Subgroups — C₃₇×#33793 #22655046

Order 9472.

538 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2254	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2240	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2233	C₂	1	4	yes	maximal, complemented
6	C₃₇×#2226	C₂	1	4	yes	maximal, complemented
7	C₃₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#532	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₃₇	—	2	1	no	—
10	C₃₇×#500	—	2	1	no	—
11	C₃₇×#500	—	2	1	no	—
12	C₃₇×#500	C₂²	1	2	yes	—
13	C₃₇×#500	C₂²	1	2	yes	—
14	C₃₇×#500	C₂²	1	1	yes	characteristic
15	C₃₇×#493	C₂²	1	1	yes	characteristic
16	C₃₇×#486	—	2	1	no	—
17	C₃₇×#466	—	2	1	no	—
18	C₃₇×#466	C₂²	1	2	yes	—
19	C₄×D₈×C₃₇	—	2	1	no	—
20	C₄×D₈×C₃₇	C₂²	1	2	yes	—
21	C₂×#107×C₃₇	—	2	1	no	—
22	C₂×#107×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#367	C₂²	1	8	yes	—
25	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₃₇×#357	C₂²	1	8	yes	—
27	C₃₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₇	C₂²	1	2	yes	—
29	C₂×#41×C₃₇	—	4	1	no	—
30	C₂²×Q₈×C₃₇	—	4	1	no	—
31	C₂×D₈×C₃₇	—	2	1	no	—
32	C₂×D₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	—	2	2	no	—
37	C₄⋊Q₈×C₃₇	—	2	1	no	—
38	C₄⋊Q₈×C₃₇	—	4	1	no	—
39	C₄⋊Q₈×C₃₇	—	4	1	no	—
40	#127×C₃₇	—	2	1	no	—
41	#124×C₃₇	—	4	1	no	—
42	#121×C₃₇	—	2	2	no	—
43	#121×C₃₇	—	4	1	no	—
44	C₄×D₄×C₃₇	—	2	2	no	—
45	C₄×D₄×C₃₇	—	4	1	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	4	1	no	—
48	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₃₇	C₂³	1	4	yes	—
50	#107×C₃₇	—	2	2	no	—
51	#107×C₃₇	—	2	2	no	—
52	#107×C₃₇	—	2	2	no	—
53	#107×C₃₇	—	2	1	no	—
54	#107×C₃₇	C₂³	1	4	yes	—
55	#107×C₃₇	C₂³	1	2	yes	—
56	#102×C₃₇	—	2	2	no	—
57	#102×C₃₇	—	2	2	no	—
58	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
62	#41×C₃₇	—	8	4	no	—
63	C₂×Q₈×C₃₇	—	4	1	no	—
64	C₂×Q₈×C₃₇	—	4	2	no	—
65	C₂×Q₈×C₃₇	—	4	2	no	—
66	C₂×Q₈×C₃₇	—	8	4	no	—
67	C₂×D₄×C₃₇	—	2	2	no	—
68	C₂×D₄×C₃₇	—	4	2	no	—
69	C₂×D₄×C₃₇	—	4	1	no	—
70	C₂×D₄×C₃₇	—	4	1	no	—
71	C₂²×C₄×C₃₇	—	4	2	no	—
72	C₂²×C₄×C₃₇	—	4	2	no	—
73	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	D₈×C₃₇	—	2	4	no	—
77	D₈×C₃₇	—	4	2	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	2	no	—
80	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₃₇	—	2	2	no	—
85	C₄⋊C₄×C₃₇	—	2	4	no	—
86	C₄⋊C₄×C₃₇	—	4	2	no	—
87	C₄⋊C₄×C₃₇	—	4	2	no	—
88	C₄⋊C₄×C₃₇	—	4	2	no	—
89	C₄⋊C₄×C₃₇	—	4	1	no	—
90	C₂²⋊C₄×C₃₇	—	4	2	no	—
91	C₂²⋊C₄×C₃₇	—	4	2	no	—
92	C₄²×C₃₇	D₈	1	1	yes	characteristic
93	C₄²×C₃₇	D₈	1	1	yes	characteristic
94	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₃₇	—	2	4	no	—
97	C₂³×C₃₇	—	4	2	no	—
98	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₃₇	—	8	4	no	—
100	Q₈×C₃₇	—	8	2	no	—
101	Q₈×C₃₇	—	8	4	no	—
102	D₄×C₃₇	—	4	2	no	—
103	D₄×C₃₇	—	4	2	no	—
104	D₄×C₃₇	—	8	4	no	—
105	D₄×C₃₇	—	8	2	no	—
106	D₄×C₃₇	—	8	2	no	—
107	C₂×C₄×C₃₇	—	2	1	no	—
108	C₂×C₄×C₃₇	—	2	1	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	8	4	no	—
113	C₂×C₄×C₃₇	—	8	4	no	—
114	C₂×C₄×C₃₇	SD₃₂	1	4	yes	—
115	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₇	—	2	1	no	—
122	C₈×C₃₇	—	2	1	no	—
123	C₈×C₃₇	#137	1	2	yes	—
124	C₈×C₃₇	2₋¹⁺⁴	1	2	yes	—
125	#33793	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₃₇	—	2	1	no	—
127	C₂²×C₃₇	—	2	1	no	—
128	C₂²×C₃₇	—	8	2	no	—
129	C₂²×C₃₇	—	8	2	no	—
130	C₂²×C₃₇	—	8	2	no	—
131	C₂²×C₃₇	C₂×SD₃₂	1	2	yes	—
132	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₃₇	—	2	1	no	—
134	C₄×C₃₇	—	2	1	no	—
135	C₄×C₃₇	—	8	2	no	—
136	C₄×C₃₇	—	8	4	no	—
137	C₄×C₃₇	C₂×SD₃₂	1	2	yes	—
138	C₄×C₃₇	C₂×SD₃₂	1	2	yes	—
139	C₄×C₃₇	#509	1	2	yes	—
140	C₄×C₃₇	#536	1	1	yes	characteristic
141	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
142	#3169	C₂×C₃₇	1	1	yes	characteristic
143	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
144	#2254	C₂×C₃₇	1	2	yes	complemented
145	#2240	C₂×C₃₇	1	2	yes	complemented
146	#2233	C₂×C₃₇	1	4	yes	complemented
147	#2226	C₂×C₃₇	1	4	yes	complemented
148	#2162	C₂×C₃₇	1	1	yes	characteristic, complemented
149	C₂×C₃₇	—	2	2	no	—
150	C₂×C₃₇	—	16	2	no	—
151	C₂×C₃₇	#3226	1	1	yes	characteristic
152	C₂×C₃₇	C₂²×SD₃₂	1	1	yes	characteristic
153	C₂×C₃₇	C₂×#509	1	1	yes	characteristic
154	#532	—	2	1	no	—
155	C₂×C₄⋊Q₈	—	2	1	no	—
156	#500	—	2	1	no	—
157	#500	—	2	1	no	—
158	#500	C₂²×C₃₇	1	2	yes	—
159	#500	C₂²×C₃₇	1	2	yes	—
160	#500	C₂²×C₃₇	1	1	yes	characteristic
161	#493	C₂²×C₃₇	1	1	yes	characteristic
162	#486	—	2	1	no	—
163	#466	—	2	1	no	—
164	#466	C₂²×C₃₇	1	2	yes	—
165	C₄×D₈	—	2	1	no	—
166	C₄×D₈	C₂²×C₃₇	1	2	yes	—
167	C₂×#107	—	2	1	no	—
168	C₂×#107	C₂²×C₃₇	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
170	#367	C₂²×C₃₇	1	8	yes	—
171	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
172	#357	C₂²×C₃₇	1	8	yes	—
173	C₂²⋊C₁₆	C₂²×C₃₇	1	2	yes	—
174	C₄×C₁₆	C₂²×C₃₇	1	2	yes	—
175	C₃₇	#33793	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×#41	—	4	1	no	—
177	C₂²×Q₈	—	4	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	C₂³×C₃₇	1	2	yes	—
180	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	4	1	no	—
186	C₄⋊Q₈	—	4	1	no	—
187	#127	—	2	1	no	—
188	#124	—	4	1	no	—
189	#121	—	2	2	no	—
190	#121	—	4	1	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	4	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	4	1	no	—
195	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
196	C₂×C₁₆	C₂³×C₃₇	1	4	yes	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	2	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂³×C₃₇	1	4	yes	—
202	#107	C₂³×C₃₇	1	2	yes	—
203	#102	—	2	2	no	—
204	#102	—	2	2	no	—
205	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
206	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
209	#41	—	8	4	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	8	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
221	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
223	D₈	—	2	4	no	—
224	D₈	—	4	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	D₈×C₃₇	1	1	yes	characteristic
240	C₄²	D₈×C₃₇	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
243	C₁₆	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
246	Q₈	—	8	4	no	—
247	Q₈	—	8	2	no	—
248	Q₈	—	8	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	SD₃₂×C₃₇	1	4	yes	—
262	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	#137×C₃₇	1	2	yes	—
271	C₈	2₋¹⁺⁴×C₃₇	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂×SD₃₂×C₃₇	1	2	yes	—
278	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂×SD₃₂×C₃₇	1	2	yes	—
284	C₄	C₂×SD₃₂×C₃₇	1	2	yes	—
285	C₄	C₃₇×#509	1	2	yes	—
286	C₄	C₃₇×#536	1	1	yes	characteristic
287	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₃₇×#3226	1	1	yes	characteristic
291	C₂	C₂²×SD₃₂×C₃₇	1	1	yes	characteristic
292	C₂	C₂×C₃₇×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
prime radical	C₃₇×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#500	C₂	semidirect
3. C₃₇×#2254	C₂	semidirect
4. C₃₇×#2240	C₂	semidirect
5. C₃₇×#2233	C₂	semidirect
6. C₃₇×#2226	C₂	semidirect
7. C₃₇×#2162	C₂	semidirect
125. #33793	C₃₇	direct (N × H)
143. C₂×#500	C₂×C₃₇	semidirect
144. #2254	C₂×C₃₇	semidirect
145. #2240	C₂×C₃₇	semidirect
146. #2233	C₂×C₃₇	semidirect
147. #2226	C₂×C₃₇	semidirect
148. #2162	C₂×C₃₇	semidirect
175. C₃₇	#33793	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×#3169	2
C₂×C₃₇×#500	2
C₃₇×#2254	2
C₃₇×#2240	2
C₃₇×#2233	2
C₃₇×#2226	2
C₃₇×#2162	2
#33793	37

Subgroups — C37×#33793 #22655046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33793 #22655046