Subgroups — #14439209

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91831	C₂	1	4	yes	maximal, complemented
3	#11756236	C₂	1	1	yes	maximal, characteristic
4	#11751193	C₂	1	2	yes	maximal, complemented
5	#11751187	C₂	1	4	yes	maximal, complemented
6	#11751186	C₂	1	2	yes	maximal, complemented
7	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
8	#10614934	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
10	C₂×#6817	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×#6718	C₂²	1	8	yes	—
17	#91831	—	2	16	no	—
18	C₂×#6717	C₂²	1	8	yes	—
19	#91819	—	2	2	no	—
20	C₂×#6710	C₂²	1	2	yes	—
21	#91764	C₂²	1	2	yes	—
22	#91763	C₂²	1	1	yes	characteristic
23	C₄×D₆₀	—	2	2	no	—
24	#91758	C₂²	1	1	yes	characteristic
25	#91087	C₂²	1	2	yes	—
26	C₄×C₃⋊#168	C₂²	1	2	yes	—
27	#83203	—	5	1	no	maximal, Hall
28	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
29	C₂×#4088	—	3	4	no	—
30	#65811	—	3	1	no	—
31	#65381	—	3	2	no	—
32	#65375	—	3	4	no	—
33	#65374	—	3	2	no	—
34	C₄×#681	—	3	1	no	—
35	C₂×C₃×C₅×Q₈	—	2	2	no	—
36	C₂×C₃×C₅×Q₈	—	2	8	no	—
37	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
38	C₂²×C₃×C₄×C₅	—	2	2	no	—
39	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×#1220	—	2	2	no	—
42	#6817	—	2	8	no	—
43	C₂×D₆₀	—	2	2	no	—
44	C₂×D₆₀	C₂³	1	2	yes	—
45	C₂×C₄×D₁₅	—	2	2	no	—
46	C₂×Dic₃₀	C₂³	1	2	yes	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	#6718	—	2	16	no	—
53	#6717	—	2	16	no	—
54	#6714	—	2	4	no	—
55	C₄⋊Dic₁₅	—	2	2	no	—
56	#6710	D₄	1	8	yes	complemented
57	C₂×C₃⋊#168	C₂³	1	4	yes	—
58	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
59	C₂×#942	—	5	4	no	—
60	#5477	—	5	1	no	—
61	#5092	—	5	2	no	—
62	#5086	—	5	4	no	—
63	#5085	—	5	2	no	—
64	C₄×#208	—	5	1	no	—
65	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
66	C₂×#709	—	3	2	no	—
67	C₅×C₄⋊Q₈	—	2	2	no	—
68	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
69	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
70	C₂×C₅×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	C₂×#689	—	3	8	no	—
73	#4088	—	6	16	no	—
74	C₂×#688	—	3	8	no	—
75	#4076	—	6	2	no	—
76	C₂×#681	—	3	2	no	—
77	#4021	—	3	2	no	—
78	#4020	—	3	1	no	—
79	C₄×D₂₀	—	6	2	no	—
80	#4015	—	3	1	no	—
81	#3935	—	3	2	no	—
82	C₄×#168	—	3	2	no	—
83	#3152	—	15	1	no	Sylow
84	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₅	—	2	2	no	—
86	C₃×C₅×Q₈	—	2	8	no	—
87	C₃×C₅×Q₈	—	4	4	no	—
88	C₂×C₃×C₄×C₅	—	2	4	no	—
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	—	4	4	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	#1220	—	4	4	no	—
97	C₂×Dic₁₅	—	2	2	no	—
98	D₆₀	—	2	4	no	—
99	D₆₀	—	4	4	no	—
100	C₄×D₁₅	—	4	4	no	—
101	Dic₃₀	—	2	4	no	—
102	C₃⋊#168	C₂×D₄	1	8	yes	—
103	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
104	C₂×#235	—	5	2	no	—
105	C₃×C₄⋊Q₈	—	2	2	no	—
106	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
107	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
108	C₂×C₃×C₄⋊C₄	—	2	2	no	—
109	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
110	C₂×#216	—	5	8	no	—
111	#942	—	10	16	no	—
112	C₂×#215	—	5	8	no	—
113	#930	—	10	2	no	—
114	C₂×#208	—	5	2	no	—
115	#875	—	5	2	no	—
116	#874	—	5	1	no	—
117	C₄×D₁₂	—	10	2	no	—
118	#869	—	5	1	no	—
119	#803	—	5	2	no	—
120	C₄×C₃⋊C₈	—	5	2	no	—
121	C₂×C₅×Q₈	—	2	2	no	—
122	C₂×C₅×Q₈	—	2	8	no	—
123	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
124	C₂²×C₄×C₅	—	2	2	no	—
125	C₂²×C₄×C₅	#67	1	2	yes	—
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₂²⋊D₅	—	6	2	no	—
128	#709	—	6	8	no	—
129	C₂×D₂₀	—	3	2	no	—
130	C₂×D₂₀	—	6	2	no	—
131	C₂×C₄×D₅	—	6	2	no	—
132	C₂×Dic₁₀	—	3	2	no	—
133	C₅×C₄⋊C₄	—	2	4	no	—
134	C₅×C₄⋊C₄	—	2	4	no	—
135	C₄²×C₅	#67	1	2	yes	—
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	#689	—	6	16	no	—
139	#688	—	6	16	no	—
140	#685	—	6	4	no	—
141	C₄⋊Dic₅	—	6	2	no	—
142	#681	—	3	8	no	—
143	C₂×#168	—	3	4	no	—
144	C₂×#137	—	15	4	no	—
145	#532	—	15	1	no	—
146	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
147	#497	—	15	4	no	—
148	#494	—	15	2	no	—
149	#492	—	15	2	no	—
150	C₄×M₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	#127	1	2	yes	—
153	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₅	—	4	2	no	—
155	C₂×D₁₅	—	4	4	no	—
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	—	4	4	no	—
158	C₃×C₄×C₅	#127	1	2	yes	—
159	C₃×C₄×C₅	#137	1	4	yes	complemented
160	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
161	Dic₁₅	—	4	2	no	—
162	C₂×C₃×Q₈	—	2	2	no	—
163	C₂×C₃×Q₈	—	2	8	no	—
164	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
165	C₂²×C₃×C₄	—	2	2	no	—
166	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
167	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
168	C₂×#67	—	10	2	no	—
169	#235	—	10	8	no	—
170	C₂×D₁₂	—	5	2	no	—
171	C₂×D₁₂	—	10	2	no	—
172	C₂×C₄×D₃	—	10	2	no	—
173	C₂×Dic₆	—	5	2	no	—
174	C₃×C₄⋊C₄	—	2	4	no	—
175	C₃×C₄⋊C₄	—	2	4	no	—
176	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	#216	—	10	16	no	—
180	#215	—	10	16	no	—
181	(C₂×D₃)⋊C₄	—	10	4	no	—
182	C₄⋊Dic₃	—	10	2	no	—
183	#208	—	5	8	no	—
184	C₂×C₃⋊C₈	—	5	4	no	—
185	C₂³×C₅	C₂×#67	1	1	yes	characteristic
186	C₂²×D₅	—	6	2	no	—
187	C₅×Q₈	—	2	8	no	—
188	C₅×Q₈	—	4	4	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	2	4	no	—
191	C₂×C₄×C₅	—	4	4	no	—
192	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
193	C₂×C₄×C₅	C₂×#67	1	2	yes	—
194	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×#67	1	2	yes	—
196	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
197	C₂²⋊D₅	—	12	4	no	—
198	C₂×Dic₅	—	6	2	no	—
199	D₂₀	—	6	4	no	—
200	D₂₀	—	12	4	no	—
201	C₄×D₅	—	12	4	no	—
202	Dic₁₀	—	6	4	no	—
203	#168	—	3	8	no	—
204	C₂×#41	—	15	2	no	—
205	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
206	#137	—	30	16	no	—
207	C₂×Q₁₆	—	15	8	no	—
208	C₂×SD₁₆	—	15	8	no	—
209	C₂×M₁₆	—	15	2	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	15	1	no	—
212	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
213	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
214	#127	—	15	1	no	—
215	#124	—	15	2	no	—
216	#121	—	30	2	no	—
217	C₄×D₄	—	30	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
220	#97	—	15	2	no	—
221	C₄×C₈	—	15	2	no	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
224	C₂×C₃×C₅	C₂×#137	1	2	yes	—
225	D₁₅	—	8	2	no	—
226	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂²×D₃	—	10	2	no	—
228	C₃×Q₈	—	2	8	no	—
229	C₃×Q₈	—	4	4	no	—
230	C₂×C₃×C₄	—	2	4	no	—
231	C₂×C₃×C₄	—	2	4	no	—
232	C₂×C₃×C₄	—	4	4	no	—
233	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
237	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
238	#67	—	20	4	no	—
239	C₂×Dic₃	—	10	2	no	—
240	D₁₂	—	10	4	no	—
241	D₁₂	—	20	4	no	—
242	C₄×D₃	—	20	4	no	—
243	Dic₆	—	10	4	no	—
244	C₃⋊C₈	—	5	8	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	#940	1	2	yes	—
247	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
248	C₂×D₅	—	12	2	no	—
249	C₂×D₅	—	12	4	no	—
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	4	4	no	—
252	C₄×C₅	#940	1	2	yes	—
253	C₄×C₅	#942	1	4	yes	complemented
254	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
255	Dic₅	—	12	2	no	—
256	#41	—	30	8	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	8	no	—
259	C₂×Q₈	—	15	2	no	—
260	C₂×Q₈	C₂²×D₁₅	1	4	yes	—
261	C₂×D₄	—	15	2	no	—
262	C₂×D₄	—	30	2	no	—
263	C₂×D₄	—	30	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	30	2	no	—
266	C₂²×C₄	#1220	1	2	yes	—
267	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
268	Q₁₆	—	30	16	no	—
269	SD₁₆	—	30	16	no	—
270	M₁₆	—	15	8	no	—
271	C₂×C₈	—	15	4	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	30	2	no	—
275	C₂²⋊C₄	—	30	4	no	—
276	C₄²	#1220	1	2	yes	—
277	C₄²	C₂²×D₁₅	1	1	yes	characteristic
278	C₄²	C₂²×D₁₅	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	#4086	1	2	yes	—
282	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
283	C₂×D₃	—	20	2	no	—
284	C₂×D₃	—	20	4	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	4	4	no	—
287	C₃×C₄	#4086	1	2	yes	—
288	C₃×C₄	#4088	1	4	yes	complemented
289	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
290	Dic₃	—	20	2	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	C₂×#940	1	1	yes	characteristic
293	C₂×C₅	C₂×#942	1	2	yes	—
294	D₅	—	24	2	no	—
295	C₂³	—	30	2	no	—
296	C₂³	C₂×#1220	1	1	yes	characteristic
297	Q₈	—	2	8	no	—
298	Q₈	—	4	4	no	—
299	Q₈	—	30	4	no	—
300	D₄	—	30	4	no	—
301	D₄	—	60	4	no	—
302	D₄	—	60	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	30	2	no	—
307	C₂×C₄	—	60	4	no	—
308	C₂×C₄	D₄×D₁₅	1	4	yes	—
309	C₂×C₄	C₂×#1220	1	2	yes	—
310	C₂×C₄	C₂×#1220	1	1	yes	characteristic
311	C₂×C₄	C₂×#1220	1	2	yes	—
312	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
313	C₈	—	15	8	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	C₂×#4086	1	1	yes	characteristic
316	C₂×C₃	C₂×#4088	1	2	yes	—
317	D₃	—	40	2	no	—
318	C₅	#83203	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	60	2	no	—
321	C₂²	—	60	4	no	—
322	C₂²	#91829	1	2	yes	—
323	C₂²	C₂²×#1220	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	60	2	no	—
327	C₄	#91829	1	2	yes	—
328	C₄	#91831	1	4	yes	complemented
329	C₄	C₂×D₄×D₁₅	1	2	yes	—
330	C₃	#10614934	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	120	2	no	—
333	C₂	C₂×#91829	1	1	yes	characteristic
334	C₂	C₂×#91831	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	#11756236	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#3152
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×#91831	C₂	semidirect
4. #11751193	C₂	semidirect
5. #11751187	C₂	semidirect
6. #11751186	C₂	semidirect
7. C₄×#6710	C₂	semidirect
28. C₂×C₅×C₄⋊Q₈	D₃	semidirect
56. #6710	D₄	semidirect
58. C₂×C₃×C₄⋊Q₈	D₅	semidirect
146. C₂×C₄⋊Q₈	D₁₅	semidirect
159. C₃×C₄×C₅	#137	semidirect
253. C₄×C₅	#942	semidirect
279. C₃×C₅	#3152	semidirect
288. C₃×C₄	#4088	semidirect
318. C₅	#83203	semidirect
328. C₄	#91831	semidirect
330. C₃	#10614934	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×#91831	2
#11756236	2
#11751193	2
#11751187	2
#11751186	2
C₄×#6710	2
#10614934	3
#83203	5