
86	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₅	C₂²×C₄	1	2	yes	—
88	C₅×Q₈	—	2	2	no	—
89	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
93	C₂×C₄×C₅	C₂⁴	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	1	yes	characteristic
97	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
98	D₂₀	—	2	4	no	—
99	D₂₀	M₁₆	1	4	yes	complemented
100	C₄×D₅	—	2	4	no	—
101	C₄×D₅	—	2	2	no	—
102	C₄×D₅	—	2	2	no	—
103	C₄×D₅	—	2	1	no	—
104	C₄×D₅	—	2	2	no	—
105	C₄×D₅	—	2	1	no	—
106	C₄×D₅	—	2	4	no	—
107	C₄×D₅	—	2	2	no	—
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	2	1	no	—
111	#168	—	2	1	no	—
112	#168	—	2	2	no	—
113	#168	—	2	1	no	—
114	C₂×#41	—	5	1	no	—
115	C₂×M₁₆	—	5	2	no	—
116	C₂²×C₈	—	5	2	no	—
117	C₂²×C₈	—	5	2	no	—
118	C₄×Q₈	—	5	1	no	—
119	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
120	C₄×D₄	—	5	1	no	—
121	C₄×D₄	—	5	2	no	—
122	C₄×D₄	—	5	1	no	—
123	C₄×D₄	—	5	2	no	—
124	#117	—	5	1	no	—
125	#117	—	5	2	no	—
126	C₂×C₄²	—	5	2	no	—
127	C₂×C₄²	—	5	1	no	—
128	C₄⋊C₈	—	5	1	no	—
129	C₄⋊C₈	—	5	2	no	—
130	C₄⋊C₈	—	5	1	no	—
131	C₄⋊C₈	—	5	2	no	—
132	C₄⋊C₈	—	5	1	no	—
133	C₄⋊C₈	C₂×D₅	1	2	yes	—
134	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
135	C₂²⋊C₈	—	5	1	no	—
136	C₂²⋊C₈	—	5	1	no	—
137	C₂²⋊C₈	—	5	2	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	#97	—	5	2	no	—
140	#97	C₂×D₅	1	2	yes	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×D₅	—	2	2	no	—
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	—	2	1	no	—
147	C₂×D₅	—	2	1	no	—
148	C₂×D₅	—	2	1	no	—
149	C₂×D₅	C₂×M₁₆	1	4	yes	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	C₂×M₁₆	1	2	yes	—
154	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
155	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₅	—	2	2	no	—
157	Dic₅	—	2	1	no	—
158	Dic₅	—	2	1	no	—
159	Dic₅	#131	1	2	yes	—
160	#41	—	10	4	no	—
161	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
162	C₂×D₄	—	5	1	no	—
163	C₂×D₄	—	5	2	no	—
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	1	no	—
166	C₂²×C₄	—	5	2	no	—
167	C₂²×C₄	—	5	1	no	—
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	—	5	1	no	—
170	M₁₆	—	20	2	no	—
171	C₂×C₈	—	5	1	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	2	no	—
174	C₂×C₈	—	20	2	no	—
175	C₂×C₈	—	20	2	no	—
176	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₅	1	2	yes	—
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	1	no	—
181	C₄⋊C₄	C₄×D₅	1	2	yes	—
182	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
183	C₂²⋊C₄	—	5	2	no	—
184	C₂²⋊C₄	—	5	2	no	—
185	C₂²⋊C₄	—	5	2	no	—
186	C₄²	—	5	2	no	—
187	C₄²	—	5	1	no	—
188	C₄²	—	5	2	no	—
189	C₄²	—	5	1	no	—
190	C₄²	—	5	1	no	—
191	C₄²	C₂²×D₅	1	1	yes	characteristic
192	C₄²	C₂²×D₅	1	2	yes	—
193	C₂×C₅	#520	1	1	yes	characteristic
194	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
195	C₂×C₅	C₂×#131	1	1	yes	characteristic
196	D₅	—	2	4	no	—
197	D₅	—	4	1	no	—
198	C₂³	—	5	2	no	—
199	C₂³	—	5	1	no	—
200	Q₈	—	2	2	no	—
201	D₄	—	5	4	no	—
202	D₄	—	10	4	no	—
203	C₂×C₄	—	5	1	no	—
204	C₂×C₄	—	5	1	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	5	1	no	—
207	C₂×C₄	—	10	4	no	—
208	C₂×C₄	—	10	2	no	—
209	C₂×C₄	—	10	4	no	—
210	C₂×C₄	—	10	2	no	—
211	C₂×C₄	—	10	1	no	—
212	C₂×C₄	—	10	2	no	—
213	C₂×C₄	—	10	1	no	—
214	C₂×C₄	—	10	2	no	—
215	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
217	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	10	1	no	—
224	C₈	—	10	1	no	—
225	C₈	—	10	2	no	—
226	C₅	#2990	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	5	4	no	—
228	C₂²	—	10	2	no	—
229	C₂²	—	10	2	no	—
230	C₂²	—	10	1	no	—
231	C₂²	—	10	1	no	—
232	C₂²	—	10	1	no	—
233	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	5	2	no	—
238	C₄	—	10	1	no	—
239	C₄	—	10	2	no	—
240	C₄	—	10	1	no	—
241	C₄	D₅×M₁₆	1	2	yes	—
242	C₄	#4147	1	1	yes	characteristic
243	C₄	#4149	1	1	yes	characteristic
244	C₂	—	10	4	no	—
245	C₂	—	20	1	no	—
246	C₂	#65908	1	1	yes	characteristic
247	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
248	C₂	D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#446	C₂
prime radical	C₂×#713	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#446	D₅
hypocenter	C₅	#2990
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₅×#446	C₂	semidirect
3. #65313	C₂	semidirect
4. #65126	C₂	semidirect
5. #65125	C₂	semidirect
6. #65123	C₂	semidirect
7. D₅×C₄⋊C₈	C₂	semidirect
8. #64995	C₂	semidirect
9. #64994	C₂	semidirect
10. C₈×D₂₀	C₂	semidirect
11. #64974	C₂	semidirect
77. #446	D₅	semidirect
99. D₂₀	M₁₆	semidirect
226. C₅	#2990	semidirect

Maximal subgroups

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

subgroup	index
C₄×#713	2
C₅×#446	2
#65313	2
#65126	2
#65125	2
#65123	2
D₅×C₄⋊C₈	2
#64995	2
#64994	2
C₈×D₂₀	2
#64974	2
#2990	5