Subgroups — #12254658

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	#10610173	C₂	1	1	yes	maximal, characteristic
3	#10602854	C₂	1	4	yes	maximal
4	#10602104	C₂	1	4	yes	maximal, complemented
5	#10602103	C₂	1	4	yes	maximal, complemented
6	C₄×#3944	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
10	#65804	—	2	1	no	—
11	C₅×#494	—	2	1	no	—
12	C₅×#494	—	2	1	no	—
13	C₅×#494	C₂²	1	2	yes	—
14	C₅×#494	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65395	—	2	1	no	—
17	#65395	C₂²	1	2	yes	—
18	C₂×#3944	C₂²	1	2	yes	—
19	#64972	—	2	1	no	—
20	#64972	C₂²	1	2	yes	—
21	#64971	C₂²	1	2	yes	—
22	#64969	—	2	1	no	—
23	#64784	C₂²	1	16	yes	—
24	#64681	C₂²	1	2	yes	—
25	C₄×#672	C₂²	1	2	yes	—
26	#33705	—	5	1	no	maximal, Sylow
27	C₂²×C₅×Q₈	—	2	2	no	—
28	C₂×C₅×Q₁₆	—	2	2	no	—
29	C₂×C₅×Q₁₆	—	2	2	no	—
30	C₂×C₅×Q₁₆	—	2	2	no	—
31	C₂×C₅×Q₁₆	—	2	8	no	—
32	C₂×C₅×Q₁₆	—	4	4	no	—
33	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	—	2	2	no	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	—	4	2	no	—
39	C₂×C₅×C₄⋊C₄	—	4	2	no	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	#4072	—	2	2	no	—
42	#4070	—	4	2	no	—
43	#4016	—	2	2	no	—
44	#4015	—	2	2	no	—
45	C₄×Dic₁₀	—	4	2	no	—
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
50	#3950	—	2	2	no	—
51	#3950	C₂³	1	4	yes	—
52	#3949	—	2	2	no	—
53	#3949	—	2	2	no	—
54	#3944	—	2	4	no	—
55	C₂×#672	C₂³	1	4	yes	—
56	#3160	—	5	1	no	—
57	C₂×#494	D₅	1	1	yes	characteristic
58	#2245	—	5	4	no	—
59	#2241	—	5	4	no	—
60	#2141	—	5	4	no	—
61	C₄×M₃₂	—	5	1	no	—
62	C₂×C₅×Q₈	—	2	4	no	—
63	C₂×C₅×Q₈	—	4	2	no	—
64	C₂×C₅×Q₈	—	4	4	no	—
65	C₂×C₅×Q₈	—	4	4	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	D₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×Dic₁₀	—	4	2	no	—
70	C₅×Q₁₆	—	2	8	no	—
71	C₅×Q₁₆	—	4	4	no	—
72	C₅×Q₁₆	—	4	4	no	—
73	C₅×Q₁₆	—	4	4	no	—
74	C₂×C₅×C₈	SD₁₆	1	4	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	#41	1	4	yes	—
79	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₄²×C₅	D₈	1	2	yes	—
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²⋊Dic₅	—	4	2	no	—
86	C₄⋊Dic₅	—	2	4	no	—
87	C₄⋊Dic₅	—	4	2	no	—
88	#683	—	4	4	no	—
89	C₄×Dic₅	—	4	2	no	—
90	#672	—	2	4	no	—
91	C₂²×Q₁₆	—	2	1	no	—
92	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
93	#533	—	10	1	no	—
94	C₂×C₄⋊Q₈	—	2	1	no	—
95	C₂×M₃₂	—	5	2	no	—
96	#500	—	5	2	no	—
97	#499	—	5	2	no	—
98	#499	—	10	1	no	—
99	#498	—	10	1	no	—
100	#494	—	2	1	no	—
101	#494	—	2	1	no	—
102	#494	C₂×D₅	1	2	yes	—
103	#494	C₂×D₅	1	2	yes	—
104	#427	—	5	2	no	—
105	#427	—	10	1	no	—
106	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	#358	—	5	16	no	—
108	#346	—	5	2	no	—
109	C₄×C₁₆	—	5	2	no	—
110	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	4	no	—
112	C₅×Q₈	—	4	4	no	—
113	C₅×Q₈	—	8	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	4	4	no	—
116	C₂×C₄×C₅	—	8	4	no	—
117	C₂×C₄×C₅	#124	1	2	yes	—
118	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₅	—	4	4	no	—
124	Dic₁₀	—	8	4	no	—
125	C₅×C₈	#124	1	2	yes	—
126	C₅×C₈	C₂×SD₁₆	1	2	yes	—
127	C₅×C₈	C₂×SD₁₆	1	2	yes	—
128	C₅×C₈	C₂×#41	1	2	yes	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	8	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	4	4	no	—
135	C₂×Q₁₆	C₂²×D₅	1	4	yes	—
136	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	4	2	no	—
141	C₄⋊Q₈	—	10	2	no	—
142	#125	—	10	2	no	—
143	C₂²⋊Q₈	—	20	2	no	—
144	C₄×Q₈	—	20	2	no	—
145	#117	—	10	2	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
148	M₃₂	—	10	4	no	—
149	C₂×C₁₆	—	5	4	no	—
150	#107	—	5	4	no	—
151	#107	—	10	2	no	—
152	#106	—	10	2	no	—
153	#106	—	10	2	no	—
154	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	#486	1	2	yes	—
160	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	8	4	no	—
163	C₄×C₅	#486	1	2	yes	—
164	C₄×C₅	#510	1	4	yes	—
165	C₄×C₅	C₂×#124	1	1	yes	characteristic
166	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
167	Dic₅	—	8	4	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	20	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	#686	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
176	Q₁₆	—	2	8	no	—
177	Q₁₆	—	4	4	no	—
178	Q₁₆	—	4	4	no	—
179	Q₁₆	—	4	4	no	—
180	C₂×C₈	#687	1	4	yes	—
181	C₂×C₈	#711	1	4	yes	—
182	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
185	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	10	4	no	—
189	C₄⋊C₄	—	20	4	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₂²⋊C₄	—	20	2	no	—
192	C₄²	—	20	2	no	—
193	C₄²	#686	1	2	yes	—
194	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₁₆	—	10	4	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	C₂×#486	1	1	yes	characteristic
199	C₂×C₅	C₂×#510	1	2	yes	—
200	C₂³	C₂×#686	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	8	4	no	—
204	Q₈	—	40	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	—	20	4	no	—
209	C₂×C₄	C₂×#686	1	2	yes	—
210	C₂×C₄	C₂×#686	1	1	yes	characteristic
211	C₂×C₄	C₂×#686	1	2	yes	—
212	C₂×C₄	C₂×#687	1	2	yes	—
213	C₂×C₄	#4082	1	2	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
215	C₈	C₂×#687	1	2	yes	—
216	C₈	C₂×#687	1	2	yes	—
217	C₈	#4082	1	2	yes	—
218	C₈	C₂×#711	1	2	yes	—
219	C₅	#33705	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	#65359	1	2	yes	—
222	C₂²	C₂²×#686	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	40	4	no	—
226	C₄	#65359	1	2	yes	—
227	C₄	#65468	1	4	yes	—
228	C₄	C₂²×#687	1	1	yes	characteristic
229	C₄	C₂×#4082	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#65359	1	1	yes	characteristic
232	C₂	C₂×#65468	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂³×C₅	C₂×D₈
hypercenter	C₂×#494	D₅
hypocenter	C₅	#33705
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
4. #10602104	C₂	semidirect
5. #10602103	C₂	semidirect
219. C₅	#33705	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
#10610173	2
#10602854	2
#10602104	2
#10602103	2
C₄×#3944	2
#33705	5