Subgroups — #12009243

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 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	#10600537	C₂	1	1	yes	maximal, characteristic
3	#10600537	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#672	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₅×C₈×Q₈	—	2	1	no	—
7	C₅×C₈×Q₈	—	2	1	no	—
8	C₅×C₈×Q₈	C₄	1	2	yes	—
9	C₅×C₈×Q₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊#672	—	2	1	no	—
15	C₂²⋊#672	—	2	1	no	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	C₂²	1	2	yes	—
18	C₄×#672	—	2	1	no	—
19	C₄×#672	C₂²	1	2	yes	—
20	#11637	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	C₈	1	2	yes	—
28	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂×#672	—	2	2	no	—
44	C₂×#672	—	2	2	no	—
45	C₂×C₈×Q₈	D₅	1	1	yes	characteristic
46	#2162	—	5	1	no	—
47	#2162	—	5	1	no	—
48	Q₈⋊C₁₆	—	5	4	no	—
49	C₂×C₅×Q₈	—	2	1	no	—
50	C₂×C₅×Q₈	—	2	2	no	—
51	C₂×C₅×Q₈	C₁₆	1	4	yes	complemented
52	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	2	1	no	—
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₅	M₁₆	1	1	yes	characteristic
75	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
76	#672	—	4	2	no	—
77	#672	—	4	2	no	—
78	C₂×C₄×Q₈	Dic₅	1	1	yes	characteristic
79	C₈×Q₈	—	2	1	no	—
80	C₈×Q₈	—	2	1	no	—
81	C₈×Q₈	Dic₅	1	2	yes	—
82	C₈×Q₈	C₂×D₅	1	2	yes	—
83	C₂×C₄⋊C₈	—	2	1	no	—
84	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
85	C₂×C₄×C₈	—	2	1	no	—
86	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	C₄⋊C₁₆	—	5	2	no	—
88	C₄⋊C₁₆	—	10	1	no	—
89	C₂²⋊C₁₆	—	10	1	no	—
90	C₂²⋊C₁₆	—	10	1	no	—
91	C₄×C₁₆	—	5	2	no	—
92	C₄×C₁₆	—	10	1	no	—
93	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
94	C₅×Q₈	—	2	4	no	—
95	C₅×Q₈	—	2	2	no	—
96	C₅×Q₈	C₂×C₁₆	1	4	yes	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
108	C₂×C₄×C₅	M₃₂	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	4	2	no	—
116	C₅×C₈	#137	1	2	yes	—
117	C₂²×Q₈	#168	1	1	yes	characteristic
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	#168	1	2	yes	—
124	C₄×Q₈	C₂×Dic₅	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
129	C₂×C₁₆	—	10	2	no	—
130	C₂×C₁₆	—	10	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	C₂²⋊C₁₆	1	2	yes	—
144	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅	—	2	4	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	4	2	no	—
150	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
151	C₄×C₅	#419	1	1	yes	characteristic
152	C₄×C₅	#421	1	1	yes	characteristic
153	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
154	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	#672	1	4	yes	complemented
158	C₂×Q₈	C₂×#168	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂×#168	1	1	yes	characteristic
162	C₂²×C₄	#681	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	C₂×#168	1	2	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂×#168	1	1	yes	characteristic
179	C₄²	#681	1	1	yes	characteristic
180	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
181	C₄²	C₂²×Dic₅	1	1	yes	characteristic
182	C₁₆	—	20	2	no	—
183	C₁₆	—	20	2	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	#1479	1	1	yes	characteristic
186	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
187	C₂×C₅	#2117	1	1	yes	characteristic
188	C₂³	C₂²⋊#168	1	1	yes	characteristic
189	Q₈	—	2	4	no	—
190	Q₈	—	2	2	no	—
191	Q₈	C₂×#672	1	4	yes	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×#672	1	2	yes	—
200	C₂×C₄	#3944	1	2	yes	—
201	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
202	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
204	C₂×C₄	C₂²×#168	1	1	yes	characteristic
205	C₂×C₄	C₂×#681	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	#4088	1	2	yes	—
212	C₅	#11637	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂²⋊#672	1	2	yes	—
216	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂²⋊#672	1	2	yes	—
223	C₄	C₂²×#672	1	1	yes	characteristic
224	C₄	C₂×#3944	1	1	yes	characteristic
225	C₄	#65514	1	1	yes	characteristic
226	C₄	#65525	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#10602414	1	1	yes	characteristic
229	C₂	C₂×C₂²⋊#672	1	1	yes	characteristic
230	C₂	#10602799	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×C₈×Q₈	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×C₈×Q₈	D₅
hypocenter	C₅	#11637
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
4. Q₈⋊#672	C₂	semidirect
51. C₂×C₅×Q₈	C₁₆	semidirect
157. C₂×Q₈	#672	semidirect
212. C₅	#11637	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
#10600537	2
Q₈⋊#672	2
#11637	5