Subgroups — #10617263

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	#65956	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	C₄×#704	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#170	C₂	1	1	yes	maximal, characteristic
6	#65709	C₂	1	2	yes	maximal, complemented
7	#65709	C₂	1	1	yes	maximal, characteristic, complemented
8	#65707	C₂	1	2	yes	maximal, complemented
9	#65703	C₂	1	1	yes	maximal, characteristic, complemented
10	#65696	C₂	1	2	yes	maximal
11	#65691	C₂	1	2	yes	maximal
12	C₂×#704	C₂²	1	1	yes	characteristic
13	C₂×#704	C₂²	1	1	yes	characteristic
14	C₂²×#170	C₂²	1	2	yes	—
15	C₅×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	Q₈×Dic₅	C₂²	1	1	yes	characteristic
17	#4090	C₄	1	1	yes	characteristic
18	#4070	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
20	#4035	C₄	1	2	yes	—
21	#4034	C₄	1	1	yes	characteristic
22	#4033	C₂²	1	1	yes	characteristic
23	#4033	C₂²	1	2	yes	—
24	#4024	C₄	1	2	yes	—
25	#4023	C₂²	1	2	yes	—
26	#4004	C₂²	1	2	yes	—
27	#4004	C₂²	1	2	yes	—
28	#4002	C₂²	1	2	yes	—
29	#4002	C₂²	1	2	yes	—
30	#4002	C₂²	1	1	yes	characteristic
31	#4002	C₂²	1	1	yes	characteristic
32	C₄⋊#170	C₂²	1	2	yes	—
33	C₄⋊#170	C₂²	1	2	yes	—
34	C₄⋊#170	C₂²	1	1	yes	characteristic
35	C₄⋊#170	C₂²	1	2	yes	—
36	C₄⋊#170	C₂²	1	1	yes	characteristic
37	C₄×#170	C₂²	1	2	yes	—
38	C₄×#170	C₂²	1	2	yes	—
39	C₄×#170	C₂²	1	1	yes	characteristic
40	C₄×#170	C₂²	1	1	yes	characteristic
41	#2962	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
48	#704	—	2	2	no	—
49	#704	Q₈	1	4	yes	complemented
50	C₂×#170	—	2	4	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	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
63	#683	C₂×C₄	1	2	yes	—
64	#683	C₂×C₄	1	2	yes	—
65	#683	C₂×C₄	1	2	yes	—
66	#683	C₂×C₄	1	2	yes	—
67	#683	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅	C₂³	1	1	yes	characteristic
71	C₄×Dic₅	C₂³	1	2	yes	—
72	C₄×Dic₅	C₂³	1	2	yes	—
73	C₄×Dic₅	C₂³	1	1	yes	characteristic
74	#533	—	5	1	no	—
75	#446	—	5	2	no	—
76	#446	—	5	1	no	—
77	#446	—	5	2	no	—
78	#446	—	5	1	no	—
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	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₅×Q₈	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₅	—	4	1	no	—
93	C₂×Dic₅	C₂²×C₄	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	2	yes	—
97	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₅	C₂×Q₈	1	2	yes	—
99	C₂×Dic₅	#41	1	2	yes	—
100	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
101	Dic₁₀	—	2	2	no	—
102	Dic₁₀	—	4	2	no	—
103	#170	—	2	2	no	—
104	#170	—	2	2	no	—
105	#170	—	2	1	no	—
106	#170	—	2	1	no	—
107	#170	C₂×Q₈	1	2	yes	—
108	#170	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	AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₄×Q₈	—	5	1	no	—
119	C₄×Q₈	—	5	1	no	—
120	C₄×Q₈	—	5	2	no	—
121	#117	—	5	2	no	—
122	C₂×C₄²	—	5	1	no	—
123	C₄⋊C₈	—	5	1	no	—
124	C₄⋊C₈	—	5	1	no	—
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	—	5	2	no	—
127	C₄⋊C₈	—	5	2	no	—
128	C₄⋊C₈	—	5	2	no	—
129	C₄⋊C₈	—	5	2	no	—
130	#97	—	5	1	no	—
131	#97	—	5	1	no	—
132	#97	—	5	2	no	—
133	#97	—	5	2	no	—
134	C₄×C₈	—	5	1	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	C₄×Q₈	1	2	yes	—
141	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	#131	1	2	yes	—
146	Dic₅	—	2	2	no	—
147	Dic₅	—	2	1	no	—
148	Dic₅	—	2	2	no	—
149	Dic₅	—	2	1	no	—
150	Dic₅	C₄×Q₈	1	2	yes	—
151	Dic₅	C₂²×Q₈	1	1	yes	characteristic
152	Dic₅	C₂×#41	1	1	yes	characteristic
153	C₂×Q₈	—	5	1	no	—
154	C₂×Q₈	—	5	2	no	—
155	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₂²×C₄	—	5	1	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	M₁₆	—	5	4	no	—
160	M₁₆	—	10	2	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	5	2	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	2	no	—
167	C₂×C₈	—	10	4	no	—
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	C₂×AGL(1,𝔽₅)	1	2	yes	—
176	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂²⋊C₄	—	5	2	no	—
178	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
179	C₄²	—	5	1	no	—
180	C₄²	—	5	2	no	—
181	C₄²	—	5	2	no	—
182	C₄²	—	5	1	no	—
183	C₄²	—	5	1	no	—
184	C₄²	—	5	1	no	—
185	C₂×C₅	—	2	2	no	—
186	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂×C₅	C₂×#131	1	1	yes	characteristic
188	C₂×C₅	#568	1	1	yes	characteristic
189	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	10	2	no	—
192	Q₈	—	20	2	no	—
193	C₂×C₄	—	2	2	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	1	no	—
199	C₂×C₄	—	5	1	no	—
200	C₂×C₄	—	5	2	no	—
201	C₂×C₄	—	5	2	no	—
202	C₂×C₄	—	20	1	no	—
203	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
205	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
206	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₈	—	5	2	no	—
208	C₈	—	5	2	no	—
209	C₈	—	10	2	no	—
210	C₈	—	10	2	no	—
211	C₈	—	10	1	no	—
212	C₈	—	10	1	no	—
213	C₅	#2962	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×AGL(1,𝔽₅)	1	2	yes	—
217	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	5	1	no	—
222	C₄	—	5	1	no	—
223	C₄	—	5	2	no	—
224	C₄	—	10	2	no	—
225	C₄	—	10	1	no	—
226	C₄	—	10	1	no	—
227	C₄	—	10	2	no	—
228	C₄	#4131	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×#4131	1	1	yes	characteristic
231	C₂	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
232	C₂	#66264	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₂²⋊Q₈	C₄
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2962
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

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
6. #65709	C₂	semidirect
7. #65709	C₂	semidirect
8. #65707	C₂	semidirect
9. #65703	C₂	semidirect
49. #704	Q₈	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
#65956	2
#65788	2
Q₈×#170	2
C₄×#704	2
C₂×C₄⋊#170	2
#65709	2
#65707	2
#65703	2
#65696	2
#65691	2
#2962	5