Subgroups — #14471481

Order 1920.

852 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65911	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11747033	C₂	1	3	yes	maximal, complemented
5	C₄×#6652	C₂	1	3	yes	maximal, complemented
6	#11746895	C₂	1	3	yes	maximal, complemented
7	#11746886	C₂	1	3	yes	maximal, complemented
8	#10615293	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	1	no	—
10	C₂×C₃×#713	—	2	1	no	—
11	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#897	—	2	3	no	—
13	C₄×C₅×D₁₂	C₂²	1	3	yes	—
14	C₅×#869	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×#6652	C₂²	1	3	yes	—
19	C₄⋊(C₃⋊#168)	C₂²	1	3	yes	—
20	C₄×C₃⋊#168	C₂²	1	3	yes	—
21	#90989	C₂²	1	3	yes	—
22	#90966	C₂²	1	1	yes	characteristic
23	#90964	C₂²	1	6	yes	—
24	#90964	C₂²	1	3	yes	—
25	#83580	—	5	1	no	maximal, Hall
26	C₅×D₄×Q₈	—	3	1	no	—
27	#65911	D₃	1	1	yes	characteristic, complemented
28	#65377	—	3	3	no	—
29	#65376	—	3	3	no	—
30	Q₈×#168	—	3	1	no	—
31	#65304	—	3	3	no	—
32	C₄×#686	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
34	C₅×D₃×Q₈	—	4	4	no	—
35	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	—	2	3	no	—
37	C₂×C₅×Dic₆	—	2	3	no	—
38	C₃×#713	—	4	4	no	—
39	C₂×C₃×D₂₀	—	2	3	no	—
40	C₂×C₃×D₂₀	—	2	3	no	—
41	C₂×C₃×D₂₀	C₂³	1	3	yes	—
42	C₂×C₃×C₄×D₅	—	2	3	no	—
43	C₃×C₅×C₄⋊C₄	D₄	1	3	yes	—
44	C₃×C₄²×C₅	C₂³	1	3	yes	—
45	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	C₂³	1	3	yes	—
47	C₅×(C₂×D₃)⋊C₄	—	2	3	no	—
48	C₅×C₄⋊Dic₃	—	2	3	no	—
49	C₅×C₄⋊Dic₃	C₂³	1	3	yes	—
50	C₃×#685	—	2	3	no	—
51	C₃×C₄⋊Dic₅	C₂³	1	1	yes	characteristic
52	#6652	—	2	6	no	—
53	C₃×#550	—	5	1	no	—
54	Q₈×D₁₂	D₅	1	1	yes	characteristic, complemented
55	#5072	—	5	3	no	—
56	#5070	—	5	3	no	—
57	Q₈×C₃⋊C₈	—	5	1	no	—
58	C₄×#213	—	5	3	no	—
59	#5007	—	5	3	no	—
60	C₂²×C₅×Q₈	—	6	1	no	—
61	C₂×#713	—	2	1	no	—
62	C₅×C₄⋊Q₈	—	3	3	no	—
63	C₅×C₂²⋊Q₈	—	6	3	no	—
64	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
65	C₄×C₅×D₄	—	3	3	no	—
66	C₂×#686	—	3	3	no	—
67	#4041	—	2	3	no	—
68	#4020	C₂×D₃	1	3	yes	—
69	C₄×D₂₀	C₂×D₃	1	3	yes	—
70	#3964	—	3	1	no	—
71	#3941	—	3	3	no	—
72	#3941	—	3	6	no	—
73	#3939	—	3	3	no	—
74	C₄⋊#168	—	3	3	no	—
75	C₄×#168	—	3	3	no	—
76	#3358	—	15	1	no	Sylow
77	C₂²×C₅×D₃	—	2	1	no	—
78	C₂²×C₃×D₅	—	2	3	no	—
79	C₃×C₅×Q₈	D₈	1	4	yes	complemented
80	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₅×Dic₃	—	2	3	no	—
84	C₅×D₁₂	—	2	1	no	—
85	C₅×D₁₂	#41	1	2	yes	complemented
86	C₄×C₅×D₃	—	4	6	no	—
87	C₅×Dic₆	—	4	6	no	—
88	C₂×C₃×Dic₅	—	2	1	no	—
89	C₃×D₂₀	—	2	3	no	—
90	C₃×D₂₀	—	2	3	no	—
91	C₃×D₂₀	—	4	6	no	—
92	C₃×D₂₀	—	8	3	no	—
93	C₃×C₄×D₅	—	4	6	no	—
94	C₃⋊#168	—	2	3	no	—
95	C₃⋊#168	—	2	1	no	—
96	C₂×C₃×#41	—	10	1	no	—
97	C₂×D₃×Q₈	—	2	1	no	—
98	C₃×#127	—	5	3	no	—
99	C₃×#121	—	10	3	no	—
100	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄×D₄	—	5	3	no	—
102	C₂×#213	—	5	3	no	—
103	#897	—	2	3	no	—
104	C₄×D₁₂	C₂×D₅	1	3	yes	—
105	#869	C₂×D₅	1	3	yes	—
106	#832	—	5	6	no	—
107	#832	—	5	3	no	—
108	#809	—	5	1	no	—
109	#807	—	5	3	no	—
110	C₄⋊(C₃⋊C₈)	—	5	3	no	—
111	C₄×C₃⋊C₈	—	5	3	no	—
112	C₂×C₅×Q₈	—	6	3	no	—
113	C₂×C₅×Q₈	—	12	4	no	—
114	C₂×C₅×Q₈	#67	1	1	yes	characteristic
115	C₂×C₅×D₄	—	3	1	no	—
116	C₂²×C₄×C₅	—	6	3	no	—
117	#713	—	4	4	no	—
118	C₂×D₂₀	—	2	3	no	—
119	C₂×D₂₀	—	2	3	no	—
120	C₂×D₂₀	C₂²×D₃	1	3	yes	—
121	C₂×C₄×D₅	—	2	3	no	—
122	C₅×C₄⋊C₄	—	3	3	no	—
123	C₅×C₄⋊C₄	—	6	3	no	—
124	C₅×C₄⋊C₄	#67	1	3	yes	—
125	C₅×C₂²⋊C₄	—	6	3	no	—
126	C₄²×C₅	C₂²×D₃	1	3	yes	—
127	#686	—	6	6	no	—
128	#685	—	2	3	no	—
129	C₄⋊Dic₅	C₂²×D₃	1	1	yes	characteristic
130	C₂×#168	—	3	1	no	—
131	C₂×#168	—	3	3	no	—
132	#550	—	5	1	no	—
133	D₄×Q₈	—	3	1	no	—
134	#486	—	15	3	no	—
135	#474	—	15	3	no	—
136	#459	—	15	3	no	—
137	C₈×Q₈	—	15	1	no	—
138	C₄×D₈	—	15	3	no	—
139	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₅×D₃	—	2	2	no	—
141	C₂×C₅×D₃	—	4	1	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₃×D₅	—	4	3	no	—
144	C₂×C₃×D₅	—	4	3	no	—
145	C₂×C₃×D₅	—	4	3	no	—
146	C₃×C₄×C₅	—	2	3	no	—
147	C₃×C₄×C₅	C₂×D₈	1	6	yes	—
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
150	C₃×Dic₅	—	4	1	no	—
151	C₅×Dic₃	—	4	3	no	—
152	C₃×#41	—	20	4	no	—
153	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₃×D₄	—	5	3	no	—
155	C₂×C₃×D₄	—	10	3	no	—
156	C₂×C₃×D₄	—	10	3	no	—
157	C₂²×C₃×C₄	—	10	3	no	—
158	D₃×Q₈	—	4	4	no	—
159	C₂×D₁₂	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₄×D₃	—	2	3	no	—
161	C₂×Dic₆	—	2	3	no	—
162	C₃×C₄⋊C₄	—	5	1	no	—
163	C₃×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
164	C₃×C₂²⋊C₄	—	10	3	no	—
165	C₃×C₄²	C₂²×D₅	1	3	yes	—
166	#213	—	10	6	no	—
167	(C₂×D₃)⋊C₄	—	2	3	no	—
168	C₄⋊Dic₃	—	2	3	no	—
169	C₄⋊Dic₃	C₂²×D₅	1	3	yes	—
170	C₂×C₃⋊C₈	—	5	1	no	—
171	C₂×C₃⋊C₈	—	5	3	no	—
172	C₂³×C₅	—	6	1	no	—
173	C₂²×D₅	—	2	3	no	—
174	C₅×Q₈	—	12	6	no	—
175	C₅×Q₈	#213	1	4	yes	complemented
176	C₅×D₄	—	3	2	no	—
177	C₅×D₄	—	6	1	no	—
178	C₂×C₄×C₅	—	6	3	no	—
179	C₂×C₄×C₅	—	12	6	no	—
180	C₂×C₄×C₅	C₂×#67	1	3	yes	—
181	C₂×C₄×C₅	C₂×#67	1	3	yes	—
182	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
183	C₂×Dic₅	—	2	1	no	—
184	D₂₀	—	2	3	no	—
185	D₂₀	—	2	3	no	—
186	D₂₀	—	4	6	no	—
187	D₂₀	—	8	3	no	—
188	C₄×D₅	—	4	6	no	—
189	#168	—	6	3	no	—
190	#168	—	6	1	no	—
191	C₂×#41	—	10	1	no	—
192	C₂²×Q₈	—	6	1	no	—
193	C₂×D₈	—	15	3	no	—
194	C₄⋊Q₈	—	3	3	no	—
195	#127	—	5	3	no	—
196	C₂²⋊Q₈	—	6	3	no	—
197	#121	—	10	3	no	—
198	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
199	C₄×D₄	—	3	3	no	—
200	C₄×D₄	—	5	3	no	—
201	#107	—	15	3	no	—
202	C₄⋊C₈	—	15	3	no	—
203	#102	—	15	3	no	—
204	#102	—	15	6	no	—
205	#102	—	15	1	no	—
206	C₄×C₈	—	15	3	no	—
207	C₂×C₃×C₅	#548	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
209	C₂×C₃×C₅	#577	1	1	yes	characteristic
210	C₃×D₅	—	8	3	no	—
211	C₅×D₃	—	4	2	no	—
212	C₂³×C₃	—	10	3	no	—
213	C₂²×D₃	—	2	1	no	—
214	C₃×Q₈	#686	1	4	yes	complemented
215	C₃×D₄	—	10	3	no	—
216	C₃×D₄	—	10	3	no	—
217	C₃×D₄	—	20	6	no	—
218	C₃×D₄	—	40	3	no	—
219	C₂×C₃×C₄	—	10	1	no	—
220	C₂×C₃×C₄	—	20	6	no	—
221	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
222	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
223	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
224	C₂×Dic₃	—	2	3	no	—
225	D₁₂	—	2	1	no	—
226	D₁₂	#713	1	2	yes	complemented
227	C₄×D₃	—	4	6	no	—
228	Dic₆	—	4	6	no	—
229	C₃⋊C₈	—	10	3	no	—
230	C₃⋊C₈	—	10	1	no	—
231	C₂²×C₅	—	6	2	no	—
232	C₂²×C₅	—	12	1	no	—
233	C₂²×C₅	—	12	1	no	—
234	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
235	C₂×D₅	—	4	3	no	—
236	C₂×D₅	—	4	3	no	—
237	C₂×D₅	—	4	3	no	—
238	C₄×C₅	—	2	3	no	—
239	C₄×C₅	—	12	3	no	—
240	C₄×C₅	C₂×#213	1	6	yes	—
241	C₄×C₅	D₃×#41	1	1	yes	characteristic
242	C₄×C₅	#1010	1	1	yes	characteristic
243	Dic₅	—	4	1	no	—
244	#41	—	20	4	no	—
245	C₂×Q₈	—	6	3	no	—
246	C₂×Q₈	—	12	4	no	—
247	C₂×Q₈	#1201	1	1	yes	characteristic
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	—	5	3	no	—
250	C₂×D₄	—	10	3	no	—
251	C₂×D₄	—	10	3	no	—
252	C₂²×C₄	—	6	3	no	—
253	C₂²×C₄	—	10	3	no	—
254	D₈	—	30	6	no	—
255	C₂×C₈	—	15	1	no	—
256	C₂×C₈	—	15	3	no	—
257	C₄⋊C₄	—	3	3	no	—
258	C₄⋊C₄	—	5	1	no	—
259	C₄⋊C₄	—	6	3	no	—
260	C₄⋊C₄	#1201	1	3	yes	—
261	C₂²⋊C₄	—	6	3	no	—
262	C₂²⋊C₄	—	10	3	no	—
263	C₄²	C₂×D₃×D₅	1	3	yes	—
264	C₃×C₅	#3358	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	—	20	3	no	—
266	C₂²×C₃	—	20	3	no	—
267	C₂²×C₃	—	20	3	no	—
268	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
269	C₂×D₃	—	2	2	no	—
270	C₂×D₃	—	4	1	no	—
271	C₂×D₃	—	4	1	no	—
272	C₃×C₄	—	2	3	no	—
273	C₃×C₄	—	20	1	no	—
274	C₃×C₄	C₂×#686	1	6	yes	—
275	C₃×C₄	C₂×#713	1	1	yes	characteristic
276	C₃×C₄	#4147	1	1	yes	characteristic
277	Dic₃	—	4	3	no	—
278	C₂×C₅	—	12	2	no	—
279	C₂×C₅	C₂²×#213	1	1	yes	characteristic
280	C₂×C₅	#5831	1	1	yes	characteristic
281	C₂×C₅	#5836	1	1	yes	characteristic
282	D₅	—	8	3	no	—
283	C₂³	—	6	1	no	—
284	C₂³	—	10	3	no	—
285	Q₈	—	12	6	no	—
286	Q₈	#6652	1	4	yes	complemented
287	D₄	—	3	2	no	—
288	D₄	—	6	1	no	—
289	D₄	—	10	3	no	—
290	D₄	—	10	3	no	—
291	D₄	—	20	6	no	—
292	D₄	—	40	3	no	—
293	C₂×C₄	—	6	3	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	12	6	no	—
296	C₂×C₄	—	20	6	no	—
297	C₂×C₄	C₂×#1201	1	3	yes	—
298	C₂×C₄	C₂×#1201	1	3	yes	—
299	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
300	C₈	—	30	1	no	—
301	C₈	—	30	3	no	—
302	C₂×C₃	—	40	3	no	—
303	C₂×C₃	C₂²×#686	1	1	yes	characteristic
304	C₂×C₃	#66151	1	1	yes	characteristic
305	C₂×C₃	#66171	1	1	yes	characteristic
306	D₃	—	4	2	no	—
307	C₅	#83580	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	6	2	no	—
309	C₂²	—	12	1	no	—
310	C₂²	—	12	1	no	—
311	C₂²	—	20	3	no	—
312	C₂²	—	20	3	no	—
313	C₂²	—	20	3	no	—
314	C₂²	C₂²×#1201	1	1	yes	characteristic
315	C₄	—	2	3	no	—
316	C₄	—	12	3	no	—
317	C₄	—	20	1	no	—
318	C₄	C₂×#6652	1	6	yes	—
319	C₄	#92028	1	1	yes	characteristic
320	C₄	D₃×#713	1	1	yes	characteristic
321	C₃	#10615293	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	12	2	no	—
323	C₂	—	40	3	no	—
324	C₂	C₂²×#6652	1	1	yes	characteristic
325	C₂	#11755247	1	1	yes	characteristic
326	C₂	#11755303	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1201
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
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#3358
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. C₅×Q₈×D₁₂	C₂	semidirect
2. C₃×#65911	C₂	semidirect
3. Q₈×C₃⋊#168	C₂	semidirect
4. #11747033	C₂	semidirect
5. C₄×#6652	C₂	semidirect
6. #11746895	C₂	semidirect
7. #11746886	C₂	semidirect
27. #65911	D₃	semidirect
54. Q₈×D₁₂	D₅	semidirect
79. C₃×C₅×Q₈	D₈	semidirect
85. C₅×D₁₂	#41	semidirect
175. C₅×Q₈	#213	semidirect
214. C₃×Q₈	#686	semidirect
226. D₁₂	#713	semidirect
264. C₃×C₅	#3358	semidirect
286. Q₈	#6652	semidirect
307. C₅	#83580	semidirect
321. C₃	#10615293	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×D₁₂	2
C₃×#65911	2
Q₈×C₃⋊#168	2
#11747033	2
C₄×#6652	2
#11746895	2
#11746886	2
#10615293	3
#83580	5