Subgroups — #14298728

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11745967	C₂	1	4	yes	maximal
5	#10600185	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊#168	C₂²	1	2	yes	—
8	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×#168	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊#168)	—	2	2	no	—
14	C₃×C₄⋊#168	—	2	2	no	—
15	#70443	—	5	1	no	maximal, Hall
16	C₄³×C₅	—	3	1	no	—
17	C₂×C₄⋊#168	—	3	1	no	—
18	C₂×C₄⋊#168	D₃	1	1	yes	characteristic
19	#64744	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
24	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	D₄	1	2	yes	—
26	C₃×C₄²×C₅	Q₈	1	2	yes	—
27	C₂×C₃⋊#168	—	2	2	no	—
28	C₂×C₃⋊#168	—	2	4	no	—
29	C₄×C₅×Dic₃	—	2	2	no	—
30	C₄×C₅×Dic₃	—	2	2	no	—
31	C₄×C₅×Dic₃	—	2	2	no	—
32	C₄×C₅×Dic₃	—	2	4	no	—
33	C₄×C₅×Dic₃	C₈	1	4	yes	complemented
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	—	2	4	no	—
36	C₂×C₃×C₄⋊C₈	—	5	1	no	—
37	C₄²×Dic₃	D₅	1	1	yes	characteristic
38	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
39	#4550	—	5	4	no	—
40	C₂×C₄²×C₅	—	3	1	no	—
41	C₂×C₄²×C₅	—	3	2	no	—
42	C₂×C₄²×C₅	—	3	1	no	—
43	C₂×C₄²×C₅	—	3	2	no	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂²×#168	—	3	2	no	—
46	C₂²×#168	C₂×D₃	1	2	yes	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	—	6	2	no	—
49	#1831	—	15	1	no	Sylow
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	#41	1	2	yes	—
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₂×C₅×Dic₃	—	2	2	no	—
61	C₂×C₅×Dic₃	—	2	2	no	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
65	C₂×C₅×Dic₃	M₁₆	1	4	yes	—
66	C₃⋊#168	—	4	2	no	—
67	C₃×#168	—	4	2	no	—
68	C₂²×C₃×C₈	—	5	2	no	—
69	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
70	C₂×C₄×Dic₃	Dic₅	1	1	yes	characteristic
71	C₂×C₄×Dic₃	Dic₅	1	2	yes	—
72	C₂×C₄×Dic₃	Dic₅	1	1	yes	characteristic
73	C₂×C₄×Dic₃	C₂×D₅	1	2	yes	—
74	C₂²×C₃⋊C₈	—	5	2	no	—
75	C₃×C₄⋊C₈	—	10	2	no	—
76	C₄⋊(C₃⋊C₈)	—	10	2	no	—
77	C₂²×C₄×C₅	—	3	2	no	—
78	C₂²×C₄×C₅	—	3	2	no	—
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	—	3	4	no	—
83	C₄²×C₅	—	6	4	no	—
84	C₄²×C₅	—	6	2	no	—
85	C₄²×C₅	—	6	2	no	—
86	C₄²×C₅	—	6	2	no	—
87	C₄²×C₅	Dic₆	1	2	yes	—
88	C₄²×C₅	#67	1	2	yes	—
89	C₂×#168	—	2	4	no	—
90	C₂×#168	—	2	2	no	—
91	C₂×#168	—	6	4	no	—
92	C₂×#168	—	6	2	no	—
93	C₂×C₄⋊C₈	—	5	1	no	—
94	C₂×C₄⋊C₈	—	15	1	no	—
95	C₄³	—	3	1	no	—
96	#336	—	15	4	no	—
97	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₃×C₅	#117	1	2	yes	—
99	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	—
104	C₃×C₄×C₅	#124	1	1	yes	characteristic
105	C₃×C₄×C₅	#124	1	1	yes	characteristic
106	C₃×C₄×C₅	#125	1	2	yes	—
107	C₅×Dic₃	—	2	4	no	—
108	C₅×Dic₃	—	2	4	no	—
109	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
113	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
114	C₂×C₃×C₈	—	10	2	no	—
115	C₂×C₃×C₈	—	10	4	no	—
116	C₃×C₄²	Dic₁₀	1	2	yes	—
117	C₃×C₄²	D₂₀	1	2	yes	—
118	C₄×Dic₃	—	2	2	no	—
119	C₄×Dic₃	—	2	2	no	—
120	C₄×Dic₃	—	2	2	no	—
121	C₄×Dic₃	—	2	4	no	—
122	C₄×Dic₃	#168	1	4	yes	complemented
123	C₂×C₃⋊C₈	—	10	2	no	—
124	C₂×C₃⋊C₈	—	10	4	no	—
125	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	3	4	no	—
128	C₂×C₄×C₅	—	3	4	no	—
129	C₂×C₄×C₅	—	6	2	no	—
130	C₂×C₄×C₅	—	6	2	no	—
131	C₂×C₄×C₅	—	6	2	no	—
132	C₂×C₄×C₅	—	6	2	no	—
133	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
134	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
135	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
138	C₂×C₄×C₅	#237	1	2	yes	—
139	C₂×C₄×C₅	#239	1	2	yes	—
140	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
141	#168	—	4	2	no	—
142	#168	—	12	2	no	—
143	C₂²×C₈	—	5	2	no	—
144	C₂²×C₈	—	15	2	no	—
145	C₂×C₄²	—	3	1	no	—
146	C₂×C₄²	—	3	2	no	—
147	C₂×C₄²	—	3	1	no	—
148	C₂×C₄²	—	3	2	no	—
149	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
150	C₄⋊C₈	—	10	2	no	—
151	C₄⋊C₈	—	30	2	no	—
152	C₂×C₃×C₅	#382	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂×C₃×C₅	#423	1	1	yes	characteristic
155	C₂×C₃×C₅	#431	1	2	yes	—
156	C₂×C₃×C₅	#432	1	2	yes	—
157	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	C₂×#168	1	2	yes	—
160	C₂×C₃×C₄	#681	1	2	yes	—
161	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
162	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
163	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
165	C₂×C₃×C₄	#709	1	2	yes	—
166	C₂×C₃×C₄	#709	1	2	yes	—
167	C₂×Dic₃	—	2	2	no	—
168	C₂×Dic₃	—	2	2	no	—
169	C₂×Dic₃	—	2	2	no	—
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	C₂×#168	1	4	yes	—
172	C₂×Dic₃	#681	1	4	yes	—
173	C₃×C₈	—	20	2	no	—
174	C₃⋊C₈	—	20	2	no	—
175	C₂²×C₅	#892	1	2	yes	—
176	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
177	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
178	C₂²×C₅	D₃×M₁₆	1	2	yes	—
179	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	6	4	no	—
182	C₄×C₅	—	6	4	no	—
183	C₄×C₅	Dic₃⋊C₈	1	4	yes	—
184	C₄×C₅	#890	1	2	yes	—
185	C₄×C₅	#936	1	1	yes	characteristic
186	C₄×C₅	#947	1	1	yes	characteristic
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
190	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
192	C₂×C₈	—	10	4	no	—
193	C₂×C₈	—	10	2	no	—
194	C₂×C₈	—	30	4	no	—
195	C₂×C₈	—	30	2	no	—
196	C₄²	—	3	4	no	—
197	C₄²	—	6	4	no	—
198	C₄²	—	6	2	no	—
199	C₄²	—	6	2	no	—
200	C₄²	—	6	2	no	—
201	C₄²	#1202	1	2	yes	—
202	C₄²	#1204	1	2	yes	—
203	C₃×C₅	#1831	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
205	C₂²×C₃	C₂×#681	1	2	yes	—
206	C₂²×C₃	C₂×#681	1	1	yes	characteristic
207	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
208	C₂²×C₃	#4072	1	2	yes	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	C₄⋊#168	1	4	yes	—
211	C₃×C₄	#4016	1	2	yes	—
212	C₃×C₄	#4021	1	1	yes	characteristic
213	C₃×C₄	#4021	1	1	yes	characteristic
214	Dic₃	—	2	4	no	—
215	Dic₃	—	2	4	no	—
216	C₂×C₅	#4833	1	2	yes	—
217	C₂×C₅	#4835	1	2	yes	—
218	C₂×C₅	#4977	1	1	yes	characteristic
219	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
220	C₂×C₅	#5118	1	1	yes	characteristic
221	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	3	4	no	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	D₃×#168	1	2	yes	—
230	C₂×C₄	#6650	1	2	yes	—
231	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
232	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
233	C₂×C₄	#6765	1	2	yes	—
234	C₂×C₄	#6769	1	2	yes	—
235	C₂×C₄	C₂×#1202	1	1	yes	characteristic
236	C₂×C₄	C₂×#1204	1	1	yes	characteristic
237	C₈	—	20	2	no	—
238	C₈	—	60	2	no	—
239	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
240	C₂×C₃	#65213	1	1	yes	characteristic
241	C₂×C₃	#65214	1	2	yes	—
242	C₂×C₃	#65215	1	2	yes	—
243	C₂×C₃	#65225	1	1	yes	characteristic
244	C₅	#70443	1	1	yes	characteristic, complemented, Sylow
245	C₂²	C₂×D₃×#168	1	1	yes	characteristic
246	C₂²	D₃×#681	1	2	yes	—
247	C₂²	C₂×#6650	1	1	yes	characteristic
248	C₂²	#91338	1	2	yes	—
249	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	6	4	no	—
252	C₄	—	6	4	no	—
253	C₄	Dic₃⋊#168	1	4	yes	—
254	C₄	#91364	1	1	yes	characteristic
255	C₄	#91374	1	1	yes	characteristic
256	C₄	#91381	1	2	yes	—
257	C₃	#10600185	1	1	yes	characteristic, complemented, Sylow
258	C₂	#11746853	1	2	yes	—
259	C₂	#11746875	1	2	yes	—
260	C₂	C₂×Dic₃⋊#168	1	1	yes	characteristic
261	C₂	#11748440	1	1	yes	characteristic
262	C₂	#11748619	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1831
LCS	G ▷ C₂²×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
33. C₄×C₅×Dic₃	C₈	semidirect
122. C₄×Dic₃	#168	semidirect
203. C₃×C₅	#1831	semidirect
244. C₅	#70443	semidirect
257. C₃	#10600185	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊#168)	2
C₄²×C₅×Dic₃	2
C₂×C₃×C₄⋊#168	2
#11745967	2
#10600185	3
#70443	5