Subgroups — #14307103

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11745968	C₂	1	4	yes	maximal
5	#10600631	—	3	1	no	maximal, Hall
6	C₂²×Dic₃₀	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₄	1	1	yes	characteristic
9	C₂×#6712	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
11	C₄×Dic₃₀	—	2	2	no	—
12	C₄×Dic₃₀	—	2	2	no	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₃×#168	C₂²	1	2	yes	—
15	C₅×#803	—	2	2	no	—
16	C₃×C₄⋊#168	—	2	2	no	—
17	#70473	—	5	1	no	maximal, Hall
18	C₂×C₄×Dic₁₀	—	3	1	no	—
19	C₂×C₅×#97	—	3	1	no	—
20	C₂×C₄⋊#168	D₃	1	1	yes	characteristic
21	#64770	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
26	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
27	C₂×Dic₃₀	—	2	2	no	—
28	C₂×Dic₃₀	—	2	4	no	—
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	2	no	—
32	#6712	—	2	4	no	—
33	C₄×Dic₁₅	—	2	2	no	—
34	C₄×Dic₁₅	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	Q₈	1	4	yes	complemented
38	C₂×C₃×#168	—	2	2	no	—
39	C₂×C₃×#168	—	2	4	no	—
40	C₂×C₄×Dic₆	—	5	1	no	—
41	C₂×C₃×C₄⋊C₈	—	5	1	no	—
42	C₂×#803	D₅	1	1	yes	characteristic
43	#4550	—	5	4	no	—
44	C₂²×Dic₁₀	—	3	1	no	—
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅	—	3	1	no	—
48	C₂×#683	—	3	2	no	—
49	C₂×C₄×Dic₅	—	3	2	no	—
50	C₂²×#168	C₂×D₃	1	2	yes	—
51	C₄×Dic₁₀	—	6	2	no	—
52	C₄×Dic₁₀	—	6	2	no	—
53	C₅×#97	—	6	2	no	—
54	C₄⋊#168	—	2	2	no	—
55	#1985	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	C₂×Dic₁₅	—	2	2	no	—
66	C₂×Dic₁₅	—	2	2	no	—
67	C₂×Dic₁₅	—	2	2	no	—
68	C₂×Dic₁₅	—	2	4	no	—
69	C₂×Dic₁₅	M₁₆	1	4	yes	—
70	Dic₃₀	—	2	8	no	—
71	C₃×#168	—	4	2	no	—
72	C₅×C₃⋊C₈	—	2	4	no	—
73	C₂²×Dic₆	—	5	1	no	—
74	C₂²×C₃×C₈	—	5	2	no	—
75	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
76	C₂×C₄⋊Dic₃	—	5	1	no	—
77	C₂×Dic₃⋊C₄	—	5	2	no	—
78	C₂×C₄×Dic₃	—	5	2	no	—
79	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
80	C₄×Dic₆	—	10	2	no	—
81	C₄×Dic₆	—	10	2	no	—
82	C₃×C₄⋊C₈	—	10	2	no	—
83	#803	—	2	2	no	—
84	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂²×Dic₅	—	3	2	no	—
88	C₂²×Dic₅	—	3	2	no	—
89	C₂×Dic₁₀	—	6	4	no	—
90	C₂×Dic₁₀	—	6	2	no	—
91	C₂×C₅×C₈	—	3	4	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₄²×C₅	D₁₂	1	2	yes	—
95	C₄²×C₅	#67	1	2	yes	—
96	C₄⋊Dic₅	—	6	2	no	—
97	#683	—	6	4	no	—
98	C₄×Dic₅	—	6	2	no	—
99	C₄×Dic₅	—	6	2	no	—
100	C₂×#168	—	2	4	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×C₄×Q₈	—	15	1	no	—
103	C₂×C₄⋊C₈	—	5	1	no	—
104	C₂×#97	—	3	1	no	—
105	#336	—	15	4	no	—
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
109	C₂²×C₃×C₅	#131	1	2	yes	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
113	C₃×C₄×C₅	#124	1	1	yes	characteristic
114	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
115	Dic₁₅	—	2	4	no	—
116	Dic₁₅	—	4	2	no	—
117	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	5	2	no	—
121	C₂²×Dic₃	—	5	2	no	—
122	C₂×Dic₆	—	10	4	no	—
123	C₂×Dic₆	—	10	2	no	—
124	C₂×C₃×C₈	—	10	2	no	—
125	C₂×C₃×C₈	—	10	4	no	—
126	C₃×C₄²	D₂₀	1	2	yes	—
127	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
128	C₄⋊Dic₃	—	10	2	no	—
129	Dic₃⋊C₄	—	10	4	no	—
130	C₄×Dic₃	—	10	2	no	—
131	C₄×Dic₃	—	10	2	no	—
132	C₂×C₃⋊C₈	—	2	2	no	—
133	C₂×C₃⋊C₈	—	2	2	no	—
134	C₂×C₃⋊C₈	Dic₁₀	1	4	yes	complemented
135	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
141	C₂×C₄×C₅	#237	1	2	yes	—
142	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
144	C₂×Dic₅	—	3	4	no	—
145	C₂×Dic₅	—	6	2	no	—
146	C₂×Dic₅	—	6	2	no	—
147	C₂×Dic₅	—	6	4	no	—
148	C₂×Dic₅	—	6	2	no	—
149	Dic₁₀	—	6	8	no	—
150	C₅×C₈	—	6	4	no	—
151	#168	—	4	2	no	—
152	C₂²×Q₈	—	15	1	no	—
153	C₂²×C₈	—	3	2	no	—
154	C₂²×C₈	—	5	2	no	—
155	C₄×Q₈	—	30	2	no	—
156	C₄×Q₈	—	30	2	no	—
157	C₂×C₄⋊C₄	—	15	2	no	—
158	C₂×C₄⋊C₄	—	15	1	no	—
159	C₂×C₄²	—	15	2	no	—
160	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
161	C₄⋊C₈	—	10	2	no	—
162	#97	—	6	2	no	—
163	C₂×C₃×C₅	#391	1	1	yes	characteristic
164	C₂×C₃×C₅	#407	1	1	yes	characteristic
165	C₂×C₃×C₅	#408	1	1	yes	characteristic
166	C₂×C₃×C₅	#446	1	4	yes	—
167	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	#685	1	2	yes	—
171	C₂×C₃×C₄	#685	1	2	yes	—
172	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
173	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
174	C₂×C₃×C₄	#709	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	4	no	—
177	C₂×Dic₃	—	10	2	no	—
178	C₂×Dic₃	—	10	2	no	—
179	C₂×Dic₃	—	10	4	no	—
180	C₂×Dic₃	—	10	2	no	—
181	Dic₆	—	10	8	no	—
182	C₃×C₈	—	20	2	no	—
183	C₃⋊C₈	—	2	4	no	—
184	C₂²×C₅	#887	1	2	yes	—
185	C₂²×C₅	#901	1	2	yes	—
186	C₂²×C₅	D₃×M₁₆	1	2	yes	—
187	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	#897	1	2	yes	—
191	C₄×C₅	#936	1	1	yes	characteristic
192	C₄×C₅	#944	1	1	yes	characteristic
193	Dic₅	—	6	4	no	—
194	Dic₅	—	12	2	no	—
195	C₂×Q₈	—	30	4	no	—
196	C₂×Q₈	—	30	2	no	—
197	C₂²×C₄	—	15	2	no	—
198	C₂²×C₄	—	15	2	no	—
199	C₂²×C₄	#1200	1	1	yes	characteristic
200	C₂²×C₄	#1200	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
202	C₂×C₈	—	3	4	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	6	2	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	—	10	2	no	—
207	C₄⋊C₄	—	30	4	no	—
208	C₄⋊C₄	—	30	2	no	—
209	C₄²	—	30	2	no	—
210	C₄²	—	30	2	no	—
211	C₄²	#1202	1	2	yes	—
212	C₄²	#1203	1	2	yes	—
213	C₃×C₅	#1985	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	C₄×Dic₁₀	1	2	yes	—
215	C₂²×C₃	D₅×M₁₆	1	2	yes	—
216	C₂²×C₃	#4053	1	2	yes	—
217	C₂²×C₃	C₂×#685	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	#4015	1	1	yes	characteristic
221	C₃×C₄	#4021	1	1	yes	characteristic
222	C₃×C₄	#4070	1	2	yes	—
223	Dic₃	—	10	4	no	—
224	Dic₃	—	20	2	no	—
225	C₂×C₅	#4828	1	4	yes	—
226	C₂×C₅	#4973	1	1	yes	characteristic
227	C₂×C₅	#5110	1	1	yes	characteristic
228	C₂×C₅	#5126	1	1	yes	characteristic
229	C₂³	C₂×#1200	1	1	yes	characteristic
230	Q₈	—	30	8	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	15	4	no	—
234	C₂×C₄	—	30	2	no	—
235	C₂×C₄	—	30	2	no	—
236	C₂×C₄	—	30	2	no	—
237	C₂×C₄	—	30	4	no	—
238	C₂×C₄	#6667	1	2	yes	—
239	C₂×C₄	#6667	1	2	yes	—
240	C₂×C₄	#6765	1	2	yes	—
241	C₂×C₄	D₃×Dic₁₀	1	2	yes	—
242	C₂×C₄	C₂×#1202	1	1	yes	characteristic
243	C₂×C₄	C₂×#1203	1	1	yes	characteristic
244	C₈	—	6	4	no	—
245	C₈	—	20	2	no	—
246	C₂×C₃	#64990	1	4	yes	—
247	C₂×C₃	#65218	1	1	yes	characteristic
248	C₂×C₃	#65415	1	1	yes	characteristic
249	C₂×C₃	#65418	1	1	yes	characteristic
250	C₅	#70473	1	1	yes	characteristic, complemented, Sylow
251	C₂²	#91290	1	2	yes	—
252	C₂²	#91292	1	2	yes	—
253	C₂²	#91340	1	2	yes	—
254	C₂²	C₂×#6667	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	30	4	no	—
258	C₄	—	60	2	no	—
259	C₄	#91364	1	1	yes	characteristic
260	C₄	#91365	1	2	yes	—
261	C₄	#91389	1	1	yes	characteristic
262	C₃	#10600631	1	1	yes	characteristic, complemented, Sylow
263	C₂	#11746856	1	4	yes	—
264	C₂	#11748349	1	1	yes	characteristic
265	C₂	#11748351	1	1	yes	characteristic
266	C₂	#11748599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator, socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1985
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
37. C₂×C₅×C₃⋊C₈	Q₈	semidirect
134. C₂×C₃⋊C₈	Dic₁₀	semidirect
213. C₃×C₅	#1985	semidirect
250. C₅	#70473	semidirect
262. C₃	#10600631	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₀	2
C₂×C₅×#803	2
C₂×C₃×C₄⋊#168	2
#11745968	2
#10600631	3
#70473	5