Subgroups — #14471959

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65911	C₂	1	1	yes	maximal, characteristic
3	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11747083	C₂	1	3	yes	maximal, complemented
5	C₄×#6657	C₂	1	3	yes	maximal, complemented
6	#11746890	C₂	1	3	yes	maximal, complemented
7	#11746881	C₂	1	3	yes	maximal, complemented
8	#10611663	—	3	1	no	maximal, Hall
9	C₂×C₃×#713	—	2	1	no	—
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₅	—	2	1	no	—
12	#91777	—	2	3	no	—
13	#91758	C₂²	1	3	yes	—
14	C₄×Dic₃₀	C₂²	1	3	yes	—
15	C₃×#4041	—	2	3	no	—
16	C₃×#4020	C₂²	1	3	yes	—
17	C₃×C₄×D₂₀	C₂²	1	3	yes	—
18	C₂×#6657	C₂²	1	3	yes	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
20	C₄×C₅×C₃⋊C₈	C₂²	1	3	yes	—
21	#90987	C₂²	1	3	yes	—
22	#90974	C₂²	1	1	yes	characteristic
23	#90965	C₂²	1	6	yes	—
24	#90965	C₂²	1	3	yes	—
25	#83586	—	5	1	no	maximal, Hall
26	#65911	D₃	1	1	yes	characteristic
27	Q₈×Dic₁₀	—	3	1	no	—
28	C₅×C₈×Q₈	—	3	1	no	—
29	#65137	—	3	3	no	—
30	#65131	—	3	3	no	—
31	#64983	—	3	3	no	—
32	C₄×#677	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
34	C₂×Dic₃₀	—	2	3	no	—
35	C₂×Dic₃₀	C₂³	1	1	yes	characteristic
36	C₃×#713	—	4	4	no	—
37	C₂×C₃×D₂₀	—	2	3	no	—
38	C₂×C₃×D₂₀	—	2	3	no	—
39	C₂×C₃×D₂₀	C₂³	1	3	yes	—
40	C₂×C₃×C₄×D₅	—	2	3	no	—
41	C₃×C₅×C₄⋊C₄	D₄	1	3	yes	—
42	C₃×C₄²×C₅	C₂³	1	3	yes	—
43	C₄⋊Dic₁₅	—	2	3	no	—
44	C₄⋊Dic₁₅	C₂³	1	3	yes	—
45	#6712	—	2	3	no	—
46	C₄×Dic₁₅	—	2	3	no	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	C₂³	1	3	yes	—
49	C₃×#685	—	2	3	no	—
50	C₃×C₄⋊Dic₅	C₂³	1	1	yes	characteristic
51	#6657	—	2	6	no	—
52	C₃×#550	—	5	1	no	—
53	Q₈×Dic₆	—	5	1	no	—
54	#5075	—	5	3	no	—
55	#5070	—	5	3	no	—
56	Q₈×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
57	C₄×#214	—	5	3	no	—
58	#5008	—	5	3	no	—
59	C₂×#713	—	2	1	no	—
60	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
61	Q₈×Dic₅	—	6	1	no	—
62	C₂×#677	—	3	3	no	—
63	#4041	—	2	3	no	—
64	#4034	—	6	3	no	—
65	#4020	C₂×D₃	1	3	yes	—
66	C₄×D₂₀	C₂×D₃	1	3	yes	—
67	#4015	—	3	3	no	—
68	C₄×Dic₁₀	—	3	3	no	—
69	C₅×C₄⋊C₈	—	3	3	no	—
70	C₄×C₅×C₈	—	3	3	no	—
71	#3953	—	3	6	no	—
72	#3953	—	3	3	no	—
73	#3949	—	3	3	no	—
74	#3948	—	3	1	no	—
75	#3362	—	15	1	no	Sylow
76	C₂²×C₃×D₅	—	2	3	no	—
77	C₃×C₅×Q₈	SD₁₆	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	1	no	—
82	C₂×Dic₁₅	—	2	3	no	—
83	Dic₃₀	—	2	1	no	—
84	Dic₃₀	—	8	3	no	—
85	Dic₃₀	#41	1	2	yes	complemented
86	C₂×C₃×Dic₅	—	2	1	no	—
87	C₃×D₂₀	—	2	3	no	—
88	C₃×D₂₀	—	2	3	no	—
89	C₃×D₂₀	—	4	6	no	—
90	C₃×D₂₀	—	8	3	no	—
91	C₃×C₄×D₅	—	4	6	no	—
92	C₅×C₃⋊C₈	—	2	3	no	—
93	C₅×C₃⋊C₈	—	2	1	no	—
94	C₂×C₃×#41	—	10	1	no	—
95	C₃×#127	—	5	3	no	—
96	C₃×#121	—	10	3	no	—
97	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×D₄	—	5	3	no	—
99	Q₈×Dic₃	—	10	1	no	—
100	C₂×#214	—	5	3	no	—
101	#888	—	10	3	no	—
102	#869	—	5	3	no	—
103	C₄×Dic₆	—	5	3	no	—
104	#832	—	5	6	no	—
105	#832	—	5	3	no	—
106	#810	—	5	1	no	—
107	#808	—	5	3	no	—
108	C₄⋊(C₃⋊C₈)	C₂×D₅	1	3	yes	—
109	C₄×C₃⋊C₈	C₂×D₅	1	3	yes	—
110	C₂×C₅×Q₈	#67	1	1	yes	characteristic
111	#713	—	4	4	no	—
112	C₂×D₂₀	—	2	3	no	—
113	C₂×D₂₀	—	2	3	no	—
114	C₂×D₂₀	C₂²×D₃	1	3	yes	—
115	C₂×C₄×D₅	—	2	3	no	—
116	C₂×Dic₁₀	—	3	1	no	—
117	C₂×Dic₁₀	—	6	3	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	—	3	3	no	—
120	C₅×C₄⋊C₄	#67	1	3	yes	—
121	C₄²×C₅	C₂²×D₃	1	3	yes	—
122	#685	—	2	3	no	—
123	C₄⋊Dic₅	—	3	3	no	—
124	C₄⋊Dic₅	—	6	3	no	—
125	C₄⋊Dic₅	C₂²×D₃	1	1	yes	characteristic
126	#683	—	6	3	no	—
127	C₄×Dic₅	—	6	3	no	—
128	#677	—	6	6	no	—
129	Q₈²	—	15	1	no	—
130	#550	—	5	1	no	—
131	#486	—	15	3	no	—
132	#476	—	15	3	no	—
133	#460	—	15	3	no	—
134	C₈×Q₈	—	3	1	no	—
135	C₄×SD₁₆	—	15	3	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	4	3	no	—
138	C₂×C₃×D₅	—	4	3	no	—
139	C₂×C₃×D₅	—	4	3	no	—
140	C₃×C₄×C₅	—	2	3	no	—
141	C₃×C₄×C₅	C₂×SD₁₆	1	6	yes	—
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₁₅	—	2	2	no	—
145	Dic₁₅	—	4	3	no	—
146	C₃×Dic₅	—	4	1	no	—
147	C₃×#41	—	20	4	no	—
148	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
149	C₂×C₃×D₄	—	5	3	no	—
150	C₂×C₃×D₄	—	10	3	no	—
151	C₂×C₃×D₄	—	10	3	no	—
152	C₂²×C₃×C₄	—	10	3	no	—
153	C₂×Dic₆	—	5	1	no	—
154	C₂×Dic₆	—	10	3	no	—
155	C₃×C₄⋊C₄	—	5	1	no	—
156	C₃×C₄⋊C₄	D₂₀	1	3	yes	—
157	C₃×C₂²⋊C₄	—	10	3	no	—
158	C₃×C₄²	C₂²×D₅	1	3	yes	—
159	#214	—	10	6	no	—
160	C₄⋊Dic₃	—	5	3	no	—
161	C₄⋊Dic₃	—	10	3	no	—
162	Dic₃⋊C₄	—	10	3	no	—
163	C₄×Dic₃	—	10	3	no	—
164	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	C₂²×D₅	1	3	yes	—
166	C₂²×D₅	—	2	3	no	—
167	C₅×Q₈	#214	1	4	yes	complemented
168	C₂×C₄×C₅	C₂×#67	1	3	yes	—
169	C₂×C₄×C₅	C₂×#67	1	3	yes	—
170	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
171	C₂×Dic₅	—	2	1	no	—
172	C₂×Dic₅	—	6	1	no	—
173	C₂×Dic₅	—	6	3	no	—
174	D₂₀	—	2	3	no	—
175	D₂₀	—	2	3	no	—
176	D₂₀	—	4	6	no	—
177	D₂₀	—	8	3	no	—
178	C₄×D₅	—	4	6	no	—
179	Dic₁₀	—	3	2	no	—
180	Dic₁₀	—	6	1	no	—
181	Dic₁₀	—	24	3	no	—
182	C₅×C₈	—	6	3	no	—
183	C₅×C₈	—	6	1	no	—
184	C₂×#41	—	10	1	no	—
185	C₂×SD₁₆	—	15	3	no	—
186	C₄⋊Q₈	—	15	3	no	—
187	C₄⋊Q₈	—	30	3	no	—
188	#127	—	5	3	no	—
189	#121	—	10	3	no	—
190	C₄×Q₈	—	15	3	no	—
191	C₄×Q₈	—	30	1	no	—
192	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
193	C₄×D₄	—	5	3	no	—
194	#106	—	15	3	no	—
195	C₄⋊C₈	—	3	3	no	—
196	#103	—	15	1	no	—
197	#102	—	15	3	no	—
198	#102	—	15	6	no	—
199	C₄×C₈	—	3	3	no	—
200	C₂×C₃×C₅	#548	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
202	C₂×C₃×C₅	#576	1	1	yes	characteristic
203	C₃×D₅	—	8	3	no	—
204	C₂³×C₃	—	10	3	no	—
205	C₃×Q₈	#677	1	4	yes	complemented
206	C₃×D₄	—	10	3	no	—
207	C₃×D₄	—	10	3	no	—
208	C₃×D₄	—	20	6	no	—
209	C₃×D₄	—	40	3	no	—
210	C₂×C₃×C₄	—	10	1	no	—
211	C₂×C₃×C₄	—	20	6	no	—
212	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
213	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
214	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
215	C₂×Dic₃	—	10	1	no	—
216	C₂×Dic₃	—	10	3	no	—
217	Dic₆	—	5	2	no	—
218	Dic₆	—	10	1	no	—
219	Dic₆	—	40	3	no	—
220	C₃⋊C₈	—	2	1	no	—
221	C₃⋊C₈	—	2	3	no	—
222	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
223	C₂×D₅	—	4	3	no	—
224	C₂×D₅	—	4	3	no	—
225	C₂×D₅	—	4	3	no	—
226	C₄×C₅	—	2	3	no	—
227	C₄×C₅	C₂×#214	1	6	yes	—
228	C₄×C₅	D₃×#41	1	1	yes	characteristic
229	C₄×C₅	#1010	1	1	yes	characteristic
230	Dic₅	—	4	1	no	—
231	Dic₅	—	6	2	no	—
232	Dic₅	—	12	3	no	—
233	#41	—	20	4	no	—
234	C₂×Q₈	—	15	1	no	—
235	C₂×Q₈	—	30	3	no	—
236	C₂×Q₈	#1202	1	1	yes	characteristic
237	C₂×D₄	—	5	3	no	—
238	C₂×D₄	—	10	3	no	—
239	C₂×D₄	—	10	3	no	—
240	C₂²×C₄	—	10	3	no	—
241	SD₁₆	—	30	6	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	3	3	no	—
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	15	3	no	—
246	C₄⋊C₄	—	30	3	no	—
247	C₄⋊C₄	—	30	3	no	—
248	C₄⋊C₄	#1202	1	3	yes	—
249	C₂²⋊C₄	—	10	3	no	—
250	C₄²	—	30	3	no	—
251	C₄²	C₂×D₃×D₅	1	3	yes	—
252	C₃×C₅	#3362	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	20	3	no	—
254	C₂²×C₃	—	20	3	no	—
255	C₂²×C₃	—	20	3	no	—
256	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
257	C₃×C₄	—	2	3	no	—
258	C₃×C₄	—	20	1	no	—
259	C₃×C₄	C₂×#677	1	6	yes	—
260	C₃×C₄	#4147	1	1	yes	characteristic
261	C₃×C₄	D₅×#41	1	1	yes	characteristic
262	Dic₃	—	10	2	no	—
263	Dic₃	—	20	3	no	—
264	C₂×C₅	C₂²×#214	1	1	yes	characteristic
265	C₂×C₅	#5831	1	1	yes	characteristic
266	C₂×C₅	#5835	1	1	yes	characteristic
267	D₅	—	8	3	no	—
268	C₂³	—	10	3	no	—
269	Q₈	—	15	2	no	—
270	Q₈	—	30	1	no	—
271	Q₈	—	120	3	no	—
272	Q₈	#6657	1	4	yes	complemented
273	D₄	—	10	3	no	—
274	D₄	—	10	3	no	—
275	D₄	—	20	6	no	—
276	D₄	—	40	3	no	—
277	C₂×C₄	—	10	1	no	—
278	C₂×C₄	—	20	6	no	—
279	C₂×C₄	—	30	1	no	—
280	C₂×C₄	—	30	3	no	—
281	C₂×C₄	C₂×#1202	1	3	yes	—
282	C₂×C₄	C₂×#1202	1	3	yes	—
283	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
284	C₈	—	6	1	no	—
285	C₈	—	6	3	no	—
286	C₂×C₃	—	40	3	no	—
287	C₂×C₃	#65910	1	1	yes	characteristic
288	C₂×C₃	C₂²×#677	1	1	yes	characteristic
289	C₂×C₃	#66086	1	1	yes	characteristic
290	C₅	#83586	1	1	yes	characteristic, complemented, Sylow
291	C₂²	—	20	3	no	—
292	C₂²	—	20	3	no	—
293	C₂²	—	20	3	no	—
294	C₂²	C₂²×#1202	1	1	yes	characteristic
295	C₄	—	2	3	no	—
296	C₄	—	20	1	no	—
297	C₄	—	30	2	no	—
298	C₄	—	60	3	no	—
299	C₄	C₂×#6657	1	6	yes	—
300	C₄	#92028	1	1	yes	characteristic
301	C₄	#92034	1	1	yes	characteristic
302	C₃	#10611663	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	40	3	no	—
304	C₂	C₂²×#6657	1	1	yes	characteristic
305	C₂	#11755257	1	1	yes	characteristic
306	C₂	#11755311	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1202
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65911	C₂
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#3362
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₀	C₂	semidirect
3. C₅×Q₈×C₃⋊C₈	C₂	semidirect
4. #11747083	C₂	semidirect
5. C₄×#6657	C₂	semidirect
6. #11746890	C₂	semidirect
7. #11746881	C₂	semidirect
56. Q₈×C₃⋊C₈	D₅	semidirect
77. C₃×C₅×Q₈	SD₁₆	semidirect
85. Dic₃₀	#41	semidirect
167. C₅×Q₈	#214	semidirect
205. C₃×Q₈	#677	semidirect
252. C₃×C₅	#3362	semidirect
272. Q₈	#6657	semidirect
290. C₅	#83586	semidirect
302. C₃	#10611663	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₀	2
C₃×#65911	2
C₅×Q₈×C₃⋊C₈	2
#11747083	2
C₄×#6657	2
#11746890	2
#11746881	2
#10611663	3
#83586	5