Subgroups — #10611694

Order 640.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₅	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	#65124	C₂	1	2	yes	maximal
5	#65122	C₂	1	1	yes	maximal, characteristic
6	#65120	C₂	1	1	yes	maximal, characteristic, complemented
7	#65117	C₂	1	2	yes	maximal, complemented
8	#64994	C₂	1	2	yes	maximal
9	#64990	C₂	1	2	yes	maximal, complemented
10	#64974	C₂	1	1	yes	maximal, characteristic
11	#64968	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅	C₄	1	1	yes	characteristic
13	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅	C₄	1	1	yes	characteristic
15	D₅×C₄⋊C₄	C₄	1	2	yes	—
16	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
17	#4033	C₄	1	1	yes	characteristic
18	#4033	C₄	1	2	yes	—
19	C₄²×D₅	C₂²	1	1	yes	characteristic
20	C₄²×D₅	C₂²	1	2	yes	—
21	C₄×Dic₁₀	C₂²	1	2	yes	—
22	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₈	C₂²	1	2	yes	—
24	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₅×#97	C₂²	1	2	yes	—
26	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	#3952	C₂²	1	2	yes	—
28	#3952	C₂²	1	1	yes	characteristic
29	#3952	C₂²	1	1	yes	characteristic
30	#3947	C₂²	1	2	yes	—
31	#3947	C₂²	1	2	yes	—
32	#3946	C₂²	1	1	yes	characteristic
33	#3946	C₂²	1	2	yes	—
34	#3946	C₂²	1	2	yes	—
35	#3946	C₂²	1	1	yes	characteristic
36	C₈×Dic₅	C₂²	1	2	yes	—
37	C₄⋊#168	C₂²	1	1	yes	characteristic
38	C₄⋊#168	C₂²	1	2	yes	—
39	#3935	C₂²	1	2	yes	—
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2983	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₅	—	2	4	no	—
44	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₅	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	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂³	1	2	yes	—
52	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₄²×C₅	C₂³	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	C₄×Dic₅	C₂×C₄	1	2	yes	—
64	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	2	yes	—
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	C₂×#168	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	1	yes	characteristic
71	C₂×#168	C₂³	1	2	yes	—
72	C₂×C₄×Q₈	—	5	1	no	—
73	#446	—	5	1	no	—
74	#446	—	5	2	no	—
75	#446	—	5	1	no	—
76	#446	—	5	2	no	—
77	#446	—	5	1	no	—
78	#446	D₅	1	1	yes	characteristic, complemented
79	#432	—	5	2	no	—
80	#432	—	5	2	no	—
81	#431	—	5	1	no	—
82	#431	—	5	1	no	—
83	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
84	C₅×Q₈	—	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×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₅	C₂²×C₄	1	2	yes	—
93	C₂×Dic₅	C₂²×C₄	1	2	yes	—
94	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
95	C₄×D₅	—	2	4	no	—
96	C₄×D₅	—	2	1	no	—
97	C₄×D₅	—	2	1	no	—
98	C₄×D₅	—	4	1	no	—
99	C₄×D₅	—	4	2	no	—
100	C₄×D₅	M₁₆	1	4	yes	—
101	Dic₁₀	—	2	2	no	—
102	Dic₁₀	—	2	4	no	—
103	C₅×C₈	—	2	1	no	—
104	C₅×C₈	—	2	2	no	—
105	C₅×C₈	—	2	1	no	—
106	#168	—	2	1	no	—
107	#168	—	2	2	no	—
108	#168	—	2	1	no	—
109	C₂²×Q₈	—	5	1	no	—
110	C₄×Q₈	—	5	1	no	—
111	C₄×Q₈	—	5	2	no	—
112	C₄×Q₈	—	5	1	no	—
113	C₄×Q₈	—	5	2	no	—
114	C₄×Q₈	—	5	1	no	—
115	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	5	1	no	—
117	C₂×C₄⋊C₄	—	5	2	no	—
118	C₂×C₄²	—	5	1	no	—
119	C₂×C₄²	—	5	2	no	—
120	C₄⋊C₈	—	5	1	no	—
121	C₄⋊C₈	—	5	1	no	—
122	C₄⋊C₈	—	5	2	no	—
123	C₄⋊C₈	—	5	2	no	—
124	C₄⋊C₈	—	5	1	no	—
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
127	C₄⋊C₈	C₂×D₅	1	2	yes	—
128	C₂²⋊C₈	—	5	2	no	—
129	C₂²⋊C₈	—	5	1	no	—
130	C₂²⋊C₈	—	5	1	no	—
131	#97	—	5	2	no	—
132	#97	—	5	2	no	—
133	#97	—	5	2	no	—
134	#97	C₂×D₅	1	2	yes	—
135	C₄×C₈	—	5	2	no	—
136	C₄×C₈	—	5	1	no	—
137	C₄×C₈	C₂×D₅	1	1	yes	characteristic
138	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₅	—	2	1	no	—
140	C₂×D₅	—	2	1	no	—
141	C₂×D₅	C₂×M₁₆	1	2	yes	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	C₂×M₁₆	1	2	yes	—
146	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₅	—	2	1	no	—
149	Dic₅	—	2	2	no	—
150	Dic₅	—	2	1	no	—
151	Dic₅	—	2	2	no	—
152	Dic₅	C₂×M₁₆	1	2	yes	—
153	C₂×Q₈	—	5	2	no	—
154	C₂×Q₈	—	5	1	no	—
155	C₂×Q₈	—	10	4	no	—
156	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	2	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	—	5	2	no	—
162	C₂×C₈	—	5	2	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	C₂²×D₅	1	2	yes	—
166	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	5	2	no	—
169	C₄⋊C₄	—	5	2	no	—
170	C₄⋊C₄	—	5	2	no	—
171	C₄⋊C₄	—	5	2	no	—
172	C₄⋊C₄	—	5	1	no	—
173	C₄⋊C₄	C₄×D₅	1	2	yes	—
174	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
175	C₄²	—	5	1	no	—
176	C₄²	—	5	2	no	—
177	C₄²	—	5	1	no	—
178	C₄²	—	5	2	no	—
179	C₄²	—	5	1	no	—
180	C₄²	—	5	2	no	—
181	C₄²	C₂²×D₅	1	1	yes	characteristic
182	C₄²	C₂²×D₅	1	2	yes	—
183	C₂×C₅	#519	1	1	yes	characteristic
184	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
185	C₂×C₅	#568	1	1	yes	characteristic
186	D₅	—	2	2	no	—
187	C₂³	—	5	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	10	4	no	—
190	Q₈	—	10	2	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	4	no	—
193	C₂×C₄	—	5	1	no	—
194	C₂×C₄	—	5	1	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	10	4	no	—
198	C₂×C₄	—	10	1	no	—
199	C₂×C₄	—	10	1	no	—
200	C₂×C₄	—	20	1	no	—
201	C₂×C₄	—	20	2	no	—
202	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	10	2	no	—
211	C₈	—	10	1	no	—
212	C₈	—	10	1	no	—
213	C₅	#2983	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	5	2	no	—
215	C₂²	—	10	1	no	—
216	C₂²	—	10	1	no	—
217	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	5	2	no	—
222	C₄	—	10	2	no	—
223	C₄	—	10	1	no	—
224	C₄	—	10	2	no	—
225	C₄	—	10	1	no	—
226	C₄	D₅×M₁₆	1	2	yes	—
227	C₄	#4147	1	1	yes	characteristic
228	C₄	#4150	1	1	yes	characteristic
229	C₂	—	10	2	no	—
230	C₂	#65906	1	1	yes	characteristic
231	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
232	C₂	#66084	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	C₂²×D₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#446	D₅
hypocenter	C₅	#2983
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
6. #65120	C₂	semidirect
7. #65117	C₂	semidirect
9. #64990	C₂	semidirect
11. #64968	C₂	semidirect
78. #446	D₅	semidirect
213. C₅	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₅	2
C₅×#446	2
#65313	2
#65124	2
#65122	2
#65120	2
#65117	2
#64994	2
#64990	2
#64974	2
#64968	2
#2983	5