Subgroups — #10611137

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×#711	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#65400	C₂	1	2	yes	maximal, complemented
4	C₂×#3946	C₂	1	2	yes	maximal, complemented
5	D₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#65126	C₂	1	1	yes	maximal, characteristic, complemented
7	#65120	C₂	1	1	yes	maximal, characteristic, complemented
8	#65019	C₂	1	2	yes	maximal, complemented
9	#65008	C₂	1	2	yes	maximal, complemented
10	#64974	C₂	1	1	yes	maximal, characteristic, complemented
11	#64968	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#711	C₄	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂²	1	2	yes	—
14	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
15	D₄×Dic₅	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₄×Dic₅	C₂²	1	2	yes	—
18	C₂²×#168	C₂²	1	2	yes	—
19	C₂×#676	C₂²	1	2	yes	—
20	#4038	C₄	1	1	yes	characteristic
21	#4033	C₄	1	1	yes	characteristic
22	#4027	C₄	1	2	yes	—
23	#4023	C₄	1	2	yes	—
24	C₄²×D₅	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₂²⋊#168	C₂²	1	2	yes	—
30	#3952	C₂²	1	1	yes	characteristic
31	#3952	C₂²	1	1	yes	characteristic
32	#3947	C₂²	1	2	yes	—
33	#3947	C₂²	1	2	yes	—
34	#3946	C₂²	1	2	yes	—
35	#3946	C₂²	1	2	yes	—
36	#3946	C₂²	1	1	yes	characteristic
37	#3946	C₂²	1	2	yes	—
38	#3946	C₂²	1	1	yes	characteristic
39	C₄⋊#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2990	—	5	1	no	maximal, Sylow
42	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅	C₂×C₄	1	2	yes	—
46	C₂²×Dic₅	C₂×C₄	1	2	yes	—
47	#711	—	2	4	no	—
48	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂³	1	2	yes	—
55	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
60	#685	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
62	#683	C₂×C₄	1	2	yes	—
63	#683	C₂×C₄	1	2	yes	—
64	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×C₄	1	2	yes	—
67	C₄×Dic₅	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂³	1	1	yes	characteristic
69	C₄×Dic₅	C₂³	1	2	yes	—
70	C₂×#168	—	4	2	no	—
71	C₂×#168	C₂³	1	1	yes	characteristic
72	C₂×#168	C₂³	1	1	yes	characteristic
73	C₂×#168	C₂³	1	2	yes	—
74	#676	—	4	2	no	—
75	C₄×#41	—	5	1	no	—
76	#446	—	5	1	no	—
77	#446	—	5	1	no	—
78	#435	—	5	2	no	—
79	#435	—	5	2	no	—
80	C₈×D₄	—	5	1	no	—
81	C₈×D₄	D₅	1	1	yes	characteristic, complemented
82	#432	—	5	2	no	—
83	#431	—	5	1	no	—
84	#423	—	5	1	no	—
85	C₂×C₄⋊C₈	—	5	2	no	—
86	C₂³×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
88	C₅×D₄	M₁₆	1	4	yes	complemented
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₅	—	2	4	no	—
96	C₂×Dic₅	—	2	4	no	—
97	C₂×Dic₅	—	2	4	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	2	yes	—
101	C₂×Dic₅	C₂²×C₄	1	2	yes	—
102	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
103	C₄×D₅	—	2	2	no	—
104	C₄×D₅	—	2	1	no	—
105	C₄×D₅	—	2	1	no	—
106	C₄×D₅	—	2	2	no	—
107	Dic₁₀	—	2	2	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	2	1	no	—
111	#168	—	2	2	no	—
112	#168	—	2	1	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	C₂×D₅	1	2	yes	—
118	C₄×Q₈	—	5	1	no	—
119	C₄×Q₈	—	5	1	no	—
120	C₄×D₄	—	5	1	no	—
121	C₄×D₄	—	5	2	no	—
122	C₄×D₄	—	5	2	no	—
123	C₄×D₄	C₂×D₅	1	1	yes	characteristic
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	2	no	—
131	C₄⋊C₈	—	5	2	no	—
132	C₄⋊C₈	—	5	1	no	—
133	C₄⋊C₈	—	5	1	no	—
134	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
135	C₂²⋊C₈	—	5	2	no	—
136	C₂²⋊C₈	—	5	1	no	—
137	C₂²⋊C₈	—	5	1	no	—
138	C₂²⋊C₈	C₂×D₅	1	2	yes	—
139	#97	—	5	2	no	—
140	#97	—	5	2	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	C₂×M₁₆	1	4	yes	—
146	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₅	—	2	1	no	—
148	C₂×D₅	—	2	1	no	—
149	C₂×D₅	—	2	1	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	C₂×M₁₆	1	2	yes	—
153	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
154	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₅	—	2	1	no	—
156	Dic₅	—	2	2	no	—
157	Dic₅	—	2	1	no	—
158	Dic₅	—	2	2	no	—
159	Dic₅	#131	1	2	yes	—
160	#41	—	10	4	no	—
161	C₂×Q₈	—	5	1	no	—
162	C₂×D₄	—	5	2	no	—
163	C₂×D₄	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	2	no	—
166	C₂²×C₄	—	5	1	no	—
167	C₂²×C₄	—	5	1	no	—
168	C₂²×C₄	—	5	1	no	—
169	C₂²×C₄	C₂²×D₅	1	2	yes	—
170	M₁₆	—	20	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	1	no	—
174	C₂×C₈	—	5	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	2	no	—
181	C₄⋊C₄	—	5	1	no	—
182	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
183	C₂²⋊C₄	—	5	2	no	—
184	C₂²⋊C₄	—	5	2	no	—
185	C₂²⋊C₄	C₄×D₅	1	2	yes	—
186	C₄²	—	5	2	no	—
187	C₄²	—	5	1	no	—
188	C₄²	—	5	1	no	—
189	C₄²	—	5	2	no	—
190	C₄²	—	5	2	no	—
191	C₄²	—	5	1	no	—
192	C₄²	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₅	—	2	4	no	—
194	C₂×C₅	#520	1	1	yes	characteristic
195	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
196	C₂×C₅	C₂×#131	1	1	yes	characteristic
197	D₅	—	4	1	no	—
198	C₂³	—	5	1	no	—
199	C₂³	C₂×C₄×D₅	1	2	yes	—
200	Q₈	—	10	2	no	—
201	D₄	—	10	4	no	—
202	D₄	#676	1	4	yes	complemented
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	5	1	no	—
206	C₂×C₄	—	5	1	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	2	no	—
210	C₂×C₄	—	10	2	no	—
211	C₂×C₄	—	10	4	no	—
212	C₂×C₄	—	10	4	no	—
213	C₂×C₄	—	10	1	no	—
214	C₂×C₄	—	10	1	no	—
215	C₂×C₄	—	10	2	no	—
216	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
217	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
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	2	no	—
222	C₈	—	2	1	no	—
223	C₈	—	10	1	no	—
224	C₈	—	10	2	no	—
225	C₈	—	10	1	no	—
226	C₅	#2990	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	10	1	no	—
230	C₂²	—	10	1	no	—
231	C₂²	—	10	1	no	—
232	C₂²	C₂×#676	1	4	yes	—
233	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	5	2	no	—
237	C₄	—	10	1	no	—
238	C₄	—	10	2	no	—
239	C₄	—	10	2	no	—
240	C₄	—	10	1	no	—
241	C₄	C₂×#676	1	2	yes	—
242	C₄	#4144	1	1	yes	characteristic
243	C₄	#4150	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	20	1	no	—
246	C₂	#65880	1	1	yes	characteristic
247	C₂	C₂²×#676	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×C₈×D₄	C₂
prime radical	C₂×#711	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₈×D₄	D₅
hypocenter	C₅	#2990
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×D₄	C₂	semidirect
3. #65400	C₂	semidirect
4. C₂×#3946	C₂	semidirect
5. D₄×#168	C₂	semidirect
6. #65126	C₂	semidirect
7. #65120	C₂	semidirect
8. #65019	C₂	semidirect
9. #65008	C₂	semidirect
10. #64974	C₂	semidirect
11. #64968	C₂	semidirect
81. C₈×D₄	D₅	semidirect
88. C₅×D₄	M₁₆	semidirect
202. D₄	#676	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×#711	2
C₅×C₈×D₄	2
#65400	2
C₂×#3946	2
D₄×#168	2
#65126	2
#65120	2
#65019	2
#65008	2
#64974	2
#64968	2
#2990	5