Subgroups — C₃₇×#28497 #22649750

Order 9472.

1106 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3151	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#472	C₂	1	4	yes	maximal, complemented
4	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2018	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1965	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	8	yes	—
9	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
10	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
11	C₂×#131×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#532	C₂²	1	1	yes	characteristic
13	C₂×#127×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#496	—	2	4	no	—
15	C₃₇×#493	—	2	2	no	—
16	C₃₇×#493	—	2	2	no	—
17	C₃₇×#493	—	2	2	no	—
18	C₃₇×#492	—	2	2	no	—
19	C₃₇×#472	—	2	16	no	—
20	C₃₇×#436	—	2	2	no	—
21	C₈×D₄×C₃₇	—	2	2	no	—
22	C₂×#108×C₃₇	C₂²	1	2	yes	—
23	C₃₇×#423	C₂²	1	1	yes	characteristic
24	C₂×#100×C₃₇	C₂²	1	8	yes	—
25	C₃₇×#408	C₂²	1	2	yes	—
26	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×#341	C₂²	1	2	yes	—
29	C₂×#41×C₃₇	C₂³	1	2	yes	—
30	C₂²×D₄×C₃₇	—	2	4	no	—
31	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
32	AGL(1,ℤ/8)×C₃₇	—	2	16	no	—
33	AGL(1,ℤ/8)×C₃₇	—	4	8	no	—
34	C₂×SD₁₆×C₃₇	—	2	8	no	—
35	C₂×D₈×C₃₇	—	2	8	no	—
36	C₂×D₈×C₃₇	—	2	8	no	—
37	C₂×D₈×C₃₇	—	2	4	no	—
38	C₂×D₈×C₃₇	—	2	2	no	—
39	C₂×D₈×C₃₇	—	2	2	no	—
40	C₂×D₈×C₃₇	—	4	4	no	—
41	#131×C₃₇	—	2	8	no	—
42	C₂×M₁₆×C₃₇	—	2	2	no	—
43	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
44	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₃₇	—	2	2	no	—
46	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
49	#127×C₃₇	—	2	2	no	—
50	#127×C₃₇	—	2	4	no	—
51	#127×C₃₇	—	2	2	no	—
52	#127×C₃₇	D₄	1	1	yes	characteristic
53	#124×C₃₇	D₄	1	2	yes	—
54	#121×C₃₇	—	2	2	no	—
55	C₄×D₄×C₃₇	—	2	2	no	—
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#108×C₃₇	D₄	1	8	yes	complemented
58	C₄⋊C₈×C₃₇	—	2	2	no	—
59	#100×C₃₇	—	2	16	no	—
60	C₂²⋊C₈×C₃₇	—	2	4	no	—
61	#97×C₃₇	—	2	2	no	—
62	C₄×C₈×C₃₇	—	2	2	no	—
63	C₄×C₈×C₃₇	D₄	1	2	yes	—
64	C₄×C₈×C₃₇	D₄	1	2	yes	—
65	C₂⁴×C₃₇	—	2	4	no	—
66	#41×C₃₇	—	2	4	no	—
67	#41×C₃₇	—	2	4	no	—
68	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
69	C₂×D₄×C₃₇	—	2	2	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	8	no	—
72	C₂×D₄×C₃₇	—	2	4	no	—
73	C₂×D₄×C₃₇	—	2	4	no	—
74	C₂×D₄×C₃₇	—	2	8	no	—
75	C₂×D₄×C₃₇	—	4	4	no	—
76	C₂×D₄×C₃₇	—	4	4	no	—
77	C₂×D₄×C₃₇	—	8	8	no	—
78	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	#41	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
82	SD₁₆×C₃₇	—	4	16	no	—
83	D₈×C₃₇	—	4	16	no	—
84	D₈×C₃₇	—	4	8	no	—
85	D₈×C₃₇	—	4	4	no	—
86	D₈×C₃₇	—	4	4	no	—
87	M₁₆×C₃₇	—	2	4	no	—
88	M₁₆×C₃₇	—	2	4	no	—
89	M₁₆×C₃₇	—	4	4	no	—
90	M₁₆×C₃₇	C₂×D₄	1	8	yes	—
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₂×C₈×C₃₇	—	2	2	no	—
93	C₂×C₈×C₃₇	—	2	2	no	—
94	C₂×C₈×C₃₇	—	4	4	no	—
95	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
99	C₄⋊C₄×C₃₇	—	2	2	no	—
100	C₂²⋊C₄×C₃₇	—	2	4	no	—
101	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₃₇	—	2	2	no	—
105	C₂³×C₃₇	—	4	4	no	—
106	C₂³×C₃₇	—	4	8	no	—
107	C₂³×C₃₇	—	8	8	no	—
108	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
109	Q₈×C₃₇	—	2	4	no	—
110	D₄×C₃₇	—	2	4	no	—
111	D₄×C₃₇	—	4	4	no	—
112	D₄×C₃₇	—	4	4	no	—
113	D₄×C₃₇	—	4	4	no	—
114	D₄×C₃₇	—	4	8	no	—
115	D₄×C₃₇	—	4	4	no	—
116	D₄×C₃₇	—	8	8	no	—
117	D₄×C₃₇	—	8	8	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	#121	1	2	yes	—
123	C₂×C₄×C₃₇	#121	1	2	yes	—
124	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
128	C₈×C₃₇	—	2	2	no	—
129	C₈×C₃₇	—	2	8	no	—
130	C₈×C₃₇	—	4	2	no	—
131	C₈×C₃₇	#127	1	2	yes	—
132	C₈×C₃₇	#127	1	2	yes	—
133	#28497	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₃₇	—	2	2	no	—
135	C₂²×C₃₇	—	4	2	no	—
136	C₂²×C₃₇	—	4	4	no	—
137	C₂²×C₃₇	—	8	4	no	—
138	C₂²×C₃₇	—	8	8	no	—
139	C₂²×C₃₇	—	16	8	no	—
140	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
141	C₂²×C₃₇	#550	1	2	yes	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	4	2	no	—
144	C₄×C₃₇	#472	1	4	yes	complemented
145	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
146	C₄×C₃₇	#535	1	1	yes	characteristic
147	C₄×C₃₇	D₄²	1	2	yes	—
148	#3151	C₂×C₃₇	1	2	yes	complemented
149	C₂×#493	C₂×C₃₇	1	1	yes	characteristic, complemented
150	C₂×#472	C₂×C₃₇	1	4	yes	complemented
151	#2931	C₂×C₃₇	1	1	yes	characteristic, complemented
152	#2018	C₂×C₃₇	1	4	yes	complemented
153	#1965	C₂×C₃₇	1	2	yes	complemented
154	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
155	C₂×C₃₇	—	2	2	no	—
156	C₂×C₃₇	—	8	2	no	—
157	C₂×C₃₇	—	16	4	no	—
158	C₂×C₃₇	#2541	1	1	yes	characteristic
159	C₂×C₃₇	C₂×#472	1	2	yes	—
160	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	8	yes	—
161	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
162	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
163	C₂×#131	C₂²×C₃₇	1	2	yes	—
164	#532	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×#127	C₂²×C₃₇	1	2	yes	—
166	#496	—	2	4	no	—
167	#493	—	2	2	no	—
168	#493	—	2	2	no	—
169	#493	—	2	2	no	—
170	#492	—	2	2	no	—
171	#472	—	2	16	no	—
172	#436	—	2	2	no	—
173	C₈×D₄	—	2	2	no	—
174	C₂×#108	C₂²×C₃₇	1	2	yes	—
175	#423	C₂²×C₃₇	1	1	yes	characteristic
176	C₂×#100	C₂²×C₃₇	1	8	yes	—
177	#408	C₂²×C₃₇	1	2	yes	—
178	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
180	#341	C₂²×C₃₇	1	2	yes	—
181	C₃₇	#28497	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#41	C₂³×C₃₇	1	2	yes	—
183	C₂²×D₄	—	2	4	no	—
184	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
185	AGL(1,ℤ/8)	—	2	16	no	—
186	AGL(1,ℤ/8)	—	4	8	no	—
187	C₂×SD₁₆	—	2	8	no	—
188	C₂×D₈	—	2	8	no	—
189	C₂×D₈	—	2	8	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	—	4	4	no	—
194	#131	—	2	8	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
197	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
201	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
202	#127	—	2	2	no	—
203	#127	—	2	4	no	—
204	#127	—	2	2	no	—
205	#127	D₄×C₃₇	1	1	yes	characteristic
206	#124	D₄×C₃₇	1	2	yes	—
207	#121	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
210	#108	D₄×C₃₇	1	8	yes	complemented
211	C₄⋊C₈	—	2	2	no	—
212	#100	—	2	16	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	#97	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	D₄×C₃₇	1	2	yes	—
217	C₄×C₈	D₄×C₃₇	1	2	yes	—
218	C₂⁴	—	2	4	no	—
219	#41	—	2	4	no	—
220	#41	—	2	4	no	—
221	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	8	8	no	—
231	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	#41×C₃₇	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
235	SD₁₆	—	4	16	no	—
236	D₈	—	4	16	no	—
237	D₈	—	4	8	no	—
238	D₈	—	4	4	no	—
239	D₈	—	4	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	4	4	no	—
243	M₁₆	C₂×D₄×C₃₇	1	8	yes	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₃₇	1	2	yes	—
256	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	8	no	—
260	C₂³	—	8	8	no	—
261	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	8	no	—
268	D₄	—	4	4	no	—
269	D₄	—	8	8	no	—
270	D₄	—	8	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#121×C₃₇	1	2	yes	—
276	C₂×C₄	#121×C₃₇	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
279	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
280	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
281	C₈	—	2	2	no	—
282	C₈	—	2	8	no	—
283	C₈	—	4	2	no	—
284	C₈	#127×C₃₇	1	2	yes	—
285	C₈	#127×C₃₇	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	8	no	—
291	C₂²	—	16	8	no	—
292	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
293	C₂²	C₃₇×#550	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	4	2	no	—
296	C₄	C₃₇×#472	1	4	yes	complemented
297	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
298	C₄	C₃₇×#535	1	1	yes	characteristic
299	C₄	D₄²×C₃₇	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	8	2	no	—
302	C₂	—	16	4	no	—
303	C₂	C₃₇×#2541	1	1	yes	characteristic
304	C₂	C₂×C₃₇×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3151	C₂	semidirect
2. C₂×C₃₇×#493	C₂	semidirect
3. C₂×C₃₇×#472	C₂	semidirect
4. C₃₇×#2931	C₂	semidirect
5. C₃₇×#2018	C₂	semidirect
6. C₃₇×#1965	C₂	semidirect
7. C₄×#108×C₃₇	C₂	semidirect
57. #108×C₃₇	D₄	semidirect
133. #28497	C₃₇	direct (N × H)
144. C₄×C₃₇	#472	semidirect
148. #3151	C₂×C₃₇	semidirect
149. C₂×#493	C₂×C₃₇	semidirect
150. C₂×#472	C₂×C₃₇	semidirect
151. #2931	C₂×C₃₇	semidirect
152. #2018	C₂×C₃₇	semidirect
153. #1965	C₂×C₃₇	semidirect
154. C₄×#108	C₂×C₃₇	semidirect
181. C₃₇	#28497	direct (N × H)
210. #108	D₄×C₃₇	semidirect
296. C₄	C₃₇×#472	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3151	2
C₂×C₃₇×#493	2
C₂×C₃₇×#472	2
C₃₇×#2931	2
C₃₇×#2018	2
C₃₇×#1965	2
C₄×#108×C₃₇	2
#28497	37

Subgroups — C37×#28497 #22649750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28497 #22649750