Subgroups — #10611691

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	#65912	C₂	1	1	yes	maximal, characteristic
2	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#65128	C₂	1	2	yes	maximal
5	D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#65120	C₂	1	1	yes	maximal, characteristic, complemented
7	#65120	C₂	1	2	yes	maximal, complemented
8	#64997	C₂	1	2	yes	maximal
9	#64990	C₂	1	2	yes	maximal, complemented
10	C₄×#676	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	#4092	C₄	1	1	yes	characteristic
14	#4090	C₄	1	1	yes	characteristic
15	C₂×#676	C₂²	1	1	yes	characteristic
16	C₂×#676	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅	C₂²	1	2	yes	—
18	#4043	C₄	1	1	yes	characteristic
19	#4042	C₄	1	2	yes	—
20	#4035	C₄	1	2	yes	—
21	#4034	C₄	1	1	yes	characteristic
22	#4018	C₂²	1	2	yes	—
23	C₄²×D₅	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀	C₂²	1	2	yes	—
26	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊C₈	C₂²	1	2	yes	—
28	C₅×#97	C₂²	1	2	yes	—
29	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	#3947	C₂²	1	1	yes	characteristic
31	#3947	C₂²	1	2	yes	—
32	#3947	C₂²	1	1	yes	characteristic
33	#3946	C₂²	1	2	yes	—
34	#3946	C₂²	1	2	yes	—
35	#3946	C₂²	1	2	yes	—
36	C₈×Dic₅	C₂²	1	2	yes	—
37	C₄⋊#168	C₂²	1	1	yes	characteristic
38	C₄⋊#168	C₂²	1	2	yes	—
39	C₄×#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	2	yes	—
41	#2962	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂³	1	2	yes	—
50	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₄²×C₅	C₂³	1	2	yes	—
55	#685	C₂×C₄	1	2	yes	—
56	#685	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
59	#683	C₂×C₄	1	2	yes	—
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	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂³	1	2	yes	—
67	C₄×Dic₅	C₂³	1	1	yes	characteristic
68	C₂×#168	C₂³	1	2	yes	—
69	C₂×#168	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	1	yes	characteristic
71	#676	—	2	2	no	—
72	#676	Q₈	1	4	yes	complemented
73	C₈×D₅	—	2	4	no	—
74	#533	—	5	1	no	—
75	#446	—	5	1	no	—
76	#446	—	5	2	no	—
77	#446	—	5	2	no	—
78	#446	D₅	1	1	yes	characteristic, complemented
79	C₈×Q₈	—	5	1	no	—
80	C₈×Q₈	—	5	1	no	—
81	#424	—	5	2	no	—
82	C₂×C₄⋊C₈	—	5	1	no	—
83	#405	—	5	2	no	—
84	C₄×M₁₆	—	5	1	no	—
85	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
86	C₅×Q₈	—	2	2	no	—
87	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂²×C₄	1	2	yes	—
95	C₂×Dic₅	C₂²×C₄	1	2	yes	—
96	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
97	C₄×D₅	—	2	2	no	—
98	C₄×D₅	—	4	1	no	—
99	C₄×D₅	C₂×Q₈	1	2	yes	—
100	C₄×D₅	#41	1	2	yes	—
101	Dic₁₀	—	2	2	no	—
102	Dic₁₀	—	4	2	no	—
103	C₅×C₈	—	2	2	no	—
104	C₅×C₈	—	2	1	no	—
105	C₅×C₈	C₂×Q₈	1	2	yes	—
106	#168	—	2	2	no	—
107	#168	—	2	1	no	—
108	#168	C₂×Q₈	1	2	yes	—
109	C₂×M₁₆	—	5	1	no	—
110	C₂×M₁₆	—	5	1	no	—
111	C₂²×C₈	—	5	2	no	—
112	C₄⋊Q₈	—	5	1	no	—
113	C₄⋊Q₈	—	5	1	no	—
114	#125	—	5	2	no	—
115	C₂²⋊Q₈	—	5	1	no	—
116	C₂²⋊Q₈	—	5	2	no	—
117	C₂²⋊Q₈	—	5	1	no	—
118	C₄×Q₈	—	5	1	no	—
119	C₄×Q₈	—	5	2	no	—
120	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
121	#117	—	5	2	no	—
122	C₂×C₄²	—	5	1	no	—
123	C₄⋊C₈	—	5	2	no	—
124	C₄⋊C₈	—	5	2	no	—
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	—	5	1	no	—
127	C₄⋊C₈	—	5	2	no	—
128	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₅	1	2	yes	—
130	#97	—	5	1	no	—
131	#97	—	5	1	no	—
132	#97	—	5	2	no	—
133	#97	C₂×D₅	1	2	yes	—
134	C₄×C₈	—	5	2	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	—	5	2	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	C₄×Q₈	1	2	yes	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	#131	1	2	yes	—
146	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₅	C₂×#41	1	1	yes	characteristic
148	Dic₅	—	2	2	no	—
149	Dic₅	—	2	1	no	—
150	Dic₅	—	2	1	no	—
151	Dic₅	—	2	2	no	—
152	Dic₅	C₄×Q₈	1	2	yes	—
153	C₂×Q₈	—	5	1	no	—
154	C₂×Q₈	—	5	2	no	—
155	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
156	C₂²×C₄	—	5	1	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	1	no	—
159	M₁₆	—	5	4	no	—
160	M₁₆	—	10	2	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	2	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	10	4	no	—
165	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₅	1	2	yes	—
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	2	no	—
173	C₄⋊C₄	—	5	2	no	—
174	C₄⋊C₄	—	5	1	no	—
175	C₄⋊C₄	C₄×D₅	1	2	yes	—
176	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
177	C₂²⋊C₄	—	5	2	no	—
178	C₂²⋊C₄	—	5	2	no	—
179	C₄²	—	5	1	no	—
180	C₄²	—	5	1	no	—
181	C₄²	—	5	2	no	—
182	C₄²	—	5	1	no	—
183	C₄²	C₂²×D₅	1	1	yes	characteristic
184	C₄²	C₂²×D₅	1	2	yes	—
185	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂×C₅	C₂×#131	1	1	yes	characteristic
187	C₂×C₅	#568	1	1	yes	characteristic
188	D₅	—	2	2	no	—
189	C₂³	—	5	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	10	2	no	—
192	Q₈	—	20	2	no	—
193	C₂×C₄	—	5	1	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	1	no	—
197	C₂×C₄	—	5	1	no	—
198	C₂×C₄	—	5	2	no	—
199	C₂×C₄	—	5	2	no	—
200	C₂×C₄	—	10	2	no	—
201	C₂×C₄	—	20	1	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	2	no	—
208	C₈	—	2	1	no	—
209	C₈	—	5	2	no	—
210	C₈	—	10	2	no	—
211	C₈	—	10	1	no	—
212	C₈	Q₈×D₅	1	2	yes	—
213	C₅	#2962	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	1	no	—
223	C₄	—	10	2	no	—
224	C₄	—	10	1	no	—
225	C₄	—	10	2	no	—
226	C₄	#4053	1	2	yes	—
227	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
228	C₄	D₅×#41	1	1	yes	characteristic
229	C₂	—	10	2	no	—
230	C₂	C₄×Q₈×D₅	1	1	yes	characteristic
231	C₂	C₂×#4053	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	#2962
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. Q₈×#168	C₂	semidirect
6. #65120	C₂	semidirect
7. #65120	C₂	semidirect
9. #64990	C₂	semidirect
11. C₈×Dic₁₀	C₂	semidirect
72. #676	Q₈	semidirect
78. #446	D₅	semidirect
213. C₅	#2962	semidirect

Maximal subgroups

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

subgroup	index
#65912	2
C₅×#446	2
Q₈×#168	2
#65128	2
D₅×C₄⋊C₈	2
#65120	2
#64997	2
#64990	2
C₄×#676	2
C₈×Dic₁₀	2
#2962	5