Subgroups — C₃₇×#20549 #22641802

Order 9472.

966 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₂×C₃₇×#423	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×#1767	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊C₈×C₃₇	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1753	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
9	C₂×#117×C₃₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	C₃₇×#432	C₄	1	8	yes	complemented
12	C₃₇×#423	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₃₇	C₂²	1	4	yes	—
18	C₂×#97×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₃₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₃₇	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#376	C₄	1	2	yes	—
25	C₄³×C₃₇	C₂²	1	1	yes	characteristic
26	C₄³×C₃₇	C₂²	1	1	yes	characteristic
27	C₃₇×#336	C₂²	1	4	yes	—
28	C₃₇×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₇	—	4	4	no	—
33	C₂×M₁₆×C₃₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
36	#117×C₃₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
52	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
53	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
55	#97×C₃₇	—	4	4	no	—
56	#97×C₃₇	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₃₇	—	4	4	no	—
58	#95×C₃₇	C₂×C₄	1	4	yes	—
59	#95×C₃₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	—	2	4	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	2	no	—
64	C₂²×C₄×C₃₇	—	4	2	no	—
65	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
66	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₃₇	—	4	8	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	—	2	4	no	—
81	C₂×C₈×C₃₇	—	2	4	no	—
82	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
84	C₄⋊C₄×C₃₇	—	2	8	no	—
85	C₂²⋊C₄×C₃₇	—	2	8	no	—
86	C₄²×C₃₇	—	2	4	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	—	2	2	no	—
89	C₄²×C₃₇	—	2	2	no	—
90	C₄²×C₃₇	—	2	2	no	—
91	C₄²×C₃₇	—	4	2	no	—
92	C₄²×C₃₇	—	4	2	no	—
93	C₄²×C₃₇	C₄²	1	4	yes	—
94	C₄²×C₃₇	C₄²	1	4	yes	—
95	C₄²×C₃₇	M₁₆	1	8	yes	complemented
96	C₄²×C₃₇	C₂²×C₄	1	2	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	C₂³×C₃₇	—	2	2	no	—
101	C₂³×C₃₇	—	2	2	no	—
102	C₂³×C₃₇	—	2	2	no	—
103	C₂³×C₃₇	C₂×C₄²	1	2	yes	—
104	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	2	2	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	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	2	4	no	—
114	C₂×C₄×C₃₇	—	2	8	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	—	4	4	no	—
119	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
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×M₁₆	1	8	yes	—
125	C₂×C₄×C₃₇	C₂×M₁₆	1	4	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	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
130	C₈×C₃₇	—	2	8	no	—
131	C₈×C₃₇	—	2	8	no	—
132	#20549	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₃₇	—	2	2	no	—
134	C₂²×C₃₇	—	2	2	no	—
135	C₂²×C₃₇	—	2	2	no	—
136	C₂²×C₃₇	—	4	4	no	—
137	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₃₇	#520	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	8	no	—
145	C₄×C₃₇	—	2	2	no	—
146	C₄×C₃₇	—	4	4	no	—
147	C₄×C₃₇	C₄×M₁₆	1	4	yes	—
148	C₄×C₃₇	#518	1	1	yes	characteristic
149	C₄×C₃₇	#519	1	1	yes	characteristic
150	C₄×C₃₇	#520	1	2	yes	—
151	C₂×#432	C₂×C₃₇	1	1	yes	characteristic
152	C₂×#423	C₂×C₃₇	1	1	yes	characteristic
153	C₄×#117	C₂×C₃₇	1	1	yes	characteristic
154	#1771	C₂×C₃₇	1	4	yes	complemented
155	#1767	C₂×C₃₇	1	2	yes	complemented
156	C₄×C₄⋊C₈	C₂×C₃₇	1	2	yes	complemented
157	#1753	C₂×C₃₇	1	4	yes	—
158	C₂×C₃₇	—	4	2	no	—
159	C₂×C₃₇	#2274	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	#2989	1	4	yes	—
163	C₂²×M₁₆	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	#423	C₄×C₃₇	1	8	yes	complemented
168	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×#97	C₂²×C₃₇	1	4	yes	—
173	C₂×#97	C₂²×C₃₇	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₃₇	1	4	yes	—
175	C₄×C₄⋊C₄	C₄×C₃₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
177	C₄×C₂²⋊C₄	C₄×C₃₇	1	2	yes	—
178	C₄×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
179	#376	C₄×C₃₇	1	2	yes	—
180	C₄³	C₂²×C₃₇	1	1	yes	characteristic
181	C₄³	C₂²×C₃₇	1	1	yes	characteristic
182	#336	C₂²×C₃₇	1	4	yes	—
183	#336	C₂²×C₃₇	1	4	yes	—
184	C₃₇	#20549	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	C₂×C₄×C₃₇	1	8	yes	—
190	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
191	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
192	#117	—	2	8	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
200	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
202	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
203	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
208	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
209	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
210	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
211	#97	—	4	4	no	—
212	#97	C₂×C₄×C₃₇	1	8	yes	—
213	C₄×C₈	—	4	4	no	—
214	#95	C₂×C₄×C₃₇	1	4	yes	—
215	#95	C₂×C₄×C₃₇	1	4	yes	—
216	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
222	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
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	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
233	M₁₆	—	4	8	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
239	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₄²×C₃₇	1	4	yes	—
250	C₄²	C₄²×C₃₇	1	4	yes	—
251	C₄²	M₁₆×C₃₇	1	8	yes	complemented
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	—	2	2	no	—
259	C₂³	C₂×C₄²×C₃₇	1	2	yes	—
260	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	2	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	C₂×M₁₆×C₃₇	1	4	yes	—
282	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
283	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
284	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
285	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₈	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
293	C₂²	C₃₇×#520	1	2	yes	—
294	C₂²	C₂²×M₁₆×C₃₇	1	2	yes	—
295	C₂²	C₃₇×#568	1	2	yes	—
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	C₄×M₁₆×C₃₇	1	4	yes	—
303	C₄	C₃₇×#518	1	1	yes	characteristic
304	C₄	C₃₇×#519	1	1	yes	characteristic
305	C₄	C₃₇×#520	1	2	yes	—
306	C₂	—	4	2	no	—
307	C₂	C₃₇×#2274	1	1	yes	characteristic
308	C₂	C₂×C₄×M₁₆×C₃₇	1	1	yes	characteristic
309	C₂	C₃₇×#2873	1	1	yes	characteristic
310	C₂	C₃₇×#2989	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
5. C₃₇×#1767	C₂	semidirect
6. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
11. C₃₇×#432	C₄	semidirect
12. C₃₇×#423	C₄	semidirect
95. C₄²×C₃₇	M₁₆	semidirect
132. #20549	C₃₇	direct (N × H)
154. #1771	C₂×C₃₇	semidirect
155. #1767	C₂×C₃₇	semidirect
156. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
166. #432	C₄×C₃₇	semidirect
167. #423	C₄×C₃₇	semidirect
184. C₃₇	#20549	direct (N × H)
251. C₄²	M₁₆×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
C₂×C₃₇×#423	2
C₄×#117×C₃₇	2
C₃₇×#1771	2
C₃₇×#1767	2
C₄×C₄⋊C₈×C₃₇	2
C₃₇×#1753	2
#20549	37

Subgroups — C37×#20549 #22641802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#20549 #22641802