Subgroups — #10617321

Order 640.

364 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	#66014	C₂	1	1	yes	maximal, characteristic
2	#65788	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	#65774	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	#65697	C₂	1	2	yes	maximal, complemented
7	#65696	C₂	1	2	yes	maximal, complemented
8	#65694	C₂	1	2	yes	maximal, complemented
9	#65692	C₂	1	2	yes	maximal, complemented
10	#65680	C₂	1	1	yes	maximal, characteristic, complemented
11	D₅⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#704	C₂²	1	2	yes	—
13	C₂²×#170	C₂²	1	2	yes	—
14	C₂×D₅⋊C₈	C₂²	1	2	yes	—
15	C₅×#124	C₄	1	1	yes	characteristic
16	#4092	C₄	1	1	yes	characteristic
17	Q₈×Dic₅	C₂²	1	1	yes	characteristic
18	#4084	C₄	1	1	yes	characteristic
19	D₄×Dic₅	C₂²	1	1	yes	characteristic
20	#4029	C₄	1	2	yes	—
21	#4027	C₂²	1	2	yes	—
22	#4025	C₄	1	2	yes	—
23	#4023	C₂²	1	2	yes	—
24	C₄²×D₅	C₂²	1	1	yes	characteristic
25	#4016	C₄	1	1	yes	characteristic
26	C₂²⋊#170	C₂²	1	2	yes	—
27	C₂²⋊#170	C₂²	1	2	yes	—
28	#4004	C₂²	1	2	yes	—
29	#4004	C₂²	1	2	yes	—
30	#4003	C₂²	1	1	yes	characteristic
31	#4003	C₂²	1	1	yes	characteristic
32	#4002	C₂²	1	2	yes	—
33	C₄⋊#170	C₂²	1	1	yes	characteristic
34	C₄⋊#170	C₂²	1	1	yes	characteristic
35	C₄⋊#170	C₂²	1	1	yes	characteristic
36	C₄⋊#170	C₂²	1	1	yes	characteristic
37	C₄⋊#170	C₂²	1	2	yes	—
38	C₄×#170	C₂²	1	1	yes	characteristic
39	C₄×#170	C₂²	1	1	yes	characteristic
40	C₄×#170	C₂²	1	2	yes	—
41	#2994	—	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×C₂²⋊D₅	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅	C₂³	1	2	yes	—
46	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
49	#704	—	4	2	no	—
50	C₂×#170	—	4	2	no	—
51	C₂×#170	C₂³	1	2	yes	—
52	C₂×#170	C₂³	1	2	yes	—
53	C₂×#170	C₂³	1	1	yes	characteristic
54	C₂×#170	C₂³	1	1	yes	characteristic
55	C₂×#170	C₂³	1	1	yes	characteristic
56	C₂×#170	C₂³	1	1	yes	characteristic
57	D₅⋊C₈	—	4	2	no	—
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	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
63	#685	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	#683	C₂×C₄	1	2	yes	—
68	#683	C₂×C₄	1	2	yes	—
69	#683	C₂×C₄	1	2	yes	—
70	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅	C₂³	1	1	yes	characteristic
72	C₄×Dic₅	C₂³	1	2	yes	—
73	C₄×Dic₅	C₂³	1	1	yes	characteristic
74	#529	—	5	1	no	—
75	#446	—	5	1	no	—
76	C₈×Q₈	—	5	1	no	—
77	#436	—	5	1	no	—
78	#435	—	5	2	no	—
79	C₈×D₄	—	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	C₂×C₄⋊C₈	—	5	1	no	—
85	C₂³×C₅	C₂²×C₄	1	2	yes	—
86	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
87	C₅×Q₈	—	2	2	no	—
88	C₅×D₄	—	2	2	no	—
89	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊D₅	—	2	4	no	—
94	C₂×Dic₅	—	2	2	no	—
95	C₂×Dic₅	—	2	2	no	—
96	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₅	C₂²×C₄	1	2	yes	—
99	C₂×Dic₅	C₂²×C₄	1	2	yes	—
100	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
101	C₄×D₅	—	2	2	no	—
102	C₄×D₅	—	2	2	no	—
103	C₄×D₅	—	2	1	no	—
104	C₄×D₅	—	2	1	no	—
105	#170	—	2	2	no	—
106	#170	—	2	2	no	—
107	#170	—	2	1	no	—
108	#170	—	2	1	no	—
109	#170	—	2	1	no	—
110	#170	—	2	1	no	—
111	C₂×M₁₆	—	5	2	no	—
112	C₂²×C₈	—	5	2	no	—
113	C₂²×C₈	—	5	2	no	—
114	#126	—	5	2	no	—
115	#125	—	5	1	no	—
116	#124	AGL(1,𝔽₅)	1	1	yes	characteristic
117	#123	—	5	2	no	—
118	C₂²⋊Q₈	—	5	1	no	—
119	#121	—	5	1	no	—
120	C₄×Q₈	—	5	1	no	—
121	C₄×D₄	—	5	1	no	—
122	C₄×D₄	—	5	2	no	—
123	#117	—	5	2	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₄⋊C₈	—	5	1	no	—
126	C₄⋊C₈	—	5	1	no	—
127	C₄⋊C₈	—	5	1	no	—
128	C₄⋊C₈	—	5	2	no	—
129	C₄⋊C₈	—	5	1	no	—
130	C₄⋊C₈	—	5	2	no	—
131	C₄⋊C₈	—	5	2	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	2	no	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×D₅	—	2	1	no	—
145	C₂×D₅	—	2	1	no	—
146	C₂×D₅	#131	1	2	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	#131	1	2	yes	—
151	Dic₅	—	2	2	no	—
152	Dic₅	—	2	1	no	—
153	Dic₅	—	2	1	no	—
154	Dic₅	—	2	2	no	—
155	Dic₅	H(5,𝔽₂)	1	1	yes	characteristic
156	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×D₄	—	5	2	no	—
159	C₂×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₂²×C₄	—	5	2	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	5	1	no	—
164	M₁₆	—	20	2	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	5	1	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	5	1	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	20	2	no	—
172	C₂×C₈	—	20	2	no	—
173	C₄⋊C₄	—	5	2	no	—
174	C₄⋊C₄	—	5	2	no	—
175	C₄⋊C₄	—	5	1	no	—
176	C₄⋊C₄	—	5	2	no	—
177	C₄⋊C₄	—	5	1	no	—
178	C₄⋊C₄	—	5	2	no	—
179	C₂²⋊C₄	—	5	2	no	—
180	C₂²⋊C₄	—	5	2	no	—
181	C₂²⋊C₄	—	5	2	no	—
182	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
183	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
184	C₄²	—	5	1	no	—
185	C₄²	—	5	2	no	—
186	C₄²	—	5	1	no	—
187	C₄²	—	5	1	no	—
188	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂×C₅	—	4	2	no	—
190	C₂×C₅	#520	1	1	yes	characteristic
191	C₂×C₅	C₂×#131	1	1	yes	characteristic
192	C₂×C₅	C₂×#131	1	1	yes	characteristic
193	D₅	—	2	2	no	—
194	C₂³	—	5	1	no	—
195	C₂³	C₂²×AGL(1,𝔽₅)	1	2	yes	—
196	Q₈	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	10	4	no	—
199	C₂×C₄	—	5	1	no	—
200	C₂×C₄	—	5	1	no	—
201	C₂×C₄	—	5	1	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	5	2	no	—
204	C₂×C₄	—	10	1	no	—
205	C₂×C₄	—	10	1	no	—
206	C₂×C₄	—	10	2	no	—
207	C₂×C₄	—	10	2	no	—
208	C₂×C₄	—	10	2	no	—
209	C₂×C₄	—	10	2	no	—
210	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
213	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₈	—	10	2	no	—
215	C₈	—	10	1	no	—
216	C₈	—	10	2	no	—
217	C₈	—	10	1	no	—
218	C₈	—	10	1	no	—
219	C₈	—	10	1	no	—
220	C₅	#2994	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	5	2	no	—
225	C₂²	—	10	1	no	—
226	C₂²	—	10	1	no	—
227	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	5	1	no	—
232	C₄	—	5	1	no	—
233	C₄	—	10	1	no	—
234	C₄	—	10	1	no	—
235	C₄	—	10	2	no	—
236	C₄	—	10	2	no	—
237	C₄	#4131	1	2	yes	—
238	C₂	—	4	2	no	—
239	C₂	—	10	2	no	—
240	C₂	C₂×#4131	1	1	yes	characteristic
241	C₂	D₅⋊#131	1	1	yes	characteristic
242	C₂	#66266	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#124	C₄
prime radical	#4084	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	#124	AGL(1,𝔽₅)
hypocenter	C₅	#2994
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. Q₈×#170	C₂	semidirect
4. #65774	C₂	semidirect
5. D₄×#170	C₂	semidirect
6. #65697	C₂	semidirect
7. #65696	C₂	semidirect
8. #65694	C₂	semidirect
9. #65692	C₂	semidirect
10. #65680	C₂	semidirect
11. D₅⋊(C₄⋊C₈)	C₂	semidirect
220. C₅	#2994	semidirect

Maximal subgroups

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

subgroup	index
#66014	2
#65788	2
Q₈×#170	2
#65774	2
D₄×#170	2
#65697	2
#65696	2
#65694	2
#65692	2
#65680	2
D₅⋊(C₄⋊C₈)	2
#2994	5