Subgroups — #14314341

Order 1920.

344 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	C₄×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#90936	C₂	1	2	yes	maximal
3	C₅×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745928	C₂	1	2	yes	maximal, complemented
6	#10600077	—	3	1	no	maximal, Hall
7	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
9	C₄²×D₁₅	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₅	C₄	1	1	yes	characteristic
12	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#168	C₄	1	1	yes	characteristic
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₃×#672	C₂²	1	2	yes	—
16	#90936	—	2	4	no	—
17	#69284	—	5	1	no	maximal, Hall
18	C₂×#3930	—	3	2	no	—
19	C₄×C₈×D₅	—	3	1	no	—
20	C₅×C₄⋊C₁₆	—	3	1	no	—
21	#64721	—	3	2	no	—
22	C₄⋊#672	D₃	1	1	yes	characteristic, complemented
23	C₂×C₄×D₁₅	C₈	1	1	yes	characteristic
24	C₂×C₄×D₁₅	C₈	1	1	yes	characteristic
25	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₅	C₈	1	1	yes	characteristic
30	C₄×Dic₁₅	C₈	1	1	yes	characteristic
31	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
34	C₈×D₁₅	—	2	2	no	—
35	C₈×D₁₅	D₄	1	2	yes	—
36	C₈×D₁₅	Q₈	1	2	yes	—
37	C₃×#672	—	2	2	no	—
38	C₅×C₃⋊C₁₆	—	2	2	no	—
39	C₂×#798	—	5	2	no	—
40	C₄×D₃×C₈	—	5	1	no	—
41	C₃×C₄⋊C₁₆	—	5	1	no	—
42	Dic₃⋊C₁₆	—	5	2	no	—
43	C₄⋊(C₃⋊C₁₆)	D₅	1	1	yes	characteristic, complemented
44	C₂×C₈×D₅	—	3	1	no	—
45	C₂×C₈×D₅	—	3	1	no	—
46	C₄²×D₅	—	3	1	no	—
47	C₂×C₅×C₁₆	—	3	2	no	—
48	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₈×Dic₅	—	3	1	no	—
50	C₈×Dic₅	—	3	1	no	—
51	C₂×#672	C₂×D₃	1	2	yes	—
52	C₄×#168	—	3	1	no	—
53	#3930	—	6	4	no	—
54	#2150	—	15	1	no	Sylow
55	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
61	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
62	C₄×D₁₅	—	2	2	no	—
63	C₄×D₁₅	—	2	2	no	—
64	C₄×D₁₅	C₄⋊C₄	1	2	yes	—
65	C₄×D₁₅	M₁₆	1	2	yes	—
66	C₃×C₅×C₈	—	2	1	no	—
67	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₃⋊#168	—	2	1	no	—
70	C₃⋊#168	C₄⋊C₄	1	2	yes	—
71	C₂×D₃×C₈	—	5	1	no	—
72	C₂×D₃×C₈	—	5	1	no	—
73	C₄²×D₃	—	5	1	no	—
74	C₂×C₃×C₁₆	—	5	2	no	—
75	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
76	C₈×Dic₃	—	5	1	no	—
77	C₈×Dic₃	—	5	1	no	—
78	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	—
79	C₄×C₃⋊C₈	—	5	1	no	—
80	#798	—	10	4	no	—
81	C₂×C₄×D₅	—	3	1	no	—
82	C₂×C₄×D₅	—	3	1	no	—
83	C₂×C₄×D₅	—	3	1	no	—
84	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
86	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
87	C₄×Dic₅	—	3	1	no	—
88	C₄×Dic₅	—	3	1	no	—
89	C₄×Dic₅	—	3	1	no	—
90	C₂×#168	—	3	1	no	—
91	C₂×#168	—	3	1	no	—
92	C₈×D₅	—	3	2	no	—
93	C₈×D₅	—	3	2	no	—
94	C₈×D₅	—	6	2	no	—
95	C₅×C₁₆	—	6	2	no	—
96	#672	—	2	2	no	—
97	C₂×M₃₂	—	15	2	no	—
98	C₂×C₄×C₈	—	15	1	no	—
99	C₄⋊C₁₆	—	3	1	no	—
100	C₄⋊C₁₆	—	5	1	no	—
101	C₄⋊C₁₆	—	15	2	no	—
102	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×D₁₅	—	2	1	no	—
104	C₂×D₁₅	—	2	1	no	—
105	C₂×D₁₅	C₄⋊C₈	1	2	yes	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	M₃₂	1	2	yes	complemented
108	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	C₄⋊C₈	1	2	yes	—
112	Dic₁₅	M₃₂	1	2	yes	complemented
113	C₂×C₄×D₃	—	5	1	no	—
114	C₂×C₄×D₃	—	5	1	no	—
115	C₂×C₄×D₃	—	5	1	no	—
116	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
118	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
119	C₄×Dic₃	—	5	1	no	—
120	C₄×Dic₃	—	5	1	no	—
121	C₄×Dic₃	—	5	1	no	—
122	C₂×C₃⋊C₈	—	5	1	no	—
123	C₂×C₃⋊C₈	—	5	1	no	—
124	D₃×C₈	—	5	2	no	—
125	D₃×C₈	—	5	2	no	—
126	D₃×C₈	—	10	2	no	—
127	C₃×C₁₆	—	10	2	no	—
128	C₃⋊C₁₆	—	2	2	no	—
129	C₂²×D₅	—	3	1	no	—
130	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×Dic₅	—	3	1	no	—
134	C₂×Dic₅	—	3	1	no	—
135	C₂×Dic₅	—	3	1	no	—
136	C₄×D₅	—	3	2	no	—
137	C₄×D₅	—	3	2	no	—
138	C₄×D₅	—	6	2	no	—
139	C₄×D₅	—	6	2	no	—
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	D₃×D₄	1	1	yes	characteristic
142	C₅×C₈	D₃×Q₈	1	1	yes	characteristic
143	#168	—	3	2	no	—
144	#168	—	6	1	no	—
145	C₂²×C₈	—	15	1	no	—
146	C₂²×C₈	—	15	1	no	—
147	C₂×C₄²	—	15	1	no	—
148	M₃₂	—	30	4	no	—
149	C₂×C₁₆	—	3	2	no	—
150	C₂×C₁₆	—	5	2	no	—
151	C₄×C₈	—	15	1	no	—
152	C₄×C₈	—	15	1	no	—
153	C₄×C₈	—	15	1	no	—
154	C₄×C₈	D₃×D₅	1	1	yes	characteristic
155	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
158	D₁₅	—	2	2	no	—
159	C₂²×D₃	—	5	1	no	—
160	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂×Dic₃	—	5	1	no	—
164	C₂×Dic₃	—	5	1	no	—
165	C₂×Dic₃	—	5	1	no	—
166	C₄×D₃	—	5	2	no	—
167	C₄×D₃	—	5	2	no	—
168	C₄×D₃	—	10	2	no	—
169	C₄×D₃	—	10	2	no	—
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	D₄×D₅	1	1	yes	characteristic
172	C₃×C₈	Q₈×D₅	1	1	yes	characteristic
173	C₃⋊C₈	—	5	2	no	—
174	C₃⋊C₈	—	10	1	no	—
175	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
176	C₂×D₅	—	3	2	no	—
177	C₂×D₅	—	6	1	no	—
178	C₂×D₅	—	6	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	#798	1	2	yes	complemented
181	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
183	Dic₅	—	3	2	no	—
184	Dic₅	—	3	2	no	—
185	Dic₅	—	6	1	no	—
186	C₂²×C₄	—	15	1	no	—
187	C₂²×C₄	—	15	1	no	—
188	C₂²×C₄	—	15	1	no	—
189	C₂×C₈	—	15	1	no	—
190	C₂×C₈	—	15	2	no	—
191	C₂×C₈	—	15	2	no	—
192	C₂×C₈	—	15	1	no	—
193	C₂×C₈	—	30	2	no	—
194	C₂×C₈	#1200	1	1	yes	characteristic
195	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
196	C₄²	—	15	1	no	—
197	C₄²	—	15	1	no	—
198	C₄²	—	15	1	no	—
199	C₄²	#1200	1	1	yes	characteristic
200	C₁₆	—	6	2	no	—
201	C₁₆	—	10	2	no	—
202	C₃×C₅	#2150	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
204	C₂×D₃	—	5	2	no	—
205	C₂×D₃	—	10	1	no	—
206	C₂×D₃	—	10	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	#3930	1	2	yes	complemented
209	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
210	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
211	Dic₃	—	5	2	no	—
212	Dic₃	—	5	2	no	—
213	Dic₃	—	10	1	no	—
214	C₂×C₅	D₃×C₄⋊C₈	1	1	yes	characteristic
215	C₂×C₅	C₂×#798	1	1	yes	characteristic
216	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
217	D₅	—	6	2	no	—
218	C₂³	—	15	1	no	—
219	C₂×C₄	—	15	1	no	—
220	C₂×C₄	—	15	1	no	—
221	C₂×C₄	—	15	2	no	—
222	C₂×C₄	—	15	2	no	—
223	C₂×C₄	—	15	1	no	—
224	C₂×C₄	—	30	2	no	—
225	C₂×C₄	—	30	2	no	—
226	C₂×C₄	#6648	1	1	yes	characteristic
227	C₂×C₄	#6648	1	1	yes	characteristic
228	C₂×C₄	C₂×#1200	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	15	2	no	—
231	C₈	—	30	1	no	—
232	C₈	D₁₅⋊Q₈	1	1	yes	characteristic
233	C₈	#6777	1	1	yes	characteristic
234	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
235	C₂×C₃	C₂×#3930	1	1	yes	characteristic
236	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
237	D₃	—	10	2	no	—
238	C₅	#69284	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	15	2	no	—
240	C₂²	—	30	1	no	—
241	C₂²	—	30	1	no	—
242	C₂²	C₂×#6648	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	15	2	no	—
245	C₄	—	15	2	no	—
246	C₄	—	30	1	no	—
247	C₄	#90936	1	2	yes	complemented
248	C₄	#91292	1	1	yes	characteristic
249	C₄	#91437	1	1	yes	characteristic
250	C₃	#10600077	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	30	2	no	—
252	C₂	#11747000	1	1	yes	characteristic
253	C₂	#11747533	1	1	yes	characteristic
254	C₂	C₂×#90936	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
prime radical	C₂²×D₁₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#2150
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₅×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
4. C₃×C₄⋊#672	C₂	semidirect
5. #11745928	C₂	semidirect
22. C₄⋊#672	D₃	semidirect
43. C₄⋊(C₃⋊C₁₆)	D₅	semidirect
107. C₃×C₄×C₅	M₃₂	semidirect
112. Dic₁₅	M₃₂	semidirect
180. C₄×C₅	#798	semidirect
202. C₃×C₅	#2150	semidirect
208. C₃×C₄	#3930	semidirect
238. C₅	#69284	semidirect
247. C₄	#90936	semidirect
250. C₃	#10600077	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×D₁₅	2
C₂×#90936	2
C₅×C₄⋊(C₃⋊C₁₆)	2
C₃×C₄⋊#672	2
#11745928	2
#10600077	3
#69284	5