Subgroups — #12014140

Order 1280.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 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	C₂²×#3944	C₂	1	2	yes	maximal
3	#10598148	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	—	2	4	no	—
8	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₂×#3944	—	2	4	no	—
11	C₂×#3944	—	2	4	no	—
12	C₂×#3944	C₄	1	8	yes	complemented
13	C₂²×#672	C₂²	1	4	yes	—
14	#12544	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	C₈	1	2	yes	—
16	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	C₈	1	4	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	4	yes	—
25	C₂²×C₅×C₈	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅	—	2	2	no	—
28	C₂×C₄²×C₅	—	2	2	no	—
29	C₂×C₄²×C₅	C₈	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₄×C₅×C₈	—	2	4	no	—
32	C₄×C₅×C₈	—	2	4	no	—
33	#3944	—	2	16	no	—
34	C₂×#672	—	2	4	no	—
35	C₂×#672	—	2	4	no	—
36	C₂×#672	C₂×C₄	1	8	yes	complemented
37	C₂²×M₃₂	—	5	2	no	—
38	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
39	#1374	—	5	4	no	—
40	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅	—	2	4	no	—
42	C₂²×C₄×C₅	—	2	8	no	—
43	C₂²×C₄×C₅	C₄²	1	2	yes	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅	M₁₆	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₄²	1	2	yes	—
56	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
59	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅	—	2	4	no	—
66	C₄²×C₅	—	2	2	no	—
67	C₄²×C₅	—	2	2	no	—
68	#672	—	2	8	no	—
69	C₂³×C₈	Dic₅	1	1	yes	characteristic
70	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₄²	Dic₅	1	1	yes	characteristic
72	C₂×M₃₂	—	5	8	no	—
73	C₂×M₃₂	—	10	4	no	—
74	C₂×M₃₂	—	10	4	no	—
75	C₂²×C₁₆	—	5	4	no	—
76	C₂×C₄×C₈	—	2	4	no	—
77	C₂×C₄×C₈	Dic₅	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
79	C₂³×C₅	—	2	4	no	—
80	C₂³×C₅	C₄×C₈	1	2	yes	—
81	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
82	C₂³×C₅	C₄⋊C₈	1	2	yes	—
83	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	16	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	#95	1	4	yes	—
88	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₅	#97	1	4	yes	—
90	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
94	C₂×C₄×C₅	M₃₂	1	8	yes	complemented
95	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₅×C₈	—	2	4	no	—
102	C₅×C₈	#95	1	4	yes	—
103	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
104	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₄	#168	1	2	yes	—
106	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	#168	1	4	yes	—
110	C₂²×C₈	Dic₁₀	1	2	yes	—
111	C₂²×C₈	C₄×D₅	1	4	yes	—
112	C₂²×C₈	D₂₀	1	2	yes	—
113	C₂²×C₈	C₂×Dic₅	1	2	yes	—
114	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
116	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
117	C₂×C₄²	—	2	2	no	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	#168	1	2	yes	—
120	C₂×C₄²	C₂×Dic₅	1	2	yes	—
121	M₃₂	—	10	16	no	—
122	C₂×C₁₆	—	5	8	no	—
123	C₂×C₁₆	—	10	4	no	—
124	C₂×C₁₆	—	10	4	no	—
125	C₄×C₈	—	2	4	no	—
126	C₄×C₈	—	2	4	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	—	2	4	no	—
130	C₂²×C₅	#336	1	4	yes	—
131	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₅	C₂×M₃₂	1	4	yes	—
135	C₄×C₅	—	2	8	no	—
136	C₄×C₅	#336	1	4	yes	—
137	C₄×C₅	C₂×#95	1	1	yes	characteristic
138	C₄×C₅	C₂×#97	1	1	yes	characteristic
139	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂⁴	C₂×#168	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	C₂×#168	1	4	yes	—
145	C₂²×C₄	C₂×#168	1	1	yes	characteristic
146	C₂²×C₄	C₂×#168	1	2	yes	—
147	C₂²×C₄	#681	1	2	yes	—
148	C₂²×C₄	#681	1	2	yes	—
149	C₂²×C₄	C₄×Dic₅	1	2	yes	—
150	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
151	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×#168	1	4	yes	—
157	C₂×C₈	C₄×Dic₅	1	2	yes	—
158	C₂×C₈	#683	1	8	yes	—
159	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
160	C₂×C₈	#685	1	8	yes	—
161	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
162	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
164	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₁₆	—	10	8	no	—
170	C₂×C₅	—	2	4	no	—
171	C₂×C₅	C₂×#336	1	1	yes	characteristic
172	C₂×C₅	C₄×M₃₂	1	2	yes	—
173	C₂×C₅	#2112	1	2	yes	—
174	C₂×C₅	#2150	1	2	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₄×#168	1	2	yes	—
177	C₂³	C₄⋊#168	1	2	yes	—
178	C₂³	C₂²⋊#168	1	2	yes	—
179	C₂³	C₂²×#168	1	1	yes	characteristic
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₄×#168	1	2	yes	—
184	C₂×C₄	#3935	1	4	yes	—
185	C₂×C₄	C₄⋊#168	1	4	yes	—
186	C₂×C₄	C₄⋊#168	1	2	yes	—
187	C₂×C₄	#3944	1	8	yes	complemented
188	C₂×C₄	C₂²⋊#168	1	4	yes	—
189	C₂×C₄	C₂²⋊#168	1	2	yes	—
190	C₂×C₄	#3963	1	4	yes	—
191	C₂×C₄	C₂²×#168	1	1	yes	characteristic
192	C₂×C₄	C₂×#681	1	1	yes	characteristic
193	C₂×C₄	C₂×#681	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	#3963	1	4	yes	—
199	C₈	C₂×#683	1	2	yes	—
200	C₈	C₂×#685	1	2	yes	—
201	C₅	#12544	1	1	yes	characteristic, complemented, Sylow
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#64744	1	4	yes	—
206	C₂²	C₂×C₄×#168	1	1	yes	characteristic
207	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
208	C₂²	C₂×#3944	1	4	yes	—
209	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	#64744	1	4	yes	—
212	C₄	C₂×#3935	1	1	yes	characteristic
213	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊#168	1	1	yes	characteristic
215	C₄	C₂×#3963	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₄×#3944	1	2	yes	—
218	C₂	#10600536	1	2	yes	—
219	C₂	C₂×#64744	1	1	yes	characteristic
220	C₂	#10602756	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#12544
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #10598148	C₂	semidirect
12. C₂×#3944	C₄	semidirect
36. C₂×#672	C₂×C₄	semidirect
94. C₂×C₄×C₅	M₃₂	semidirect
187. C₂×C₄	#3944	semidirect
201. C₅	#12544	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
C₂²×#3944	2
#10598148	2
#12544	5