Subgroups — C₃₇×#28814 #22650067

Order 9472.

718 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×#499	C₂	1	2	yes	maximal
2	C₃₇×#2552	C₂	1	2	yes	maximal
3	C₃₇×#2059	C₂	1	4	yes	maximal
4	C₃₇×#1948	C₂	1	2	yes	maximal
5	C₃₇×#1942	C₂	1	2	yes	maximal
6	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₃₇	C₂²	1	4	yes	—
10	C₃₇×#499	—	2	8	no	—
11	C₂×#107×C₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₃₇×#398	—	2	4	no	—
20	C₃₇×#389	—	2	2	no	—
21	C₃₇×#389	C₂²	1	4	yes	—
22	C₃₇×#384	—	2	2	no	—
23	C₃₇×#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#340	C₂²	1	4	yes	—
27	C₃₇×#340	C₂²	1	2	yes	—
28	C₃₇×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
32	#125×C₃₇	—	2	8	no	—
33	#125×C₃₇	—	4	4	no	—
34	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₇	—	2	2	no	—
45	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#107×C₃₇	—	2	4	no	—
48	#107×C₃₇	—	4	2	no	—
49	#107×C₃₇	D₄	1	4	yes	—
50	#107×C₃₇	Q₈	1	4	yes	—
51	#106×C₃₇	—	2	8	no	—
52	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
53	C₄×C₈×C₃₇	—	2	4	no	—
54	C₄×C₈×C₃₇	Q₈	1	4	yes	—
55	#95×C₃₇	—	2	2	no	—
56	#95×C₃₇	—	2	4	no	—
57	#95×C₃₇	D₄	1	2	yes	—
58	C₂²×C₄×C₃₇	—	2	2	no	—
59	C₂²×C₄×C₃₇	—	2	2	no	—
60	C₂²×C₄×C₃₇	—	2	4	no	—
61	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₃₇	—	2	2	no	—
67	C₂×C₈×C₃₇	—	2	1	no	—
68	C₂×C₈×C₃₇	—	2	1	no	—
69	C₂×C₈×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
75	C₄⋊C₄×C₃₇	—	2	2	no	—
76	C₄⋊C₄×C₃₇	—	2	2	no	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	—	4	8	no	—
82	C₄⋊C₄×C₃₇	—	4	8	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₄⋊C₄×C₃₇	—	8	2	no	—
85	C₄⋊C₄×C₃₇	—	8	4	no	—
86	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₃₇	C₂×Q₈	1	4	yes	—
88	C₄²×C₃₇	—	2	2	no	—
89	C₄²×C₃₇	—	2	2	no	—
90	C₄²×C₃₇	—	4	4	no	—
91	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	—	2	1	no	—
95	C₂×C₄×C₃₇	—	2	1	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	4	no	—
99	C₂×C₄×C₃₇	—	2	4	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	—	2	2	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	4	no	—
109	C₂×C₄×C₃₇	—	4	4	no	—
110	C₂×C₄×C₃₇	—	4	4	no	—
111	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₃₇	#135	1	2	yes	—
114	C₂×C₄×C₃₇	#135	1	2	yes	—
115	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
121	C₈×C₃₇	—	2	2	no	—
122	C₈×C₃₇	—	2	4	no	—
123	C₈×C₃₇	C₄⋊Q₈	1	4	yes	—
124	#28814	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₃₇	D₄²	1	1	yes	characteristic
127	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
128	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
129	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
130	C₂²×C₃₇	#576	1	1	yes	characteristic
131	C₂²×C₃₇	#578	1	1	yes	characteristic
132	C₄×C₃₇	—	2	2	no	—
133	C₄×C₃₇	—	4	4	no	—
134	C₄×C₃₇	—	4	2	no	—
135	C₄×C₃₇	—	4	4	no	—
136	C₄×C₃₇	—	4	2	no	—
137	C₄×C₃₇	—	8	4	no	—
138	C₄×C₃₇	#499	1	4	yes	complemented
139	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₄×C₃₇	#544	1	1	yes	characteristic
141	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
142	C₄×C₃₇	Q₈²	1	1	yes	characteristic
143	C₂×#499	C₂×C₃₇	1	2	yes	—
144	#2552	C₂×C₃₇	1	2	yes	—
145	#2059	C₂×C₃₇	1	4	yes	—
146	#1948	C₂×C₃₇	1	2	yes	—
147	#1942	C₂×C₃₇	1	2	yes	—
148	#1845	C₂×C₃₇	1	1	yes	characteristic
149	C₄×#107	C₂×C₃₇	1	1	yes	characteristic
150	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
151	C₂×C₃₇	#2551	1	1	yes	characteristic
152	C₂×C₃₇	C₂×#499	1	1	yes	characteristic
153	C₂×C₃₇	#3164	1	1	yes	characteristic
154	C₂×C₃₇	#3283	1	1	yes	characteristic
155	C₂×C₃₇	#3287	1	1	yes	characteristic
156	C₂×C₃₇	#3380	1	1	yes	characteristic
157	C₂×C₃₇	#3383	1	1	yes	characteristic
158	C₂×#125	C₂²×C₃₇	1	4	yes	—
159	#499	—	2	8	no	—
160	C₂×#107	C₂²×C₃₇	1	2	yes	—
161	C₂×#107	C₂²×C₃₇	1	2	yes	—
162	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
163	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
164	C₂×#106	C₂²×C₃₇	1	4	yes	—
165	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
168	#398	—	2	4	no	—
169	#389	—	2	2	no	—
170	#389	C₂²×C₃₇	1	4	yes	—
171	#384	—	2	2	no	—
172	#384	C₂²×C₃₇	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
175	#340	C₂²×C₃₇	1	2	yes	—
176	#340	C₂²×C₃₇	1	4	yes	—
177	#336	C₂²×C₃₇	1	2	yes	—
178	C₃₇	#28814	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
182	#125	—	2	8	no	—
183	#125	—	4	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	4	2	no	—
199	#107	D₄×C₃₇	1	4	yes	—
200	#107	Q₈×C₃₇	1	4	yes	—
201	#106	—	2	8	no	—
202	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	Q₈×C₃₇	1	4	yes	—
205	#95	—	2	2	no	—
206	#95	—	2	4	no	—
207	#95	D₄×C₃₇	1	2	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
223	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
224	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₄⋊C₄	—	8	4	no	—
236	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
237	C₄⋊C₄	C₂×Q₈×C₃₇	1	4	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
242	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
243	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
262	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
263	C₂×C₄	#135×C₃₇	1	2	yes	—
264	C₂×C₄	#135×C₃₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
270	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	4	no	—
273	C₈	C₄⋊Q₈×C₃₇	1	4	yes	—
274	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
275	C₂²	D₄²×C₃₇	1	1	yes	characteristic
276	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
277	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
278	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
279	C₂²	C₃₇×#576	1	1	yes	characteristic
280	C₂²	C₃₇×#578	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₃₇×#499	1	4	yes	complemented
288	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
289	C₄	C₃₇×#544	1	1	yes	characteristic
290	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
291	C₄	Q₈²×C₃₇	1	1	yes	characteristic
292	C₂	C₃₇×#2551	1	1	yes	characteristic
293	C₂	C₂×C₃₇×#499	1	1	yes	characteristic
294	C₂	C₃₇×#3164	1	1	yes	characteristic
295	C₂	C₃₇×#3283	1	1	yes	characteristic
296	C₂	C₃₇×#3287	1	1	yes	characteristic
297	C₂	C₃₇×#3380	1	1	yes	characteristic
298	C₂	C₃₇×#3383	1	1	yes	characteristic

	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
124. #28814	C₃₇	direct (N × H)
138. C₄×C₃₇	#499	semidirect
178. C₃₇	#28814	direct (N × H)
287. C₄	C₃₇×#499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₃₇×#2552	2
C₃₇×#2059	2
C₃₇×#1948	2
C₃₇×#1942	2
C₃₇×#1845	2
C₄×#107×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28814	37

Subgroups — C37×#28814 #22650067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28814 #22650067