Subgroups — C₃₇×#3200 #17992018

Order 4736.

410 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#536	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#535	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#485	C₂	1	2	yes	maximal, complemented
4	C₃₇×#482	C₂	1	2	yes	maximal, complemented
5	C₃₇×#468	C₂	1	2	yes	maximal, complemented
6	C₃₇×#465	C₂	1	2	yes	maximal, complemented
7	C₃₇×#450	C₂	1	2	yes	maximal, complemented
8	C₃₇×#448	C₂	1	2	yes	maximal, complemented
9	C₃₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₃₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
17	C₄⋊Q₈×C₃₇	—	2	1	no	—
18	#127×C₃₇	—	2	1	no	—
19	#124×C₃₇	—	2	1	no	—
20	#124×C₃₇	—	2	1	no	—
21	#123×C₃₇	—	2	1	no	—
22	#123×C₃₇	—	2	1	no	—
23	C₂²⋊Q₈×C₃₇	C₂²	1	2	yes	—
24	C₂²⋊Q₈×C₃₇	C₂²	1	2	yes	complemented
25	#121×C₃₇	C₂²	1	2	yes	—
26	#121×C₃₇	C₂²	1	2	yes	—
27	#120×C₃₇	—	2	1	no	—
28	#120×C₃₇	—	2	1	no	—
29	#117×C₃₇	C₂²	1	1	yes	characteristic
30	#117×C₃₇	C₂²	1	1	yes	characteristic, complemented
31	#107×C₃₇	C₂²	1	2	yes	complemented
32	#106×C₃₇	C₂²	1	2	yes	complemented
33	#103×C₃₇	C₂²	1	2	yes	complemented
34	#103×C₃₇	C₂²	1	2	yes	complemented
35	#102×C₃₇	C₂²	1	2	yes	—
36	#102×C₃₇	C₂²	1	2	yes	—
37	C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
38	C₂²⋊C₈×C₃₇	C₂²	1	2	yes	complemented
39	C₂⁴×C₃₇	—	2	1	no	—
40	#41×C₃₇	—	4	1	no	—
41	#41×C₃₇	—	4	1	no	—
42	C₂×Q₈×C₃₇	—	2	1	no	—
43	C₂×Q₈×C₃₇	—	4	1	no	—
44	C₂×Q₈×C₃₇	—	4	1	no	—
45	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
47	C₂×D₄×C₃₇	—	2	2	no	—
48	C₂×D₄×C₃₇	—	2	2	no	—
49	C₂×D₄×C₃₇	—	2	1	no	—
50	C₂×D₄×C₃₇	—	4	1	no	—
51	C₂×D₄×C₃₇	—	4	1	no	—
52	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
53	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
54	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
55	C₂×D₄×C₃₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₃₇	—	2	1	no	—
57	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
60	SD₁₆×C₃₇	—	4	2	no	—
61	SD₁₆×C₃₇	—	4	2	no	—
62	M₁₆×C₃₇	—	4	1	no	—
63	C₂×C₈×C₃₇	—	4	1	no	—
64	C₂×C₈×C₃₇	C₂³	1	2	yes	—
65	C₂×C₈×C₃₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₃₇	—	2	2	no	—
67	C₄⋊C₄×C₃₇	—	2	2	no	—
68	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
69	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₃₇	—	2	1	no	—
71	C₂²⋊C₄×C₃₇	—	2	1	no	—
72	C₂²⋊C₄×C₃₇	—	2	2	no	—
73	C₂²⋊C₄×C₃₇	—	2	2	no	—
74	C₂²⋊C₄×C₃₇	—	2	2	no	—
75	C₂²⋊C₄×C₃₇	—	2	2	no	—
76	C₄²×C₃₇	—	2	1	no	—
77	C₄²×C₃₇	—	2	1	no	—
78	C₂³×C₃₇	—	2	2	no	—
79	C₂³×C₃₇	—	4	2	no	—
80	C₂³×C₃₇	—	8	2	no	—
81	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₃₇	—	2	2	no	—
85	Q₈×C₃₇	—	2	2	no	—
86	Q₈×C₃₇	—	4	1	no	—
87	Q₈×C₃₇	—	4	2	no	—
88	D₄×C₃₇	—	2	2	no	—
89	D₄×C₃₇	—	2	2	no	—
90	D₄×C₃₇	—	4	1	no	—
91	D₄×C₃₇	—	4	1	no	—
92	D₄×C₃₇	—	4	1	no	—
93	D₄×C₃₇	—	4	2	no	—
94	D₄×C₃₇	—	8	2	no	—
95	D₄×C₃₇	—	8	2	no	—
96	C₂×C₄×C₃₇	—	2	1	no	—
97	C₂×C₄×C₃₇	—	2	1	no	—
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	—	2	1	no	—
100	C₂×C₄×C₃₇	—	2	1	no	—
101	C₂×C₄×C₃₇	—	2	1	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
110	C₈×C₃₇	—	2	2	no	—
111	C₈×C₃₇	—	2	2	no	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	—	2	1	no	—
115	C₂²×C₃₇	—	2	2	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	—	2	2	no	—
118	C₂²×C₃₇	—	4	2	no	—
119	C₂²×C₃₇	—	4	2	no	—
120	C₂²×C₃₇	—	4	2	no	—
121	C₂²×C₃₇	—	8	4	no	—
122	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₃₇	—	2	1	no	—
124	C₄×C₃₇	—	2	1	no	—
125	C₄×C₃₇	—	2	1	no	—
126	C₄×C₃₇	—	4	2	no	—
127	C₄×C₃₇	—	4	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
131	#3200	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₃₇	—	4	1	no	—
133	C₂×C₃₇	—	4	1	no	—
134	C₂×C₃₇	—	4	1	no	—
135	C₂×C₃₇	—	8	2	no	—
136	C₂×C₃₇	#534	1	1	yes	characteristic
137	C₂×C₃₇	#577	1	2	yes	—
138	#536	C₂×C₃₇	1	1	yes	characteristic, complemented
139	#535	C₂×C₃₇	1	1	yes	characteristic
140	#485	C₂×C₃₇	1	2	yes	complemented
141	#482	C₂×C₃₇	1	2	yes	complemented
142	#468	C₂×C₃₇	1	2	yes	complemented
143	#465	C₂×C₃₇	1	2	yes	complemented
144	#450	C₂×C₃₇	1	2	yes	complemented
145	#448	C₂×C₃₇	1	2	yes	complemented
146	#408	C₂×C₃₇	1	1	yes	characteristic, complemented
147	C₃₇	#3200	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#41	C₂²×C₃₇	1	1	yes	characteristic
149	C₂²×Q₈	C₂²×C₃₇	1	1	yes	characteristic, complemented
150	C₂²×D₄	C₂²×C₃₇	1	1	yes	characteristic
151	C₂×SD₁₆	C₂²×C₃₇	1	2	yes	—
152	C₂×SD₁₆	C₂²×C₃₇	1	2	yes	—
153	C₂×M₁₆	C₂²×C₃₇	1	1	yes	characteristic, complemented
154	C₂²×C₈	C₂²×C₃₇	1	1	yes	characteristic, complemented
155	C₄⋊Q₈	—	2	1	no	—
156	#127	—	2	1	no	—
157	#124	—	2	1	no	—
158	#124	—	2	1	no	—
159	#123	—	2	1	no	—
160	#123	—	2	1	no	—
161	C₂²⋊Q₈	C₂²×C₃₇	1	2	yes	—
162	C₂²⋊Q₈	C₂²×C₃₇	1	2	yes	complemented
163	#121	C₂²×C₃₇	1	2	yes	—
164	#121	C₂²×C₃₇	1	2	yes	—
165	#120	—	2	1	no	—
166	#120	—	2	1	no	—
167	#117	C₂²×C₃₇	1	1	yes	characteristic
168	#117	C₂²×C₃₇	1	1	yes	characteristic, complemented
169	#107	C₂²×C₃₇	1	2	yes	complemented
170	#106	C₂²×C₃₇	1	2	yes	complemented
171	#103	C₂²×C₃₇	1	2	yes	complemented
172	#103	C₂²×C₃₇	1	2	yes	complemented
173	#102	C₂²×C₃₇	1	2	yes	—
174	#102	C₂²×C₃₇	1	2	yes	—
175	C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
176	C₂²⋊C₈	C₂²×C₃₇	1	2	yes	complemented
177	C₂⁴	—	2	1	no	—
178	#41	—	4	1	no	—
179	#41	—	4	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	D₄×C₃₇	1	1	yes	characteristic
184	C₂×Q₈	C₂³×C₃₇	1	2	yes	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	D₄×C₃₇	1	1	yes	characteristic
191	C₂×D₄	D₄×C₃₇	1	1	yes	characteristic
192	C₂×D₄	D₄×C₃₇	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₃₇	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
198	SD₁₆	—	4	2	no	—
199	SD₁₆	—	4	2	no	—
200	M₁₆	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	C₂³×C₃₇	1	2	yes	—
203	C₂×C₈	C₂³×C₃₇	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
207	C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	2	no	—
219	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
220	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
221	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	4	1	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	1	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂²×D₄×C₃₇	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
268	C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	1	no	—
272	C₂	—	8	2	no	—
273	C₂	C₃₇×#534	1	1	yes	characteristic
274	C₂	C₃₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×#41×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#536	C₂	semidirect
3. C₃₇×#485	C₂	semidirect
4. C₃₇×#482	C₂	semidirect
5. C₃₇×#468	C₂	semidirect
6. C₃₇×#465	C₂	semidirect
7. C₃₇×#450	C₂	semidirect
8. C₃₇×#448	C₂	semidirect
9. C₃₇×#408	C₂	semidirect
11. C₂²×Q₈×C₃₇	C₂²	semidirect
15. C₂×M₁₆×C₃₇	C₂²	semidirect
16. C₂²×C₈×C₃₇	C₂²	semidirect
24. C₂²⋊Q₈×C₃₇	C₂²	semidirect
30. #117×C₃₇	C₂²	semidirect
31. #107×C₃₇	C₂²	semidirect
32. #106×C₃₇	C₂²	semidirect
33. #103×C₃₇	C₂²	semidirect
34. #103×C₃₇	C₂²	semidirect
38. C₂²⋊C₈×C₃₇	C₂²	semidirect
131. #3200	C₃₇	direct (N × H)
138. #536	C₂×C₃₇	semidirect
140. #485	C₂×C₃₇	semidirect
141. #482	C₂×C₃₇	semidirect
142. #468	C₂×C₃₇	semidirect
143. #465	C₂×C₃₇	semidirect
144. #450	C₂×C₃₇	semidirect
145. #448	C₂×C₃₇	semidirect
146. #408	C₂×C₃₇	semidirect
147. C₃₇	#3200	direct (N × H)
149. C₂²×Q₈	C₂²×C₃₇	semidirect
153. C₂×M₁₆	C₂²×C₃₇	semidirect
154. C₂²×C₈	C₂²×C₃₇	semidirect
162. C₂²⋊Q₈	C₂²×C₃₇	semidirect
168. #117	C₂²×C₃₇	semidirect
169. #107	C₂²×C₃₇	semidirect
170. #106	C₂²×C₃₇	semidirect
171. #103	C₂²×C₃₇	semidirect
172. #103	C₂²×C₃₇	semidirect
176. C₂²⋊C₈	C₂²×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#536	2
C₃₇×#535	2
C₃₇×#485	2
C₃₇×#482	2
C₃₇×#468	2
C₃₇×#465	2
C₃₇×#450	2
C₃₇×#448	2
C₃₇×#408	2
#3200	37

Subgroups — C37×#3200 #17992018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#3200 #17992018