Subgroups — #18003510

Order 4736.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₃₇	C₂	1	1	yes	maximal, characteristic
2	#14558298	C₂	1	2	yes	maximal, complemented
3	#14557656	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557655	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557654	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557648	C₂	1	2	yes	maximal, complemented
7	#14557647	C₂	1	2	yes	maximal, complemented
8	#108×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557548	C₂	1	2	yes	maximal, complemented
10	#14557490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₃₇	C₂²	1	2	yes	—
13	#11924559	C₄	1	1	yes	characteristic, complemented
14	#11924558	C₂²	1	1	yes	characteristic
15	#11924557	C₄	1	2	yes	complemented
16	#11924556	C₂²	1	2	yes	—
17	#11924555	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₃₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₃₇	C₄	1	2	yes	complemented
20	D₈×D₃₇	C₄	1	1	yes	characteristic, complemented
21	#11924544	—	2	2	no	—
22	#11924544	C₂²	1	2	yes	complemented
23	#11924543	C₂²	1	2	yes	complemented
24	#11924542	—	2	2	no	—
25	#11924542	C₂²	1	2	yes	complemented
26	C₂×#10596929	—	2	2	no	—
27	C₂×#10596929	C₂²	1	1	yes	characteristic
28	M₁₆×D₃₇	—	2	2	no	—
29	M₁₆×D₃₇	C₂²	1	2	yes	—
30	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
31	#11924493	C₂²	1	1	yes	characteristic
32	#11924472	C₂²	1	1	yes	characteristic, complemented
33	#11924471	C₂²	1	2	yes	complemented
34	#11924464	C₂²	1	2	yes	complemented
35	#11924463	C₂²	1	2	yes	complemented
36	#11924462	C₂²	1	2	yes	complemented
37	#11924446	C₂²	1	1	yes	characteristic, complemented
38	C₄×#64436	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
40	#135×C₃₇	C₂²	1	1	yes	characteristic
41	#131×C₃₇	C₂²	1	2	yes	complemented
42	#108×C₃₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₃₇	C₂²	1	2	yes	complemented
44	#97×C₃₇	C₂²	1	1	yes	characteristic, complemented
45	#41×C₃₇	C₂³	1	2	yes	—
46	#10596966	C₂×C₄	1	2	yes	—
47	Q₈×D₃₇	C₂×C₄	1	2	yes	—
48	#10596964	—	2	2	no	—
49	#10596964	C₂×C₄	1	2	yes	—
50	D₄×D₃₇	—	2	2	no	—
51	D₄×D₃₇	C₂×C₄	1	2	yes	—
52	#10596962	—	2	2	no	—
53	#10596962	C₂³	1	2	yes	—
54	C₂×C₄×D₃₇	—	2	2	no	—
55	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
56	Q₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₃₇	C₂×C₄	1	2	yes	—
58	D₈×C₃₇	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₃₇	—	2	2	no	—
60	M₁₆×C₃₇	C₂³	1	2	yes	—
61	C₂×C₈×C₃₇	—	2	2	no	—
62	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
64	C₄²×C₃₇	C₂³	1	1	yes	characteristic
65	#10596942	C₂×C₄	1	2	yes	—
66	#10596941	C₂×C₄	1	2	yes	—
67	#10596940	C₂×C₄	1	2	yes	—
68	#10596939	C₂×C₄	1	2	yes	—
69	#10596938	—	2	1	no	—
70	#10596936	—	2	1	no	—
71	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
72	#10596934	—	2	2	no	—
73	#10596934	C₂³	1	2	yes	—
74	C₂×#64436	—	2	2	no	—
75	C₂×#64436	C₂³	1	1	yes	characteristic
76	C₂×#64436	C₂³	1	1	yes	characteristic
77	Dic₁₄₈	C₂×C₄	1	1	yes	characteristic
78	D₂₉₆	C₂×C₄	1	1	yes	characteristic
79	#10596930	C₂×C₄	1	2	yes	—
80	#10596929	—	2	2	no	—
81	#10596929	—	2	2	no	—
82	#10596929	—	2	2	no	—
83	#10596929	D₄	1	2	yes	—
84	#10596929	D₄	1	2	yes	—
85	C₈×D₃₇	—	2	2	no	—
86	C₈×D₃₇	—	2	2	no	—
87	C₈×D₃₇	—	2	2	no	—
88	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
89	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
90	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
91	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
92	C₂²×D₃₇	—	2	2	no	—
93	C₂²×D₃₇	#41	1	1	yes	characteristic
94	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
95	D₄×C₃₇	—	2	2	no	—
96	D₄×C₃₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₃₇	—	2	2	no	—
101	C₂²⋊D₃₇	—	2	2	no	—
102	C₂²⋊D₃₇	—	2	2	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	C₂×Dic₃₇	—	2	1	no	—
105	C₂×Dic₃₇	#41	1	1	yes	characteristic
106	D₁₄₈	—	2	2	no	—
107	D₁₄₈	C₂²×C₄	1	2	yes	—
108	C₄×D₃₇	—	2	2	no	—
109	C₄×D₃₇	—	2	2	no	—
110	C₄×D₃₇	—	2	2	no	—
111	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
114	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
115	Dic₇₄	—	2	2	no	—
116	Dic₇₄	C₂²×C₄	1	2	yes	—
117	C₈×C₃₇	—	2	2	no	—
118	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₃₇	C₂×D₄	1	2	yes	—
121	#64436	—	2	2	no	—
122	#64436	C₂²×C₄	1	1	yes	characteristic
123	#64436	C₂²×C₄	1	1	yes	characteristic
124	#64436	C₂×D₄	1	2	yes	—
125	C₂²×C₃₇	—	2	2	no	—
126	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₃₇	—	2	2	no	—
128	C₂×D₃₇	—	4	1	no	—
129	C₂×D₃₇	—	8	2	no	—
130	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
131	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₃₇	—	2	2	no	—
133	C₄×C₃₇	—	4	1	no	—
134	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
136	Dic₃₇	—	2	2	no	—
137	Dic₃₇	—	4	1	no	—
138	Dic₃₇	C₄×D₄	1	1	yes	characteristic
139	Dic₃₇	C₄×D₄	1	1	yes	characteristic
140	#2957	—	37	1	no	maximal, Sylow
141	C₂×C₃₇	—	2	1	no	—
142	C₂×C₃₇	—	4	2	no	—
143	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
144	D₃₇	—	2	1	no	—
145	D₃₇	—	2	1	no	—
146	D₃₇	—	4	2	no	—
147	C₂×#135	—	37	1	no	—
148	#568	—	37	2	no	—
149	#444	—	37	1	no	—
150	#444	—	37	1	no	—
151	#444	—	37	1	no	—
152	#444	D₃₇	1	1	yes	characteristic, complemented
153	#443	—	37	2	no	—
154	#443	—	37	2	no	—
155	C₂×#108	—	37	1	no	—
156	#421	—	37	2	no	—
157	#405	—	37	1	no	—
158	C₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	37	2	no	—
160	#135	—	37	2	no	—
161	#135	—	37	1	no	—
162	#135	—	37	2	no	—
163	#135	—	37	1	no	—
164	#135	—	37	1	no	—
165	#135	C₂×D₃₇	1	1	yes	characteristic
166	C₂×Q₁₆	—	37	1	no	—
167	C₂×SD₁₆	—	37	2	no	—
168	C₂×D₈	—	37	1	no	—
169	#131	—	37	2	no	—
170	#131	—	37	2	no	—
171	#131	—	37	2	no	—
172	#131	—	74	2	no	—
173	#131	—	74	2	no	—
174	#131	C₂×D₃₇	1	2	yes	complemented
175	C₂×M₁₆	—	37	1	no	—
176	C₂×M₁₆	—	37	2	no	—
177	C₂×M₁₆	—	74	2	no	—
178	C₂×M₁₆	—	74	2	no	—
179	C₂²×C₈	—	37	1	no	—
180	#117	—	37	1	no	—
181	#108	—	37	2	no	—
182	#108	—	37	1	no	—
183	#108	C₂×D₃₇	1	1	yes	characteristic, complemented
184	#104	—	37	2	no	—
185	#104	—	37	2	no	—
186	#104	—	37	2	no	—
187	#104	C₂×D₃₇	1	2	yes	complemented
188	#97	—	37	1	no	—
189	#97	C₂×D₃₇	1	1	yes	characteristic, complemented
190	C₄×C₈	—	37	1	no	—
191	C₄×C₈	—	37	1	no	—
192	#41	—	37	2	no	—
193	#41	—	37	2	no	—
194	#41	—	37	2	no	—
195	#41	—	74	2	no	—
196	#41	—	74	2	no	—
197	#41	C₂²×D₃₇	1	2	yes	—
198	C₂×Q₈	—	37	2	no	—
199	C₂×D₄	—	37	2	no	—
200	C₂×D₄	—	74	2	no	—
201	C₂²×C₄	—	37	1	no	—
202	C₂²×C₄	—	74	2	no	—
203	Q₁₆	—	37	2	no	—
204	Q₁₆	—	37	1	no	—
205	Q₁₆	C₄×D₃₇	1	1	yes	characteristic
206	SD₁₆	—	37	2	no	—
207	SD₁₆	—	37	2	no	—
208	SD₁₆	—	37	2	no	—
209	SD₁₆	C₄×D₃₇	1	2	yes	—
210	D₈	—	37	2	no	—
211	D₈	—	37	1	no	—
212	D₈	C₄×D₃₇	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	37	2	no	—
215	M₁₆	—	37	2	no	—
216	M₁₆	—	37	2	no	—
217	M₁₆	—	74	2	no	—
218	M₁₆	—	74	2	no	—
219	M₁₆	—	74	2	no	—
220	M₁₆	—	74	2	no	—
221	M₁₆	C₂²×D₃₇	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	37	1	no	—
224	C₂×C₈	—	37	1	no	—
225	C₂×C₈	—	37	1	no	—
226	C₂×C₈	—	37	1	no	—
227	C₂×C₈	—	37	1	no	—
228	C₂×C₈	—	37	1	no	—
229	C₂×C₈	—	74	2	no	—
230	C₂×C₈	—	74	2	no	—
231	C₂×C₈	—	74	2	no	—
232	C₂×C₈	—	74	2	no	—
233	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
235	C₄⋊C₄	—	74	1	no	—
236	C₂²⋊C₄	—	74	1	no	—
237	C₄²	—	37	1	no	—
238	C₄²	C₂²×D₃₇	1	1	yes	characteristic
239	C₂³	—	37	1	no	—
240	C₂³	—	74	2	no	—
241	Q₈	—	37	2	no	—
242	Q₈	—	74	2	no	—
243	Q₈	C₂×C₄×D₃₇	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	37	2	no	—
246	D₄	—	74	2	no	—
247	D₄	—	74	2	no	—
248	D₄	—	74	2	no	—
249	D₄	—	74	2	no	—
250	D₄	C₂×C₄×D₃₇	1	2	yes	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	37	1	no	—
254	C₂×C₄	—	37	1	no	—
255	C₂×C₄	—	37	1	no	—
256	C₂×C₄	—	37	1	no	—
257	C₂×C₄	—	37	1	no	—
258	C₂×C₄	—	74	2	no	—
259	C₂×C₄	—	74	2	no	—
260	C₂×C₄	—	74	2	no	—
261	C₂×C₄	—	74	2	no	—
262	C₂×C₄	—	74	1	no	—
263	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	37	1	no	—
266	C₈	—	37	1	no	—
267	C₈	—	37	2	no	—
268	C₈	—	74	2	no	—
269	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
270	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
271	C₈	D₄×D₃₇	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	37	1	no	—
274	C₂²	—	37	1	no	—
275	C₂²	—	296	2	no	—
276	C₂²	—	74	2	no	—
277	C₂²	—	148	1	no	—
278	C₂²	#41×D₃₇	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	37	1	no	—
282	C₄	—	37	1	no	—
283	C₄	—	74	2	no	—
284	C₄	—	148	1	no	—
285	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
286	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	74	1	no	—
290	C₂	—	74	1	no	—
291	C₂	—	148	2	no	—
292	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#444	C₂
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#135×D₃₇	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#444	D₃₇
hypocenter	C₃₇	#2957
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #14558298	C₂	semidirect
3. #14557656	C₂	semidirect
4. #14557655	C₂	semidirect
5. #14557654	C₂	semidirect
6. #14557648	C₂	semidirect
7. #14557647	C₂	semidirect
8. #108×D₃₇	C₂	semidirect
9. #14557548	C₂	semidirect
10. #14557490	C₂	semidirect
11. C₃₇×#444	C₂	semidirect
13. #11924559	C₄	semidirect
15. #11924557	C₄	semidirect
17. #11924555	C₄	semidirect
18. Q₁₆×D₃₇	C₄	semidirect
19. SD₁₆×D₃₇	C₄	semidirect
20. D₈×D₃₇	C₄	semidirect
22. #11924544	C₂²	semidirect
23. #11924543	C₂²	semidirect
25. #11924542	C₂²	semidirect
32. #11924472	C₂²	semidirect
33. #11924471	C₂²	semidirect
34. #11924464	C₂²	semidirect
35. #11924463	C₂²	semidirect
36. #11924462	C₂²	semidirect
37. #11924446	C₂²	semidirect
38. C₄×#64436	C₂²	semidirect
39. C₈×Dic₃₇	C₂²	semidirect
41. #131×C₃₇	C₂²	semidirect
42. #108×C₃₇	C₂²	semidirect
43. #104×C₃₇	C₂²	semidirect
44. #97×C₃₇	C₂²	semidirect
152. #444	D₃₇	semidirect
158. C₃₇	#2957	semidirect
174. #131	C₂×D₃₇	semidirect
183. #108	C₂×D₃₇	semidirect
187. #104	C₂×D₃₇	semidirect
189. #97	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#135×D₃₇	2
#14558298	2
#14557656	2
#14557655	2
#14557654	2
#14557648	2
#14557647	2
#108×D₃₇	2
#14557548	2
#14557490	2
C₃₇×#444	2
#2957	37