Subgroups — C₃₇×#28712 #22649965

Order 9472.

846 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#436	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2547	C₂	1	2	yes	maximal
4	C₃₇×#2076	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1988	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1809	C₂	1	4	yes	maximal, complemented
7	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
10	C₂×#124×C₃₇	—	2	2	no	—
11	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#500	—	2	4	no	—
13	C₃₇×#436	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#397	—	2	4	no	—
21	C₃₇×#391	—	2	2	no	—
22	C₃₇×#380	C₂²	1	8	yes	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#340	C₂²	1	8	yes	—
25	C₃₇×#340	C₂²	1	2	yes	—
26	C₂²×Q₈×C₃₇	—	2	2	no	—
27	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
29	C₂×M₁₆×C₃₇	—	2	4	no	—
30	C₂×M₁₆×C₃₇	—	2	8	no	—
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₃₇	—	2	4	no	—
35	C₄⋊Q₈×C₃₇	—	2	4	no	—
36	#124×C₃₇	—	4	8	no	—
37	C₄×D₄×C₃₇	—	2	2	no	—
38	C₄×D₄×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₃₇	—	2	2	no	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#107×C₃₇	—	2	4	no	—
48	#107×C₃₇	—	2	4	no	—
49	#106×C₃₇	—	2	4	no	—
50	C₄⋊C₈×C₃₇	—	2	2	no	—
51	C₂²⋊C₈×C₃₇	—	2	4	no	—
52	C₄×C₈×C₃₇	D₄	1	4	yes	—
53	#95×C₃₇	—	2	2	no	—
54	#95×C₃₇	—	2	8	no	—
55	C₂⁴×C₃₇	C₂×D₄	1	2	yes	—
56	C₂×Q₈×C₃₇	—	4	4	no	—
57	C₂×Q₈×C₃₇	—	4	4	no	—
58	C₂×Q₈×C₃₇	—	4	4	no	—
59	C₂×D₄×C₃₇	—	2	2	no	—
60	C₂×D₄×C₃₇	—	2	4	no	—
61	C₂²×C₄×C₃₇	—	2	2	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	—	2	4	no	—
65	C₂²×C₄×C₃₇	—	2	4	no	—
66	C₂²×C₄×C₃₇	—	2	4	no	—
67	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₃₇	—	4	8	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	2	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₃₇	#41	1	4	yes	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	—	2	4	no	—
82	C₄⋊C₄×C₃₇	—	2	4	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₂²⋊C₄×C₃₇	—	2	4	no	—
85	C₂²⋊C₄×C₃₇	—	4	16	no	—
86	C₄²×C₃₇	—	4	4	no	—
87	C₄²×C₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₃₇	#41	1	2	yes	—
89	C₂³×C₃₇	—	2	2	no	—
90	C₂³×C₃₇	—	2	4	no	—
91	C₂³×C₃₇	—	2	4	no	—
92	C₂³×C₃₇	—	2	4	no	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₃₇	—	8	8	no	—
95	D₄×C₃₇	—	4	4	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	4	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	4	2	no	—
101	C₂×C₄×C₃₇	—	4	4	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	—	4	4	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	4	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	4	4	no	—
108	C₂×C₄×C₃₇	#121	1	2	yes	—
109	C₂×C₄×C₃₇	#121	1	2	yes	—
110	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
113	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₃₇	—	2	2	no	—
116	C₈×C₃₇	—	4	2	no	—
117	C₈×C₃₇	#124	1	4	yes	complemented
118	#28712	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₃₇	—	4	8	no	—
120	C₂²×C₃₇	—	4	2	no	—
121	C₂²×C₃₇	—	4	2	no	—
122	C₂²×C₃₇	—	4	2	no	—
123	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
124	C₂²×C₃₇	#547	1	2	yes	—
125	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
126	C₂²×C₃₇	#578	1	2	yes	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	—	8	4	no	—
130	C₄×C₃₇	—	8	4	no	—
131	C₄×C₃₇	#469	1	4	yes	—
132	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
133	C₄×C₃₇	#536	1	1	yes	characteristic
134	C₄×C₃₇	#556	1	2	yes	—
135	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
136	C₂×#436	C₂×C₃₇	1	1	yes	characteristic
137	#2547	C₂×C₃₇	1	2	yes	—
138	#2076	C₂×C₃₇	1	4	yes	complemented
139	#1988	C₂×C₃₇	1	2	yes	complemented
140	#1809	C₂×C₃₇	1	4	yes	complemented
141	C₄×#106	C₂×C₃₇	1	1	yes	characteristic, complemented
142	C₂×C₃₇	—	8	2	no	—
143	C₂×C₃₇	#2547	1	1	yes	characteristic
144	C₂×C₃₇	C₂×#469	1	1	yes	characteristic
145	C₂×C₃₇	#3059	1	1	yes	characteristic
146	C₂×C₃₇	#3353	1	4	yes	—
147	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
148	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
149	C₂×#124	—	2	2	no	—
150	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
151	#500	—	2	4	no	—
152	#436	—	2	4	no	—
153	C₂×#107	C₂²×C₃₇	1	2	yes	—
154	C₂×#107	C₂²×C₃₇	1	2	yes	—
155	C₂×#106	C₂²×C₃₇	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
159	#397	—	2	4	no	—
160	#391	—	2	2	no	—
161	#380	C₂²×C₃₇	1	8	yes	—
162	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
163	#340	C₂²×C₃₇	1	8	yes	—
164	#340	C₂²×C₃₇	1	2	yes	—
165	C₃₇	#28712	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	2	4	no	—
176	#124	—	4	8	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	8	no	—
184	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#107	—	2	4	no	—
189	#106	—	2	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₄×C₈	D₄×C₃₇	1	4	yes	—
193	#95	—	2	2	no	—
194	#95	—	2	8	no	—
195	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
211	M₁₆	—	4	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
217	C₂×C₈	#41×C₃₇	1	4	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	4	16	no	—
226	C₄²	—	4	4	no	—
227	C₄²	C₂×D₄×C₃₇	1	2	yes	—
228	C₄²	#41×C₃₇	1	2	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	2	4	no	—
233	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
234	Q₈	—	8	8	no	—
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#121×C₃₇	1	2	yes	—
249	C₂×C₄	#121×C₃₇	1	2	yes	—
250	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
251	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
253	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
254	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	2	no	—
257	C₈	#124×C₃₇	1	4	yes	complemented
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	8	no	—
262	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
263	C₂²	C₃₇×#547	1	2	yes	—
264	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
265	C₂²	C₃₇×#578	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	4	no	—
270	C₄	C₃₇×#469	1	4	yes	—
271	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
272	C₄	C₃₇×#536	1	1	yes	characteristic
273	C₄	C₃₇×#556	1	2	yes	—
274	C₂	—	8	2	no	—
275	C₂	C₃₇×#2547	1	1	yes	characteristic
276	C₂	C₂×C₃₇×#469	1	1	yes	characteristic
277	C₂	C₃₇×#3059	1	1	yes	characteristic
278	C₂	C₃₇×#3353	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×#500	C₂	semidirect
4. C₃₇×#2076	C₂	semidirect
5. C₃₇×#1988	C₂	semidirect
6. C₃₇×#1809	C₂	semidirect
7. C₄×#106×C₃₇	C₂	semidirect
117. C₈×C₃₇	#124	semidirect
118. #28712	C₃₇	direct (N × H)
135. C₂×#500	C₂×C₃₇	semidirect
138. #2076	C₂×C₃₇	semidirect
139. #1988	C₂×C₃₇	semidirect
140. #1809	C₂×C₃₇	semidirect
141. C₄×#106	C₂×C₃₇	semidirect
165. C₃₇	#28712	direct (N × H)
257. C₈	#124×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#500	2
C₂×C₃₇×#436	2
C₃₇×#2547	2
C₃₇×#2076	2
C₃₇×#1988	2
C₃₇×#1809	2
C₄×#106×C₃₇	2
#28712	37

Subgroups — C37×#28712 #22649965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28712 #22649965