Subgroups — #12014628

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×#64732	C₂	1	2	yes	maximal
3	#10600542	C₂	1	1	yes	maximal, characteristic
4	#10600536	C₂	1	1	yes	maximal, characteristic
5	#10598302	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₂²	1	1	yes	characteristic
9	C₂×#3947	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	#65221	C₄	1	1	yes	characteristic
14	C₂×#3935	C₄	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	C₄	1	4	yes	complemented
16	#64732	—	2	4	no	—
17	C₄⋊#672	C₄	1	4	yes	complemented
18	#12657	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂×C₄	1	1	yes	characteristic
26	C₅×M₃₂	—	2	2	no	—
27	C₅×M₃₂	—	2	2	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
31	#3963	—	2	2	no	—
32	#3947	—	2	1	no	—
33	#3947	—	2	1	no	—
34	#3947	D₄	1	2	yes	—
35	#3947	Q₈	1	2	yes	—
36	C₈×Dic₅	—	2	2	no	—
37	C₈×Dic₅	—	2	2	no	—
38	#3944	—	2	2	no	—
39	#3944	—	2	2	no	—
40	C₂×#672	C₂×C₄	1	4	yes	—
41	#3935	—	2	2	no	—
42	#2150	—	5	1	no	—
43	#2150	D₅	1	1	yes	characteristic
44	C₂×#347	—	5	2	no	—
45	#1726	—	5	1	no	—
46	#1375	—	5	2	no	—
47	C₂²×C₄×C₅	M₁₆	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅	—	2	1	no	—
50	C₂²×Dic₅	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	C₄²	1	2	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₂×C₅×C₈	#41	1	1	yes	characteristic
60	C₄²×C₅	C₄²	1	2	yes	—
61	C₄²×C₅	M₁₆	1	2	yes	—
62	C₄×Dic₅	—	2	1	no	—
63	C₄×Dic₅	—	2	1	no	—
64	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
65	C₄×Dic₅	M₁₆	1	2	yes	—
66	C₂×#168	—	2	1	no	—
67	C₂×#168	—	2	1	no	—
68	C₂×#168	—	2	1	no	—
69	C₂×#168	—	2	1	no	—
70	C₂×#168	C₄⋊C₄	1	2	yes	—
71	C₂×#168	M₁₆	1	2	yes	—
72	C₅×C₁₆	—	2	4	no	—
73	#672	—	2	4	no	—
74	C₂×M₃₂	—	5	2	no	—
75	C₂×M₃₂	C₂×D₅	1	2	yes	—
76	C₂×#97	—	5	1	no	—
77	C₂×#97	—	5	1	no	—
78	C₂×#97	—	5	1	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	#376	—	5	1	no	—
83	C₄⋊C₁₆	—	5	4	no	—
84	C₄⋊C₁₆	Dic₅	1	4	yes	complemented
85	#347	—	10	4	no	—
86	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	#97	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	#117	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
95	C₂×Dic₅	—	2	2	no	—
96	C₂×Dic₅	—	2	2	no	—
97	C₂×Dic₅	—	4	2	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₅×C₈	—	2	2	no	—
100	C₅×C₈	C₄×D₄	1	1	yes	characteristic
101	C₅×C₈	C₄×D₄	1	1	yes	characteristic
102	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
103	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
104	#168	—	2	2	no	—
105	#168	—	2	2	no	—
106	C₂²×C₈	—	5	1	no	—
107	C₂²×C₈	—	5	1	no	—
108	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
110	C₂×C₄²	—	5	1	no	—
111	C₂×C₄²	—	5	1	no	—
112	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
113	M₃₂	—	2	2	no	—
114	M₃₂	—	2	2	no	—
115	M₃₂	—	10	2	no	—
116	M₃₂	—	10	2	no	—
117	C₂×C₁₆	—	5	4	no	—
118	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
119	#97	—	5	2	no	—
120	#97	—	5	2	no	—
121	#97	—	10	1	no	—
122	#97	—	10	1	no	—
123	#97	—	10	2	no	—
124	C₄×C₈	—	10	2	no	—
125	C₄×C₈	—	10	2	no	—
126	C₄×C₈	C₄×D₅	1	2	yes	—
127	C₄×C₈	C₂×Dic₅	1	2	yes	—
128	#95	—	10	2	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	C₂×#97	1	1	yes	characteristic
131	C₂²×C₅	#423	1	1	yes	characteristic
132	C₂²×C₅	#432	1	1	yes	characteristic
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
136	C₄×C₅	#423	1	1	yes	characteristic
137	C₄×C₅	#432	1	1	yes	characteristic
138	Dic₅	—	4	2	no	—
139	Dic₅	—	4	2	no	—
140	C₂²×C₄	—	10	1	no	—
141	C₂²×C₄	—	10	1	no	—
142	C₂²×C₄	#676	1	2	yes	—
143	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	5	2	no	—
147	C₂×C₈	—	5	2	no	—
148	C₂×C₈	—	10	1	no	—
149	C₂×C₈	—	10	1	no	—
150	C₂×C₈	—	10	1	no	—
151	C₂×C₈	—	10	1	no	—
152	C₂×C₈	C₄×Dic₅	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
154	C₂×C₈	D₄×D₅	1	1	yes	characteristic
155	C₂×C₈	#711	1	1	yes	characteristic
156	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
157	C₂×C₈	#713	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
159	C₄²	—	5	2	no	—
160	C₄²	—	5	2	no	—
161	C₄²	—	10	1	no	—
162	C₄²	—	10	1	no	—
163	C₄²	#676	1	2	yes	—
164	C₄²	C₄×Dic₅	1	2	yes	—
165	C₁₆	—	2	4	no	—
166	C₁₆	—	10	4	no	—
167	C₂×C₅	—	2	1	no	—
168	C₂×C₅	—	2	1	no	—
169	C₂×C₅	#1767	1	1	yes	characteristic
170	C₂×C₅	#2110	1	2	yes	—
171	C₂³	C₂×#676	1	1	yes	characteristic
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	10	2	no	—
174	C₂×C₄	—	10	2	no	—
175	C₂×C₄	—	20	2	no	—
176	C₂×C₄	—	20	2	no	—
177	C₂×C₄	#3947	1	4	yes	—
178	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄	#4038	1	1	yes	characteristic
180	C₂×C₄	C₂×#676	1	1	yes	characteristic
181	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
182	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	10	2	no	—
186	C₈	—	10	2	no	—
187	C₈	#4033	1	1	yes	characteristic
188	C₈	#4039	1	1	yes	characteristic
189	C₈	D₄×Dic₅	1	1	yes	characteristic
190	C₈	Q₈×Dic₅	1	1	yes	characteristic
191	C₅	#12657	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	#65124	1	1	yes	characteristic
194	C₂²	#65126	1	1	yes	characteristic
195	C₂²	C₂×#3947	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	20	2	no	—
198	C₄	—	20	2	no	—
199	C₄	#65124	1	1	yes	characteristic
200	C₄	#65126	1	1	yes	characteristic
201	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
202	C₄	M₁₆×Dic₅	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#10601234	1	1	yes	characteristic
206	C₂	#10602218	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2150	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2150	D₅
hypocenter	C₅	#12657
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
15. C₅×C₄⋊C₁₆	C₄	semidirect
17. C₄⋊#672	C₄	semidirect
84. C₄⋊C₁₆	Dic₅	semidirect
191. C₅	#12657	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2150	2
C₂×#64732	2
#10600542	2
#10600536	2
#10598302	2
#12657	5