Subgroups — #18004766

Order 4736.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558062	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557867	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557850	C₂	1	1	yes	maximal, characteristic
5	#14557841	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557840	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557752	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557687	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557670	C₂	1	1	yes	maximal, characteristic
10	#14557640	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557637	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557634	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557630	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557589	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
17	C₂×#10596942	C₂²	1	1	yes	characteristic
18	C₂×#10596940	C₂²	1	1	yes	characteristic
19	#11924529	—	2	1	no	—
20	#11924528	C₂²	1	1	yes	characteristic
21	#11924518	—	2	1	no	—
22	#11924512	C₂²	1	1	yes	characteristic
23	#11924507	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₇₄	—	2	1	no	—
27	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
28	#11924501	—	2	1	no	—
29	#11924501	C₂²	1	1	yes	characteristic
30	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
31	#11924496	—	2	1	no	—
32	#11924492	—	2	1	no	—
33	C₄⋊C₄×D₃₇	—	2	1	no	—
34	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
35	C₄²×D₃₇	—	2	1	no	—
36	C₄²×D₃₇	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	C₄⋊#64436	C₂²	1	1	yes	characteristic
42	#11924461	C₂²	1	1	yes	characteristic
43	#11924461	C₂²	1	1	yes	characteristic
44	#11924460	C₂²	1	1	yes	characteristic
45	#11924460	C₂²	1	1	yes	characteristic
46	#11924460	C₂²	1	1	yes	characteristic
47	#11924457	C₂²	1	1	yes	characteristic
48	#11924448	C₂²	1	1	yes	characteristic
49	#11924448	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	#11924446	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	#124×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×Q₈×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×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₃₇	—	2	1	no	—
63	C₂²×Dic₃₇	—	2	1	no	—
64	Q₈×D₃₇	—	2	2	no	—
65	Q₈×D₃₇	—	2	2	no	—
66	C₂×C₄×D₃₇	—	2	1	no	—
67	C₂×C₄×D₃₇	—	2	1	no	—
68	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
69	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
70	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₇₄	—	2	1	no	—
72	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
73	Q₁₆×C₃₇	—	2	2	no	—
74	SD₁₆×C₃₇	—	2	2	no	—
75	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₃₇	—	2	1	no	—
78	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₃₇	—	2	1	no	—
80	C₂²⋊C₄×C₃₇	—	2	1	no	—
81	C₄²×C₃₇	—	2	1	no	—
82	C₄²×C₃₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₃₇	—	2	1	no	—
84	C₂²⋊Dic₃₇	—	2	1	no	—
85	#10596942	—	2	2	no	—
86	#10596940	—	2	2	no	—
87	#10596938	—	2	1	no	—
88	C₄⋊Dic₃₇	—	2	1	no	—
89	C₄⋊Dic₃₇	—	2	1	no	—
90	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	#10596936	—	2	1	no	—
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
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	D₄	1	1	yes	characteristic
104	C₄×Dic₃₇	D₄	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	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₃₇	—	2	1	no	—
111	Q₈×C₃₇	—	2	1	no	—
112	Q₈×C₃₇	—	2	1	no	—
113	Q₈×C₃₇	#41	1	2	yes	—
114	D₄×C₃₇	—	2	1	no	—
115	D₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₃₇	—	4	2	no	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	4	2	no	—
128	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₃₇	—	2	4	no	—
132	C₄×D₃₇	—	2	2	no	—
133	C₄×D₃₇	—	2	1	no	—
134	C₄×D₃₇	—	2	1	no	—
135	C₄×D₃₇	—	4	1	no	—
136	C₄×D₃₇	—	4	2	no	—
137	Dic₇₄	—	2	4	no	—
138	Dic₇₄	—	2	1	no	—
139	Dic₇₄	#41	1	2	yes	—
140	C₈×C₃₇	—	2	1	no	—
141	C₈×C₃₇	—	2	1	no	—
142	#64436	—	2	1	no	—
143	#64436	—	2	1	no	—
144	C₂²×C₃₇	—	4	1	no	—
145	C₂²×C₃₇	—	4	1	no	—
146	C₂²×C₃₇	—	4	1	no	—
147	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₃₇	—	2	1	no	—
149	C₂×D₃₇	—	2	1	no	—
150	C₂×D₃₇	#137	1	2	yes	—
151	C₄×C₃₇	—	2	2	no	—
152	C₄×C₃₇	—	2	1	no	—
153	C₄×C₃₇	—	2	1	no	—
154	C₄×C₃₇	—	4	1	no	—
155	C₄×C₃₇	—	4	1	no	—
156	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
157	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
158	Dic₃₇	—	2	1	no	—
159	Dic₃₇	—	2	2	no	—
160	Dic₃₇	—	2	1	no	—
161	Dic₃₇	—	2	1	no	—
162	Dic₃₇	—	4	1	no	—
163	Dic₃₇	—	4	1	no	—
164	Dic₃₇	—	4	1	no	—
165	#3117	—	37	1	no	maximal, Sylow
166	C₂×C₃₇	—	8	1	no	—
167	C₂×C₃₇	#525	1	1	yes	characteristic
168	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
169	C₂×C₃₇	#575	1	1	yes	characteristic
170	D₃₇	—	2	2	no	—
171	#529	—	37	1	no	—
172	C₂×C₄×Q₈	—	37	1	no	—
173	#484	—	37	1	no	—
174	#481	—	37	1	no	—
175	#479	—	37	1	no	—
176	#479	—	37	1	no	—
177	#464	—	37	1	no	—
178	#464	D₃₇	1	1	yes	characteristic, complemented
179	#451	—	37	1	no	—
180	#448	—	37	1	no	—
181	#441	—	37	1	no	—
182	#441	—	37	1	no	—
183	#440	—	37	1	no	—
184	#440	—	37	1	no	—
185	#432	—	37	1	no	—
186	C₃₇	#3117	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	37	1	no	—
188	C₂×Q₁₆	—	37	1	no	—
189	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
190	C₂×SD₁₆	—	37	1	no	—
191	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
192	#126	—	74	1	no	—
193	#125	—	37	1	no	—
194	#124	C₂×D₃₇	1	1	yes	characteristic
195	#123	—	74	1	no	—
196	C₂²⋊Q₈	—	37	1	no	—
197	#121	—	37	1	no	—
198	C₄×Q₈	—	37	1	no	—
199	C₄×Q₈	—	37	1	no	—
200	C₄×Q₈	—	37	1	no	—
201	C₄×Q₈	—	37	1	no	—
202	C₄×Q₈	—	74	1	no	—
203	C₄×Q₈	—	74	1	no	—
204	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
205	C₄×D₄	—	37	1	no	—
206	C₄×D₄	—	74	1	no	—
207	#117	—	74	1	no	—
208	C₂×C₄⋊C₄	—	37	1	no	—
209	C₂×C₄⋊C₄	—	74	1	no	—
210	C₂×C₄²	—	37	1	no	—
211	C₂×C₄²	—	74	1	no	—
212	#107	—	37	1	no	—
213	#107	—	37	1	no	—
214	#106	—	37	1	no	—
215	#106	—	37	1	no	—
216	C₄⋊C₈	—	37	1	no	—
217	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
218	#103	—	37	1	no	—
219	#103	—	37	1	no	—
220	#103	—	37	1	no	—
221	#103	—	37	1	no	—
222	#103	—	37	1	no	—
223	#103	C₂×D₃₇	1	1	yes	characteristic
224	#102	—	37	1	no	—
225	#102	C₂×D₃₇	1	1	yes	characteristic
226	C₂²⋊C₈	—	37	1	no	—
227	C₂²⋊C₈	—	37	1	no	—
228	#97	—	37	1	no	—
229	#97	—	37	1	no	—
230	C₂×Q₈	—	37	1	no	—
231	C₂×Q₈	—	74	1	no	—
232	C₂×Q₈	—	74	2	no	—
233	C₂×Q₈	—	74	2	no	—
234	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
236	C₂×D₄	—	74	1	no	—
237	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
238	C₂²×C₄	—	37	1	no	—
239	C₂²×C₄	—	37	1	no	—
240	C₂²×C₄	—	37	1	no	—
241	C₂²×C₄	—	74	1	no	—
242	C₂²×C₄	—	74	1	no	—
243	C₂²×C₄	—	74	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	74	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	74	2	no	—
248	C₂×C₈	—	37	1	no	—
249	C₂×C₈	—	37	1	no	—
250	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	37	1	no	—
254	C₄⋊C₄	—	37	1	no	—
255	C₄⋊C₄	—	37	1	no	—
256	C₄⋊C₄	—	74	1	no	—
257	C₄⋊C₄	—	74	1	no	—
258	C₄⋊C₄	—	74	1	no	—
259	C₄⋊C₄	—	74	1	no	—
260	C₄⋊C₄	—	74	1	no	—
261	C₄⋊C₄	—	74	1	no	—
262	C₄⋊C₄	—	74	1	no	—
263	C₄⋊C₄	—	74	1	no	—
264	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	74	1	no	—
268	C₂²⋊C₄	—	74	1	no	—
269	C₂²⋊C₄	—	74	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	37	1	no	—
272	C₄²	—	37	1	no	—
273	C₄²	—	37	1	no	—
274	C₄²	—	74	1	no	—
275	C₄²	—	74	1	no	—
276	C₄²	—	74	1	no	—
277	C₄²	—	74	1	no	—
278	C₄²	C₂²×D₃₇	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	37	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	37	2	no	—
285	Q₈	—	74	4	no	—
286	Q₈	—	74	1	no	—
287	Q₈	#10596964	1	2	yes	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	148	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	37	1	no	—
295	C₂×C₄	—	37	1	no	—
296	C₂×C₄	—	37	1	no	—
297	C₂×C₄	—	74	2	no	—
298	C₂×C₄	—	74	4	no	—
299	C₂×C₄	—	74	1	no	—
300	C₂×C₄	—	74	1	no	—
301	C₂×C₄	—	74	1	no	—
302	C₂×C₄	—	74	1	no	—
303	C₂×C₄	—	74	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	—	148	1	no	—
306	C₂×C₄	—	148	2	no	—
307	C₂×C₄	—	148	2	no	—
308	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
309	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	74	1	no	—
314	C₈	—	74	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	37	2	no	—
319	C₂²	—	74	1	no	—
320	C₂²	—	74	1	no	—
321	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	74	1	no	—
328	C₄	—	74	2	no	—
329	C₄	—	74	1	no	—
330	C₄	—	74	1	no	—
331	C₄	—	148	1	no	—
332	C₄	—	148	1	no	—
333	C₄	—	148	1	no	—
334	C₄	#41×D₃₇	1	1	yes	characteristic
335	C₄	C₂×#10596964	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	74	2	no	—
338	C₂	#14558043	1	1	yes	characteristic
339	C₂	#137×D₃₇	1	1	yes	characteristic
340	C₂	#14558330	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×#464	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	#11924507	C₂²
hypercenter	#464	D₃₇
hypocenter	C₃₇	#3117
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₃₇	C₂	semidirect
3. #14557867	C₂	semidirect
5. #14557841	C₂	semidirect
6. #14557840	C₂	semidirect
7. #14557752	C₂	semidirect
8. #14557687	C₂	semidirect
10. #14557640	C₂	semidirect
11. #14557637	C₂	semidirect
12. #14557634	C₂	semidirect
13. #14557630	C₂	semidirect
14. #14557589	C₂	semidirect
15. C₃₇×#464	C₂	semidirect
178. #464	D₃₇	semidirect
186. C₃₇	#3117	semidirect

Maximal subgroups

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

subgroup	index
#14558062	2
C₄×Q₈×D₃₇	2
#14557867	2
#14557850	2
#14557841	2
#14557840	2
#14557752	2
#14557687	2
#14557670	2
#14557640	2
#14557637	2
#14557634	2
#14557630	2
#14557589	2
C₃₇×#464	2
#3117	37