Subgroups — C₃₇×#28625 #22649878

Order 9472.

790 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#491	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2642	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2327	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2056	C₂	1	4	yes	maximal
6	C₃₇×#1991	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#1960	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1878	C₂	1	4	yes	maximal
9	C₃₇×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#494	—	2	4	no	—
16	C₃₇×#491	—	2	4	no	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₃₇	C₂²	1	4	yes	—
20	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#400	—	2	4	no	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#391	C₂²	1	2	yes	—
25	C₃₇×#391	C₂²	1	2	yes	—
26	C₃₇×#387	—	2	4	no	—
27	C₃₇×#382	C₂²	1	1	yes	characteristic
28	C₃₇×#376	C₂²	1	1	yes	characteristic
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₃₇	—	2	8	no	—
34	C₂×Q₁₆×C₃₇	—	2	8	no	—
35	C₂×Q₁₆×C₃₇	—	4	4	no	—
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₃₇	—	2	2	no	—
40	C₄⋊Q₈×C₃₇	—	2	2	no	—
41	C₄⋊Q₈×C₃₇	D₄	1	4	yes	—
42	#125×C₃₇	—	2	2	no	—
43	#125×C₃₇	—	2	2	no	—
44	C₄×Q₈×C₃₇	—	4	4	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₇	—	2	2	no	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₃₇	—	2	4	no	—
56	#103×C₃₇	—	2	8	no	—
57	#103×C₃₇	—	4	4	no	—
58	#97×C₃₇	—	2	2	no	—
59	C₄×C₈×C₃₇	—	2	2	no	—
60	#95×C₃₇	—	2	2	no	—
61	#95×C₃₇	—	2	4	no	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	D₄	1	4	yes	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	4	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	4	no	—
70	C₂×Q₈×C₃₇	—	2	4	no	—
71	C₂×Q₈×C₃₇	—	2	4	no	—
72	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₇	—	4	16	no	—
82	C₂×C₈×C₃₇	—	2	2	no	—
83	C₂×C₈×C₃₇	—	2	2	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	#41	1	4	yes	—
88	C₄⋊C₄×C₃₇	—	2	8	no	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
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	#41	1	2	yes	—
100	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₃₇	—	2	8	no	—
102	Q₈×C₃₇	—	4	4	no	—
103	Q₈×C₃₇	—	4	4	no	—
104	Q₈×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	4	no	—
110	C₂×C₄×C₃₇	—	2	4	no	—
111	C₂×C₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	4	2	no	—
118	C₂×C₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	4	4	no	—
120	C₂×C₄×C₃₇	#121	1	4	yes	—
121	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
125	C₈×C₃₇	—	2	4	no	—
126	C₈×C₃₇	—	4	2	no	—
127	#28625	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
129	C₂²×C₃₇	#532	1	1	yes	characteristic
130	C₂²×C₃₇	#535	1	1	yes	characteristic
131	C₂²×C₃₇	#578	1	4	yes	—
132	C₄×C₃₇	—	2	4	no	—
133	C₄×C₃₇	—	4	2	no	—
134	C₄×C₃₇	—	4	2	no	—
135	C₄×C₃₇	—	4	4	no	—
136	C₄×C₃₇	—	8	2	no	—
137	C₄×C₃₇	—	8	2	no	—
138	C₄×C₃₇	—	8	2	no	—
139	C₄×C₃₇	#546	1	2	yes	—
140	C₄×C₃₇	#559	1	2	yes	—
141	C₂×#494	C₂×C₃₇	1	1	yes	characteristic
142	C₂×#491	C₂×C₃₇	1	1	yes	characteristic
143	#2642	C₂×C₃₇	1	1	yes	characteristic
144	#2327	C₂×C₃₇	1	1	yes	characteristic
145	#2056	C₂×C₃₇	1	4	yes	—
146	#1991	C₂×C₃₇	1	1	yes	characteristic
147	#1960	C₂×C₃₇	1	1	yes	characteristic
148	#1878	C₂×C₃₇	1	4	yes	—
149	#1768	C₂×C₃₇	1	1	yes	characteristic
150	C₂×C₃₇	#2544	1	1	yes	characteristic
151	C₂×C₃₇	#3044	1	2	yes	—
152	C₂×C₃₇	#3177	1	2	yes	—
153	C₂×C₃₇	#3223	1	2	yes	—
154	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
155	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
157	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
159	#494	—	2	4	no	—
160	#491	—	2	4	no	—
161	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
162	C₂×#103	C₂²×C₃₇	1	4	yes	—
163	C₂×#103	C₂²×C₃₇	1	4	yes	—
164	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
166	#400	—	2	4	no	—
167	#391	C₂²×C₃₇	1	2	yes	—
168	#391	C₂²×C₃₇	1	2	yes	—
169	#391	C₂²×C₃₇	1	2	yes	—
170	#387	—	2	4	no	—
171	#382	C₂²×C₃₇	1	1	yes	characteristic
172	#376	C₂²×C₃₇	1	1	yes	characteristic
173	#340	C₂²×C₃₇	1	4	yes	—
174	#336	C₂²×C₃₇	1	2	yes	—
175	C₃₇	#28625	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
177	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
178	C₂×Q₁₆	—	2	8	no	—
179	C₂×Q₁₆	—	2	8	no	—
180	C₂×Q₁₆	—	4	4	no	—
181	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
182	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	D₄×C₃₇	1	4	yes	—
187	#125	—	2	2	no	—
188	#125	—	2	2	no	—
189	C₄×Q₈	—	4	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	#103	—	2	8	no	—
202	#103	—	4	4	no	—
203	#97	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	—	2	4	no	—
209	#95	D₄×C₃₇	1	4	yes	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
226	Q₁₆	—	4	16	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	#41×C₃₇	1	4	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	4	no	—
243	C₄²	C₂×D₄×C₃₇	1	2	yes	—
244	C₄²	#41×C₃₇	1	2	yes	—
245	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
246	Q₈	—	2	8	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	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	#121×C₃₇	1	4	yes	—
266	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
268	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
273	C₂²	C₃₇×#532	1	1	yes	characteristic
274	C₂²	C₃₇×#535	1	1	yes	characteristic
275	C₂²	C₃₇×#578	1	4	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	C₃₇×#546	1	2	yes	—
284	C₄	C₃₇×#559	1	2	yes	—
285	C₂	C₃₇×#2544	1	1	yes	characteristic
286	C₂	C₃₇×#3044	1	2	yes	—
287	C₂	C₃₇×#3177	1	2	yes	—
288	C₂	C₃₇×#3223	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 ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
127. #28625	C₃₇	direct (N × H)
175. C₃₇	#28625	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×C₃₇×#494	2
C₂×C₃₇×#491	2
C₃₇×#2642	2
C₃₇×#2327	2
C₃₇×#2056	2
C₃₇×#1991	2
C₃₇×#1960	2
C₃₇×#1878	2
C₃₇×#1768	2
#28625	37

Subgroups — C37×#28625 #22649878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28625 #22649878