Subgroups — #18005123

Order 4736.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558163	C₂	1	1	yes	maximal, characteristic
2	#14558160	C₂	1	1	yes	maximal, characteristic
3	#14557942	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557941	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557940	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557934	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557932	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557930	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557929	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557574	C₂	1	1	yes	maximal, characteristic
11	#14557572	C₂	1	1	yes	maximal, characteristic
12	#14557570	C₂	1	1	yes	maximal, characteristic
13	#106×D₃₇	C₂	1	1	yes	maximal, characteristic
14	#14557490	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596929	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
18	#11924536	C₂²	1	1	yes	characteristic
19	#11924535	—	2	1	no	—
20	#11924535	C₂²	1	1	yes	characteristic
21	#11924535	C₂²	1	1	yes	characteristic
22	#11924534	—	2	1	no	—
23	#11924528	C₂²	1	1	yes	characteristic
24	#11924527	C₂²	1	1	yes	characteristic
25	#11924527	C₂²	1	1	yes	characteristic
26	#11924526	—	2	1	no	—
27	#11924526	—	2	1	no	—
28	#11924516	—	2	1	no	—
29	#11924514	—	2	1	no	—
30	#11924514	—	2	1	no	—
31	#11924512	—	2	1	no	—
32	#11924494	C₂²	1	1	yes	characteristic
33	#11924494	C₂²	1	1	yes	characteristic
34	#11924493	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924470	C₂²	1	1	yes	characteristic
39	#11924469	C₂²	1	1	yes	characteristic
40	#11924469	C₂²	1	1	yes	characteristic
41	#11924468	C₂²	1	1	yes	characteristic
42	#11924468	C₂²	1	1	yes	characteristic
43	#11924467	C₂²	1	1	yes	characteristic
44	#11924467	C₂²	1	1	yes	characteristic
45	#11924467	C₂²	1	1	yes	characteristic
46	#11924467	C₂²	1	1	yes	characteristic
47	#11924467	C₂²	1	1	yes	characteristic
48	#11924467	C₂²	1	1	yes	characteristic
49	#11924446	C₂²	1	1	yes	characteristic
50	C₄×#64436	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
53	#125×C₃₇	C₂²	1	1	yes	characteristic
54	#107×C₃₇	C₂²	1	1	yes	characteristic
55	#107×C₃₇	C₂²	1	1	yes	characteristic
56	#106×C₃₇	C₂²	1	1	yes	characteristic
57	#106×C₃₇	C₂²	1	1	yes	characteristic
58	#97×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	—	2	1	no	—
60	C₂×C₄×D₃₇	—	2	1	no	—
61	C₂×C₄×D₃₇	—	2	1	no	—
62	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₇₄	—	2	1	no	—
64	C₂×Dic₇₄	—	2	1	no	—
65	C₂×Dic₇₄	—	2	1	no	—
66	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₇	—	2	1	no	—
69	C₄⋊C₄×C₃₇	—	2	1	no	—
70	C₄⋊C₄×C₃₇	—	2	1	no	—
71	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
75	C₄²×C₃₇	C₂³	1	1	yes	characteristic
76	#10596938	—	2	1	no	—
77	#10596938	—	2	1	no	—
78	#10596938	D₄	1	1	yes	characteristic
79	#10596938	D₄	1	1	yes	characteristic
80	C₄⋊Dic₃₇	—	2	1	no	—
81	C₄⋊Dic₃₇	—	2	1	no	—
82	C₄⋊Dic₃₇	—	2	1	no	—
83	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
87	#10596936	—	2	1	no	—
88	#10596936	—	2	1	no	—
89	#10596936	—	2	1	no	—
90	#10596936	—	2	1	no	—
91	#10596936	—	2	1	no	—
92	#10596936	—	2	1	no	—
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	#10596936	D₄	1	1	yes	characteristic
96	#10596936	D₄	1	1	yes	characteristic
97	C₄×Dic₃₇	—	2	1	no	—
98	C₄×Dic₃₇	—	2	1	no	—
99	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
100	C₂×#64436	C₂³	1	1	yes	characteristic
101	C₂×#64436	C₂³	1	1	yes	characteristic
102	#10596929	Q₈	1	4	yes	complemented
103	C₈×D₃₇	Q₈	1	4	yes	complemented
104	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₃₇	—	8	1	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	2	1	no	—
108	C₂×C₄×C₃₇	—	2	1	no	—
109	C₂×C₄×C₃₇	—	2	1	no	—
110	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₃₇	—	2	1	no	—
114	C₂×Dic₃₇	—	2	1	no	—
115	C₂×Dic₃₇	—	2	1	no	—
116	C₂×Dic₃₇	—	2	1	no	—
117	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
120	C₄×D₃₇	—	8	1	no	—
121	C₄×D₃₇	—	8	1	no	—
122	C₄×D₃₇	C₂×Q₈	1	2	yes	—
123	C₄×D₃₇	C₂×Q₈	1	2	yes	—
124	Dic₇₄	—	8	1	no	—
125	Dic₇₄	—	8	1	no	—
126	Dic₇₄	—	8	1	no	—
127	C₈×C₃₇	C₂×Q₈	1	2	yes	—
128	C₈×C₃₇	C₂×Q₈	1	2	yes	—
129	#64436	C₂×Q₈	1	2	yes	—
130	#64436	C₂×Q₈	1	2	yes	—
131	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₃₇	—	2	1	no	—
133	C₂×D₃₇	—	2	1	no	—
134	C₂×D₃₇	C₄⋊Q₈	1	2	yes	—
135	C₄×C₃₇	—	2	1	no	—
136	C₄×C₃₇	—	2	1	no	—
137	C₄×C₃₇	—	4	1	no	—
138	C₄×C₃₇	—	4	1	no	—
139	C₄×C₃₇	—	4	1	no	—
140	C₄×C₃₇	—	4	1	no	—
141	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
143	Dic₃₇	—	2	1	no	—
144	Dic₃₇	—	2	1	no	—
145	Dic₃₇	—	4	1	no	—
146	Dic₃₇	—	4	1	no	—
147	Dic₃₇	—	4	1	no	—
148	Dic₃₇	—	4	1	no	—
149	Dic₃₇	C₄⋊Q₈	1	2	yes	—
150	#3163	—	37	1	no	maximal, Sylow
151	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₃₇	#577	1	1	yes	characteristic
153	C₂×C₃₇	#577	1	1	yes	characteristic
154	D₃₇	—	2	2	no	—
155	#544	—	37	1	no	—
156	#544	—	37	1	no	—
157	#501	—	37	1	no	—
158	#501	—	37	1	no	—
159	#501	—	37	1	no	—
160	#501	D₃₇	1	1	yes	characteristic, complemented
161	#499	—	37	1	no	—
162	#499	—	37	1	no	—
163	#498	—	37	1	no	—
164	#498	—	37	1	no	—
165	#428	—	37	1	no	—
166	#428	—	37	1	no	—
167	#427	—	37	1	no	—
168	C₂×#106	—	37	1	no	—
169	#405	—	37	1	no	—
170	C₃₇	#3163	1	1	yes	characteristic, complemented, Sylow
171	C₂×M₁₆	—	37	1	no	—
172	C₂²×C₈	—	37	1	no	—
173	C₄⋊Q₈	—	37	1	no	—
174	C₄⋊Q₈	—	37	1	no	—
175	C₄⋊Q₈	—	37	1	no	—
176	C₄⋊Q₈	—	74	1	no	—
177	C₄⋊Q₈	—	74	1	no	—
178	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
179	#125	—	37	1	no	—
180	#125	—	37	1	no	—
181	#125	—	37	1	no	—
182	#125	—	74	1	no	—
183	#125	—	74	1	no	—
184	#125	C₂×D₃₇	1	1	yes	characteristic
185	C₂²⋊Q₈	—	74	1	no	—
186	C₂²⋊Q₈	—	74	1	no	—
187	C₂²⋊Q₈	—	74	1	no	—
188	C₂²⋊Q₈	—	74	1	no	—
189	#117	—	37	1	no	—
190	#117	—	37	1	no	—
191	#117	—	37	1	no	—
192	C₂×C₄⋊C₄	—	37	1	no	—
193	C₂×C₄⋊C₄	—	37	1	no	—
194	#107	—	37	1	no	—
195	#107	—	37	1	no	—
196	#107	—	37	1	no	—
197	#107	—	37	1	no	—
198	#107	C₂×D₃₇	1	1	yes	characteristic
199	#107	C₂×D₃₇	1	1	yes	characteristic
200	#106	—	37	1	no	—
201	#106	—	37	1	no	—
202	#106	—	37	1	no	—
203	#106	—	37	1	no	—
204	#106	—	37	1	no	—
205	#106	—	37	1	no	—
206	#106	—	37	1	no	—
207	#106	—	37	1	no	—
208	#106	C₂×D₃₇	1	1	yes	characteristic
209	#106	C₂×D₃₇	1	1	yes	characteristic
210	#97	—	37	1	no	—
211	#97	C₂×D₃₇	1	1	yes	characteristic
212	C₄×C₈	—	37	1	no	—
213	C₄×C₈	—	37	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	74	1	no	—
216	C₂×Q₈	—	74	1	no	—
217	C₂×Q₈	—	74	1	no	—
218	C₂²×C₄	—	37	1	no	—
219	C₂²×C₄	—	74	1	no	—
220	C₂²×C₄	—	74	1	no	—
221	M₁₆	—	37	4	no	—
222	C₂×C₈	—	37	1	no	—
223	C₂×C₈	—	37	1	no	—
224	C₂×C₈	—	37	4	no	—
225	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	37	1	no	—
231	C₄⋊C₄	—	37	1	no	—
232	C₄⋊C₄	—	37	1	no	—
233	C₄⋊C₄	—	37	1	no	—
234	C₄⋊C₄	—	37	1	no	—
235	C₄⋊C₄	—	37	1	no	—
236	C₄⋊C₄	—	74	1	no	—
237	C₄⋊C₄	—	74	1	no	—
238	C₄⋊C₄	—	74	1	no	—
239	C₄⋊C₄	—	74	1	no	—
240	C₄⋊C₄	—	74	1	no	—
241	C₄⋊C₄	—	74	1	no	—
242	C₄⋊C₄	—	74	1	no	—
243	C₄⋊C₄	—	74	1	no	—
244	C₄⋊C₄	—	74	1	no	—
245	C₄⋊C₄	—	74	1	no	—
246	C₄⋊C₄	—	74	1	no	—
247	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
251	C₂²⋊C₄	—	37	1	no	—
252	C₂²⋊C₄	—	37	1	no	—
253	C₂²⋊C₄	—	74	1	no	—
254	C₂²⋊C₄	—	74	1	no	—
255	C₄²	—	37	1	no	—
256	C₄²	—	74	1	no	—
257	C₄²	—	74	1	no	—
258	C₄²	C₂²×D₃₇	1	1	yes	characteristic
259	C₂³	—	37	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	296	1	no	—
262	Q₈	—	296	1	no	—
263	Q₈	—	296	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	37	1	no	—
269	C₂×C₄	—	37	2	no	—
270	C₂×C₄	—	37	2	no	—
271	C₂×C₄	—	37	1	no	—
272	C₂×C₄	—	37	1	no	—
273	C₂×C₄	—	296	1	no	—
274	C₂×C₄	—	296	1	no	—
275	C₂×C₄	—	74	1	no	—
276	C₂×C₄	—	74	1	no	—
277	C₂×C₄	—	74	1	no	—
278	C₂×C₄	—	74	1	no	—
279	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
280	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
282	C₈	—	37	2	no	—
283	C₈	—	37	2	no	—
284	C₈	Q₈×D₃₇	1	2	yes	—
285	C₈	Q₈×D₃₇	1	2	yes	—
286	C₂²	—	37	2	no	—
287	C₂²	—	74	1	no	—
288	C₂²	—	74	1	no	—
289	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	37	2	no	—
297	C₄	—	74	1	no	—
298	C₄	—	74	1	no	—
299	C₄	—	148	1	no	—
300	C₄	—	148	1	no	—
301	C₄	—	148	1	no	—
302	C₄	—	148	1	no	—
303	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
305	C₂	—	74	2	no	—
306	C₂	C₄⋊Q₈×D₃₇	1	1	yes	characteristic
307	C₂	#14558342	1	1	yes	characteristic
308	C₂	#14558343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#501	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#501	D₃₇
hypocenter	C₃₇	#3163
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #14557942	C₂	semidirect
4. #14557941	C₂	semidirect
5. #14557940	C₂	semidirect
6. #14557934	C₂	semidirect
7. #14557932	C₂	semidirect
8. #14557930	C₂	semidirect
9. #14557929	C₂	semidirect
15. C₃₇×#501	C₂	semidirect
102. #10596929	Q₈	semidirect
103. C₈×D₃₇	Q₈	semidirect
160. #501	D₃₇	semidirect
170. C₃₇	#3163	semidirect

Maximal subgroups

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

subgroup	index
#14558163	2
#14558160	2
#14557942	2
#14557941	2
#14557940	2
#14557934	2
#14557932	2
#14557930	2
#14557929	2
#14557574	2
#14557572	2
#14557570	2
#106×D₃₇	2
#14557490	2
C₃₇×#501	2
#3163	37