Subgroups — #14287194

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#215	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
3	#11747008	C₂	1	1	yes	maximal, characteristic
4	#11746356	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745869	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745836	C₂	1	1	yes	maximal, characteristic
7	#11745833	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601691	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#215	C₄	1	1	yes	characteristic
11	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×#6650	—	2	1	no	—
13	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₅×#835	C₄	1	1	yes	characteristic
16	C₅×#809	C₄	1	1	yes	characteristic
17	C₅×#808	C₄	1	1	yes	characteristic
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊#168	—	2	1	no	—
20	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	—	2	1	no	—
22	C₃×C₄×#168	C₂²	1	1	yes	characteristic
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#70936	—	5	1	no	maximal, Hall
25	C₄×C₅×SD₁₆	—	3	1	no	—
26	Q₈×#168	D₃	1	1	yes	characteristic, complemented
27	#65301	—	3	1	no	—
28	Q₈⋊#168	—	3	1	no	—
29	D₄⋊#168	—	3	1	no	—
30	#64678	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₅×#215	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
43	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	—	2	1	no	—
47	C₂×C₃×#168	D₄	1	1	yes	characteristic
48	C₂×C₃×#168	D₄	1	1	yes	characteristic
49	#6650	—	4	2	no	—
50	C₃×C₈×Q₈	—	5	1	no	—
51	C₄×#215	D₅	1	1	yes	characteristic
52	#4688	—	5	1	no	—
53	#4535	—	5	1	no	—
54	#4483	—	5	1	no	—
55	#4479	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₂×C₅×SD₁₆	—	3	1	no	—
58	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₅×D₄	—	3	1	no	—
60	C₂×#681	—	6	1	no	—
61	C₅×#106	—	3	1	no	—
62	C₅×#103	—	3	1	no	—
63	C₅×#102	—	3	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₂²⋊#168	—	6	1	no	—
66	C₄⋊#168	—	2	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
69	C₄×#168	—	2	1	no	—
70	C₄×#168	—	3	1	no	—
71	C₄×#168	C₂×D₃	1	1	yes	characteristic
72	#1588	—	15	1	no	Sylow
73	C₂²×C₅×D₃	—	2	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	M₁₆	1	2	yes	complemented
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₅×D₁₂	—	2	1	no	—
84	C₄×C₅×D₃	—	4	1	no	—
85	C₄×C₅×D₃	—	4	1	no	—
86	C₃⋊#168	—	2	2	no	—
87	C₃⋊#168	—	4	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₃×#168	—	4	1	no	—
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	—	2	1	no	—
93	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#215	Dic₅	1	1	yes	characteristic
95	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
96	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	10	1	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	—	10	1	no	—
101	#835	Dic₅	1	1	yes	characteristic
102	(C₂×D₃)⋊C₈	—	10	1	no	—
103	#809	Dic₅	1	1	yes	characteristic
104	#808	Dic₅	1	1	yes	characteristic
105	C₄⋊(C₃⋊C₈)	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	3	1	no	—
110	C₂²×C₄×C₅	—	6	1	no	—
111	C₅×SD₁₆	—	6	2	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	3	1	no	—
116	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	—	6	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	#681	—	12	2	no	—
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	6	1	no	—
123	C₂×#168	—	6	1	no	—
124	C₂×#168	#67	1	1	yes	characteristic
125	C₂×#168	#67	1	1	yes	characteristic
126	C₈×Q₈	—	5	1	no	—
127	C₄×SD₁₆	—	3	1	no	—
128	#436	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	—	15	1	no	—
132	#322	—	15	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	4	1	no	—
135	C₂×C₅×D₃	—	4	1	no	—
136	C₂×C₅×D₃	—	4	1	no	—
137	C₃×C₄×C₅	—	2	2	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	#131	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₃×C₄×C₅	#137	1	1	yes	characteristic
144	C₅×Dic₃	—	4	1	no	—
145	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
146	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
147	C₂×C₄×D₃	—	2	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	10	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
153	C₃×C₄²	—	2	1	no	—
154	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
155	#215	—	2	2	no	—
156	(C₂×D₃)⋊C₄	—	2	1	no	—
157	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
158	C₂×C₃⋊C₈	—	10	1	no	—
159	C₂×C₃⋊C₈	—	10	1	no	—
160	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
162	ΓL(1,𝔽₂₅)	—	20	2	no	—
163	C₂³×C₅	—	6	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
166	C₅×D₄	—	6	1	no	—
167	C₅×D₄	—	6	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	—	12	1	no	—
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	#235	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
176	C₅×C₈	—	6	1	no	—
177	C₅×C₈	—	6	1	no	—
178	#168	—	2	1	no	—
179	#168	—	2	1	no	—
180	#168	—	4	1	no	—
181	#168	—	6	2	no	—
182	#168	—	12	1	no	—
183	C₂×SD₁₆	—	3	1	no	—
184	C₂×M₁₆	—	30	1	no	—
185	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
186	C₄×D₄	—	3	1	no	—
187	#106	—	3	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	10	1	no	—
190	C₄⋊C₈	—	15	1	no	—
191	#103	—	3	1	no	—
192	#102	—	3	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	10	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₂×C₃×C₅	#435	1	1	yes	characteristic
199	C₂×C₃×C₅	#440	1	1	yes	characteristic
200	C₂×C₃×C₅	#443	1	1	yes	characteristic
201	C₅×D₃	—	8	1	no	—
202	C₂²×D₃	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	#681	1	2	yes	complemented
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	#711	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×Dic₃	—	2	1	no	—
211	D₁₂	—	2	1	no	—
212	D₁₂	—	2	1	no	—
213	C₄×D₃	—	4	1	no	—
214	C₄×D₃	—	4	1	no	—
215	C₃×C₈	—	10	1	no	—
216	C₃×C₈	—	10	1	no	—
217	C₃×C₈	—	20	1	no	—
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	2	1	no	—
220	C₃⋊C₈	—	10	2	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	—	12	1	no	—
223	C₂²×C₅	—	12	1	no	—
224	C₂²×C₅	—	12	1	no	—
225	C₂²×C₅	C₄×#67	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	—	12	1	no	—
230	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
231	C₄×C₅	#901	1	1	yes	characteristic
232	C₄×C₅	#942	1	1	yes	characteristic
233	C₄×C₅	#949	1	1	yes	characteristic
234	C₂×Q₈	D₃×Dic₅	1	1	yes	characteristic
235	C₂×D₄	—	3	1	no	—
236	C₂²×C₄	—	6	1	no	—
237	SD₁₆	—	6	2	no	—
238	M₁₆	—	60	2	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	5	1	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	10	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	30	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₄⋊C₄	D₃×Dic₅	1	1	yes	characteristic
249	C₂²⋊C₄	—	6	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#1588	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
254	C₂×D₃	—	4	1	no	—
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	—	4	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	#4060	1	1	yes	characteristic
261	C₃×C₄	#4061	1	1	yes	characteristic
262	C₃×C₄	C₂×#681	1	1	yes	characteristic
263	C₃×C₄	#4094	1	1	yes	characteristic
264	Dic₃	—	4	1	no	—
265	C₂×C₅	—	24	1	no	—
266	C₂×C₅	#5026	1	1	yes	characteristic
267	C₂×C₅	#5111	1	1	yes	characteristic
268	C₂×C₅	#5144	1	1	yes	characteristic
269	C₂³	—	6	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	#6650	1	2	yes	complemented
272	D₄	—	6	1	no	—
273	D₄	—	6	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	—	12	1	no	—
279	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
280	C₂×C₄	#6782	1	1	yes	characteristic
281	C₂×C₄	D₅×#67	1	1	yes	characteristic
282	C₈	—	6	1	no	—
283	C₈	—	6	1	no	—
284	C₈	—	10	1	no	—
285	C₈	—	10	1	no	—
286	C₈	—	20	1	no	—
287	C₈	—	30	2	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	#65300	1	1	yes	characteristic
290	C₂×C₃	#65453	1	1	yes	characteristic
291	C₂×C₃	#65485	1	1	yes	characteristic
292	D₃	—	8	1	no	—
293	C₅	#70936	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	12	1	no	—
295	C₂²	—	12	1	no	—
296	C₂²	—	12	1	no	—
297	C₂²	Dic₅×#67	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	12	1	no	—
302	C₄	#91286	1	1	yes	characteristic
303	C₄	C₂×#6650	1	1	yes	characteristic
304	C₄	#91502	1	1	yes	characteristic
305	C₄	#91513	1	1	yes	characteristic
306	C₃	#10601691	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	24	1	no	—
308	C₂	#11748500	1	1	yes	characteristic
309	C₂	#11749223	1	1	yes	characteristic
310	C₂	#11749432	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×#67
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₁₂	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#1588
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×#168	C₂	semidirect
4. #11746356	C₂	semidirect
5. #11745869	C₂	semidirect
7. #11745833	C₂	semidirect
26. Q₈×#168	D₃	semidirect
75. C₃×C₅×Q₈	M₁₆	semidirect
165. C₅×Q₈	ΓL(1,𝔽₂₅)	semidirect
204. C₃×Q₈	#681	semidirect
252. C₃×C₅	#1588	semidirect
271. Q₈	#6650	semidirect
293. C₅	#70936	semidirect
306. C₃	#10601691	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#215	2
C₃×Q₈×#168	2
#11747008	2
#11746356	2
#11745869	2
#11745836	2
#11745833	2
#10601691	3
#70936	5