Subgroups — C₃₇×#28509 #22649762

Order 9472.

1090 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#476	C₂	1	4	yes	maximal, complemented
2	C₃₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2094	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1957	C₂	1	2	yes	maximal
6	C₃₇×#1844	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₃₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
10	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₇	—	2	2	no	—
12	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#476	—	2	16	no	—
14	C₂×#106×C₃₇	C₂²	1	8	yes	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#102×C₃₇	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
18	C₃₇×#390	—	2	2	no	—
19	C₃₇×#390	C₂²	1	2	yes	—
20	C₃₇×#384	—	2	2	no	—
21	C₃₇×#384	C₂²	1	2	yes	—
22	C₃₇×#384	C₂²	1	1	yes	characteristic
23	C₃₇×#384	C₂²	1	2	yes	—
24	C₄³×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#336	C₂²	1	2	yes	—
26	C₂²×Q₈×C₃₇	—	2	2	no	—
27	C₂²×D₄×C₃₇	—	2	2	no	—
28	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₇	—	2	2	no	—
30	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₃₇	—	2	8	no	—
32	C₄⋊Q₈×C₃₇	—	4	8	no	—
33	#127×C₃₇	—	2	2	no	—
34	#127×C₃₇	Q₈	1	4	yes	complemented
35	C₄×D₄×C₃₇	—	2	8	no	—
36	C₄×D₄×C₃₇	—	4	8	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
45	C₂×C₄²×C₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₃₇	D₄	1	2	yes	—
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#106×C₃₇	—	2	16	no	—
51	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
52	#102×C₃₇	—	2	16	no	—
53	C₄×C₈×C₃₇	—	2	4	no	—
54	C₂⁴×C₃₇	—	2	2	no	—
55	C₂×Q₈×C₃₇	—	4	8	no	—
56	C₂×Q₈×C₃₇	—	4	4	no	—
57	C₂×D₄×C₃₇	—	2	8	no	—
58	C₂×D₄×C₃₇	—	2	4	no	—
59	C₂×D₄×C₃₇	—	2	2	no	—
60	C₂×D₄×C₃₇	—	2	4	no	—
61	C₂×D₄×C₃₇	—	4	4	no	—
62	C₂×D₄×C₃₇	—	4	4	no	—
63	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	4	no	—
66	C₂²×C₄×C₃₇	—	4	4	no	—
67	C₂²×C₄×C₃₇	—	4	8	no	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
76	C₄⋊C₄×C₃₇	—	2	8	no	—
77	C₄⋊C₄×C₃₇	—	2	4	no	—
78	C₄⋊C₄×C₃₇	—	4	8	no	—
79	C₄⋊C₄×C₃₇	—	4	4	no	—
80	C₄⋊C₄×C₃₇	—	4	4	no	—
81	C₄⋊C₄×C₃₇	—	4	8	no	—
82	C₂²⋊C₄×C₃₇	—	4	4	no	—
83	C₂²⋊C₄×C₃₇	—	4	8	no	—
84	C₄²×C₃₇	—	2	4	no	—
85	C₄²×C₃₇	—	2	2	no	—
86	C₄²×C₃₇	SD₁₆	1	8	yes	—
87	C₄²×C₃₇	C₂×D₄	1	4	yes	—
88	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₃₇	#41	1	2	yes	—
90	C₂³×C₃₇	—	2	4	no	—
91	C₂³×C₃₇	—	4	2	no	—
92	C₂³×C₃₇	—	4	2	no	—
93	C₂³×C₃₇	—	4	4	no	—
94	C₂³×C₃₇	—	4	4	no	—
95	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₃₇	—	8	8	no	—
97	D₄×C₃₇	—	2	8	no	—
98	D₄×C₃₇	—	4	4	no	—
99	D₄×C₃₇	—	4	16	no	—
100	C₂×C₄×C₃₇	—	2	2	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	4	no	—
106	C₂×C₄×C₃₇	—	4	8	no	—
107	C₂×C₄×C₃₇	—	8	8	no	—
108	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₃₇	#127	1	2	yes	—
110	C₂×C₄×C₃₇	#127	1	2	yes	—
111	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
112	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
113	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
114	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
119	C₈×C₃₇	—	2	8	no	—
120	#28509	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	—	4	4	no	—
122	C₂²×C₃₇	—	4	8	no	—
123	C₂²×C₃₇	—	8	2	no	—
124	C₂²×C₃₇	—	8	2	no	—
125	C₂²×C₃₇	—	8	4	no	—
126	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
127	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
128	C₂²×C₃₇	#536	1	1	yes	characteristic
129	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₃₇	—	2	2	no	—
132	C₄×C₃₇	—	2	4	no	—
133	C₄×C₃₇	—	8	2	no	—
134	C₄×C₃₇	—	8	4	no	—
135	C₄×C₃₇	#476	1	8	yes	complemented
136	C₄×C₃₇	#492	1	4	yes	—
137	C₄×C₃₇	#547	1	2	yes	—
138	C₄×C₃₇	D₄×Q₈	1	2	yes	—
139	C₂×#476	C₂×C₃₇	1	4	yes	complemented
140	#2868	C₂×C₃₇	1	1	yes	characteristic, complemented
141	C₄×#127	C₂×C₃₇	1	1	yes	characteristic
142	#2094	C₂×C₃₇	1	4	yes	complemented
143	#1957	C₂×C₃₇	1	2	yes	—
144	#1844	C₂×C₃₇	1	2	yes	complemented
145	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
146	C₂×C₃₇	—	8	4	no	—
147	C₂×C₃₇	#2542	1	1	yes	characteristic
148	C₂×C₃₇	C₂×#476	1	2	yes	—
149	C₂×C₃₇	C₂×#492	1	1	yes	characteristic
150	C₂×C₃₇	#3151	1	1	yes	characteristic
151	C₂×C₃₇	#3227	1	2	yes	—
152	C₂×C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
154	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
155	C₂×C₄×D₄	—	2	2	no	—
156	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
157	#476	—	2	16	no	—
158	C₂×#106	C₂²×C₃₇	1	8	yes	—
159	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
160	C₂×#102	C₂²×C₃₇	1	8	yes	—
161	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
162	#390	—	2	2	no	—
163	#390	C₂²×C₃₇	1	2	yes	—
164	#384	—	2	2	no	—
165	#384	C₂²×C₃₇	1	1	yes	characteristic
166	#384	C₂²×C₃₇	1	2	yes	—
167	#384	C₂²×C₃₇	1	2	yes	—
168	C₄³	C₂²×C₃₇	1	1	yes	characteristic
169	#336	C₂²×C₃₇	1	2	yes	—
170	C₃₇	#28509	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₈	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
176	C₄⋊Q₈	—	2	8	no	—
177	C₄⋊Q₈	—	4	8	no	—
178	#127	—	2	2	no	—
179	#127	Q₈×C₃₇	1	4	yes	complemented
180	C₄×D₄	—	2	8	no	—
181	C₄×D₄	—	4	8	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
187	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	D₄×C₃₇	1	2	yes	—
191	C₂×C₄²	D₄×C₃₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
195	#106	—	2	16	no	—
196	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
197	#102	—	2	16	no	—
198	C₄×C₈	—	2	4	no	—
199	C₂⁴	—	2	2	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×D₄	—	2	8	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	8	no	—
213	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	8	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	4	8	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	SD₁₆×C₃₇	1	8	yes	—
232	C₄²	C₂×D₄×C₃₇	1	4	yes	—
233	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
234	C₄²	#41×C₃₇	1	2	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
241	Q₈	—	8	8	no	—
242	D₄	—	2	8	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	16	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	8	8	no	—
253	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
254	C₂×C₄	#127×C₃₇	1	2	yes	—
255	C₂×C₄	#127×C₃₇	1	2	yes	—
256	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
257	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
258	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
259	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
260	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
261	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
263	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₂²	—	4	4	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	4	no	—
270	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
271	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
272	C₂²	C₃₇×#536	1	1	yes	characteristic
273	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
274	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	4	no	—
277	C₄	—	8	2	no	—
278	C₄	—	8	4	no	—
279	C₄	C₃₇×#476	1	8	yes	complemented
280	C₄	C₃₇×#492	1	4	yes	—
281	C₄	C₃₇×#547	1	2	yes	—
282	C₄	D₄×Q₈×C₃₇	1	2	yes	—
283	C₂	—	8	4	no	—
284	C₂	C₃₇×#2542	1	1	yes	characteristic
285	C₂	C₂×C₃₇×#476	1	2	yes	—
286	C₂	C₂×C₃₇×#492	1	1	yes	characteristic
287	C₂	C₃₇×#3151	1	1	yes	characteristic
288	C₂	C₃₇×#3227	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#476	C₂	semidirect
2. C₃₇×#2868	C₂	semidirect
4. C₃₇×#2094	C₂	semidirect
6. C₃₇×#1844	C₂	semidirect
7. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
34. #127×C₃₇	Q₈	semidirect
51. C₄⋊C₈×C₃₇	D₄	semidirect
120. #28509	C₃₇	direct (N × H)
135. C₄×C₃₇	#476	semidirect
139. C₂×#476	C₂×C₃₇	semidirect
140. #2868	C₂×C₃₇	semidirect
142. #2094	C₂×C₃₇	semidirect
144. #1844	C₂×C₃₇	semidirect
145. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
170. C₃₇	#28509	direct (N × H)
179. #127	Q₈×C₃₇	semidirect
196. C₄⋊C₈	D₄×C₃₇	semidirect
279. C₄	C₃₇×#476	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#476	2
C₃₇×#2868	2
C₄×#127×C₃₇	2
C₃₇×#2094	2
C₃₇×#1957	2
C₃₇×#1844	2
C₄×C₄⋊C₈×C₃₇	2
#28509	37

Subgroups — C37×#28509 #22649762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28509 #22649762