Subgroups — C₃₇×#28531 #22649784

Order 9472.

854 conjugacy classes of subgroups.

The subgroup lattice has 300 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
2	C₂×C₃₇×#477	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2567	C₂	1	2	yes	maximal
5	C₃₇×#2095	C₂	1	4	yes	maximal
6	C₃₇×#1969	C₂	1	2	yes	maximal
7	C₃₇×#1845	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#500	—	2	4	no	—
13	C₃₇×#477	—	2	8	no	—
14	C₄×Q₁₆×C₃₇	—	2	4	no	—
15	C₂×#107×C₃₇	C₂²	1	2	yes	—
16	C₂×#107×C₃₇	C₂²	1	4	yes	—
17	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₃₇	C₂²	1	2	yes	—
21	C₂×#103×C₃₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#395	—	2	4	no	—
25	C₃₇×#391	C₂²	1	4	yes	—
26	C₃₇×#389	—	2	4	no	—
27	C₃₇×#384	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	—	2	2	no	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂×Q₁₆×C₃₇	—	2	2	no	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	complemented
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₇	—	2	2	no	—
49	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	#107×C₃₇	—	2	2	no	—
52	#107×C₃₇	—	2	8	no	—
53	#107×C₃₇	—	2	4	no	—
54	#107×C₃₇	—	2	4	no	—
55	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
56	#103×C₃₇	—	2	4	no	—
57	#103×C₃₇	—	2	8	no	—
58	C₄×C₈×C₃₇	—	2	4	no	—
59	C₄×C₈×C₃₇	D₄	1	4	yes	—
60	#95×C₃₇	—	2	8	no	—
61	#95×C₃₇	D₄	1	2	yes	—
62	C₂×Q₈×C₃₇	—	2	2	no	—
63	C₂×Q₈×C₃₇	—	2	2	no	—
64	C₂×Q₈×C₃₇	—	2	2	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	4	4	no	—
68	C₂×Q₈×C₃₇	—	8	4	no	—
69	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	4	no	—
72	C₂²×C₄×C₃₇	—	2	2	no	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₃₇	—	2	8	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	1	no	—
80	C₂×C₈×C₃₇	—	2	1	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₃₇	#41	1	2	yes	—
87	C₄⋊C₄×C₃₇	—	2	2	no	—
88	C₄⋊C₄×C₃₇	—	2	4	no	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	4	no	—
91	C₄⋊C₄×C₃₇	—	4	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	8	8	no	—
94	C₄⋊C₄×C₃₇	Q₁₆	1	8	yes	complemented
95	C₄²×C₃₇	—	2	2	no	—
96	C₄²×C₃₇	—	2	2	no	—
97	C₄²×C₃₇	—	4	4	no	—
98	C₄²×C₃₇	C₂×D₄	1	2	yes	—
99	C₄²×C₃₇	C₂×D₄	1	2	yes	—
100	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₃₇	—	2	4	no	—
102	Q₈×C₃₇	—	2	4	no	—
103	Q₈×C₃₇	—	4	2	no	—
104	Q₈×C₃₇	—	4	2	no	—
105	Q₈×C₃₇	—	8	8	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	—	2	4	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	4	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	4	no	—
114	C₂×C₄×C₃₇	—	4	4	no	—
115	C₂×C₄×C₃₇	—	4	8	no	—
116	C₂×C₄×C₃₇	#127	1	2	yes	—
117	C₂×C₄×C₃₇	#127	1	2	yes	—
118	C₂×C₄×C₃₇	C₂×Q₁₆	1	8	yes	—
119	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₈×C₃₇	—	2	2	no	—
127	C₈×C₃₇	—	2	4	no	—
128	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
129	#28531	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
131	C₂²×C₃₇	#547	1	1	yes	characteristic
132	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₃₇	C₂²×Q₁₆	1	2	yes	—
134	C₂²×C₃₇	#576	1	2	yes	—
135	C₄×C₃₇	—	2	8	no	—
136	C₄×C₃₇	—	2	2	no	—
137	C₄×C₃₇	—	4	4	no	—
138	C₄×C₃₇	—	8	2	no	—
139	C₄×C₃₇	—	8	4	no	—
140	C₄×C₃₇	#494	1	4	yes	—
141	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₄×C₃₇	#536	1	1	yes	characteristic
143	C₄×C₃₇	#547	1	1	yes	characteristic
144	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
145	C₂×#500	C₂×C₃₇	1	1	yes	characteristic
146	C₂×#477	C₂×C₃₇	1	2	yes	—
147	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
148	#2567	C₂×C₃₇	1	2	yes	—
149	#2095	C₂×C₃₇	1	4	yes	—
150	#1969	C₂×C₃₇	1	2	yes	—
151	#1845	C₂×C₃₇	1	2	yes	—
152	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
153	C₂×C₃₇	#2542	1	1	yes	characteristic
154	C₂×C₃₇	C₂×#494	1	1	yes	characteristic
155	C₂×C₃₇	#3155	1	1	yes	characteristic
156	C₂×C₃₇	#3338	1	2	yes	—
157	C₂×C₃₇	Q₈×Q₁₆	1	2	yes	—
158	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
160	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
161	#500	—	2	4	no	—
162	#477	—	2	8	no	—
163	C₄×Q₁₆	—	2	4	no	—
164	C₂×#107	C₂²×C₃₇	1	4	yes	—
165	C₂×#107	C₂²×C₃₇	1	2	yes	—
166	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×#107	C₂²×C₃₇	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×#103	C₂²×C₃₇	1	4	yes	—
170	C₂×#103	C₂²×C₃₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
173	#395	—	2	4	no	—
174	#391	C₂²×C₃₇	1	4	yes	—
175	#389	—	2	4	no	—
176	#384	C₂²×C₃₇	1	4	yes	—
177	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
178	#336	C₂²×C₃₇	1	2	yes	—
179	C₃₇	#28531	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	—	2	2	no	—
181	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
185	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
200	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
201	#107	—	2	2	no	—
202	#107	—	2	8	no	—
203	#107	—	2	4	no	—
204	#107	—	2	4	no	—
205	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
206	#103	—	2	4	no	—
207	#103	—	2	8	no	—
208	C₄×C₈	—	2	4	no	—
209	C₄×C₈	D₄×C₃₇	1	4	yes	—
210	#95	—	2	8	no	—
211	#95	D₄×C₃₇	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	8	4	no	—
219	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
226	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
227	Q₁₆	—	2	8	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
234	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
235	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
236	C₂×C₈	#41×C₃₇	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	8	8	no	—
244	C₄⋊C₄	Q₁₆×C₃₇	1	8	yes	complemented
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₄²	C₂×D₄×C₃₇	1	2	yes	—
249	C₄²	C₂×D₄×C₃₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	8	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	#127×C₃₇	1	2	yes	—
267	C₂×C₄	#127×C₃₇	1	2	yes	—
268	C₂×C₄	C₂×Q₁₆×C₃₇	1	8	yes	—
269	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
270	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
274	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
275	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
279	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
280	C₂²	C₃₇×#547	1	1	yes	characteristic
281	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
282	C₂²	C₂²×Q₁₆×C₃₇	1	2	yes	—
283	C₂²	C₃₇×#576	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	4	no	—
287	C₄	—	8	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₃₇×#494	1	4	yes	—
290	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
291	C₄	C₃₇×#536	1	1	yes	characteristic
292	C₄	C₃₇×#547	1	1	yes	characteristic
293	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
294	C₂	C₃₇×#2542	1	1	yes	characteristic
295	C₂	C₂×C₃₇×#494	1	1	yes	characteristic
296	C₂	C₃₇×#3155	1	1	yes	characteristic
297	C₂	C₃₇×#3338	1	2	yes	—
298	C₂	Q₈×Q₁₆×C₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
34. C₂×Q₁₆×C₃₇	Q₈	semidirect
94. C₄⋊C₄×C₃₇	Q₁₆	semidirect
129. #28531	C₃₇	direct (N × H)
179. C₃₇	#28531	direct (N × H)
184. C₂×Q₁₆	Q₈×C₃₇	semidirect
244. C₄⋊C₄	Q₁₆×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×#477	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2567	2
C₃₇×#2095	2
C₃₇×#1969	2
C₃₇×#1845	2
C₈×C₄⋊C₄×C₃₇	2
#28531	37

Subgroups — C37×#28531 #22649784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28531 #22649784