Subgroups — #11993044

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
2	#10601236	C₂	1	2	yes	maximal
3	C₂×#64700	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₄²×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	C₅×#335	—	2	2	no	—
16	#64770	—	2	4	no	—
17	#64700	—	2	4	no	—
18	#64678	—	2	2	no	—
19	#8807	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	D₄	1	2	yes	—
27	C₂×C₄×Dic₅	Q₈	1	2	yes	—
28	C₂²×#168	—	2	2	no	—
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₄×C₅×C₈	D₄	1	2	yes	—
34	C₄×C₅×C₈	Q₈	1	2	yes	—
35	C₈×Dic₅	—	2	2	no	—
36	C₈×Dic₅	—	2	2	no	—
37	C₈×Dic₅	—	2	2	no	—
38	C₈×Dic₅	C₈	1	4	yes	complemented
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	—	2	2	no	—
41	C₄×#168	D₄	1	2	yes	—
42	C₄×#168	Q₈	1	2	yes	—
43	#1831	—	5	2	no	—
44	C₄²×C₈	—	5	1	no	—
45	C₂×#335	—	5	1	no	—
46	C₂×#335	D₅	1	1	yes	characteristic
47	C₂×#334	—	5	2	no	—
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	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
52	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
53	C₂×C₅×C₈	—	2	1	no	—
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	4	no	—
57	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
60	C₂×C₅×C₈	M₁₆	1	2	yes	—
61	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₅	—	2	1	no	—
66	C₄×Dic₅	—	2	1	no	—
67	C₄×Dic₅	—	2	2	no	—
68	C₄×Dic₅	—	2	2	no	—
69	C₄×Dic₅	—	2	4	no	—
70	C₄×Dic₅	C₂×C₈	1	2	yes	—
71	C₄×Dic₅	M₁₆	1	2	yes	—
72	C₂×#168	—	2	1	no	—
73	C₂×#168	—	2	1	no	—
74	C₂×#168	—	4	2	no	—
75	C₂×#168	—	4	4	no	—
76	C₂×#168	C₄⋊C₄	1	2	yes	—
77	C₂×#168	C₄⋊C₄	1	2	yes	—
78	C₂×#168	C₂×C₈	1	2	yes	—
79	C₂×#168	M₁₆	1	2	yes	—
80	C₂×C₄⋊C₈	—	5	2	no	—
81	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
82	C₂×C₄×C₈	—	5	1	no	—
83	C₂×C₄×C₈	—	5	1	no	—
84	C₂×C₄×C₈	—	5	1	no	—
85	C₂×C₄×C₈	—	5	1	no	—
86	C₂×C₄×C₈	—	5	1	no	—
87	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	C₄³	—	5	1	no	—
89	#336	—	10	4	no	—
90	#335	—	2	2	no	—
91	#335	—	10	2	no	—
92	#334	—	10	4	no	—
93	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	#127	1	1	yes	characteristic
98	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅	#135	1	2	yes	—
106	C₂×C₄×C₅	#135	1	2	yes	—
107	C₂×Dic₅	—	2	2	no	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	2	2	no	—
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	C₄⋊C₈	1	4	yes	—
112	C₂×Dic₅	#108	1	4	yes	—
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	8	2	no	—
115	C₅×C₈	C₄⋊C₈	1	4	yes	—
116	#168	—	2	2	no	—
117	#168	—	8	2	no	—
118	#168	C₄⋊C₈	1	4	yes	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	5	1	no	—
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	—	10	2	no	—
123	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
124	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	—	5	1	no	—
127	C₂×C₄²	—	5	2	no	—
128	C₂×C₄²	—	5	2	no	—
129	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	10	2	no	—
133	C₄⋊C₈	—	10	2	no	—
134	C₄×C₈	—	5	4	no	—
135	C₄×C₈	—	5	2	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	10	2	no	—
138	C₄×C₈	—	10	2	no	—
139	C₄×C₈	—	10	2	no	—
140	C₄×C₈	Dic₁₀	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅	1	2	yes	—
142	C₂²×C₅	#384	1	1	yes	characteristic
143	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₅	#423	1	1	yes	characteristic
145	C₂²×C₅	#427	1	2	yes	—
146	C₂²×C₅	C₂×#108	1	2	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₄×C₅	#423	1	1	yes	characteristic
152	C₄×C₅	#423	1	1	yes	characteristic
153	C₄×C₅	#495	1	1	yes	characteristic
154	C₄×C₅	#495	1	1	yes	characteristic
155	C₄×C₅	#499	1	2	yes	—
156	Dic₅	—	2	4	no	—
157	Dic₅	—	2	4	no	—
158	C₂²×C₄	—	5	2	no	—
159	C₂²×C₄	—	5	2	no	—
160	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
162	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	5	2	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	10	1	no	—
172	C₂×C₈	—	10	1	no	—
173	C₂×C₈	—	20	4	no	—
174	C₂×C₈	—	20	2	no	—
175	C₂×C₈	C₈×D₅	1	2	yes	—
176	C₂×C₈	#676	1	2	yes	—
177	C₂×C₈	#683	1	2	yes	—
178	C₂×C₈	#683	1	2	yes	—
179	C₄²	—	5	2	no	—
180	C₄²	—	5	2	no	—
181	C₄²	—	10	4	no	—
182	C₄²	—	10	1	no	—
183	C₄²	—	10	1	no	—
184	C₄²	—	10	2	no	—
185	C₄²	—	10	2	no	—
186	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
187	C₄²	D₄×D₅	1	1	yes	characteristic
188	C₄²	Q₈×D₅	1	1	yes	characteristic
189	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₅	#1564	1	2	yes	—
191	C₂×C₅	#1568	1	2	yes	—
192	C₂×C₅	#1842	1	1	yes	characteristic
193	C₂×C₅	#1846	1	1	yes	characteristic
194	C₂×C₅	#1848	1	1	yes	characteristic
195	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	5	4	no	—
198	C₂×C₄	—	5	4	no	—
199	C₂×C₄	—	10	2	no	—
200	C₂×C₄	—	10	2	no	—
201	C₂×C₄	—	10	2	no	—
202	C₂×C₄	—	10	2	no	—
203	C₂×C₄	#4034	1	1	yes	characteristic
204	C₂×C₄	#4034	1	1	yes	characteristic
205	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
207	C₂×C₄	C₂×#676	1	1	yes	characteristic
208	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
209	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
210	C₂×C₄	#4058	1	2	yes	—
211	C₂×C₄	#4065	1	2	yes	—
212	C₂×C₄	C₂×#683	1	1	yes	characteristic
213	C₂×C₄	#4086	1	1	yes	characteristic
214	C₂×C₄	#4090	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	5	4	no	—
217	C₈	—	8	2	no	—
218	C₈	—	10	2	no	—
219	C₈	—	40	2	no	—
220	C₈	#3946	1	4	yes	—
221	C₅	#8807	1	1	yes	characteristic, complemented, Sylow
222	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
223	C₂²	#65126	1	1	yes	characteristic
224	C₂²	#65169	1	2	yes	—
225	C₂²	D₅×#108	1	2	yes	—
226	C₂²	#65265	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	10	4	no	—
230	C₄	—	10	4	no	—
231	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
232	C₄	#65126	1	1	yes	characteristic
233	C₄	#65167	1	2	yes	—
234	C₄	C₂×#3946	1	1	yes	characteristic
235	C₄	#65407	1	1	yes	characteristic
236	C₄	#65444	1	1	yes	characteristic
237	C₄	#65480	1	1	yes	characteristic
238	C₂	#10599360	1	2	yes	—
239	C₂	#10599363	1	2	yes	—
240	C₂	#10601238	1	1	yes	characteristic
241	C₂	#10601400	1	1	yes	characteristic
242	C₂	#10601486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#335	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#335	D₅
hypocenter	C₅	#8807
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
38. C₈×Dic₅	C₈	semidirect
221. C₅	#8807	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#335	2
#10601236	2
C₂×#64700	2
C₄×C₈×Dic₅	2
C₂×#64678	2
#8807	5