Subgroups — #18007240

Order 4736.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558252	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558242	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557887	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557886	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557882	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557842	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557841	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557821	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557746	C₂	1	1	yes	maximal, characteristic
10	#14557736	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557657	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#10596940	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#463	C₂	1	1	yes	maximal, characteristic
16	C₂×#10596940	C₂²	1	1	yes	characteristic
17	#11924535	C₂²	1	1	yes	characteristic
18	#11924534	C₂²	1	1	yes	characteristic
19	#11924529	—	2	1	no	—
20	#11924529	—	2	1	no	—
21	#11924528	C₂²	1	1	yes	characteristic
22	#11924527	C₂²	1	1	yes	characteristic
23	#11924523	C₂²	1	1	yes	characteristic
24	#11924515	—	2	1	no	—
25	#11924513	—	2	1	no	—
26	#11924513	—	2	1	no	—
27	#11924513	—	2	1	no	—
28	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
31	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
33	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
34	#11924492	—	2	1	no	—
35	C₂×#10596936	—	2	1	no	—
36	#11924470	C₂²	1	1	yes	characteristic
37	#11924468	C₂²	1	1	yes	characteristic
38	#11924467	C₂²	1	1	yes	characteristic
39	#11924465	C₂²	1	1	yes	characteristic
40	#11924465	C₂²	1	1	yes	characteristic
41	#11924461	C₂²	1	1	yes	characteristic
42	#11924461	C₂²	1	1	yes	characteristic
43	#11924460	C₂²	1	1	yes	characteristic
44	#11924460	C₂²	1	1	yes	characteristic
45	#11924459	C₂²	1	1	yes	characteristic
46	#11924457	C₂²	1	1	yes	characteristic
47	#11924457	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	C₄×#64436	C₂²	1	1	yes	characteristic
50	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₃₇	C₂²	1	1	yes	characteristic
54	#124×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	—	2	1	no	—
63	C₂²×Dic₃₇	—	2	1	no	—
64	C₂²×Dic₃₇	—	2	1	no	—
65	C₂²×Dic₃₇	—	2	1	no	—
66	C₂×Dic₇₄	—	2	1	no	—
67	C₂×Dic₇₄	D₄	1	1	yes	characteristic
68	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
69	SD₁₆×C₃₇	—	2	2	no	—
70	D₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₃₇	—	2	1	no	—
75	C₂²⋊C₄×C₃₇	—	2	1	no	—
76	C₂²⋊C₄×C₃₇	—	2	1	no	—
77	C₄²×C₃₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₃₇	—	2	1	no	—
79	C₂²⋊Dic₃₇	—	2	1	no	—
80	C₂²⋊Dic₃₇	—	2	1	no	—
81	C₂²⋊Dic₃₇	—	2	1	no	—
82	C₂²⋊Dic₃₇	—	2	1	no	—
83	#10596940	—	2	2	no	—
84	C₄⋊Dic₃₇	—	2	1	no	—
85	C₄⋊Dic₃₇	—	2	1	no	—
86	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
87	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
96	#10596936	—	2	1	no	—
97	#10596936	D₄	1	1	yes	characteristic
98	#10596936	D₄	1	1	yes	characteristic
99	C₄×Dic₃₇	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
103	C₂×#64436	C₂³	1	1	yes	characteristic
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	C₂³×C₃₇	—	2	1	no	—
106	C₂³×C₃₇	—	2	1	no	—
107	Q₈×C₃₇	—	2	1	no	—
108	Q₈×C₃₇	—	2	1	no	—
109	D₄×C₃₇	—	2	1	no	—
110	D₄×C₃₇	—	2	1	no	—
111	D₄×C₃₇	—	2	1	no	—
112	D₄×C₃₇	#41	1	2	yes	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	—	8	1	no	—
116	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	—	2	1	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	—	2	1	no	—
123	C₂×Dic₃₇	—	4	2	no	—
124	C₂×Dic₃₇	—	4	2	no	—
125	C₂×Dic₃₇	—	8	1	no	—
126	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	Dic₇₄	—	2	2	no	—
131	Dic₇₄	—	2	1	no	—
132	Dic₇₄	—	2	1	no	—
133	Dic₇₄	—	8	1	no	—
134	C₈×C₃₇	—	2	1	no	—
135	C₈×C₃₇	—	2	1	no	—
136	#64436	—	2	1	no	—
137	#64436	—	2	1	no	—
138	C₂²×C₃₇	—	2	2	no	—
139	C₂²×C₃₇	—	4	1	no	—
140	C₂²×C₃₇	—	4	1	no	—
141	C₂²×C₃₇	—	4	1	no	—
142	C₂²×C₃₇	—	4	1	no	—
143	C₂²×C₃₇	—	4	1	no	—
144	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₃₇	—	2	1	no	—
146	C₄×C₃₇	—	2	1	no	—
147	C₄×C₃₇	—	4	1	no	—
148	C₄×C₃₇	—	4	1	no	—
149	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
150	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₃₇	—	2	1	no	—
152	Dic₃₇	—	2	1	no	—
153	Dic₃₇	—	2	1	no	—
154	Dic₃₇	—	4	1	no	—
155	Dic₃₇	—	4	1	no	—
156	Dic₃₇	—	4	1	no	—
157	Dic₃₇	—	4	1	no	—
158	Dic₃₇	#135	1	2	yes	—
159	#3360	—	37	1	no	maximal, Sylow
160	C₂×C₃₇	—	4	2	no	—
161	C₂×C₃₇	—	8	1	no	—
162	C₂×C₃₇	#548	1	1	yes	characteristic
163	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
164	C₂×C₃₇	#577	1	1	yes	characteristic
165	#556	—	37	1	no	—
166	#554	—	37	1	no	—
167	#488	—	37	1	no	—
168	#488	—	37	1	no	—
169	#487	—	37	1	no	—
170	#479	—	37	1	no	—
171	#479	—	37	1	no	—
172	#476	—	37	1	no	—
173	#463	—	37	1	no	—
174	#463	D₃₇	1	1	yes	characteristic
175	#461	—	37	1	no	—
176	C₈×Q₈	—	37	1	no	—
177	C₄×SD₁₆	—	37	1	no	—
178	C₄×SD₁₆	—	37	1	no	—
179	C₄×D₈	—	37	1	no	—
180	C₃₇	#3360	1	1	yes	characteristic, complemented, Sylow
181	C₂×SD₁₆	—	37	1	no	—
182	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
183	C₂×D₈	C₂×D₃₇	1	1	yes	characteristic
184	C₄⋊Q₈	—	37	1	no	—
185	C₄⋊Q₈	—	37	1	no	—
186	#126	—	74	1	no	—
187	#126	—	74	1	no	—
188	#125	—	37	1	no	—
189	#125	—	37	1	no	—
190	#124	—	37	1	no	—
191	#124	C₂×D₃₇	1	1	yes	characteristic
192	C₂²⋊Q₈	—	74	1	no	—
193	C₂²⋊Q₈	—	74	1	no	—
194	C₂²⋊Q₈	—	74	1	no	—
195	C₂²⋊Q₈	—	74	1	no	—
196	C₄×Q₈	—	37	1	no	—
197	C₄×Q₈	—	37	1	no	—
198	C₄×Q₈	—	37	1	no	—
199	C₄×D₄	—	37	1	no	—
200	C₄×D₄	—	37	1	no	—
201	C₄×D₄	—	37	1	no	—
202	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
203	#117	—	74	1	no	—
204	C₂×C₄⋊C₄	—	74	1	no	—
205	#107	—	37	1	no	—
206	#106	—	37	1	no	—
207	#106	—	37	1	no	—
208	C₄⋊C₈	—	37	1	no	—
209	C₄⋊C₈	—	37	1	no	—
210	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
211	#103	—	37	1	no	—
212	#103	—	37	1	no	—
213	#103	—	37	1	no	—
214	#103	—	37	1	no	—
215	#103	—	37	1	no	—
216	#103	C₂×D₃₇	1	1	yes	characteristic
217	#102	—	37	1	no	—
218	#102	—	37	1	no	—
219	#102	—	37	1	no	—
220	#102	C₂×D₃₇	1	1	yes	characteristic
221	C₄×C₈	—	37	1	no	—
222	C₄×C₈	—	37	1	no	—
223	C₄×C₈	—	37	1	no	—
224	C₂×Q₈	—	37	1	no	—
225	C₂×Q₈	—	37	1	no	—
226	C₂×Q₈	—	74	1	no	—
227	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
228	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
229	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	74	1	no	—
232	C₂²×C₄	—	74	1	no	—
233	C₂²×C₄	—	74	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	74	2	no	—
236	D₈	—	2	2	no	—
237	C₂×C₈	—	37	1	no	—
238	C₂×C₈	—	37	1	no	—
239	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
241	C₄⋊C₄	—	37	1	no	—
242	C₄⋊C₄	—	37	1	no	—
243	C₄⋊C₄	—	37	1	no	—
244	C₄⋊C₄	—	37	1	no	—
245	C₄⋊C₄	—	37	1	no	—
246	C₄⋊C₄	—	37	1	no	—
247	C₄⋊C₄	—	74	1	no	—
248	C₄⋊C₄	—	74	1	no	—
249	C₄⋊C₄	—	74	1	no	—
250	C₄⋊C₄	—	74	1	no	—
251	C₄⋊C₄	—	74	1	no	—
252	C₄⋊C₄	—	74	1	no	—
253	C₄⋊C₄	—	74	1	no	—
254	C₄⋊C₄	—	74	1	no	—
255	C₄⋊C₄	—	74	1	no	—
256	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	74	1	no	—
261	C₂²⋊C₄	—	74	1	no	—
262	C₂²⋊C₄	—	74	1	no	—
263	C₂²⋊C₄	—	74	1	no	—
264	C₂²⋊C₄	—	74	1	no	—
265	C₄²	—	37	1	no	—
266	C₄²	—	37	1	no	—
267	C₄²	—	74	1	no	—
268	C₄²	—	74	1	no	—
269	C₄²	C₂²×D₃₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	296	1	no	—
275	Q₈	—	74	2	no	—
276	Q₈	—	74	1	no	—
277	Q₈	—	74	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	#10596964	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	37	1	no	—
286	C₂×C₄	—	37	1	no	—
287	C₂×C₄	—	37	1	no	—
288	C₂×C₄	—	37	1	no	—
289	C₂×C₄	—	296	1	no	—
290	C₂×C₄	—	74	1	no	—
291	C₂×C₄	—	74	1	no	—
292	C₂×C₄	—	74	1	no	—
293	C₂×C₄	—	74	1	no	—
294	C₂×C₄	—	148	2	no	—
295	C₂×C₄	—	148	2	no	—
296	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
297	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	74	1	no	—
302	C₈	—	74	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	37	2	no	—
315	C₄	—	74	1	no	—
316	C₄	—	74	1	no	—
317	C₄	—	74	1	no	—
318	C₄	—	148	1	no	—
319	C₄	—	148	1	no	—
320	C₄	—	148	1	no	—
321	C₄	—	148	1	no	—
322	C₄	C₂×#10596964	1	1	yes	characteristic
323	C₄	#11924583	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	8	1	no	—
326	C₂	#14558189	1	1	yes	characteristic
327	C₂	#135×D₃₇	1	1	yes	characteristic
328	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×#463	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×D₈×C₃₇	C₂²
hypercenter	#463	D₃₇
hypocenter	C₃₇	#3360
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #14558252	C₂	semidirect
2. #14558242	C₂	semidirect
3. #14557887	C₂	semidirect
4. #14557886	C₂	semidirect
5. #14557882	C₂	semidirect
6. #14557842	C₂	semidirect
7. #14557841	C₂	semidirect
8. #14557821	C₂	semidirect
10. #14557736	C₂	semidirect
11. #14557657	C₂	semidirect
12. SD₁₆×Dic₃₇	C₂	semidirect
13. C₄×#10596940	C₂	semidirect
180. C₃₇	#3360	semidirect

Maximal subgroups

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

subgroup	index
#14558252	2
#14558242	2
#14557887	2
#14557886	2
#14557882	2
#14557842	2
#14557841	2
#14557821	2
#14557746	2
#14557736	2
#14557657	2
SD₁₆×Dic₃₇	2
C₄×#10596940	2
D₈×Dic₃₇	2
C₃₇×#463	2
#3360	37