Subgroups — #12098362

Order 1280.

914 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	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×#168	C₂	1	1	yes	maximal, characteristic
4	#10601259	C₂	1	3	yes	maximal
5	#10601234	C₂	1	3	yes	maximal
6	#10600550	C₂	1	3	yes	maximal
7	#10600543	C₂	1	3	yes	maximal
8	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₅	C₄	1	2	yes	—
10	C₅×C₈×Q₈	C₄	1	8	yes	complemented
11	C₂×C₅×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
13	C₂×#3947	C₂²	1	6	yes	—
14	Q₈×#168	C₄	1	8	yes	complemented
15	C₄⋊C₄×Dic₅	C₄	1	6	yes	—
16	C₄×C₄⋊Dic₅	C₂²	1	3	yes	—
17	C₄²×Dic₅	C₂²	1	3	yes	—
18	C₂×C₄⋊#168	C₂²	1	3	yes	—
19	C₂×C₄×#168	C₂²	1	3	yes	—
20	#64770	C₂²	1	3	yes	—
21	#64770	C₂²	1	3	yes	—
22	#64770	C₂²	1	1	yes	characteristic
23	#64770	C₂²	1	3	yes	—
24	#20572	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	3	yes	—
28	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
29	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₅	C₂³	1	3	yes	—
32	Q₈×Dic₅	—	2	8	no	—
33	C₂×C₄⋊Dic₅	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
36	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₅	C₂³	1	3	yes	—
38	C₂²×#168	C₂³	1	1	yes	characteristic
39	C₂²×#168	C₂³	1	3	yes	—
40	C₅×C₄⋊C₈	C₂×C₄	1	12	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
42	#3963	C₂×C₄	1	6	yes	—
43	#3947	—	4	6	no	—
44	C₄⋊#168	C₂×C₄	1	12	yes	—
45	C₄×#168	C₂×C₄	1	12	yes	—
46	C₂×C₈×Q₈	—	5	1	no	—
47	C₂×C₈×Q₈	D₅	1	1	yes	characteristic
48	C₄²×Q₈	—	5	1	no	—
49	#1770	—	5	3	no	—
50	#1770	—	5	3	no	—
51	#1767	—	5	3	no	—
52	#1744	—	5	3	no	—
53	C₂×C₅×Q₈	C₄²	1	4	yes	—
54	C₂×C₅×Q₈	M₁₆	1	8	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅	C₂²×C₄	1	6	yes	—
59	C₂²×Dic₅	C₂²×C₄	1	2	yes	—
60	C₂×C₅×C₈	—	2	3	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	3	no	—
64	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
66	C₅×C₄⋊C₄	C₄²	1	12	yes	—
67	C₄²×C₅	C₂²×C₄	1	6	yes	—
68	C₄²×C₅	C₂²×C₄	1	6	yes	—
69	C₄⋊Dic₅	—	2	12	no	—
70	C₄×Dic₅	—	2	12	no	—
71	C₄×Dic₅	—	2	3	no	—
72	C₄×Dic₅	—	2	1	no	—
73	C₄×Dic₅	—	2	3	no	—
74	C₄×Dic₅	—	2	1	no	—
75	C₄×Dic₅	—	2	12	no	—
76	C₂×#168	—	2	3	no	—
77	C₂×#168	—	2	1	no	—
78	C₂×#168	—	2	3	no	—
79	C₂×#168	—	2	1	no	—
80	C₂×#168	C₂²×C₄	1	2	yes	—
81	C₂×#168	C₂²×C₄	1	6	yes	—
82	C₂×C₄×Q₈	—	5	2	no	—
83	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
84	C₈×Q₈	—	5	8	no	—
85	C₈×Q₈	Dic₅	1	8	yes	complemented
86	C₂×C₄⋊C₈	—	5	3	no	—
87	C₂×C₄⋊C₈	C₂×D₅	1	3	yes	—
88	C₂×#97	—	5	6	no	—
89	C₂×C₄×C₈	—	5	3	no	—
90	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
91	C₄×C₄⋊C₄	—	5	6	no	—
92	C₄×C₄⋊C₄	—	5	3	no	—
93	C₄³	—	5	3	no	—
94	#336	—	5	1	no	—
95	#336	—	5	3	no	—
96	#336	—	5	3	no	—
97	#336	—	5	3	no	—
98	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
99	C₅×Q₈	#97	1	16	yes	complemented
100	C₂×C₄×C₅	—	2	6	no	—
101	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
102	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
103	C₂×C₄×C₅	C₂×M₁₆	1	12	yes	—
104	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	—	2	6	no	—
113	C₂×Dic₅	—	2	6	no	—
114	C₂×Dic₅	—	2	6	no	—
115	C₂×Dic₅	#131	1	4	yes	—
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	—	2	6	no	—
118	#168	—	2	2	no	—
119	#168	—	2	6	no	—
120	C₂²×Q₈	C₄×D₅	1	1	yes	characteristic
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	—	5	3	no	—
123	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₅	1	3	yes	—
125	C₄×Q₈	—	10	8	no	—
126	C₄×Q₈	C₄×D₅	1	4	yes	—
127	C₄×Q₈	C₂×Dic₅	1	4	yes	—
128	C₂×C₄⋊C₄	—	5	6	no	—
129	C₂×C₄⋊C₄	C₄×D₅	1	3	yes	—
130	C₂×C₄²	—	5	6	no	—
131	C₂×C₄²	—	5	6	no	—
132	C₂×C₄²	—	5	1	no	—
133	C₂×C₄²	—	5	3	no	—
134	C₂×C₄²	C₂²×D₅	1	3	yes	—
135	C₄⋊C₈	—	5	12	no	—
136	C₄⋊C₈	C₂×Dic₅	1	12	yes	—
137	#97	—	20	6	no	—
138	C₄×C₈	—	5	12	no	—
139	C₄×C₈	C₂×Dic₅	1	12	yes	—
140	#95	—	5	6	no	—
141	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
142	C₂²×C₅	#519	1	1	yes	characteristic
143	C₂²×C₅	#520	1	1	yes	characteristic
144	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
145	C₂²×C₅	C₂×#131	1	2	yes	—
146	C₄×C₅	—	2	6	no	—
147	C₄×C₅	C₂×#97	1	12	yes	—
148	C₄×C₅	#519	1	1	yes	characteristic
149	C₄×C₅	#519	1	1	yes	characteristic
150	C₄×C₅	#520	1	1	yes	characteristic
151	C₄×C₅	#520	1	1	yes	characteristic
152	Dic₅	—	2	4	no	—
153	Dic₅	—	4	6	no	—
154	C₂×Q₈	#676	1	8	yes	—
155	C₂×Q₈	C₄×Dic₅	1	4	yes	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	—	5	6	no	—
158	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
159	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
160	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	3	no	—
164	C₂×C₈	—	2	3	no	—
165	C₂×C₈	—	5	2	no	—
166	C₂×C₈	—	5	6	no	—
167	C₂×C₈	—	10	1	no	—
168	C₂×C₈	—	10	3	no	—
169	C₂×C₈	—	10	3	no	—
170	C₂×C₈	—	10	1	no	—
171	C₂×C₈	C₂²×Dic₅	1	2	yes	—
172	C₂×C₈	C₂²×Dic₅	1	6	yes	—
173	C₄⋊C₄	—	10	12	no	—
174	C₄⋊C₄	C₄×Dic₅	1	12	yes	—
175	C₄²	—	10	12	no	—
176	C₄²	—	10	1	no	—
177	C₄²	—	10	3	no	—
178	C₄²	—	10	3	no	—
179	C₄²	—	10	1	no	—
180	C₄²	—	10	12	no	—
181	C₄²	C₂×C₄×D₅	1	6	yes	—
182	C₄²	C₂²×Dic₅	1	6	yes	—
183	C₂×C₅	#2276	1	1	yes	characteristic
184	C₂×C₅	C₂²×#97	1	1	yes	characteristic
185	C₂×C₅	C₄×#131	1	1	yes	characteristic
186	C₂×C₅	#2984	1	2	yes	—
187	C₂×C₅	#2991	1	2	yes	—
188	C₂³	C₂²×C₄×D₅	1	1	yes	characteristic
189	Q₈	#3947	1	16	yes	complemented
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	5	4	no	—
192	C₂×C₄	—	10	2	no	—
193	C₂×C₄	—	10	2	no	—
194	C₂×C₄	—	10	6	no	—
195	C₂×C₄	—	10	6	no	—
196	C₂×C₄	—	10	6	no	—
197	C₂×C₄	C₂×#676	1	12	yes	—
198	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
199	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
200	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
201	C₂×C₄	#4147	1	1	yes	characteristic
202	C₂×C₄	#4147	1	1	yes	characteristic
203	C₂×C₄	#4149	1	1	yes	characteristic
204	C₂×C₄	#4150	1	1	yes	characteristic
205	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	6	no	—
208	C₈	—	10	2	no	—
209	C₈	—	10	6	no	—
210	C₅	#20572	1	1	yes	characteristic, complemented, Sylow
211	C₂²	#65906	1	1	yes	characteristic
212	C₂²	#65908	1	1	yes	characteristic
213	C₂²	C₂²×#676	1	2	yes	—
214	C₂²	D₅×#131	1	2	yes	—
215	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
216	C₄	—	2	6	no	—
217	C₄	—	10	4	no	—
218	C₄	—	20	6	no	—
219	C₄	C₂×#3947	1	12	yes	—
220	C₄	#65906	1	1	yes	characteristic
221	C₄	#65908	1	1	yes	characteristic
222	C₄	#66146	1	1	yes	characteristic
223	C₄	#66161	1	1	yes	characteristic
224	C₂	#10611638	1	2	yes	—
225	C₂	#10611652	1	2	yes	—
226	C₂	#10614181	1	1	yes	characteristic
227	C₂	C₂²×#3947	1	1	yes	characteristic
228	C₂	Dic₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂²×C₄²
Fitting	C₂×C₅×C₈×Q₈	C₂
socle, prime radical	C₂³×C₅	C₂³×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅
hypocenter	C₅	#20572
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₅×C₈×Q₈	C₄	semidirect
14. Q₈×#168	C₄	semidirect
85. C₈×Q₈	Dic₅	semidirect
99. C₅×Q₈	#97	semidirect
189. Q₈	#3947	semidirect
210. C₅	#20572	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×Q₈	2
C₄×Q₈×Dic₅	2
C₂×Q₈×#168	2
#10601259	2
#10601234	2
#10600550	2
#10600543	2
#20572	5