Subgroups — #18004881

Order 4736.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558136	C₂	1	1	yes	maximal, characteristic
2	#14558011	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557858	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557854	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557842	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557838	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557731	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557731	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557672	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557636	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557634	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557618	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557591	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#448	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596941	C₂²	1	1	yes	characteristic
17	C₂×#10596941	C₂²	1	1	yes	characteristic
18	#11924532	C₂²	1	1	yes	characteristic
19	#11924527	C₂²	1	1	yes	characteristic
20	#11924520	—	2	1	no	—
21	#11924518	—	2	1	no	—
22	#11924510	C₂²	1	1	yes	characteristic
23	#11924508	—	2	1	no	—
24	#11924507	—	2	1	no	—
25	Q₈×Dic₃₇	—	2	1	no	—
26	Q₈×Dic₃₇	—	2	1	no	—
27	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
28	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
29	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
30	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
31	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	#11924498	C₂²	1	1	yes	characteristic
33	#11924495	—	2	1	no	—
34	#11924495	—	2	1	no	—
35	#11924495	C₂²	1	1	yes	characteristic
36	#11924495	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924469	C₂²	1	1	yes	characteristic
39	#11924468	C₂²	1	1	yes	characteristic
40	#11924467	C₂²	1	1	yes	characteristic
41	#11924465	C₂²	1	1	yes	characteristic
42	#11924465	C₂²	1	1	yes	characteristic
43	#11924460	C₂²	1	1	yes	characteristic
44	#11924460	C₂²	1	1	yes	characteristic
45	#11924458	C₂²	1	1	yes	characteristic
46	#11924458	C₂²	1	1	yes	characteristic
47	#11924457	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	C₂²⋊#64436	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
55	#121×C₃₇	C₂²	1	1	yes	characteristic
56	#103×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	—	2	2	no	—
60	C₂×Q₈×C₃₇	—	2	2	no	—
61	C₂×Q₈×C₃₇	—	2	1	no	—
62	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₃₇	—	2	1	no	—
65	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	—	2	1	no	—
67	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₃₇	—	2	1	no	—
69	C₂×C₂²⋊D₃₇	—	2	1	no	—
70	C₂×C₂²⋊D₃₇	—	2	1	no	—
71	C₂²×Dic₃₇	—	2	1	no	—
72	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₃₇	—	2	1	no	—
74	SD₁₆×C₃₇	—	2	2	no	—
75	SD₁₆×C₃₇	—	2	2	no	—
76	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₃₇	—	2	1	no	—
80	C₂²⋊Dic₃₇	—	2	1	no	—
81	C₂²⋊Dic₃₇	—	2	1	no	—
82	C₂²⋊Dic₃₇	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₃₇	D₄	1	1	yes	characteristic
84	#10596941	—	2	2	no	—
85	#10596941	—	2	2	no	—
86	#10596938	—	2	1	no	—
87	#10596938	—	2	1	no	—
88	C₄⋊Dic₃₇	—	2	1	no	—
89	C₄⋊Dic₃₇	—	2	1	no	—
90	C₄⋊Dic₃₇	—	2	1	no	—
91	C₄⋊Dic₃₇	—	2	1	no	—
92	C₄⋊Dic₃₇	—	2	1	no	—
93	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
94	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
95	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
98	#10596936	—	2	1	no	—
99	C₄×Dic₃₇	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	—	2	1	no	—
103	C₄×Dic₃₇	—	2	1	no	—
104	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
105	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
106	C₂×#64436	C₂³	1	1	yes	characteristic
107	C₂×#64436	C₂³	1	1	yes	characteristic
108	C₂³×C₃₇	—	2	1	no	—
109	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₃₇	—	2	1	no	—
111	Q₈×C₃₇	—	2	4	no	—
112	Q₈×C₃₇	—	2	1	no	—
113	Q₈×C₃₇	—	2	1	no	—
114	Q₈×C₃₇	#41	1	2	yes	—
115	Q₈×C₃₇	#41	1	2	yes	—
116	D₄×C₃₇	—	2	1	no	—
117	D₄×C₃₇	—	2	1	no	—
118	D₄×C₃₇	—	8	1	no	—
119	C₂×C₄×C₃₇	—	2	4	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	1	no	—
123	C₂×C₄×C₃₇	—	2	1	no	—
124	C₂×C₄×C₃₇	—	2	1	no	—
125	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₃₇	—	8	1	no	—
128	C₂²⋊D₃₇	—	8	1	no	—
129	C₂²⋊D₃₇	—	8	1	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	2	1	no	—
133	C₂×Dic₃₇	—	8	1	no	—
134	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
138	D₁₄₈	—	2	1	no	—
139	D₁₄₈	—	2	1	no	—
140	C₄×D₃₇	—	8	1	no	—
141	C₈×C₃₇	—	2	1	no	—
142	C₈×C₃₇	—	2	1	no	—
143	#64436	—	2	1	no	—
144	#64436	—	2	1	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	2	1	no	—
147	C₂²×C₃₇	—	2	1	no	—
148	C₂²×C₃₇	—	4	1	no	—
149	C₂²×C₃₇	—	4	1	no	—
150	C₂²×C₃₇	—	4	1	no	—
151	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₃₇	—	4	1	no	—
153	C₂×D₃₇	—	4	1	no	—
154	C₂×D₃₇	—	4	1	no	—
155	C₄×C₃₇	—	2	2	no	—
156	C₄×C₃₇	—	2	2	no	—
157	C₄×C₃₇	—	2	1	no	—
158	C₄×C₃₇	—	4	1	no	—
159	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
160	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
161	Dic₃₇	—	2	1	no	—
162	Dic₃₇	—	2	1	no	—
163	Dic₃₇	—	2	1	no	—
164	Dic₃₇	—	2	1	no	—
165	Dic₃₇	—	4	1	no	—
166	Dic₃₇	—	4	1	no	—
167	Dic₃₇	—	4	1	no	—
168	#3126	—	37	1	no	maximal, Sylow
169	C₂×C₃₇	—	4	1	no	—
170	C₂×C₃₇	—	8	1	no	—
171	C₂×C₃₇	#525	1	1	yes	characteristic
172	C₂×C₃₇	#577	1	1	yes	characteristic
173	C₂×C₃₇	#577	1	1	yes	characteristic
174	D₃₇	—	8	1	no	—
175	#540	—	37	1	no	—
176	#519	—	37	1	no	—
177	#482	—	37	1	no	—
178	#482	—	37	1	no	—
179	#479	—	37	1	no	—
180	#479	—	37	1	no	—
181	#460	—	37	1	no	—
182	#460	—	37	1	no	—
183	#448	—	37	1	no	—
184	#448	D₃₇	1	1	yes	characteristic, complemented
185	#440	—	37	1	no	—
186	#440	—	37	1	no	—
187	C₄×SD₁₆	—	37	1	no	—
188	C₄×SD₁₆	—	37	1	no	—
189	#433	—	37	1	no	—
190	C₃₇	#3126	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
192	C₂×SD₁₆	—	37	1	no	—
193	C₂×SD₁₆	—	37	1	no	—
194	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
195	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
196	#127	—	37	1	no	—
197	#125	—	37	1	no	—
198	#123	—	74	1	no	—
199	#123	—	74	1	no	—
200	#121	—	37	1	no	—
201	#121	—	74	1	no	—
202	#121	—	74	1	no	—
203	#121	C₂×D₃₇	1	1	yes	characteristic
204	C₄×Q₈	—	37	1	no	—
205	C₄×Q₈	—	37	1	no	—
206	C₄×Q₈	—	37	1	no	—
207	C₄×Q₈	—	37	1	no	—
208	C₄×Q₈	—	74	1	no	—
209	C₄×Q₈	—	74	1	no	—
210	C₄×D₄	—	37	1	no	—
211	C₄×D₄	—	37	1	no	—
212	#117	—	37	1	no	—
213	#117	—	37	1	no	—
214	#117	—	74	1	no	—
215	#117	—	74	1	no	—
216	#107	—	37	1	no	—
217	#107	—	37	1	no	—
218	#106	—	37	1	no	—
219	#106	—	37	1	no	—
220	C₄⋊C₈	—	37	1	no	—
221	C₄⋊C₈	—	37	1	no	—
222	#103	—	37	1	no	—
223	#103	—	37	1	no	—
224	#103	C₂×D₃₇	1	1	yes	characteristic
225	#103	C₂×D₃₇	1	1	yes	characteristic
226	#102	—	37	1	no	—
227	#102	—	37	1	no	—
228	#102	—	37	1	no	—
229	#102	—	37	1	no	—
230	C₂²⋊C₈	—	37	1	no	—
231	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
232	#97	—	37	1	no	—
233	C₄×C₈	—	37	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
238	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	37	1	no	—
241	C₂×D₄	—	74	1	no	—
242	C₂×D₄	—	74	1	no	—
243	C₂×D₄	—	74	1	no	—
244	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	74	1	no	—
247	C₂²×C₄	—	74	1	no	—
248	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	74	2	no	—
252	SD₁₆	—	74	2	no	—
253	C₂×C₈	—	37	1	no	—
254	C₂×C₈	—	37	1	no	—
255	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
257	C₄⋊C₄	—	37	1	no	—
258	C₄⋊C₄	—	37	1	no	—
259	C₄⋊C₄	—	37	1	no	—
260	C₄⋊C₄	—	37	1	no	—
261	C₄⋊C₄	—	37	1	no	—
262	C₄⋊C₄	—	74	1	no	—
263	C₄⋊C₄	—	74	1	no	—
264	C₄⋊C₄	—	74	1	no	—
265	C₄⋊C₄	—	74	1	no	—
266	C₄⋊C₄	—	74	1	no	—
267	C₄⋊C₄	—	74	1	no	—
268	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	37	1	no	—
271	C₂²⋊C₄	—	37	1	no	—
272	C₂²⋊C₄	—	74	1	no	—
273	C₂²⋊C₄	—	74	1	no	—
274	C₂²⋊C₄	—	74	1	no	—
275	C₂²⋊C₄	—	74	1	no	—
276	C₄²	—	37	1	no	—
277	C₄²	—	37	1	no	—
278	C₄²	—	74	1	no	—
279	C₄²	—	74	1	no	—
280	C₄²	—	74	1	no	—
281	C₄²	—	74	1	no	—
282	C₄²	—	74	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	74	1	no	—
285	C₂³	D₄×D₃₇	1	1	yes	characteristic
286	Q₈	—	2	4	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	#10596964	1	2	yes	—
290	Q₈	#10596964	1	2	yes	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	8	1	no	—
294	D₄	—	296	1	no	—
295	D₄	—	296	1	no	—
296	D₄	—	296	1	no	—
297	D₄	—	74	1	no	—
298	D₄	—	74	1	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	37	1	no	—
306	C₂×C₄	—	37	1	no	—
307	C₂×C₄	—	37	1	no	—
308	C₂×C₄	—	37	1	no	—
309	C₂×C₄	—	296	1	no	—
310	C₂×C₄	—	296	1	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	1	no	—
313	C₂×C₄	—	74	1	no	—
314	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	74	1	no	—
319	C₈	—	74	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	148	1	no	—
327	C₂²	—	148	1	no	—
328	C₂²	—	148	1	no	—
329	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	74	1	no	—
335	C₄	—	74	1	no	—
336	C₄	—	74	1	no	—
337	C₄	—	74	1	no	—
338	C₄	—	148	1	no	—
339	C₄	—	148	1	no	—
340	C₄	—	148	1	no	—
341	C₄	C₂×#10596964	1	1	yes	characteristic
342	C₄	C₂×#10596964	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	8	1	no	—
345	C₂	—	296	1	no	—
346	C₂	#14558039	1	1	yes	characteristic
347	C₂	#14558339	1	1	yes	characteristic
348	C₂	#14558342	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×#448	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	#14558136	C₂
hypercenter	#448	D₃₇
hypocenter	C₃₇	#3126
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
2. #14558011	C₂	semidirect
3. #14557858	C₂	semidirect
4. #14557854	C₂	semidirect
5. #14557842	C₂	semidirect
6. #14557838	C₂	semidirect
7. #14557731	C₂	semidirect
8. #14557731	C₂	semidirect
9. #14557672	C₂	semidirect
10. #14557636	C₂	semidirect
11. #14557634	C₂	semidirect
12. SD₁₆×Dic₃₇	C₂	semidirect
13. #14557618	C₂	semidirect
14. #14557591	C₂	semidirect
15. C₃₇×#448	C₂	semidirect
184. #448	D₃₇	semidirect
190. C₃₇	#3126	semidirect

Maximal subgroups

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

subgroup	index
#14558136	2
#14558011	2
#14557858	2
#14557854	2
#14557842	2
#14557838	2
#14557731	2
#14557672	2
#14557636	2
#14557634	2
SD₁₆×Dic₃₇	2
#14557618	2
#14557591	2
C₃₇×#448	2
#3126	37