Subgroups — #14299034

Order 1920.

600 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⋊(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	C₂×Dic₃⋊#168	C₂	1	4	yes	maximal
5	#10600187	—	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	Dic₃⋊#168	—	2	8	no	—
16	#70445	—	5	1	no	maximal, Hall
17	C₄³×C₅	—	3	1	no	—
18	C₂×C₄⋊#168	—	3	4	no	—
19	C₂×C₄⋊#168	—	3	1	no	—
20	C₂×C₄⋊#168	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
25	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	D₄	1	2	yes	—
27	C₃×C₄²×C₅	Q₈	1	2	yes	—
28	C₂×C₃⋊#168	—	2	4	no	—
29	C₂×C₃⋊#168	—	2	2	no	—
30	C₄×C₅×Dic₃	—	2	2	no	—
31	C₄×C₅×Dic₃	—	2	4	no	—
32	C₄×C₅×Dic₃	C₈	1	4	yes	complemented
33	C₄×C₅×Dic₃	D₄	1	4	yes	—
34	C₄×C₅×Dic₃	Q₈	1	4	yes	—
35	C₂×C₃×#168	—	2	4	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×C₄⋊C₈	—	5	1	no	—
38	C₂×Dic₃⋊C₈	—	5	4	no	—
39	C₄²×Dic₃	D₅	1	1	yes	characteristic
40	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
41	C₂×C₄²×C₅	—	3	1	no	—
42	C₂×C₄²×C₅	—	3	2	no	—
43	C₂×C₄²×C₅	—	3	2	no	—
44	C₂×C₄²×C₅	—	3	1	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂²×#168	—	3	2	no	—
47	C₂²×#168	C₂×D₃	1	2	yes	—
48	C₄⋊#168	—	2	2	no	—
49	C₄⋊#168	—	6	8	no	—
50	C₄⋊#168	—	6	2	no	—
51	#1842	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
67	C₂×C₅×Dic₃	M₁₆	1	4	yes	—
68	C₃⋊#168	—	4	2	no	—
69	C₃×#168	—	4	2	no	—
70	C₂²×C₃×C₈	—	5	2	no	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄×Dic₃	Dic₅	1	1	yes	characteristic
73	C₂×C₄×Dic₃	Dic₅	1	2	yes	—
74	C₂×C₄×Dic₃	Dic₅	1	1	yes	characteristic
75	C₂×C₄×Dic₃	C₂×D₅	1	2	yes	—
76	C₂²×C₃⋊C₈	—	5	2	no	—
77	C₃×C₄⋊C₈	—	10	2	no	—
78	Dic₃⋊C₈	—	10	8	no	—
79	C₄⋊(C₃⋊C₈)	—	10	2	no	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₄²×C₅	—	3	4	no	—
86	C₄²×C₅	—	3	4	no	—
87	C₄²×C₅	—	3	4	no	—
88	C₄²×C₅	—	6	4	no	—
89	C₄²×C₅	—	6	2	no	—
90	C₄²×C₅	Dic₆	1	2	yes	—
91	C₄²×C₅	#67	1	2	yes	—
92	C₂×#168	—	2	4	no	—
93	C₂×#168	—	2	2	no	—
94	C₂×#168	—	6	4	no	—
95	C₂×#168	—	6	2	no	—
96	C₂×C₄⋊C₈	—	5	1	no	—
97	C₂×C₄⋊C₈	—	15	4	no	—
98	C₂×C₄⋊C₈	—	15	1	no	—
99	C₄³	—	3	1	no	—
100	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
101	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	complemented
107	C₃×C₄×C₅	#127	1	1	yes	characteristic
108	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
109	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₅×Dic₃	—	2	4	no	—
111	C₅×Dic₃	C₄⋊C₈	1	8	yes	complemented
112	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
116	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
117	C₂×C₃×C₈	—	10	4	no	—
118	C₂×C₃×C₈	—	10	2	no	—
119	C₃×C₄²	Dic₁₀	1	2	yes	—
120	C₃×C₄²	D₂₀	1	2	yes	—
121	C₄×Dic₃	—	2	2	no	—
122	C₄×Dic₃	—	2	4	no	—
123	C₄×Dic₃	#168	1	4	yes	complemented
124	C₄×Dic₃	Dic₁₀	1	4	yes	—
125	C₄×Dic₃	D₂₀	1	4	yes	—
126	C₂×C₃⋊C₈	—	10	2	no	—
127	C₂×C₃⋊C₈	—	10	4	no	—
128	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	3	4	no	—
131	C₂×C₄×C₅	—	3	4	no	—
132	C₂×C₄×C₅	—	3	4	no	—
133	C₂×C₄×C₅	—	3	4	no	—
134	C₂×C₄×C₅	—	6	2	no	—
135	C₂×C₄×C₅	—	6	2	no	—
136	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
137	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
138	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
141	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
142	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
144	#168	—	4	2	no	—
145	#168	—	12	2	no	—
146	C₂²×C₈	—	5	2	no	—
147	C₂²×C₈	—	15	2	no	—
148	C₂×C₄²	—	3	1	no	—
149	C₂×C₄²	—	3	2	no	—
150	C₂×C₄²	—	3	2	no	—
151	C₂×C₄²	—	3	1	no	—
152	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
153	C₄⋊C₈	—	10	2	no	—
154	C₄⋊C₈	—	30	8	no	—
155	C₄⋊C₈	—	30	2	no	—
156	C₂×C₃×C₅	#384	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
158	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂×C₃×C₅	#423	1	2	yes	—
160	C₂×C₃×C₅	#423	1	1	yes	characteristic
161	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	C₂×#168	1	2	yes	—
164	C₂×C₃×C₄	#681	1	2	yes	—
165	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
166	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
167	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
168	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
170	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
171	C₂×Dic₃	—	2	2	no	—
172	C₂×Dic₃	—	2	2	no	—
173	C₂×Dic₃	C₂×#168	1	4	yes	—
174	C₂×Dic₃	#681	1	4	yes	—
175	C₂×Dic₃	C₄⋊Dic₅	1	4	yes	—
176	C₂×Dic₃	C₄⋊Dic₅	1	4	yes	—
177	C₃×C₈	—	20	2	no	—
178	C₃⋊C₈	—	20	2	no	—
179	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
180	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
181	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
182	C₂²×C₅	D₃×M₁₆	1	2	yes	—
183	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	3	8	no	—
186	C₄×C₅	—	6	4	no	—
187	C₄×C₅	Dic₃⋊C₈	1	4	yes	complemented
188	C₄×C₅	#888	1	2	yes	—
189	C₄×C₅	#940	1	1	yes	characteristic
190	C₄×C₅	#944	1	1	yes	characteristic
191	C₂²×C₄	—	3	2	no	—
192	C₂²×C₄	—	3	2	no	—
193	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
194	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	—	10	2	no	—
198	C₂×C₈	—	30	4	no	—
199	C₂×C₈	—	30	2	no	—
200	C₄²	—	3	4	no	—
201	C₄²	—	3	4	no	—
202	C₄²	—	3	4	no	—
203	C₄²	—	6	4	no	—
204	C₄²	—	6	2	no	—
205	C₄²	#1202	1	2	yes	—
206	C₄²	#1204	1	2	yes	—
207	C₃×C₅	#1842	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
209	C₂²×C₃	C₂×#681	1	2	yes	—
210	C₂²×C₃	C₂×#681	1	1	yes	characteristic
211	C₂²×C₃	C₂×C₄⋊Dic₅	1	2	yes	—
212	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	C₄⋊#168	1	4	yes	complemented
215	C₃×C₄	#4015	1	2	yes	—
216	C₃×C₄	#4015	1	1	yes	characteristic
217	C₃×C₄	#4020	1	1	yes	characteristic
218	Dic₃	—	2	4	no	—
219	Dic₃	C₄⋊#168	1	8	yes	complemented
220	C₂×C₅	D₃×C₄⋊C₈	1	2	yes	—
221	C₂×C₅	#4837	1	2	yes	—
222	C₂×C₅	#4975	1	1	yes	characteristic
223	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
224	C₂×C₅	#5118	1	1	yes	characteristic
225	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	3	4	no	—
228	C₂×C₄	—	3	4	no	—
229	C₂×C₄	—	3	4	no	—
230	C₂×C₄	—	3	4	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	D₃×#168	1	2	yes	—
234	C₂×C₄	#6650	1	2	yes	—
235	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
236	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
237	C₂×C₄	D₃×Dic₁₀	1	2	yes	—
238	C₂×C₄	D₃×D₂₀	1	2	yes	—
239	C₂×C₄	C₂×#1202	1	1	yes	characteristic
240	C₂×C₄	C₂×#1204	1	1	yes	characteristic
241	C₈	—	20	2	no	—
242	C₈	—	60	2	no	—
243	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
244	C₂×C₃	C₂×C₄⋊#168	1	2	yes	—
245	C₂×C₃	#65213	1	1	yes	characteristic
246	C₂×C₃	#65213	1	2	yes	—
247	C₂×C₃	#65224	1	1	yes	characteristic
248	C₅	#70445	1	1	yes	characteristic, complemented, Sylow
249	C₂²	C₂×D₃×#168	1	1	yes	characteristic
250	C₂²	D₃×#681	1	2	yes	—
251	C₂²	C₂×#6650	1	1	yes	characteristic
252	C₂²	D₃×C₄⋊Dic₅	1	2	yes	—
253	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	3	8	no	—
256	C₄	—	6	4	no	—
257	C₄	Dic₃⋊#168	1	4	yes	complemented
258	C₄	#91389	1	1	yes	characteristic
259	C₄	#91458	1	1	yes	characteristic
260	C₄	#91469	1	2	yes	—
261	C₃	#10600187	1	1	yes	characteristic, complemented, Sylow
262	C₂	D₃×C₄⋊#168	1	2	yes	—
263	C₂	#11747011	1	2	yes	—
264	C₂	C₂×Dic₃⋊#168	1	1	yes	characteristic
265	C₂	#11748440	1	1	yes	characteristic
266	C₂	#11748861	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	#1842
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
32. C₄×C₅×Dic₃	C₈	semidirect
106. C₃×C₄×C₅	C₄⋊C₈	semidirect
111. C₅×Dic₃	C₄⋊C₈	semidirect
123. C₄×Dic₃	#168	semidirect
187. C₄×C₅	Dic₃⋊C₈	semidirect
207. C₃×C₅	#1842	semidirect
214. C₃×C₄	C₄⋊#168	semidirect
219. Dic₃	C₄⋊#168	semidirect
248. C₅	#70445	semidirect
257. C₄	Dic₃⋊#168	semidirect
261. C₃	#10600187	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
C₂×Dic₃⋊#168	2
#10600187	3
#70445	5