Subgroups — #11988769

Order 1280.

670 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	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65215	C₂	1	2	yes	maximal
3	#10598145	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	—	2	4	no	—
7	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₂²⋊#168	—	2	2	no	—
10	#65215	—	2	4	no	—
11	#65215	C₄	1	8	yes	—
12	C₂×C₄⋊#168	C₂²	1	4	yes	—
13	C₂×#3935	—	2	2	no	—
14	#7875	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	—	2	8	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	4	yes	—
25	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₂²×#168	—	2	4	no	—
28	C₄×C₅×C₈	—	2	4	no	—
29	C₄×C₅×C₈	—	2	4	no	—
30	C₂²⋊#168	—	2	8	no	—
31	C₄⋊#168	—	2	4	no	—
32	C₄⋊#168	C₂×C₄	1	8	yes	—
33	#3935	—	2	8	no	—
34	C₂×#432	—	5	2	no	—
35	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
36	#1277	—	5	4	no	—
37	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	—	2	4	no	—
39	C₂²×C₄×C₅	—	2	4	no	—
40	C₂²×C₄×C₅	—	2	4	no	—
41	C₂²×C₄×C₅	C₄²	1	2	yes	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₅×C₈	—	2	16	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	M₁₆	1	8	yes	—
55	C₄²×C₅	C₄²	1	2	yes	—
56	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅	C₂²×C₄	1	2	yes	—
60	C₄²×C₅	C₂×D₄	1	2	yes	—
61	C₂×#168	—	2	8	no	—
62	C₂×#168	—	4	8	no	—
63	C₂³×C₈	Dic₅	1	2	yes	—
64	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
65	#432	—	5	8	no	—
66	#432	—	10	4	no	—
67	C₂×C₄⋊C₈	—	5	4	no	—
68	C₂×C₂²⋊C₈	—	10	2	no	—
69	C₂×#97	—	10	2	no	—
70	C₂×C₄×C₈	—	2	4	no	—
71	C₂×C₄×C₈	Dic₅	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
73	C₂³×C₅	—	2	2	no	—
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	#95	1	2	yes	—
76	C₂³×C₅	#108	1	4	yes	—
77	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	8	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	#95	1	2	yes	—
87	C₂×C₄×C₅	#95	1	2	yes	—
88	C₂×C₄×C₅	#95	1	2	yes	—
89	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
94	C₂×C₄×C₅	#135	1	4	yes	—
95	C₅×C₈	—	2	8	no	—
96	#168	—	4	8	no	—
97	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
98	C₂³×C₄	D₂₀	1	1	yes	characteristic
99	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	—	10	4	no	—
103	C₂²×C₈	C₂×Dic₅	1	4	yes	—
104	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
105	C₂×C₄²	C₄×D₅	1	4	yes	—
106	C₂×C₄²	D₂₀	1	1	yes	characteristic
107	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
108	C₂×C₄²	C₂²⋊D₅	1	4	yes	—
109	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
110	C₄⋊C₈	—	5	8	no	—
111	C₄⋊C₈	—	10	4	no	—
112	C₂²⋊C₈	—	10	8	no	—
113	#97	—	10	8	no	—
114	C₄×C₈	—	2	4	no	—
115	C₄×C₈	—	2	4	no	—
116	C₂²×C₅	—	2	8	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	#341	1	4	yes	—
120	C₂²×C₅	C₂×#95	1	1	yes	characteristic
121	C₂²×C₅	#423	1	2	yes	—
122	C₂²×C₅	#427	1	2	yes	—
123	C₂²×C₅	C₂×#108	1	2	yes	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	C₄×M₁₆	1	2	yes	—
129	C₄×C₅	#407	1	2	yes	—
130	C₄×C₅	#416	1	4	yes	—
131	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₄×Dic₅	1	2	yes	—
136	C₂²×C₄	#683	1	4	yes	—
137	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
138	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
140	C₂²×C₄	#685	1	4	yes	—
141	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
145	C₂×C₈	—	2	16	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	10	8	no	—
149	C₂×C₈	—	20	8	no	—
150	C₂×C₈	#681	1	8	yes	—
151	C₄²	C₄×Dic₅	1	2	yes	—
152	C₄²	#683	1	4	yes	—
153	C₄²	#685	1	4	yes	—
154	C₄²	C₂²⋊Dic₅	1	2	yes	—
155	C₄²	C₂×C₄×D₅	1	2	yes	—
156	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
157	C₂×C₅	—	2	4	no	—
158	C₂×C₅	#1568	1	4	yes	—
159	C₂×C₅	#1728	1	1	yes	characteristic
160	C₂×C₅	#1735	1	1	yes	characteristic
161	C₂×C₅	C₂×#341	1	1	yes	characteristic
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	#3951	1	4	yes	—
165	C₂³	#3963	1	2	yes	—
166	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	#3963	1	2	yes	—
176	C₂×C₄	#3963	1	2	yes	—
177	C₂×C₄	#3963	1	2	yes	—
178	C₂×C₄	#4050	1	4	yes	—
179	C₂×C₄	C₂×#681	1	4	yes	—
180	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
181	C₂×C₄	C₂×#683	1	2	yes	—
182	C₂×C₄	C₂×#685	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
184	C₈	—	2	8	no	—
185	C₈	—	20	8	no	—
186	C₅	#7875	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#64772	1	4	yes	—
191	C₂²	#65213	1	2	yes	—
192	C₂²	#65395	1	2	yes	—
193	C₂²	C₂×#3951	1	2	yes	—
194	C₂²	C₂×#3963	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	C₄×#681	1	2	yes	—
200	C₄	#65402	1	4	yes	—
201	C₄	#65496	1	2	yes	—
202	C₂	—	2	4	no	—
203	C₂	#10600525	1	4	yes	—
204	C₂	#10602320	1	1	yes	characteristic
205	C₂	#10602762	1	1	yes	characteristic
206	C₂	C₂×#64772	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#7875
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #10598145	C₂	semidirect
186. C₅	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×#65215	2
#10598145	2
#7875	5