Subgroups — #10611104

Order 640.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65856	C₂	1	1	yes	maximal, characteristic
2	C₅×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#65289	C₂	1	1	yes	maximal, characteristic, complemented
4	#65128	C₂	1	2	yes	maximal, complemented
5	#65124	C₂	1	2	yes	maximal, complemented
6	#65016	C₂	1	2	yes	maximal
7	#65008	C₂	1	2	yes	maximal, complemented
8	#64998	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	#64979	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#676	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#709	C₂²	1	1	yes	characteristic
13	C₅×#117	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#676	C₂²	1	2	yes	—
16	C₂×C₈×D₅	C₂²	1	2	yes	—
17	#4039	C₄	1	2	yes	—
18	#4033	C₄	1	2	yes	—
19	#4027	C₄	1	2	yes	—
20	#4027	C₄	1	2	yes	—
21	#4018	C₂²	1	1	yes	characteristic
22	#4018	C₂²	1	1	yes	characteristic
23	C₄²×D₅	C₂²	1	1	yes	characteristic
24	C₄²×D₅	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₈	C₂²	1	2	yes	—
26	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₅×#97	C₂²	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₂²⋊#168	C₂²	1	2	yes	—
30	#3952	C₂²	1	2	yes	—
31	#3952	C₂²	1	2	yes	—
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	C₄⋊#168	C₂²	1	2	yes	—
37	#3935	C₂²	1	1	yes	characteristic
38	C₄×#168	C₂²	1	1	yes	characteristic
39	#2923	—	5	1	no	maximal, Sylow
40	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₅	C₂×C₄	1	2	yes	—
44	#709	—	2	4	no	—
45	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂³	1	2	yes	—
52	C₂×C₅×C₈	C₂³	1	2	yes	—
53	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₄²×C₅	C₂³	1	1	yes	characteristic
57	#685	C₂×C₄	1	2	yes	—
58	#685	C₂×C₄	1	2	yes	—
59	#683	C₂×C₄	1	2	yes	—
60	#683	C₂×C₄	1	2	yes	—
61	C₄×Dic₅	C₂×C₄	1	2	yes	—
62	C₄×Dic₅	C₂×C₄	1	2	yes	—
63	C₄×Dic₅	C₂×C₄	1	2	yes	—
64	C₄×Dic₅	C₂³	1	1	yes	characteristic
65	C₄×Dic₅	C₂³	1	1	yes	characteristic
66	C₂×#168	C₂³	1	2	yes	—
67	C₂×#168	C₂³	1	2	yes	—
68	#676	—	2	4	no	—
69	C₈×D₅	—	2	4	no	—
70	C₄×#41	—	5	1	no	—
71	#433	—	5	1	no	—
72	#433	—	5	1	no	—
73	#433	—	5	1	no	—
74	#433	D₅	1	1	yes	characteristic, complemented
75	#432	—	5	2	no	—
76	#432	—	5	2	no	—
77	#424	—	5	2	no	—
78	#408	—	5	2	no	—
79	C₄×M₁₆	—	5	1	no	—
80	C₂×#97	—	5	1	no	—
81	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	—	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	C₂⁴	1	1	yes	characteristic
89	C₂²⋊D₅	—	2	2	no	—
90	C₂²⋊D₅	—	2	2	no	—
91	C₂×Dic₅	—	4	2	no	—
92	C₂×Dic₅	C₂²×C₄	1	2	yes	—
93	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂²×C₄	1	2	yes	—
95	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
96	D₂₀	—	2	2	no	—
97	C₄×D₅	—	2	2	no	—
98	C₄×D₅	—	4	2	no	—
99	C₄×D₅	—	4	2	no	—
100	C₄×D₅	#41	1	4	yes	—
101	Dic₁₀	—	2	2	no	—
102	C₅×C₈	—	2	2	no	—
103	C₅×C₈	—	2	2	no	—
104	#168	—	2	2	no	—
105	#168	—	2	2	no	—
106	C₂×#41	—	5	1	no	—
107	C₂×M₁₆	—	5	2	no	—
108	C₂²×C₈	—	5	2	no	—
109	C₄×Q₈	—	5	2	no	—
110	C₄×D₄	—	5	2	no	—
111	C₄×D₄	—	5	2	no	—
112	C₄×D₄	—	5	2	no	—
113	#117	—	5	1	no	—
114	#117	—	5	1	no	—
115	#117	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄²	—	5	1	no	—
117	C₂×C₄²	—	5	1	no	—
118	C₂×C₄²	—	5	1	no	—
119	C₄⋊C₈	—	5	2	no	—
120	C₄⋊C₈	—	5	2	no	—
121	C₄⋊C₈	—	5	2	no	—
122	C₄⋊C₈	C₂×D₅	1	2	yes	—
123	C₂²⋊C₈	—	5	2	no	—
124	C₂²⋊C₈	—	5	2	no	—
125	C₂²⋊C₈	—	5	2	no	—
126	C₂²⋊C₈	C₂×D₅	1	2	yes	—
127	#97	—	5	2	no	—
128	#97	—	5	2	no	—
129	#97	—	5	1	no	—
130	#97	C₂×D₅	1	1	yes	characteristic
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	C₂×D₅	1	1	yes	characteristic
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	—	2	2	no	—
135	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₅	—	2	2	no	—
137	C₂×D₅	—	2	1	no	—
138	C₂×D₅	—	2	2	no	—
139	C₂×D₅	#117	1	2	yes	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	C₂×#41	1	2	yes	—
144	Dic₅	—	2	1	no	—
145	Dic₅	—	2	2	no	—
146	Dic₅	—	2	2	no	—
147	Dic₅	#117	1	2	yes	—
148	#41	—	10	4	no	—
149	C₂×Q₈	—	5	1	no	—
150	C₂×D₄	—	5	1	no	—
151	C₂×D₄	—	5	1	no	—
152	C₂×D₄	—	5	1	no	—
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	5	2	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	—	5	2	no	—
158	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
159	M₁₆	—	10	4	no	—
160	C₂×C₈	—	5	2	no	—
161	C₂×C₈	—	5	2	no	—
162	C₂×C₈	—	10	4	no	—
163	C₂×C₈	C₂²×D₅	1	2	yes	—
164	C₂×C₈	C₂²×D₅	1	2	yes	—
165	C₄⋊C₄	—	5	2	no	—
166	C₄⋊C₄	—	5	2	no	—
167	C₄⋊C₄	C₄×D₅	1	2	yes	—
168	C₂²⋊C₄	—	5	2	no	—
169	C₂²⋊C₄	—	5	2	no	—
170	C₂²⋊C₄	C₄×D₅	1	2	yes	—
171	C₄²	—	5	2	no	—
172	C₄²	—	5	2	no	—
173	C₄²	—	5	2	no	—
174	C₄²	—	5	1	no	—
175	C₄²	—	5	1	no	—
176	C₄²	C₂²×D₅	1	1	yes	characteristic
177	C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₅	—	4	1	no	—
179	C₂×C₅	C₂×#117	1	1	yes	characteristic
180	C₂×C₅	#568	1	2	yes	—
181	D₅	—	2	2	no	—
182	D₅	—	4	1	no	—
183	C₂³	—	5	1	no	—
184	C₂³	—	5	1	no	—
185	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
186	Q₈	—	10	2	no	—
187	D₄	—	10	2	no	—
188	D₄	—	10	2	no	—
189	D₄	—	10	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	4	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	1	no	—
196	C₂×C₄	—	10	2	no	—
197	C₂×C₄	—	20	2	no	—
198	C₂×C₄	—	20	2	no	—
199	C₂×C₄	—	20	2	no	—
200	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
203	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	10	2	no	—
207	C₈	—	10	2	no	—
208	C₅	#2923	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	5	2	no	—
212	C₂²	—	10	2	no	—
213	C₂²	—	10	2	no	—
214	C₂²	—	10	1	no	—
215	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	5	2	no	—
220	C₄	—	10	2	no	—
221	C₄	—	10	2	no	—
222	C₄	—	10	1	no	—
223	C₄	D₅×#41	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	10	2	no	—
226	C₂	—	20	1	no	—
227	C₂	D₅×#117	1	1	yes	characteristic
228	C₂	#66084	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#433	C₂
prime radical	C₂×#709	C₂²
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#433	D₅
hypocenter	C₅	#2923
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₅×#433	C₂	semidirect
3. #65289	C₂	semidirect
4. #65128	C₂	semidirect
5. #65124	C₂	semidirect
7. #65008	C₂	semidirect
8. #64998	C₂	semidirect
9. D₅×#97	C₂	semidirect
10. #64979	C₂	semidirect
11. C₄×#676	C₂	semidirect
74. #433	D₅	semidirect
208. C₅	#2923	semidirect

Maximal subgroups

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

subgroup	index
#65856	2
C₅×#433	2
#65289	2
#65128	2
#65124	2
#65016	2
#65008	2
#64998	2
D₅×#97	2
#64979	2
C₄×#676	2
#2923	5