Subgroups — C₃₇×#20479 #22641732

Order 9472.

1034 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1771	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1752	C₂	1	2	yes	maximal
6	C₈×C₂²⋊C₄×C₃₇	C₂	1	2	yes	maximal
7	C₃₇×#1745	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₃₇	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
10	C₂×#117×C₃₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#432	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×#97×C₃₇	C₄	1	8	yes	complemented
16	C₂×#97×C₃₇	C₂²	1	4	yes	—
17	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₃₇	C₂²	1	2	yes	—
20	C₂×#97×C₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₃₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₃₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
27	C₃₇×#376	C₄	1	2	yes	—
28	C₄³×C₃₇	C₂²	1	2	yes	—
29	C₃₇×#336	C₂²	1	4	yes	—
30	C₃₇×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	—	2	8	no	—
35	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
39	#117×C₃₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
55	#97×C₃₇	—	2	8	no	—
56	#97×C₃₇	C₂×C₄	1	4	yes	—
57	#97×C₃₇	C₂×C₄	1	8	yes	—
58	#97×C₃₇	C₂×C₄	1	4	yes	—
59	#97×C₃₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₃₇	—	2	8	no	—
61	#95×C₃₇	C₂×C₄	1	4	yes	—
62	#95×C₃₇	C₂×C₄	1	2	yes	—
63	#95×C₃₇	C₂×C₄	1	2	yes	—
64	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	—	2	4	no	—
66	C₂²×C₄×C₃₇	—	2	4	no	—
67	C₂²×C₄×C₃₇	—	4	2	no	—
68	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
69	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₃₇	—	2	4	no	—
81	C₂×C₈×C₃₇	—	2	4	no	—
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	—	2	16	no	—
84	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₃₇	#41	1	8	yes	—
88	C₄⋊C₄×C₃₇	—	2	8	no	—
89	C₂²⋊C₄×C₃₇	—	2	4	no	—
90	C₂²⋊C₄×C₃₇	M₁₆	1	8	yes	—
91	C₄²×C₃₇	—	2	8	no	—
92	C₄²×C₃₇	—	2	4	no	—
93	C₄²×C₃₇	—	2	4	no	—
94	C₄²×C₃₇	—	4	4	no	—
95	C₄²×C₃₇	C₄²	1	4	yes	—
96	C₄²×C₃₇	C₄²	1	4	yes	—
97	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
101	C₂³×C₃₇	—	2	2	no	—
102	C₂³×C₃₇	—	2	2	no	—
103	C₂³×C₃₇	C₂×C₄²	1	2	yes	—
104	C₂³×C₃₇	C₂×M₁₆	1	4	yes	—
105	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
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	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	4	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	2	8	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	2	4	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	4	4	no	—
120	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₃₇	C₂×M₁₆	1	8	yes	—
126	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₇	C₂×#41	1	4	yes	—
129	C₈×C₃₇	—	2	4	no	—
130	C₈×C₃₇	—	2	8	no	—
131	C₈×C₃₇	#117	1	8	yes	complemented
132	#20479	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₃₇	—	2	2	no	—
134	C₂²×C₃₇	—	2	8	no	—
135	C₂²×C₃₇	—	2	2	no	—
136	C₂²×C₃₇	C₄×M₁₆	1	4	yes	—
137	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₃₇	C₄×#41	1	2	yes	—
139	C₂²×C₃₇	C₂²×M₁₆	1	2	yes	—
140	C₂²×C₃₇	#568	1	2	yes	—
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	2	2	no	—
144	C₄×C₃₇	—	2	2	no	—
145	C₄×C₃₇	—	2	8	no	—
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	4	4	no	—
148	C₄×C₃₇	C₂×#117	1	2	yes	—
149	C₄×C₃₇	C₄×#41	1	2	yes	—
150	C₂×#432	C₂×C₃₇	1	1	yes	characteristic
151	C₂²×#97	C₂×C₃₇	1	1	yes	characteristic
152	C₄×#117	C₂×C₃₇	1	1	yes	characteristic
153	#1771	C₂×C₃₇	1	4	yes	complemented
154	#1752	C₂×C₃₇	1	2	yes	—
155	C₈×C₂²⋊C₄	C₂×C₃₇	1	2	yes	—
156	#1745	C₂×C₃₇	1	2	yes	complemented
157	C₄×#97	C₂×C₃₇	1	2	yes	complemented
158	C₂×C₃₇	—	2	4	no	—
159	C₂×C₃₇	C₄×#117	1	1	yes	characteristic
160	C₂×C₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
161	C₂×C₃₇	#2873	1	1	yes	characteristic
162	C₂×C₃₇	#2970	1	4	yes	—
163	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
164	C₂×#117	C₄×C₃₇	1	2	yes	—
165	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
166	#432	C₄×C₃₇	1	8	yes	complemented
167	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
169	C₂×#97	C₄×C₃₇	1	8	yes	complemented
170	C₂×#97	C₂²×C₃₇	1	4	yes	—
171	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
172	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
173	C₂×#97	C₂²×C₃₇	1	2	yes	—
174	C₂×#97	C₂²×C₃₇	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₃₇	1	4	yes	—
176	C₄×C₄⋊C₄	C₄×C₃₇	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
178	C₄×C₂²⋊C₄	C₄×C₃₇	1	2	yes	—
179	C₄×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
180	C₄×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
181	#376	C₄×C₃₇	1	2	yes	—
182	C₄³	C₂²×C₃₇	1	2	yes	—
183	#336	C₂²×C₃₇	1	4	yes	—
184	#336	C₂²×C₃₇	1	4	yes	—
185	C₃₇	#20479	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
189	C₂²×C₈	—	2	8	no	—
190	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	—
191	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
193	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
194	#117	—	2	8	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
200	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
203	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
204	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
209	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
210	#97	—	2	8	no	—
211	#97	C₂×C₄×C₃₇	1	4	yes	—
212	#97	C₂×C₄×C₃₇	1	8	yes	—
213	#97	C₂×C₄×C₃₇	1	4	yes	—
214	#97	C₂×C₄×C₃₇	1	8	yes	—
215	C₄×C₈	—	2	8	no	—
216	#95	C₂×C₄×C₃₇	1	4	yes	—
217	#95	C₂×C₄×C₃₇	1	2	yes	—
218	#95	C₂×C₄×C₃₇	1	2	yes	—
219	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
224	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
241	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
242	C₂×C₈	#41×C₃₇	1	8	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	M₁₆×C₃₇	1	8	yes	—
246	C₄²	—	2	8	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	4	4	no	—
250	C₄²	C₄²×C₃₇	1	4	yes	—
251	C₄²	C₄²×C₃₇	1	4	yes	—
252	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
253	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
254	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
255	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄²×C₃₇	1	2	yes	—
259	C₂³	C₂×M₁₆×C₃₇	1	4	yes	—
260	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
278	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
279	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
280	C₂×C₄	C₂×M₁₆×C₃₇	1	8	yes	—
281	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
282	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₃₇	1	4	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	2	8	no	—
286	C₈	#117×C₃₇	1	8	yes	complemented
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₄×M₁₆×C₃₇	1	4	yes	—
291	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
292	C₂²	C₄×#41×C₃₇	1	2	yes	—
293	C₂²	C₂²×M₁₆×C₃₇	1	2	yes	—
294	C₂²	C₃₇×#568	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	8	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	C₂×#117×C₃₇	1	2	yes	—
303	C₄	C₄×#41×C₃₇	1	2	yes	—
304	C₂	—	2	4	no	—
305	C₂	C₄×#117×C₃₇	1	1	yes	characteristic
306	C₂	C₂×C₄×M₁₆×C₃₇	1	1	yes	characteristic
307	C₂	C₃₇×#2873	1	1	yes	characteristic
308	C₂	C₃₇×#2970	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂³×C₄
prime radical	C₂⁴×C₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#1771	C₂	semidirect
7. C₃₇×#1745	C₂	semidirect
8. C₄×#97×C₃₇	C₂	semidirect
12. C₃₇×#432	C₄	semidirect
15. C₂×#97×C₃₇	C₄	semidirect
131. C₈×C₃₇	#117	semidirect
132. #20479	C₃₇	direct (N × H)
153. #1771	C₂×C₃₇	semidirect
156. #1745	C₂×C₃₇	semidirect
157. C₄×#97	C₂×C₃₇	semidirect
166. #432	C₄×C₃₇	semidirect
169. C₂×#97	C₄×C₃₇	semidirect
185. C₃₇	#20479	direct (N × H)
286. C₈	#117×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×#432	2
C₂²×#97×C₃₇	2
C₄×#117×C₃₇	2
C₃₇×#1771	2
C₃₇×#1752	2
C₈×C₂²⋊C₄×C₃₇	2
C₃₇×#1745	2
C₄×#97×C₃₇	2
#20479	37

Subgroups — C37×#20479 #22641732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#20479 #22641732