Subgroups — #14280723

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#676	C₂	1	1	yes	maximal, characteristic
2	C₄×#6649	C₂	1	1	yes	maximal, characteristic
3	C₄×#6649	C₂	1	1	yes	maximal, characteristic
4	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598868	—	3	1	no	maximal, Hall
9	C₂×C₃×#676	C₄	1	2	yes	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₂×#6649	C₄	1	2	yes	—
12	C₂×#6649	C₄	1	2	yes	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×#3947	C₄	1	2	yes	—
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90952	C₄	1	2	yes	—
21	#90952	C₄	1	2	yes	—
22	#69849	—	5	1	no	maximal, Hall
23	C₄×#676	—	3	1	no	—
24	C₄×#676	—	3	1	no	—
25	C₄×#676	D₃	1	1	yes	characteristic
26	C₅×#322	—	3	1	no	—
27	#64674	—	3	1	no	—
28	#64674	—	3	1	no	—
29	C₈×#168	—	3	1	no	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₃×#676	—	2	4	no	—
44	#6649	—	2	4	no	—
45	#6649	—	2	4	no	—
46	C₃×C₄×M₁₆	—	5	1	no	—
47	C₄×#208	—	5	1	no	—
48	C₄×#208	—	5	1	no	—
49	#4455	—	5	1	no	—
50	#4454	—	5	1	no	—
51	#4454	D₅	1	1	yes	characteristic, complemented
52	C₈×C₃⋊C₈	—	5	1	no	—
53	C₂×#676	—	3	2	no	—
54	C₂×#676	—	3	2	no	—
55	C₂×#676	Dic₃	1	2	yes	—
56	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
57	C₄×C₅×C₈	—	3	1	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	#3947	—	3	2	no	—
61	#3947	—	3	2	no	—
62	#3947	Dic₃	1	2	yes	—
63	C₄×#168	—	3	1	no	—
64	C₄×#168	—	3	1	no	—
65	C₄×#168	C₂×D₃	1	1	yes	characteristic
66	#1455	—	15	1	no	Sylow
67	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
73	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
74	C₃×C₄×D₅	—	2	2	no	—
75	C₃×C₄×D₅	—	2	2	no	—
76	C₃×C₄×D₅	—	2	2	no	—
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃⋊#168	—	2	2	no	—
79	C₃⋊#168	M₁₆	1	4	yes	complemented
80	C₃×#168	M₁₆	1	4	yes	complemented
81	C₅×C₃⋊C₈	—	2	2	no	—
82	C₅×C₃⋊C₈	M₁₆	1	4	yes	complemented
83	C₂×C₃×M₁₆	—	5	2	no	—
84	C₂×C₃×C₄²	—	5	1	no	—
85	C₂×#208	—	5	2	no	—
86	C₂×#208	—	5	2	no	—
87	C₃×#97	—	5	2	no	—
88	C₃×C₄×C₈	—	5	1	no	—
89	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	#803	—	5	2	no	—
91	#803	—	5	2	no	—
92	C₄×C₃⋊C₈	—	5	1	no	—
93	C₄×C₃⋊C₈	—	5	1	no	—
94	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
95	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
97	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
98	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	—	3	2	no	—
101	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
102	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
104	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
105	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
106	C₂×#168	—	3	2	no	—
107	C₂×#168	—	3	2	no	—
108	C₂×#168	C₂×Dic₃	1	2	yes	—
109	#676	—	2	4	no	—
110	#676	—	6	4	no	—
111	#676	—	6	4	no	—
112	C₄×M₁₆	—	5	1	no	—
113	C₄×M₁₆	—	15	1	no	—
114	C₄×M₁₆	—	15	1	no	—
115	#322	—	3	1	no	—
116	#322	—	15	1	no	—
117	#322	—	15	1	no	—
118	C₈²	—	15	1	no	—
119	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₃×D₅	—	2	1	no	—
121	C₂×C₃×D₅	—	2	1	no	—
122	C₂×C₃×D₅	—	2	1	no	—
123	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
125	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
126	C₃×Dic₅	—	2	1	no	—
127	C₃×Dic₅	—	2	1	no	—
128	C₃×Dic₅	—	2	1	no	—
129	C₂²×C₃×C₄	—	5	1	no	—
130	C₂²×C₃×C₄	—	5	1	no	—
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₃×M₁₆	—	10	4	no	—
133	C₂×C₃×C₈	—	5	2	no	—
134	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
135	C₃×C₄²	—	5	1	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	—	5	1	no	—
138	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
139	#208	—	10	4	no	—
140	#208	—	10	4	no	—
141	C₂×C₃⋊C₈	—	5	2	no	—
142	C₂×C₃⋊C₈	—	5	2	no	—
143	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
144	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
145	C₂²×D₅	C₄×Dic₃	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
149	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
150	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
151	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
152	C₄×D₅	—	2	2	no	—
153	C₄×D₅	—	2	2	no	—
154	C₄×D₅	—	2	2	no	—
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	3	4	no	—
157	C₅×C₈	—	6	2	no	—
158	#168	—	3	4	no	—
159	#168	—	6	2	no	—
160	#168	#208	1	4	yes	complemented
161	C₂×M₁₆	—	5	2	no	—
162	C₂×M₁₆	—	15	2	no	—
163	C₂×M₁₆	—	15	2	no	—
164	C₂×C₄²	—	5	1	no	—
165	#97	—	5	2	no	—
166	#97	—	15	2	no	—
167	#97	—	15	2	no	—
168	C₄×C₈	—	3	1	no	—
169	C₄×C₈	—	3	1	no	—
170	C₄×C₈	—	5	1	no	—
171	C₄×C₈	—	15	1	no	—
172	C₄×C₈	—	15	1	no	—
173	C₄×C₈	D₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
175	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
176	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
177	C₃×D₅	—	4	1	no	—
178	C₂³×C₃	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	10	2	no	—
183	C₂×C₃×C₄	—	10	2	no	—
184	C₂×C₃×C₄	—	10	2	no	—
185	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	5	4	no	—
190	C₃⋊C₈	—	2	2	no	—
191	C₃⋊C₈	—	5	4	no	—
192	C₃⋊C₈	—	10	2	no	—
193	C₃⋊C₈	#676	1	4	yes	complemented
194	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
195	C₂×D₅	—	2	1	no	—
196	C₂×D₅	—	2	1	no	—
197	C₂×D₅	—	2	1	no	—
198	C₄×C₅	D₃×M₁₆	1	2	yes	—
199	C₄×C₅	D₃×M₁₆	1	2	yes	—
200	C₄×C₅	C₂×#208	1	2	yes	—
201	Dic₅	—	2	1	no	—
202	Dic₅	—	2	1	no	—
203	Dic₅	—	2	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	1	no	—
207	M₁₆	—	10	4	no	—
208	M₁₆	—	30	4	no	—
209	M₁₆	—	30	4	no	—
210	C₂×C₈	—	3	2	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	—	5	2	no	—
213	C₂×C₈	—	15	2	no	—
214	C₂×C₈	—	15	2	no	—
215	C₂×C₈	D₅×Dic₃	1	2	yes	—
216	C₄²	—	5	1	no	—
217	C₄²	—	5	1	no	—
218	C₄²	—	5	1	no	—
219	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	10	1	no	—
222	C₂²×C₃	—	10	1	no	—
223	C₂²×C₃	—	10	1	no	—
224	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
225	C₃×C₄	—	10	1	no	—
226	C₃×C₄	—	10	1	no	—
227	C₃×C₄	—	10	1	no	—
228	C₃×C₄	C₂×#676	1	2	yes	—
229	C₃×C₄	D₅×M₁₆	1	2	yes	—
230	C₃×C₄	D₅×M₁₆	1	2	yes	—
231	C₂×C₅	C₄×#208	1	1	yes	characteristic
232	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
233	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
234	D₅	—	4	1	no	—
235	C₂³	—	5	1	no	—
236	C₂×C₄	—	5	1	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	5	1	no	—
239	C₂×C₄	—	10	2	no	—
240	C₂×C₄	—	10	2	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
243	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
244	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	3	4	no	—
247	C₈	—	5	4	no	—
248	C₈	—	6	2	no	—
249	C₈	—	15	4	no	—
250	C₈	—	30	2	no	—
251	C₂×C₃	—	20	1	no	—
252	C₂×C₃	C₄×#676	1	1	yes	characteristic
253	C₂×C₃	#64996	1	1	yes	characteristic
254	C₂×C₃	#64996	1	1	yes	characteristic
255	C₅	#69849	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	10	1	no	—
257	C₂²	—	10	1	no	—
258	C₂²	—	10	1	no	—
259	C₂²	C₄×D₅×Dic₃	1	1	yes	characteristic
260	C₄	—	10	1	no	—
261	C₄	—	10	1	no	—
262	C₄	—	10	1	no	—
263	C₄	D₃×#676	1	2	yes	—
264	C₄	#91246	1	2	yes	—
265	C₄	D₅×#208	1	2	yes	—
266	C₃	#10598868	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	20	1	no	—
268	C₂	#11747003	1	1	yes	characteristic
269	C₂	Dic₃×#676	1	1	yes	characteristic
270	C₂	#11747601	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₄×C₅×C₈	C₂²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1455
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11746275	C₂	semidirect
5. C₅×#4454	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
51. #4454	D₅	semidirect
79. C₃⋊#168	M₁₆	semidirect
80. C₃×#168	M₁₆	semidirect
82. C₅×C₃⋊C₈	M₁₆	semidirect
160. #168	#208	semidirect
193. C₃⋊C₈	#676	semidirect
220. C₃×C₅	#1455	semidirect
255. C₅	#69849	semidirect
266. C₃	#10598868	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#676	2
C₄×#6649	2
#11746275	2
C₅×#4454	2
#11745831	2
C₃⋊C₈×#168	2
#10598868	3
#69849	5