Subgroups — #14294654

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₅	C₂	1	2	yes	maximal
2	C₂²×#6705	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746379	C₂	1	2	yes	maximal, complemented
5	#11746373	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599731	—	3	1	no	maximal, Hall
7	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	M₁₆×D₁₅	—	2	4	no	—
13	M₁₆×D₁₅	C₄	1	8	yes	—
14	C₂×#6705	—	2	4	no	—
15	C₂×#6705	—	2	2	no	—
16	C₂×#6705	C₄	1	4	yes	—
17	C₂×C₈×D₁₅	—	2	2	no	—
18	#68938	—	5	1	no	maximal, Hall
19	C₂×D₅×M₁₆	—	3	2	no	—
20	C₂²×#676	—	3	1	no	—
21	C₅×#341	D₃	1	1	yes	characteristic, complemented
22	#64778	—	3	2	no	—
23	#64772	—	3	1	no	—
24	C₂³×D₁₅	Q₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₅	Q₈	1	1	yes	characteristic
27	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₅	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₅	D₄	1	2	yes	—
30	C₂×C₄×D₁₅	D₄	1	4	yes	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	#6710	—	2	2	no	—
37	#6710	C₂×C₄	1	4	yes	—
38	C₂×C₃⋊#168	—	2	2	no	—
39	C₂×C₃⋊#168	—	2	2	no	—
40	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
41	#6705	—	2	8	no	—
42	#6705	—	2	8	no	—
43	C₈×D₁₅	—	2	8	no	—
44	C₂×D₃×M₁₆	—	5	2	no	—
45	C₂²×ΓL(1,𝔽₂₅)	—	5	1	no	—
46	C₃×#341	D₅	1	1	yes	characteristic, complemented
47	#4558	—	5	2	no	—
48	#4552	—	5	1	no	—
49	C₂²×C₄×D₅	—	3	1	no	—
50	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×#681	—	3	2	no	—
53	C₂²×#168	—	3	1	no	—
54	D₅×M₁₆	—	3	8	no	—
55	D₅×M₁₆	—	6	4	no	—
56	C₂×#676	—	3	4	no	—
57	C₂×#676	—	6	4	no	—
58	C₂×#676	—	6	2	no	—
59	C₂×C₈×D₅	—	6	2	no	—
60	#1738	—	15	1	no	Sylow
61	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×D₁₅	—	4	2	no	—
63	C₂²×D₁₅	C₄⋊C₄	1	2	yes	—
64	C₂²×D₁₅	C₄⋊C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₅	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
71	C₄×D₁₅	C₄²	1	4	yes	—
72	C₄×D₁₅	C₂²⋊C₄	1	4	yes	—
73	C₄×D₁₅	C₂²⋊C₄	1	4	yes	—
74	C₄×D₁₅	C₂²⋊C₄	1	4	yes	—
75	C₃×C₅×C₈	—	2	4	no	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃⋊#168	—	2	4	no	—
78	C₃⋊#168	—	2	2	no	—
79	C₂²×C₄×D₃	—	5	1	no	—
80	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
81	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×#208	—	5	2	no	—
83	C₂²×C₃⋊C₈	—	5	1	no	—
84	D₃×M₁₆	—	5	8	no	—
85	D₃×M₁₆	—	10	4	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
87	C₂×ΓL(1,𝔽₂₅)	—	10	4	no	—
88	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
89	C₂×D₃×C₈	—	10	2	no	—
90	C₂³×D₅	—	3	1	no	—
91	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
92	C₂²×Dic₅	—	3	1	no	—
93	C₂×C₄×D₅	—	3	2	no	—
94	C₂×C₄×D₅	—	3	4	no	—
95	C₂×C₄×D₅	—	3	2	no	—
96	C₂×C₄×D₅	—	3	4	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	C₄×D₃	1	4	yes	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	#681	—	3	4	no	—
103	#681	—	6	2	no	—
104	C₂×#168	—	3	2	no	—
105	C₂×#168	—	6	2	no	—
106	C₂×#168	—	6	2	no	—
107	#676	—	6	8	no	—
108	#676	—	6	8	no	—
109	C₈×D₅	—	6	8	no	—
110	C₂²×M₁₆	—	15	1	no	—
111	C₂²×M₁₆	—	15	2	no	—
112	#341	—	15	2	no	—
113	#341	—	15	1	no	—
114	#341	D₁₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×D₁₅	—	2	4	no	—
119	C₂×D₁₅	—	4	4	no	—
120	C₂×D₁₅	#95	1	4	yes	—
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	Dic₁₅	#95	1	4	yes	—
125	C₂³×D₃	—	5	1	no	—
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₂²×Dic₃	—	5	1	no	—
128	C₂×C₄×D₃	—	5	2	no	—
129	C₂×C₄×D₃	—	5	4	no	—
130	C₂×C₄×D₃	—	5	2	no	—
131	C₂×C₄×D₃	—	5	4	no	—
132	C₃×M₁₆	—	2	2	no	—
133	C₃×M₁₆	C₄×D₅	1	4	yes	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
137	#208	—	5	4	no	—
138	#208	—	10	2	no	—
139	C₂×C₃⋊C₈	—	5	2	no	—
140	C₂×C₃⋊C₈	—	10	2	no	—
141	C₂×C₃⋊C₈	—	10	2	no	—
142	ΓL(1,𝔽₂₅)	—	10	8	no	—
143	ΓL(1,𝔽₂₅)	—	10	8	no	—
144	D₃×C₈	—	10	8	no	—
145	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
146	C₂²×D₅	—	3	2	no	—
147	C₂²×D₅	—	3	4	no	—
148	C₂²×D₅	—	12	2	no	—
149	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
151	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
152	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
153	C₂×Dic₅	—	3	4	no	—
154	C₂×Dic₅	—	3	2	no	—
155	C₄×D₅	—	3	4	no	—
156	C₄×D₅	—	3	4	no	—
157	C₄×D₅	—	3	4	no	—
158	C₄×D₅	—	3	4	no	—
159	C₅×C₈	—	2	4	no	—
160	C₅×C₈	—	2	2	no	—
161	#168	—	6	4	no	—
162	#168	—	6	2	no	—
163	C₂³×C₄	—	15	1	no	—
164	C₂×M₁₆	—	15	4	no	—
165	C₂×M₁₆	—	15	2	no	—
166	C₂×M₁₆	—	15	8	no	—
167	C₂×M₁₆	—	30	4	no	—
168	C₂×M₁₆	—	30	2	no	—
169	C₂×M₁₆	—	30	4	no	—
170	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
171	C₂²×C₈	—	15	1	no	—
172	C₂²×C₈	—	30	2	no	—
173	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
174	C₂×C₃×C₅	—	2	2	no	—
175	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
176	C₂×C₃×C₅	#430	1	2	yes	—
177	D₁₅	—	2	4	no	—
178	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
179	C₂²×D₃	—	5	2	no	—
180	C₂²×D₃	—	5	4	no	—
181	C₂²×D₃	—	20	2	no	—
182	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
184	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
185	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
186	C₂×Dic₃	—	5	4	no	—
187	C₂×Dic₃	—	5	2	no	—
188	C₄×D₃	—	5	4	no	—
189	C₄×D₃	—	5	4	no	—
190	C₄×D₃	—	5	4	no	—
191	C₄×D₃	—	5	4	no	—
192	C₃×C₈	—	2	4	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃⋊C₈	—	10	4	no	—
195	C₃⋊C₈	—	10	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
198	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂×D₅	—	3	4	no	—
200	C₂×D₅	—	6	4	no	—
201	C₂×D₅	—	12	4	no	—
202	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
203	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
204	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
205	Dic₅	—	3	4	no	—
206	C₂⁴	—	15	1	no	—
207	C₂²×C₄	—	15	1	no	—
208	C₂²×C₄	—	15	4	no	—
209	C₂²×C₄	—	15	2	no	—
210	C₂²×C₄	—	15	4	no	—
211	C₂²×C₄	—	15	2	no	—
212	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	15	4	no	—
215	M₁₆	—	30	8	no	—
216	M₁₆	—	30	8	no	—
217	M₁₆	—	30	2	no	—
218	M₁₆	C₄×D₁₅	1	4	yes	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	15	2	no	—
222	C₂×C₈	—	30	8	no	—
223	C₂×C₈	—	30	2	no	—
224	C₂×C₈	—	30	2	no	—
225	C₂×C₈	C₄×D₁₅	1	2	yes	—
226	C₃×C₅	#1738	1	1	yes	characteristic, complemented, Hall
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
229	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
230	C₂×D₃	—	5	4	no	—
231	C₂×D₃	—	10	4	no	—
232	C₂×D₃	—	20	4	no	—
233	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
234	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₃×C₄	D₅×C₂²⋊C₄	1	2	yes	—
236	Dic₃	—	5	4	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	D₃×#95	1	1	yes	characteristic
239	C₂×C₅	#4893	1	2	yes	—
240	D₅	—	6	4	no	—
241	C₂³	—	15	2	no	—
242	C₂³	—	15	4	no	—
243	C₂³	—	60	2	no	—
244	C₂³	Q₈×D₁₅	1	1	yes	characteristic
245	C₂×C₄	—	15	4	no	—
246	C₂×C₄	—	15	2	no	—
247	C₂×C₄	—	15	4	no	—
248	C₂×C₄	—	15	4	no	—
249	C₂×C₄	—	15	4	no	—
250	C₂×C₄	—	15	4	no	—
251	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
253	C₂×C₄	D₄×D₁₅	1	2	yes	—
254	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	4	no	—
257	C₈	—	30	2	no	—
258	C₈	—	30	4	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	D₅×#95	1	1	yes	characteristic
261	C₂×C₃	#65182	1	2	yes	—
262	D₃	—	10	4	no	—
263	C₅	#68938	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	15	4	no	—
266	C₂²	—	30	4	no	—
267	C₂²	—	60	4	no	—
268	C₂²	C₄⋊C₄×D₁₅	1	2	yes	—
269	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
270	C₄	—	15	4	no	—
271	C₄	C₄²×D₁₅	1	1	yes	characteristic
272	C₄	C₂²⋊C₄×D₁₅	1	1	yes	characteristic
273	C₄	C₂²⋊C₄×D₁₅	1	2	yes	—
274	C₃	#10599731	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	30	4	no	—
277	C₂	D₁₅×#95	1	1	yes	characteristic
278	C₂	#11750994	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×D₁₅	Q₈
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#341	D₁₅
hypocenter	C₃×C₅	#1738
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#341	C₂	semidirect
4. #11746379	C₂	semidirect
5. #11746373	C₂	semidirect
21. C₅×#341	D₃	semidirect
46. C₃×#341	D₅	semidirect
114. #341	D₁₅	semidirect
226. C₃×C₅	#1738	semidirect
263. C₅	#68938	semidirect
274. C₃	#10599731	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₅	2
C₂²×#6705	2
C₃×C₅×#341	2
#11746379	2
#11746373	2
#10599731	3
#68938	5