Subgroups — #18003317

Order 4736.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924542	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558044	C₂	1	1	yes	maximal, characteristic
4	#14557608	C₂	1	2	yes	maximal, complemented
5	#14557607	C₂	1	2	yes	maximal, complemented
6	#14557601	C₂	1	2	yes	maximal, complemented
7	#14557596	C₂	1	2	yes	maximal, complemented
8	#14557559	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557499	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924542	—	2	4	no	—
14	C₂×#10596934	—	2	4	no	—
15	C₂×#10596934	C₂²	1	1	yes	characteristic
16	C₂×#10596934	C₂²	1	2	yes	—
17	C₂×#10596934	C₂²	1	1	yes	characteristic
18	C₂×#10596929	C₂²	1	2	yes	complemented
19	C₂²×#64436	C₂²	1	2	yes	—
20	C₂×C₈×D₃₇	C₂²	1	2	yes	complemented
21	#11924519	C₄	1	2	yes	—
22	#11924515	C₄	1	1	yes	characteristic
23	#11924515	C₄	1	1	yes	characteristic
24	#11924510	C₄	1	1	yes	characteristic
25	#11924510	C₄	1	1	yes	characteristic
26	#11924505	C₄	1	2	yes	—
27	C₄×C₂²⋊D₃₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₃₇	C₂²	1	2	yes	—
29	#11924495	C₂²	1	1	yes	characteristic
30	#11924465	C₂²	1	2	yes	complemented
31	#11924465	C₂²	1	2	yes	complemented
32	#11924448	C₂²	1	2	yes	complemented
33	#11924448	C₂²	1	2	yes	complemented
34	#11924446	C₂²	1	2	yes	complemented
35	C₈×Dic₃₇	C₂²	1	2	yes	complemented
36	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
43	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₇	—	2	2	no	—
45	C₂²×C₄×C₃₇	—	2	2	no	—
46	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
55	#10596962	—	2	2	no	—
56	#10596962	—	2	2	no	—
57	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₃₇	C₂³	1	2	yes	—
59	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₇	—	2	2	no	—
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	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
69	#10596938	C₂×C₄	1	2	yes	—
70	#10596938	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
73	#10596936	C₂×C₄	1	2	yes	—
74	#10596936	C₂×C₄	1	2	yes	—
75	C₄×Dic₃₇	C₂³	1	2	yes	—
76	#10596934	—	2	2	no	—
77	#10596934	—	2	4	no	—
78	#10596934	D₄	1	4	yes	complemented
79	C₂×#64436	—	2	4	no	—
80	C₂×#64436	C₂³	1	2	yes	—
81	C₂×#64436	C₂³	1	2	yes	—
82	#10596929	—	2	4	no	—
83	C₈×D₃₇	—	2	4	no	—
84	C₂³×C₃₇	—	2	2	no	—
85	C₂³×C₃₇	—	2	2	no	—
86	C₂³×C₃₇	—	4	1	no	—
87	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	1	no	—
95	C₂×C₄×C₃₇	—	2	1	no	—
96	C₂×C₄×C₃₇	—	2	1	no	—
97	C₂×C₄×C₃₇	—	2	1	no	—
98	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₃₇	#41	1	2	yes	—
103	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₃₇	—	2	4	no	—
105	C₂²⋊D₃₇	—	2	4	no	—
106	C₂²⋊D₃₇	—	4	2	no	—
107	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
108	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
109	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
110	D₁₄₈	—	2	2	no	—
111	C₄×D₃₇	—	2	2	no	—
112	C₄×D₃₇	—	2	2	no	—
113	Dic₇₄	—	2	2	no	—
114	C₈×C₃₇	—	2	1	no	—
115	C₈×C₃₇	—	2	1	no	—
116	C₈×C₃₇	—	2	1	no	—
117	C₈×C₃₇	—	2	1	no	—
118	#64436	—	2	2	no	—
119	#64436	C₂×D₄	1	4	yes	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	—	2	1	no	—
122	C₂²×C₃₇	—	2	1	no	—
123	C₂²×C₃₇	—	2	1	no	—
124	C₂²×C₃₇	—	2	1	no	—
125	C₂²×C₃₇	—	2	1	no	—
126	C₂²×C₃₇	—	2	1	no	—
127	C₂²×C₃₇	—	4	4	no	—
128	C₂²×C₃₇	C₄×D₄	1	2	yes	—
129	C₂²×C₃₇	#131	1	2	yes	complemented
130	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₃₇	—	2	2	no	—
132	C₂×D₃₇	—	2	2	no	—
133	C₂×D₃₇	—	2	2	no	—
134	C₄×C₃₇	—	2	1	no	—
135	C₄×C₃₇	—	2	1	no	—
136	C₄×C₃₇	C₄×D₄	1	2	yes	—
137	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
139	Dic₃₇	—	2	2	no	—
140	Dic₃₇	—	2	2	no	—
141	Dic₃₇	—	2	2	no	—
142	#2933	—	37	1	no	maximal, Sylow
143	C₂×C₃₇	—	2	2	no	—
144	C₂×C₃₇	—	2	2	no	—
145	C₂×C₃₇	—	4	1	no	—
146	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
148	C₂×C₃₇	#568	1	1	yes	characteristic
149	D₃₇	—	4	2	no	—
150	C₂×#131	—	37	1	no	—
151	C₂²×M₁₆	—	37	1	no	—
152	#525	—	37	1	no	—
153	#436	—	37	2	no	—
154	#435	—	37	2	no	—
155	#435	—	37	2	no	—
156	C₈×D₄	—	37	2	no	—
157	#424	—	37	1	no	—
158	#408	—	37	1	no	—
159	C₂×C₂²⋊C₈	D₃₇	1	1	yes	characteristic, complemented
160	#405	—	37	1	no	—
161	C₃₇	#2933	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	37	1	no	—
163	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
164	#131	—	74	4	no	—
165	C₂×M₁₆	—	37	1	no	—
166	C₂×M₁₆	—	37	1	no	—
167	C₂×M₁₆	—	37	2	no	—
168	C₂×M₁₆	—	37	2	no	—
169	C₂×M₁₆	—	74	4	no	—
170	C₂²×C₈	—	37	2	no	—
171	C₂²×C₈	—	37	2	no	—
172	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
173	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
174	#123	—	37	2	no	—
175	C₂²⋊Q₈	—	37	1	no	—
176	C₂²⋊Q₈	—	37	1	no	—
177	#121	—	37	1	no	—
178	#121	—	37	1	no	—
179	#120	—	37	2	no	—
180	C₄×D₄	—	37	2	no	—
181	C₄×D₄	—	37	2	no	—
182	#117	—	37	1	no	—
183	C₄⋊C₈	—	37	2	no	—
184	C₄⋊C₈	—	37	2	no	—
185	C₂²⋊C₈	—	37	2	no	—
186	C₂²⋊C₈	—	37	2	no	—
187	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
188	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
189	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
190	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
191	#97	—	37	2	no	—
192	C₄×C₈	—	37	2	no	—
193	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
194	#41	—	74	2	no	—
195	#41	—	74	2	no	—
196	C₂×Q₈	—	37	1	no	—
197	C₂×D₄	—	37	2	no	—
198	C₂×D₄	—	37	2	no	—
199	C₂×D₄	—	37	2	no	—
200	C₂×D₄	—	37	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	37	2	no	—
204	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
210	M₁₆	—	37	4	no	—
211	M₁₆	—	74	4	no	—
212	M₁₆	—	74	4	no	—
213	M₁₆	—	74	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	37	2	no	—
217	C₂×C₈	—	37	2	no	—
218	C₂×C₈	—	74	4	no	—
219	C₂×C₈	—	74	4	no	—
220	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
224	C₄⋊C₄	—	37	2	no	—
225	C₄⋊C₄	—	37	2	no	—
226	C₄⋊C₄	—	37	1	no	—
227	C₄⋊C₄	—	37	1	no	—
228	C₂²⋊C₄	—	37	2	no	—
229	C₂²⋊C₄	—	37	2	no	—
230	C₂²⋊C₄	—	37	2	no	—
231	C₂²⋊C₄	—	37	2	no	—
232	C₂²⋊C₄	—	37	2	no	—
233	C₄²	—	37	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	37	2	no	—
238	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
240	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
241	Q₈	—	74	2	no	—
242	D₄	—	74	4	no	—
243	D₄	—	74	4	no	—
244	D₄	—	74	2	no	—
245	D₄	—	148	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	37	2	no	—
254	C₂×C₄	—	37	2	no	—
255	C₂×C₄	—	37	2	no	—
256	C₂×C₄	—	74	2	no	—
257	C₂×C₄	—	74	2	no	—
258	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
261	C₂×C₄	D₄×D₃₇	1	2	yes	—
262	C₂×C₄	#10596964	1	2	yes	—
263	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	37	4	no	—
269	C₈	—	74	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	74	2	no	—
279	C₂²	—	74	2	no	—
280	C₂²	—	74	2	no	—
281	C₂²	#11924496	1	2	yes	—
282	C₂²	#11924542	1	2	yes	complemented
283	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	74	2	no	—
287	C₄	—	74	2	no	—
288	C₄	—	74	2	no	—
289	C₄	#11924496	1	2	yes	—
290	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
291	C₄	C₂×#10596964	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	148	2	no	—
296	C₂	C₂×#11924496	1	1	yes	characteristic
297	C₂	C₂×#11924542	1	1	yes	characteristic
298	C₂	#14558299	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₃₇	C₂
prime radical	#14558044	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃₇
hypocenter	C₃₇	#2933
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11924542	C₂	semidirect
2. C₂²×#10596934	C₂	semidirect
4. #14557608	C₂	semidirect
5. #14557607	C₂	semidirect
6. #14557601	C₂	semidirect
7. #14557596	C₂	semidirect
8. #14557559	C₂	semidirect
9. #14557499	C₂	semidirect
10. #14557488	C₂	semidirect
11. C₂×C₂²⋊C₈×C₃₇	C₂	semidirect
18. C₂×#10596929	C₂²	semidirect
20. C₂×C₈×D₃₇	C₂²	semidirect
30. #11924465	C₂²	semidirect
31. #11924465	C₂²	semidirect
32. #11924448	C₂²	semidirect
33. #11924448	C₂²	semidirect
34. #11924446	C₂²	semidirect
35. C₈×Dic₃₇	C₂²	semidirect
78. #10596934	D₄	semidirect
129. C₂²×C₃₇	#131	semidirect
159. C₂×C₂²⋊C₈	D₃₇	semidirect
161. C₃₇	#2933	semidirect
282. C₂²	#11924542	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924542	2
C₂²×#10596934	2
#14558044	2
#14557608	2
#14557607	2
#14557601	2
#14557596	2
#14557559	2
#14557499	2
#14557488	2
C₂×C₂²⋊C₈×C₃₇	2
#2933	37