Subgroups — C₃₇×#28489 #22649742

Order 9472.

1094 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#470	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₇×#436	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2542	C₂	1	2	yes	maximal
5	C₃₇×#2001	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1981	C₂	1	2	yes	maximal, complemented
7	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₃₇	C₂²	1	8	yes	—
13	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#494	—	2	4	no	—
15	C₃₇×#470	—	2	16	no	—
16	C₃₇×#436	—	2	4	no	—
17	C₂×#106×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₃₇	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#399	—	2	4	no	—
23	C₃₇×#390	—	2	2	no	—
24	C₃₇×#384	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₃₇×#340	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
28	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₃₇	—	2	8	no	—
31	C₂×Q₁₆×C₃₇	—	2	4	no	—
32	C₂×Q₁₆×C₃₇	—	2	4	no	—
33	C₂×Q₁₆×C₃₇	—	4	4	no	—
34	C₂×M₁₆×C₃₇	—	2	4	no	—
35	C₂×M₁₆×C₃₇	—	2	8	no	—
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	C₂²⋊Q₈×C₃₇	—	2	16	no	—
42	C₂²⋊Q₈×C₃₇	—	4	8	no	—
43	C₄×D₄×C₃₇	—	2	2	no	—
44	C₄×D₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
51	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	—	2	2	no	—
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	#106×C₃₇	D₄	1	8	yes	complemented
55	C₄⋊C₈×C₃₇	—	2	2	no	—
56	#103×C₃₇	—	2	16	no	—
57	C₂²⋊C₈×C₃₇	—	2	4	no	—
58	C₄×C₈×C₃₇	D₄	1	4	yes	—
59	#95×C₃₇	—	2	2	no	—
60	C₂⁴×C₃₇	C₂×D₄	1	2	yes	—
61	C₂×Q₈×C₃₇	—	2	8	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	4	no	—
64	C₂×Q₈×C₃₇	—	2	8	no	—
65	C₂×D₄×C₃₇	—	2	2	no	—
66	C₂×D₄×C₃₇	—	2	4	no	—
67	C₂²×C₄×C₃₇	—	2	4	no	—
68	C₂²×C₄×C₃₇	—	2	2	no	—
69	C₂²×C₄×C₃₇	—	2	4	no	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	4	no	—
72	C₂²×C₄×C₃₇	—	2	4	no	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₃₇	—	4	8	no	—
78	Q₁₆×C₃₇	—	4	8	no	—
79	M₁₆×C₃₇	—	4	8	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₃₇	—	2	2	no	—
87	C₄⋊C₄×C₃₇	—	2	4	no	—
88	C₄⋊C₄×C₃₇	—	4	16	no	—
89	C₄⋊C₄×C₃₇	—	4	4	no	—
90	C₄⋊C₄×C₃₇	—	4	4	no	—
91	C₄⋊C₄×C₃₇	C₂×D₄	1	8	yes	—
92	C₂²⋊C₄×C₃₇	—	2	4	no	—
93	C₂²⋊C₄×C₃₇	—	4	16	no	—
94	C₄²×C₃₇	—	4	4	no	—
95	C₄²×C₃₇	C₂×D₄	1	2	yes	—
96	C₄²×C₃₇	#41	1	2	yes	—
97	C₂³×C₃₇	—	2	2	no	—
98	C₂³×C₃₇	—	2	4	no	—
99	C₂³×C₃₇	—	2	4	no	—
100	C₂³×C₃₇	—	2	4	no	—
101	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₃₇	—	4	4	no	—
103	Q₈×C₃₇	—	4	4	no	—
104	Q₈×C₃₇	—	4	8	no	—
105	D₄×C₃₇	—	4	4	no	—
106	C₂×C₄×C₃₇	—	2	2	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	8	no	—
109	C₂×C₄×C₃₇	—	2	4	no	—
110	C₂×C₄×C₃₇	—	4	4	no	—
111	C₂×C₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	4	no	—
115	C₂×C₄×C₃₇	—	4	4	no	—
116	C₂×C₄×C₃₇	—	4	8	no	—
117	C₂×C₄×C₃₇	#121	1	2	yes	—
118	C₂×C₄×C₃₇	#121	1	2	yes	—
119	C₂×C₄×C₃₇	#137	1	4	yes	—
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
124	C₈×C₃₇	—	2	2	no	—
125	C₈×C₃₇	—	4	2	no	—
126	C₈×C₃₇	#127	1	4	yes	—
127	#28489	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₃₇	—	4	8	no	—
129	C₂²×C₃₇	—	4	2	no	—
130	C₂²×C₃₇	—	4	2	no	—
131	C₂²×C₃₇	—	4	2	no	—
132	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₃₇	D₄²	1	2	yes	—
134	C₂²×C₃₇	C₂×#137	1	2	yes	—
135	C₂²×C₃₇	#578	1	2	yes	—
136	C₄×C₃₇	—	2	2	no	—
137	C₄×C₃₇	—	4	8	no	—
138	C₄×C₃₇	—	4	2	no	—
139	C₄×C₃₇	—	8	4	no	—
140	C₄×C₃₇	#470	1	4	yes	complemented
141	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
142	C₄×C₃₇	#535	1	1	yes	characteristic
143	C₄×C₃₇	#550	1	2	yes	—
144	C₂×#494	C₂×C₃₇	1	1	yes	characteristic, complemented
145	C₂×#470	C₂×C₃₇	1	4	yes	complemented
146	C₂×#436	C₂×C₃₇	1	1	yes	characteristic
147	#2542	C₂×C₃₇	1	2	yes	—
148	#2001	C₂×C₃₇	1	4	yes	complemented
149	#1981	C₂×C₃₇	1	2	yes	complemented
150	C₄×#106	C₂×C₃₇	1	1	yes	characteristic, complemented
151	C₂×C₃₇	—	8	2	no	—
152	C₂×C₃₇	#2541	1	1	yes	characteristic
153	C₂×C₃₇	C₂×#470	1	1	yes	characteristic
154	C₂×C₃₇	#3059	1	1	yes	characteristic
155	C₂×C₃₇	#3276	1	4	yes	—
156	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
157	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
159	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
160	C₂×C₂²⋊Q₈	C₂²×C₃₇	1	8	yes	—
161	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
162	#494	—	2	4	no	—
163	#470	—	2	16	no	—
164	#436	—	2	4	no	—
165	C₂×#106	C₂²×C₃₇	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×#103	C₂²×C₃₇	1	8	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
170	#399	—	2	4	no	—
171	#390	—	2	2	no	—
172	#384	—	2	2	no	—
173	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
174	#340	C₂²×C₃₇	1	2	yes	—
175	C₃₇	#28489	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
177	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
179	C₂×Q₁₆	—	2	8	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	4	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	8	no	—
185	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₂²⋊Q₈	—	2	16	no	—
191	C₂²⋊Q₈	—	4	8	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	8	no	—
200	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
203	#106	D₄×C₃₇	1	8	yes	complemented
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	16	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	C₄×C₈	D₄×C₃₇	1	4	yes	—
208	#95	—	2	2	no	—
209	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	8	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
226	Q₁₆	—	4	8	no	—
227	Q₁₆	—	4	8	no	—
228	M₁₆	—	4	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	16	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	C₂×D₄×C₃₇	1	8	yes	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	16	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂×D₄×C₃₇	1	2	yes	—
245	C₄²	#41×C₃₇	1	2	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
251	Q₈	—	4	4	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	4	8	no	—
254	D₄	—	4	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	#121×C₃₇	1	2	yes	—
267	C₂×C₄	#121×C₃₇	1	2	yes	—
268	C₂×C₄	#137×C₃₇	1	4	yes	—
269	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
271	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
272	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	4	2	no	—
275	C₈	#127×C₃₇	1	4	yes	—
276	C₂²	—	4	8	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
281	C₂²	D₄²×C₃₇	1	2	yes	—
282	C₂²	C₂×#137×C₃₇	1	2	yes	—
283	C₂²	C₃₇×#578	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	4	8	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	4	no	—
288	C₄	C₃₇×#470	1	4	yes	complemented
289	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
290	C₄	C₃₇×#535	1	1	yes	characteristic
291	C₄	C₃₇×#550	1	2	yes	—
292	C₂	—	8	2	no	—
293	C₂	C₃₇×#2541	1	1	yes	characteristic
294	C₂	C₂×C₃₇×#470	1	1	yes	characteristic
295	C₂	C₃₇×#3059	1	1	yes	characteristic
296	C₂	C₃₇×#3276	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄×D₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#494	C₂	semidirect
2. C₂×C₃₇×#470	C₂	semidirect
5. C₃₇×#2001	C₂	semidirect
6. C₃₇×#1981	C₂	semidirect
7. C₄×#106×C₃₇	C₂	semidirect
54. #106×C₃₇	D₄	semidirect
127. #28489	C₃₇	direct (N × H)
140. C₄×C₃₇	#470	semidirect
144. C₂×#494	C₂×C₃₇	semidirect
145. C₂×#470	C₂×C₃₇	semidirect
148. #2001	C₂×C₃₇	semidirect
149. #1981	C₂×C₃₇	semidirect
150. C₄×#106	C₂×C₃₇	semidirect
175. C₃₇	#28489	direct (N × H)
203. #106	D₄×C₃₇	semidirect
288. C₄	C₃₇×#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#494	2
C₂×C₃₇×#470	2
C₂×C₃₇×#436	2
C₃₇×#2542	2
C₃₇×#2001	2
C₃₇×#1981	2
C₄×#106×C₃₇	2
#28489	37

Subgroups — C37×#28489 #22649742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28489 #22649742