Subgroups — #10614973

Order 640.

556 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4090	C₂	1	2	yes	maximal
3	C₂×#4034	C₂	1	1	yes	maximal, characteristic
4	#65518	C₂	1	2	yes	maximal
5	#65268	C₂	1	2	yes	maximal
6	#65265	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
8	#65262	C₂	1	1	yes	maximal, characteristic
9	#65261	C₂	1	2	yes	maximal
10	C₄×#683	C₂	1	1	yes	maximal, characteristic
11	#65217	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₅×Q₈	C₂²	1	2	yes	—
13	C₂²×Dic₁₀	C₂²	1	2	yes	—
14	C₅×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
19	#4090	—	2	8	no	—
20	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₂×#683	C₂²	1	2	yes	—
23	C₂×#683	C₂²	1	2	yes	—
24	C₂×#683	C₂²	1	2	yes	—
25	C₂×#683	C₂²	1	2	yes	—
26	C₂×#683	C₂²	1	2	yes	—
27	C₂×#683	C₂²	1	1	yes	characteristic
28	C₂×#683	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₅	C₂²	1	2	yes	—
31	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
32	#4034	—	2	4	no	—
33	#3963	C₂²	1	2	yes	—
34	#3963	C₂²	1	1	yes	characteristic
35	#3963	C₂²	1	1	yes	characteristic
36	#3963	C₂²	1	2	yes	—
37	#3963	C₂²	1	2	yes	—
38	#2551	—	5	1	no	maximal, Sylow
39	C₂×C₅×Q₈	—	2	4	no	—
40	C₂×C₅×Q₈	—	2	4	no	—
41	C₂×C₅×Q₈	—	4	2	no	—
42	C₂²×C₄×C₅	C₂³	1	2	yes	—
43	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂³	1	2	yes	—
45	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂²×Dic₅	C₂³	1	2	yes	—
48	C₂²×Dic₅	C₂³	1	1	yes	characteristic
49	C₂²×Dic₅	C₂³	1	1	yes	characteristic
50	C₂²×Dic₅	C₂³	1	1	yes	characteristic
51	C₂²×Dic₅	C₂³	1	2	yes	—
52	C₂²×Dic₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₀	—	2	4	no	—
54	C₂×Dic₁₀	—	2	4	no	—
55	C₂×Dic₁₀	—	4	2	no	—
56	C₅×C₄⋊C₄	—	2	4	no	—
57	C₅×C₄⋊C₄	—	2	2	no	—
58	C₅×C₄⋊C₄	D₄	1	4	yes	—
59	C₄²×C₅	—	2	2	no	—
60	C₄⋊Dic₅	—	2	2	no	—
61	C₄⋊Dic₅	Q₈	1	4	yes	complemented
62	#683	—	2	4	no	—
63	#683	—	2	4	no	—
64	#683	—	2	4	no	—
65	#683	—	2	4	no	—
66	#683	—	4	2	no	—
67	#683	D₄	1	4	yes	—
68	#683	Q₈	1	4	yes	complemented
69	C₄×Dic₅	—	2	2	no	—
70	C₄×Dic₅	—	2	4	no	—
71	C₄×Dic₅	Q₈	1	4	yes	complemented
72	C₂×C₄⋊Q₈	—	5	2	no	—
73	C₂×C₄⋊Q₈	—	5	1	no	—
74	C₂×C₄⋊Q₈	D₅	1	1	yes	characteristic
75	#395	—	5	2	no	—
76	#395	—	5	2	no	—
77	#389	—	5	2	no	—
78	#389	—	5	1	no	—
79	#389	—	5	1	no	—
80	#384	—	5	1	no	—
81	C₄×C₄⋊C₄	—	5	1	no	—
82	C₄×C₄⋊C₄	—	5	1	no	—
83	C₂³×C₅	C₂⁴	1	1	yes	characteristic
84	C₅×Q₈	—	4	8	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
100	C₂×Dic₅	—	2	2	no	—
101	C₂×Dic₅	—	2	1	no	—
102	C₂×Dic₅	—	2	1	no	—
103	C₂×Dic₅	—	2	1	no	—
104	C₂×Dic₅	—	2	1	no	—
105	C₂×Dic₅	—	2	1	no	—
106	C₂×Dic₅	—	2	1	no	—
107	C₂×Dic₅	—	2	2	no	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	2	2	no	—
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	C₂×D₄	1	2	yes	—
112	C₂×Dic₅	C₂×D₄	1	2	yes	—
113	C₂×Dic₅	C₂×Q₈	1	2	yes	—
114	C₂×Dic₅	C₂×Q₈	1	2	yes	—
115	C₂×Dic₅	C₂×Q₈	1	4	yes	—
116	C₂×Dic₅	C₂×Q₈	1	2	yes	—
117	C₂×Dic₅	C₂×Q₈	1	2	yes	—
118	Dic₁₀	—	4	8	no	—
119	C₂²×Q₈	—	5	2	no	—
120	C₂²×Q₈	C₂×D₅	1	2	yes	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	10	8	no	—
123	C₄⋊Q₈	—	10	4	no	—
124	C₂×C₄⋊C₄	—	5	1	no	—
125	C₂×C₄⋊C₄	—	5	1	no	—
126	C₂×C₄⋊C₄	—	5	1	no	—
127	C₂×C₄⋊C₄	—	5	2	no	—
128	C₂×C₄⋊C₄	—	5	1	no	—
129	C₂×C₄⋊C₄	—	5	2	no	—
130	C₂×C₄⋊C₄	—	5	2	no	—
131	C₂×C₄⋊C₄	—	5	2	no	—
132	C₂×C₄⋊C₄	—	5	2	no	—
133	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
135	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
136	C₂×C₄²	—	5	1	no	—
137	C₂×C₄²	—	5	2	no	—
138	C₂×C₄²	—	5	1	no	—
139	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
140	#95	—	5	1	no	—
141	#95	—	5	2	no	—
142	#95	—	5	1	no	—
143	#95	—	5	2	no	—
144	#95	—	5	2	no	—
145	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
148	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
149	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
150	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄×C₅	—	2	4	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	4	2	no	—
156	C₄×C₅	C₄⋊Q₈	1	4	yes	complemented
157	Dic₅	—	2	2	no	—
158	Dic₅	—	2	2	no	—
159	Dic₅	—	2	4	no	—
160	Dic₅	—	2	2	no	—
161	Dic₅	—	4	2	no	—
162	Dic₅	C₄⋊Q₈	1	4	yes	complemented
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	10	4	no	—
167	C₂×Q₈	—	10	4	no	—
168	C₂×Q₈	—	20	2	no	—
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	—	5	1	no	—
171	C₂²×C₄	—	5	2	no	—
172	C₂²×C₄	—	5	1	no	—
173	C₂²×C₄	—	5	1	no	—
174	C₂²×C₄	—	5	2	no	—
175	C₂²×C₄	C₂²×D₅	1	2	yes	—
176	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₅	1	2	yes	—
179	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	5	4	no	—
183	C₄⋊C₄	—	5	4	no	—
184	C₄⋊C₄	—	5	4	no	—
185	C₄⋊C₄	—	10	4	no	—
186	C₄⋊C₄	—	10	4	no	—
187	C₄⋊C₄	—	10	4	no	—
188	C₄⋊C₄	—	10	4	no	—
189	C₄⋊C₄	—	10	2	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄⋊C₄	C₂²⋊D₅	1	4	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	5	4	no	—
194	C₄²	—	10	4	no	—
195	C₄²	—	10	2	no	—
196	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
198	C₂×C₅	#544	1	1	yes	characteristic
199	C₂×C₅	D₄²	1	1	yes	characteristic
200	C₂×C₅	D₄×Q₈	1	1	yes	characteristic
201	C₂×C₅	D₄×Q₈	1	1	yes	characteristic
202	C₂×C₅	Q₈²	1	1	yes	characteristic
203	C₂³	C₂³×D₅	1	1	yes	characteristic
204	Q₈	—	4	8	no	—
205	Q₈	—	20	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	5	2	no	—
216	C₂×C₄	—	5	2	no	—
217	C₂×C₄	—	5	2	no	—
218	C₂×C₄	—	5	2	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	—	5	4	no	—
221	C₂×C₄	—	5	2	no	—
222	C₂×C₄	—	10	2	no	—
223	C₂×C₄	—	10	1	no	—
224	C₂×C₄	—	10	1	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	—	10	1	no	—
227	C₂×C₄	—	10	1	no	—
228	C₂×C₄	—	10	1	no	—
229	C₂×C₄	—	10	1	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	D₄×D₅	1	2	yes	—
234	C₂×C₄	Q₈×D₅	1	2	yes	—
235	C₂×C₄	Q₈×D₅	1	2	yes	—
236	C₂×C₄	Q₈×D₅	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₅	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
239	C₅	#2551	1	1	yes	characteristic, complemented, Sylow
240	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
241	C₂²	#4144	1	1	yes	characteristic
242	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
243	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
244	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
245	C₂²	#4147	1	1	yes	characteristic
246	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	5	4	no	—
252	C₄	—	10	4	no	—
253	C₄	—	10	2	no	—
254	C₄	—	10	2	no	—
255	C₄	—	10	2	no	—
256	C₄	—	20	2	no	—
257	C₄	#4090	1	4	yes	complemented
258	C₂	#66056	1	1	yes	characteristic
259	C₂	D₅×C₄⋊Q₈	1	1	yes	characteristic
260	C₂	#66061	1	1	yes	characteristic
261	C₂	#66063	1	1	yes	characteristic
262	C₂	D₄×C₂²⋊D₅	1	1	yes	characteristic
263	C₂	C₂×#4090	1	1	yes	characteristic
264	C₂	Q₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₅×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₅
hypocenter	C₅	#2551
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
61. C₄⋊Dic₅	Q₈	semidirect
68. #683	Q₈	semidirect
71. C₄×Dic₅	Q₈	semidirect
156. C₄×C₅	C₄⋊Q₈	semidirect
162. Dic₅	C₄⋊Q₈	semidirect
239. C₅	#2551	semidirect
257. C₄	#4090	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊Q₈	2
C₂×#4090	2
C₂×#4034	2
#65518	2
#65268	2
#65265	2
C₄⋊C₄×Dic₅	2
#65262	2
#65261	2
C₄×#683	2
#65217	2
#2551	5