Subgroups — #14312878

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#901	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊#672	C₂	1	4	yes	maximal, complemented
5	#10602797	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₅×#901	—	2	2	no	—
10	C₅×#901	—	2	2	no	—
11	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₂×C₃⋊#672	—	2	2	no	—
14	C₂×C₃⋊#672	C₂²	1	2	yes	—
15	C₃×#3944	—	2	2	no	—
16	C₂×C₃×#672	C₂²	1	2	yes	—
17	#71432	—	5	1	no	maximal, Hall
18	C₂×C₅×#131	—	3	1	no	—
19	C₂×#3944	D₃	1	1	yes	characteristic, complemented
20	C₂²×#672	—	3	1	no	—
21	C₂²⋊#672	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×#67	C₈	1	2	yes	—
24	C₅×#235	—	2	2	no	—
25	C₅×#235	—	2	2	no	—
26	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
27	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
28	C₂×C₅×Dic₆	C₈	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×#208	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
36	C₅×D₃×C₈	—	2	2	no	—
37	C₅×D₃×C₈	—	2	2	no	—
38	C₃⋊#672	—	2	2	no	—
39	C₃×#672	—	2	2	no	—
40	C₂×#901	D₅	1	1	yes	characteristic
41	C₂×C₃×M₃₂	—	5	1	no	—
42	C₂²×C₃⋊C₁₆	—	5	1	no	—
43	(C₂×D₃)⋊C₁₆	—	5	4	no	—
44	C₂×C₅×#41	—	3	1	no	—
45	C₅×#131	—	6	2	no	—
46	C₅×#131	—	6	2	no	—
47	C₂×C₅×M₁₆	—	3	1	no	—
48	C₂×C₅×M₁₆	—	3	2	no	—
49	C₂²×C₅×C₈	—	3	2	no	—
50	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#3944	—	2	2	no	—
52	C₂×#672	—	3	2	no	—
53	C₂×#672	—	6	2	no	—
54	C₂×#672	C₂×D₃	1	2	yes	—
55	#2113	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₅×#67	—	2	4	no	—
63	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
64	C₅×D₁₂	—	2	2	no	—
65	C₄×C₅×D₃	—	2	2	no	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₅×Dic₆	—	2	2	no	—
68	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₃⋊C₈	—	2	2	no	—
72	C₂×#235	Dic₅	1	1	yes	characteristic
73	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×#208	Dic₅	1	1	yes	characteristic
75	#901	—	2	2	no	—
76	#901	—	2	2	no	—
77	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	2	yes	—
78	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
79	C₃×M₃₂	—	10	2	no	—
80	C₂×C₃×C₁₆	—	5	2	no	—
81	C₂×C₃⋊C₁₆	—	5	2	no	—
82	C₂×C₃⋊C₁₆	—	10	2	no	—
83	C₅×#41	—	6	2	no	—
84	C₅×#41	—	6	2	no	—
85	C₂×C₅×Q₈	—	3	1	no	—
86	C₂×C₅×D₄	—	3	2	no	—
87	C₂×C₅×D₄	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	2	no	—
89	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
90	C₅×M₁₆	—	6	2	no	—
91	C₅×M₁₆	—	6	4	no	—
92	C₂×C₅×C₈	—	3	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	—	6	2	no	—
95	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₅×C₈	D₁₂	1	2	yes	—
97	C₂×C₅×C₈	#67	1	2	yes	—
98	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
99	#672	—	2	2	no	—
100	#672	—	6	2	no	—
101	C₂×#131	—	3	1	no	—
102	C₂×M₃₂	—	5	1	no	—
103	C₂²×C₁₆	—	15	1	no	—
104	C₂²⋊C₁₆	—	15	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
107	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	2	no	—
109	C₂×C₅×D₃	—	2	4	no	—
110	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
111	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₅×Dic₃	—	2	2	no	—
114	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
115	C₂×#67	#168	1	2	yes	—
116	#235	—	2	2	no	—
117	#235	—	2	2	no	—
118	C₂×D₁₂	#168	1	1	yes	characteristic
119	C₂×C₄×D₃	C₂×Dic₅	1	2	yes	—
120	C₂×Dic₆	#168	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
124	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
125	#208	—	2	2	no	—
126	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
127	ΓL(1,𝔽₂₅)	—	2	4	no	—
128	D₃×C₈	—	2	2	no	—
129	D₃×C₈	—	2	2	no	—
130	C₃×C₁₆	—	10	2	no	—
131	C₃⋊C₁₆	—	10	2	no	—
132	C₂³×C₅	—	3	2	no	—
133	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
134	C₅×Q₈	—	6	2	no	—
135	C₅×D₄	—	6	4	no	—
136	C₅×D₄	—	6	2	no	—
137	C₂×C₄×C₅	—	3	2	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	—	6	2	no	—
140	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
142	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₅×C₈	—	6	2	no	—
145	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
146	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
147	C₅×C₈	C₂×#67	1	1	yes	characteristic
148	C₂×#41	—	3	1	no	—
149	#131	—	6	2	no	—
150	#131	—	6	2	no	—
151	C₂×M₁₆	—	3	1	no	—
152	C₂×M₁₆	—	3	2	no	—
153	C₂²×C₈	—	3	2	no	—
154	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
155	M₃₂	—	10	2	no	—
156	C₂×C₁₆	—	5	2	no	—
157	C₂×C₁₆	—	15	2	no	—
158	C₂×C₁₆	—	30	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₃×C₅	#504	1	2	yes	—
162	C₅×D₃	—	4	2	no	—
163	C₂³×C₃	C₂×#168	1	1	yes	characteristic
164	C₂²×D₃	C₂×#168	1	2	yes	—
165	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
166	C₂×C₃×C₄	#681	1	2	yes	—
167	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
168	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
169	#67	—	2	4	no	—
170	C₂×Dic₃	C₂×#168	1	2	yes	—
171	D₁₂	—	2	2	no	—
172	C₄×D₃	—	2	2	no	—
173	C₄×D₃	—	2	2	no	—
174	Dic₆	—	2	2	no	—
175	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
176	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
178	C₃⋊C₈	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	6	4	no	—
181	C₂²×C₅	—	6	2	no	—
182	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
183	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
186	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
187	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	#41	—	6	2	no	—
189	#41	—	6	2	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×D₄	—	3	2	no	—
192	C₂×D₄	—	3	1	no	—
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
195	M₁₆	—	6	2	no	—
196	M₁₆	—	6	4	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	6	2	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
201	C₂×C₈	#1201	1	2	yes	—
202	C₂×C₈	#1203	1	2	yes	—
203	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
204	C₁₆	—	10	2	no	—
205	C₁₆	—	30	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	C₂²⋊#168	1	2	yes	—
209	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
210	C₂×D₃	—	2	4	no	—
211	C₂×D₃	—	2	2	no	—
212	C₃×C₄	C₂²⋊#168	1	2	yes	—
213	C₃×C₄	C₂×#681	1	1	yes	characteristic
214	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
215	Dic₃	—	2	2	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	12	2	no	—
218	C₂×C₅	#4907	1	2	yes	—
219	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
220	C₂³	—	3	2	no	—
221	C₂³	D₃×#168	1	1	yes	characteristic
222	Q₈	—	6	2	no	—
223	D₄	—	6	4	no	—
224	D₄	—	6	2	no	—
225	C₂×C₄	—	3	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	D₃×#168	1	1	yes	characteristic
229	C₂×C₄	#6650	1	2	yes	—
230	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
231	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
232	C₈	—	6	2	no	—
233	C₈	(C₂×D₃)⋊Dic₅	1	2	yes	—
234	C₈	C₂×#1201	1	1	yes	characteristic
235	C₈	C₂×#1203	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	#65429	1	2	yes	—
238	C₂×C₃	C₂×C₂²⋊#168	1	1	yes	characteristic
239	D₃	—	4	2	no	—
240	C₅	#71432	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	6	4	no	—
243	C₂²	—	6	2	no	—
244	C₂²	(C₂×D₃)⋊#168	1	2	yes	—
245	C₂²	C₂×D₃×#168	1	1	yes	characteristic
246	C₄	—	6	2	no	—
247	C₄	(C₂×D₃)⋊#168	1	2	yes	—
248	C₄	C₂×#6650	1	1	yes	characteristic
249	C₄	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
250	C₃	#10602797	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	12	2	no	—
253	C₂	#11747529	1	2	yes	—
254	C₂	C₂×(C₂×D₃)⋊#168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₅×C₈	C₂²
prime radical	C₂×C₅×#235	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#672	C₂	semidirect
3. C₂×C₃×#3944	C₂	semidirect
4. (C₂×D₃)⋊#672	C₂	semidirect
19. C₂×#3944	D₃	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	#71432	semidirect
250. C₃	#10602797	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#901	2
C₂²×C₃⋊#672	2
C₂×C₃×#3944	2
(C₂×D₃)⋊#672	2
#10602797	3
#71432	5