Subgroups — #12009255

Order 1280.

296 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	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	C₅×#533	C₄	1	1	yes	characteristic
6	C₅×#446	—	2	1	no	—
7	C₅×#446	C₄	1	2	yes	—
8	C₅×C₈×Q₈	—	2	1	no	—
9	C₅×C₈×Q₈	C₂²	1	2	yes	—
10	C₅×#424	—	2	1	no	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#405	—	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	#11640	—	5	1	no	maximal, Sylow
21	C₂×C₅×M₁₆	—	2	1	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
26	C₅×#125	—	2	1	no	—
27	C₅×C₂²⋊Q₈	—	2	1	no	—
28	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
29	C₄×C₅×Q₈	—	2	1	no	—
30	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
31	C₅×#117	—	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	C₅×#97	—	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	#2960	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	1	no	—
50	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
54	C₅×M₁₆	—	4	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	4	1	no	—
58	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	2	1	no	—
63	C₅×C₄⋊C₄	—	2	1	no	—
64	C₅×C₄⋊C₄	—	2	1	no	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₅×C₂²⋊C₄	—	2	2	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅	M₁₆	1	1	yes	characteristic
73	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
74	#672	—	4	2	no	—
75	#672	—	4	2	no	—
76	#533	Dic₅	1	1	yes	characteristic
77	#446	—	2	1	no	—
78	#446	Dic₅	1	2	yes	—
79	C₈×Q₈	—	2	1	no	—
80	C₈×Q₈	C₂×D₅	1	2	yes	—
81	#424	—	2	1	no	—
82	#423	Dic₅	1	1	yes	characteristic
83	#405	—	2	1	no	—
84	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
85	C₄⋊C₁₆	—	5	2	no	—
86	C₄⋊C₁₆	—	10	1	no	—
87	C₂²⋊C₁₆	—	10	1	no	—
88	C₂²⋊C₁₆	—	10	1	no	—
89	#346	—	10	1	no	—
90	C₄×C₁₆	—	5	2	no	—
91	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
92	C₅×Q₈	—	2	4	no	—
93	C₅×Q₈	—	4	2	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	4	1	no	—
100	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	—	4	2	no	—
110	C₅×C₈	#137	1	2	yes	—
111	C₂×M₁₆	—	2	1	no	—
112	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
114	C₄⋊Q₈	#168	1	1	yes	characteristic
115	C₄⋊Q₈	#168	1	1	yes	characteristic
116	#125	—	2	1	no	—
117	C₂²⋊Q₈	—	2	1	no	—
118	C₂²⋊Q₈	#168	1	2	yes	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	C₂×Dic₅	1	2	yes	—
121	#117	—	2	1	no	—
122	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
123	C₂×C₁₆	—	10	2	no	—
124	C₂×C₁₆	—	10	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
134	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
135	C₂²×C₅	—	2	1	no	—
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	4	2	no	—
144	C₄×C₅	—	4	2	no	—
145	C₄×C₅	#419	1	1	yes	characteristic
146	C₄×C₅	#421	1	1	yes	characteristic
147	C₄×C₅	#504	1	2	yes	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂×#168	1	2	yes	—
150	C₂²×C₄	C₂×#168	1	1	yes	characteristic
151	C₂²×C₄	#681	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	C₂×#168	1	2	yes	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂×#168	1	1	yes	characteristic
170	C₄²	#681	1	1	yes	characteristic
171	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
172	C₄²	C₂²×Dic₅	1	1	yes	characteristic
173	C₁₆	—	20	2	no	—
174	C₁₆	—	20	2	no	—
175	C₂×C₅	—	4	1	no	—
176	C₂×C₅	#1479	1	1	yes	characteristic
177	C₂×C₅	#2113	1	1	yes	characteristic
178	C₂×C₅	#2117	1	1	yes	characteristic
179	C₂³	C₂²⋊#168	1	1	yes	characteristic
180	Q₈	—	2	4	no	—
181	Q₈	—	4	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
189	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
191	C₂×C₄	C₂²×#168	1	1	yes	characteristic
192	C₂×C₄	C₂×#681	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	4	2	no	—
198	C₈	#4088	1	2	yes	—
199	C₅	#11640	1	1	yes	characteristic, complemented, Sylow
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	#65429	1	2	yes	—
211	C₄	#65514	1	1	yes	characteristic
212	C₄	#65525	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	#10602414	1	1	yes	characteristic
215	C₂	#10602797	1	1	yes	characteristic
216	C₂	#10602799	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	C₅×#2960	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#2960	D₅
hypocenter	C₅	#11640
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
4. Q₈⋊#672	C₂	semidirect
199. C₅	#11640	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2960	2
#10600538	2
#10600537	2
Q₈⋊#672	2
#11640	5