Subgroups — C₃₇×#8672 #22629925

Order 9472.

386 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×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#1489	C₂	1	2	yes	maximal
5	C₃₇×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₃₇	C₂²	1	1	yes	characteristic
7	C₃₇×#446	C₄	1	1	yes	characteristic
8	C₃₇×#446	C₄	1	1	yes	characteristic
9	C₃₇×#436	C₄	1	1	yes	characteristic
10	C₃₇×#436	C₄	1	1	yes	characteristic
11	C₃₇×#435	—	2	1	no	—
12	C₃₇×#435	—	2	1	no	—
13	C₃₇×#431	—	2	1	no	—
14	C₃₇×#431	C₂²	1	1	yes	characteristic
15	C₂×#108×C₃₇	—	2	1	no	—
16	C₃₇×#423	C₂²	1	2	yes	—
17	C₄×M₁₆×C₃₇	—	2	1	no	—
18	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#341	—	2	1	no	—
21	C₃₇×#334	—	2	1	no	—
22	C₃₇×#334	—	2	1	no	—
23	C₃₇×Q₈⋊C₈	—	2	2	no	—
24	C₃₇×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₃₇	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₃₇	—	2	1	no	—
27	C₂×M₁₆×C₃₇	—	2	1	no	—
28	C₂×M₁₆×C₃₇	—	2	2	no	—
29	C₂×M₁₆×C₃₇	—	2	2	no	—
30	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₃₇	D₄	1	1	yes	characteristic
34	C₄×D₄×C₃₇	—	2	1	no	—
35	C₄×D₄×C₃₇	—	2	1	no	—
36	C₄×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₃₇	D₄	1	1	yes	characteristic
38	#117×C₃₇	—	2	1	no	—
39	#117×C₃₇	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	—	2	1	no	—
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	#108×C₃₇	—	2	4	no	—
43	C₄⋊C₈×C₃₇	—	2	1	no	—
44	C₄⋊C₈×C₃₇	—	2	1	no	—
45	C₄⋊C₈×C₃₇	—	2	2	no	—
46	C₄⋊C₈×C₃₇	—	2	2	no	—
47	C₄⋊C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊C₈×C₃₇	—	2	1	no	—
50	C₂²⋊C₈×C₃₇	—	2	1	no	—
51	C₂²⋊C₈×C₃₇	—	2	1	no	—
52	#97×C₃₇	—	2	1	no	—
53	#97×C₃₇	—	2	1	no	—
54	#97×C₃₇	—	2	2	no	—
55	C₄×C₈×C₃₇	—	2	1	no	—
56	C₄×C₈×C₃₇	—	2	2	no	—
57	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
61	#41×C₃₇	—	4	2	no	—
62	C₂×Q₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₃₇	—	2	1	no	—
64	C₂×D₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	—	2	1	no	—
66	C₂²×C₄×C₃₇	—	2	1	no	—
67	C₂²×C₄×C₃₇	—	2	1	no	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
71	M₁₆×C₃₇	—	2	4	no	—
72	M₁₆×C₃₇	—	4	2	no	—
73	M₁₆×C₃₇	—	4	2	no	—
74	M₁₆×C₃₇	—	4	2	no	—
75	M₁₆×C₃₇	—	8	2	no	—
76	C₂×C₈×C₃₇	—	2	2	no	—
77	C₂×C₈×C₃₇	—	2	1	no	—
78	C₂×C₈×C₃₇	—	2	1	no	—
79	C₂×C₈×C₃₇	—	4	1	no	—
80	C₂×C₈×C₃₇	—	4	2	no	—
81	C₂×C₈×C₃₇	—	4	2	no	—
82	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₃₇	—	2	1	no	—
87	C₄⋊C₄×C₃₇	—	2	1	no	—
88	C₄⋊C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₃₇	—	2	1	no	—
91	C₂²⋊C₄×C₃₇	—	2	1	no	—
92	C₂²⋊C₄×C₃₇	—	2	1	no	—
93	C₄²×C₃₇	—	2	1	no	—
94	C₄²×C₃₇	—	2	1	no	—
95	C₄²×C₃₇	—	2	1	no	—
96	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₃₇	#41	1	1	yes	characteristic
100	C₂³×C₃₇	—	2	1	no	—
101	C₂³×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₃₇	—	2	2	no	—
103	D₄×C₃₇	—	2	2	no	—
104	D₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	2	1	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	1	no	—
110	C₂×C₄×C₃₇	—	2	1	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	1	no	—
114	C₂×C₄×C₃₇	—	4	1	no	—
115	C₂×C₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₃₇	—	2	1	no	—
123	C₈×C₃₇	—	2	1	no	—
124	C₈×C₃₇	—	2	1	no	—
125	C₈×C₃₇	—	4	4	no	—
126	C₈×C₃₇	—	4	1	no	—
127	C₈×C₃₇	—	4	1	no	—
128	C₈×C₃₇	—	8	2	no	—
129	C₈×C₃₇	#104	1	2	yes	—
130	#8672	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₃₇	—	2	1	no	—
132	C₂²×C₃₇	—	2	1	no	—
133	C₂²×C₃₇	—	2	1	no	—
134	C₂²×C₃₇	—	4	1	no	—
135	C₂²×C₃₇	—	4	1	no	—
136	C₂²×C₃₇	—	4	1	no	—
137	C₂²×C₃₇	#380	1	1	yes	characteristic
138	C₄×C₃₇	—	2	1	no	—
139	C₄×C₃₇	—	2	1	no	—
140	C₄×C₃₇	—	2	1	no	—
141	C₄×C₃₇	—	4	1	no	—
142	C₄×C₃₇	—	4	1	no	—
143	C₄×C₃₇	—	4	1	no	—
144	C₄×C₃₇	C₂×#104	1	1	yes	characteristic
145	C₄×C₃₇	#421	1	1	yes	characteristic
146	C₄×C₃₇	#430	1	2	yes	—
147	C₄×C₃₇	#471	1	2	yes	—
148	#2965	C₂×C₃₇	1	1	yes	characteristic
149	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
150	#1566	C₂×C₃₇	1	1	yes	characteristic, complemented
151	#1489	C₂×C₃₇	1	2	yes	—
152	#1299	C₂×C₃₇	1	2	yes	complemented
153	C₂×C₃₇	—	4	1	no	—
154	C₂×C₃₇	—	8	1	no	—
155	C₂×C₃₇	#1806	1	1	yes	characteristic
156	C₂×C₃₇	#1811	1	1	yes	characteristic
157	C₂×C₃₇	#1815	1	1	yes	characteristic
158	C₄×#41	C₂²×C₃₇	1	1	yes	characteristic
159	#446	C₄×C₃₇	1	1	yes	characteristic
160	#446	C₄×C₃₇	1	1	yes	characteristic
161	#436	C₄×C₃₇	1	1	yes	characteristic
162	#436	C₄×C₃₇	1	1	yes	characteristic
163	#435	—	2	1	no	—
164	#435	—	2	1	no	—
165	#431	—	2	1	no	—
166	#431	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×#108	—	2	1	no	—
168	#423	C₂²×C₃₇	1	2	yes	—
169	C₄×M₁₆	—	2	1	no	—
170	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
172	#341	—	2	1	no	—
173	#334	—	2	1	no	—
174	#334	—	2	1	no	—
175	Q₈⋊C₈	—	2	2	no	—
176	D₄⋊C₈	—	2	2	no	—
177	C₃₇	#8672	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#41	Q₈×C₃₇	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
185	C₄×Q₈	C₂×C₄×C₃₇	1	1	yes	characteristic
186	C₄×Q₈	D₄×C₃₇	1	1	yes	characteristic
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	C₂×C₄×C₃₇	1	1	yes	characteristic
190	C₄×D₄	D₄×C₃₇	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	#117	Q₈×C₃₇	1	1	yes	characteristic
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
195	#108	—	2	4	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	2	1	no	—
207	#97	—	2	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
211	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
212	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
213	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
214	#41	—	4	2	no	—
215	C₂×Q₈	C₄⋊C₄×C₃₇	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
223	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	8	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
242	C₄⋊C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
252	C₄²	#41×C₃₇	1	1	yes	characteristic
253	C₂³	—	2	1	no	—
254	C₂³	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
271	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
272	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
273	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
274	C₂×C₄	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	4	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	8	2	no	—
282	C₈	#104×C₃₇	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₃₇×#380	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#104×C₃₇	1	1	yes	characteristic
297	C₄	C₃₇×#421	1	1	yes	characteristic
298	C₄	C₃₇×#430	1	2	yes	—
299	C₄	C₃₇×#471	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₃₇×#1806	1	1	yes	characteristic
303	C₂	C₃₇×#1811	1	1	yes	characteristic
304	C₂	C₃₇×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₇	#121
commutator	C₄²	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
socle	C₂²×C₃₇	#380
prime radical	C₂×#41×C₃₇	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#108×C₃₇	C₂	semidirect
3. C₃₇×#1566	C₂	semidirect
5. C₃₇×#1299	C₂	semidirect
130. #8672	C₃₇	direct (N × H)
149. C₄×#108	C₂×C₃₇	semidirect
150. #1566	C₂×C₃₇	semidirect
152. #1299	C₂×C₃₇	semidirect
177. C₃₇	#8672	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2965	2
C₄×#108×C₃₇	2
C₃₇×#1566	2
C₃₇×#1489	2
C₃₇×#1299	2
#8672	37

Subgroups — C37×#8672 #22629925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8672 #22629925