Subgroups — C₃₇×#8648 #22629901

Order 9472.

526 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#1802	C₂	1	2	yes	maximal
4	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#1296	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₇	—	2	1	no	—
7	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₃₇	—	2	2	no	—
9	C₈×D₄×C₃₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#102×C₃₇	—	2	2	no	—
13	C₂×C₂²⋊C₈×C₃₇	—	2	1	no	—
14	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#384	C₂²	1	2	yes	—
16	C₃₇×#340	—	2	1	no	—
17	C₃₇×#340	—	2	1	no	—
18	C₃₇×#340	—	2	2	no	—
19	C₃₇×#335	—	2	2	no	—
20	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₇	—	2	1	no	—
22	C₂²×C₈×C₃₇	—	2	1	no	—
23	C₂²×C₈×C₃₇	—	2	4	no	—
24	C₂²×C₈×C₃₇	—	2	2	no	—
25	C₂²×C₈×C₃₇	—	4	4	no	—
26	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
28	C₄×D₄×C₃₇	—	2	1	no	—
29	C₄×D₄×C₃₇	—	2	1	no	—
30	C₄×D₄×C₃₇	C₂×C₄	1	2	yes	—
31	C₄×D₄×C₃₇	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
37	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₃₇	—	2	1	no	—
39	C₄⋊C₈×C₃₇	—	2	2	no	—
40	C₄⋊C₈×C₃₇	—	2	2	no	—
41	C₄⋊C₈×C₃₇	C₂×C₄	1	2	yes	—
42	#102×C₃₇	—	4	4	no	—
43	C₂²⋊C₈×C₃₇	—	2	4	no	—
44	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₃₇	D₄	1	2	yes	—
46	C₂⁴×C₃₇	—	2	1	no	—
47	C₂×D₄×C₃₇	—	2	1	no	—
48	C₂×D₄×C₃₇	—	2	1	no	—
49	C₂×D₄×C₃₇	—	2	1	no	—
50	C₂×D₄×C₃₇	C₄⋊C₄	1	2	yes	—
51	C₂×D₄×C₃₇	D₈	1	2	yes	—
52	C₂×D₄×C₃₇	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₃₇	—	2	1	no	—
54	C₂²×C₄×C₃₇	—	2	2	no	—
55	C₂²×C₄×C₃₇	—	2	2	no	—
56	C₂²×C₄×C₃₇	—	4	2	no	—
57	C₂²×C₄×C₃₇	—	4	2	no	—
58	C₂²×C₄×C₃₇	—	4	2	no	—
59	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
62	C₂×C₈×C₃₇	—	2	2	no	—
63	C₂×C₈×C₃₇	—	2	2	no	—
64	C₂×C₈×C₃₇	—	2	2	no	—
65	C₂×C₈×C₃₇	—	4	8	no	—
66	C₂×C₈×C₃₇	—	4	2	no	—
67	C₂×C₈×C₃₇	—	4	2	no	—
68	C₂×C₈×C₃₇	—	4	2	no	—
69	C₂×C₈×C₃₇	—	4	2	no	—
70	C₂×C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
72	C₄⋊C₄×C₃₇	—	2	1	no	—
73	C₄⋊C₄×C₃₇	—	4	2	no	—
74	C₄⋊C₄×C₃₇	—	4	2	no	—
75	C₄⋊C₄×C₃₇	—	4	2	no	—
76	C₄⋊C₄×C₃₇	—	4	2	no	—
77	C₄⋊C₄×C₃₇	—	8	2	no	—
78	C₄⋊C₄×C₃₇	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₃₇	—	2	2	no	—
80	C₂²⋊C₄×C₃₇	—	2	2	no	—
81	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₇	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₃₇	#41	1	1	yes	characteristic
85	C₂³×C₃₇	—	2	2	no	—
86	C₂³×C₃₇	—	2	2	no	—
87	C₂³×C₃₇	—	2	2	no	—
88	C₂³×C₃₇	—	4	2	no	—
89	C₂³×C₃₇	—	4	2	no	—
90	C₂³×C₃₇	#121	1	1	yes	characteristic
91	D₄×C₃₇	—	2	2	no	—
92	D₄×C₃₇	—	2	2	no	—
93	D₄×C₃₇	—	2	2	no	—
94	D₄×C₃₇	#107	1	4	yes	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	4	2	no	—
98	C₂×C₄×C₃₇	—	4	2	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	8	2	no	—
102	C₂×C₄×C₃₇	—	8	2	no	—
103	C₂×C₄×C₃₇	—	8	2	no	—
104	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₃₇	#135	1	2	yes	—
113	C₂×C₄×C₃₇	#135	1	1	yes	characteristic
114	C₂×C₄×C₃₇	#135	1	1	yes	characteristic
115	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₈×C₃₇	—	2	2	no	—
117	C₈×C₃₇	—	2	2	no	—
118	C₈×C₃₇	—	4	2	no	—
119	C₈×C₃₇	—	8	2	no	—
120	#8648	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	—	2	4	no	—
122	C₂²×C₃₇	—	4	2	no	—
123	C₂²×C₃₇	—	4	2	no	—
124	C₂²×C₃₇	—	4	4	no	—
125	C₂²×C₃₇	—	4	4	no	—
126	C₂²×C₃₇	#380	1	1	yes	characteristic
127	C₂²×C₃₇	#463	1	2	yes	—
128	C₂²×C₃₇	#466	1	1	yes	characteristic
129	C₂²×C₃₇	#467	1	1	yes	characteristic
130	C₂²×C₃₇	#471	1	2	yes	—
131	C₄×C₃₇	—	2	1	no	—
132	C₄×C₃₇	—	2	1	no	—
133	C₄×C₃₇	—	4	2	no	—
134	C₄×C₃₇	—	16	2	no	—
135	C₄×C₃₇	#416	1	1	yes	characteristic
136	C₄×C₃₇	#418	1	1	yes	characteristic
137	C₄×C₃₇	C₂×#107	1	1	yes	characteristic
138	C₄×C₃₇	#427	1	1	yes	characteristic
139	C₄×C₃₇	#466	1	1	yes	characteristic
140	C₄×C₃₇	#467	1	1	yes	characteristic
141	C₄×C₃₇	#478	1	2	yes	—
142	C₂×C₈×D₄	C₂×C₃₇	1	1	yes	characteristic
143	#1834	C₂×C₃₇	1	1	yes	characteristic, complemented
144	#1802	C₂×C₃₇	1	2	yes	—
145	C₂×#335	C₂×C₃₇	1	1	yes	characteristic, complemented
146	#1296	C₂×C₃₇	1	2	yes	complemented
147	C₂×C₃₇	—	4	4	no	—
148	C₂×C₃₇	#1681	1	2	yes	—
149	C₂×C₃₇	#1683	1	2	yes	—
150	C₂×C₃₇	#1804	1	1	yes	characteristic
151	C₂×C₃₇	#1808	1	1	yes	characteristic
152	C₂×C₃₇	#1815	1	1	yes	characteristic
153	C₂³×C₈	—	2	1	no	—
154	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
155	C₈×D₄	—	2	2	no	—
156	C₈×D₄	C₄×C₃₇	1	4	yes	complemented
157	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×#102	—	2	2	no	—
160	C₂×C₂²⋊C₈	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
162	#384	C₂²×C₃₇	1	2	yes	—
163	#340	—	2	1	no	—
164	#340	—	2	1	no	—
165	#340	—	2	2	no	—
166	#335	—	2	2	no	—
167	C₃₇	#8648	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
169	C₂³×C₄	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	4	4	no	—
174	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	C₂×C₄×C₃₇	1	2	yes	—
179	C₄×D₄	Q₈×C₃₇	1	2	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	—	2	1	no	—
185	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	C₂×C₄×C₃₇	1	2	yes	—
190	#102	—	4	4	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
193	C₄×C₈	D₄×C₃₇	1	2	yes	—
194	C₂⁴	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	C₄⋊C₄×C₃₇	1	2	yes	—
199	C₂×D₄	D₈×C₃₇	1	2	yes	—
200	C₂×D₄	Q₁₆×C₃₇	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₃₇	1	2	yes	—
219	C₂×C₈	C₂²⋊C₄×C₃₇	1	2	yes	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	8	2	no	—
226	C₄⋊C₄	C₄⋊C₄×C₃₇	1	2	yes	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
231	C₄²	C₂×Q₈×C₃₇	1	1	yes	characteristic
232	C₄²	#41×C₃₇	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	#121×C₃₇	1	1	yes	characteristic
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	#107×C₃₇	1	4	yes	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
254	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
255	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₁₆×C₃₇	1	1	yes	characteristic
260	C₂×C₄	#135×C₃₇	1	2	yes	—
261	C₂×C₄	#135×C₃₇	1	1	yes	characteristic
262	C₂×C₄	#135×C₃₇	1	1	yes	characteristic
263	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	8	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₃₇×#380	1	1	yes	characteristic
274	C₂²	C₃₇×#463	1	2	yes	—
275	C₂²	C₃₇×#466	1	1	yes	characteristic
276	C₂²	C₃₇×#467	1	1	yes	characteristic
277	C₂²	C₃₇×#471	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	16	2	no	—
282	C₄	C₃₇×#416	1	1	yes	characteristic
283	C₄	C₃₇×#418	1	1	yes	characteristic
284	C₄	C₂×#107×C₃₇	1	1	yes	characteristic
285	C₄	C₃₇×#427	1	1	yes	characteristic
286	C₄	C₃₇×#466	1	1	yes	characteristic
287	C₄	C₃₇×#467	1	1	yes	characteristic
288	C₄	C₃₇×#478	1	2	yes	—
289	C₂	—	4	4	no	—
290	C₂	C₃₇×#1681	1	2	yes	—
291	C₂	C₃₇×#1683	1	2	yes	—
292	C₂	C₃₇×#1804	1	1	yes	characteristic
293	C₂	C₃₇×#1808	1	1	yes	characteristic
294	C₂	C₃₇×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₇	#121
commutator	C₄²	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
prime radical	C₂²×D₄×C₃₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#1834	C₂	semidirect
4. C₂×C₃₇×#335	C₂	semidirect
5. C₃₇×#1296	C₂	semidirect
9. C₈×D₄×C₃₇	C₄	semidirect
120. #8648	C₃₇	direct (N × H)
143. #1834	C₂×C₃₇	semidirect
145. C₂×#335	C₂×C₃₇	semidirect
146. #1296	C₂×C₃₇	semidirect
156. C₈×D₄	C₄×C₃₇	semidirect
167. C₃₇	#8648	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₇	2
C₃₇×#1834	2
C₃₇×#1802	2
C₂×C₃₇×#335	2
C₃₇×#1296	2
#8648	37

Subgroups — C37×#8648 #22629901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8648 #22629901