Subgroups — #12009251

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 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	#10600538	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	#64681	—	2	1	no	—
19	C₄×#672	C₂²	1	2	yes	—
20	#11639	—	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	#2163	—	5	1	no	—
47	#2162	—	5	1	no	—
48	Q₈⋊C₁₆	—	5	4	no	—
49	C₂×C₅×Q₈	—	2	4	no	—
50	C₂×C₅×Q₈	—	2	1	no	—
51	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	2	1	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₅	M₁₆	1	1	yes	characteristic
74	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
75	#672	—	4	2	no	—
76	#672	—	4	2	no	—
77	C₂×C₄×Q₈	Dic₅	1	1	yes	characteristic
78	C₈×Q₈	—	2	1	no	—
79	C₈×Q₈	—	2	1	no	—
80	C₈×Q₈	Dic₅	1	2	yes	—
81	C₈×Q₈	C₂×D₅	1	2	yes	—
82	C₂×C₄⋊C₈	—	2	1	no	—
83	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
84	C₂×C₄×C₈	—	2	1	no	—
85	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
86	C₄⋊C₁₆	—	5	2	no	—
87	C₄⋊C₁₆	—	10	1	no	—
88	C₂²⋊C₁₆	—	10	1	no	—
89	C₂²⋊C₁₆	—	10	1	no	—
90	#346	—	10	1	no	—
91	C₄×C₁₆	—	5	2	no	—
92	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
93	C₅×Q₈	—	2	4	no	—
94	C₅×Q₈	—	2	2	no	—
95	C₅×Q₈	M₃₂	1	4	yes	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	2	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	—	2	2	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×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	4	2	no	—
114	C₅×C₈	#137	1	2	yes	—
115	C₂²×Q₈	#168	1	1	yes	characteristic
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
118	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	#168	1	2	yes	—
122	C₄×Q₈	C₂×Dic₅	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
127	C₂×C₁₆	—	10	2	no	—
128	C₂×C₁₆	—	10	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	#350	1	2	yes	—
142	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₅	—	2	4	no	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	#419	1	1	yes	characteristic
150	C₄×C₅	#421	1	1	yes	characteristic
151	C₄×C₅	C₂×M₃₂	1	2	yes	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂×#168	1	2	yes	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	C₂×#168	1	1	yes	characteristic
158	C₂²×C₄	#681	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂×#168	1	2	yes	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×#168	1	1	yes	characteristic
175	C₄²	#681	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
177	C₄²	C₂²×Dic₅	1	1	yes	characteristic
178	C₁₆	—	20	2	no	—
179	C₁₆	—	20	2	no	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	#1479	1	1	yes	characteristic
182	C₂×C₅	#2112	1	1	yes	characteristic
183	C₂×C₅	C₂×#350	1	1	yes	characteristic
184	C₂³	C₂²⋊#168	1	1	yes	characteristic
185	Q₈	—	2	4	no	—
186	Q₈	—	2	2	no	—
187	Q₈	#3944	1	4	yes	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
199	C₂×C₄	C₂²×#168	1	1	yes	characteristic
200	C₂×C₄	C₂×#681	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	4	2	no	—
206	C₈	#4088	1	2	yes	—
207	C₅	#11639	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#64781	1	2	yes	—
211	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	C₂×#3944	1	2	yes	—
219	C₄	#65514	1	1	yes	characteristic
220	C₄	#65525	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	#10602414	1	1	yes	characteristic
223	C₂	#10602756	1	1	yes	characteristic
224	C₂	C₂×#64781	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
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	#11639
LCS	G ▷ C₂×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
207. C₅	#11639	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
#10600538	2
#10600537	2
Q₈⋊#672	2
#11639	5