Subgroups — C₃₇×#20568 #22641821

Order 9472.

1086 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×#436	C₂	1	2	yes	maximal
2	C₄²×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1770	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#1753	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1749	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1728	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₃₇	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#436	C₄	1	16	yes	complemented
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
16	C₂×#97×C₃₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₃₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₃₇	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
23	C₄³×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#336	C₂²	1	1	yes	characteristic
25	C₃₇×#336	C₂²	1	2	yes	—
26	C₃₇×#336	C₂²	1	4	yes	—
27	C₃₇×#336	C₂²	1	1	yes	characteristic
28	C₃₇×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₃₇	—	4	4	no	—
33	C₂×M₁₆×C₃₇	—	4	4	no	—
34	C₂×M₁₆×C₃₇	C₂×C₄	1	16	yes	complemented
35	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
38	C₄×D₄×C₃₇	—	2	8	no	—
39	C₄×D₄×C₃₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₇	—	4	4	no	—
45	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	complemented
53	C₂²⋊C₈×C₃₇	C₂×C₄	1	16	yes	complemented
54	#97×C₃₇	—	4	4	no	—
55	C₄×C₈×C₃₇	—	4	2	no	—
56	C₄×C₈×C₃₇	C₂×C₄	1	8	yes	complemented
57	#95×C₃₇	C₂×C₄	1	2	yes	—
58	#95×C₃₇	C₂×C₄	1	4	yes	—
59	C₂⁴×C₃₇	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₃₇	—	2	4	no	—
61	C₂×D₄×C₃₇	C₄²	1	4	yes	—
62	C₂²×C₄×C₃₇	—	2	4	no	—
63	C₂²×C₄×C₃₇	—	2	2	no	—
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	8	no	—
66	C₂²×C₄×C₃₇	—	4	4	no	—
67	C₂²×C₄×C₃₇	—	4	4	no	—
68	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	8	yes	—
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	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₃₇	—	4	16	no	—
77	C₂×C₈×C₃₇	—	2	4	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	—	2	2	no	—
83	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₃₇	—	2	4	no	—
87	C₄⋊C₄×C₃₇	C₄²	1	4	yes	—
88	C₂²⋊C₄×C₃₇	—	2	8	no	—
89	C₂²⋊C₄×C₃₇	C₄²	1	8	yes	—
90	C₄²×C₃₇	—	2	1	no	—
91	C₄²×C₃₇	—	2	4	no	—
92	C₄²×C₃₇	—	2	2	no	—
93	C₄²×C₃₇	—	2	1	no	—
94	C₄²×C₃₇	—	4	4	no	—
95	C₄²×C₃₇	—	4	2	no	—
96	C₄²×C₃₇	M₁₆	1	8	yes	complemented
97	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
98	C₂³×C₃₇	—	2	2	no	—
99	C₂³×C₃₇	—	2	2	no	—
100	C₂³×C₃₇	—	2	4	no	—
101	C₂³×C₃₇	—	2	4	no	—
102	C₂³×C₃₇	C₂×C₄²	1	4	yes	—
103	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
104	D₄×C₃₇	—	2	8	no	—
105	C₂×C₄×C₃₇	—	2	2	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	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	8	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	—	4	16	no	—
118	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₃₇	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₃₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₃₇	—	2	8	no	—
129	C₈×C₃₇	—	2	4	no	—
130	C₈×C₃₇	—	2	4	no	—
131	#20568	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₃₇	—	2	4	no	—
133	C₂²×C₃₇	—	2	4	no	—
134	C₂²×C₃₇	—	2	4	no	—
135	C₂²×C₃₇	—	4	4	no	—
136	C₂²×C₃₇	—	4	4	no	—
137	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₃₇	#518	1	1	yes	characteristic
139	C₂²×C₃₇	#520	1	1	yes	characteristic
140	C₂²×C₃₇	C₂²×M₁₆	1	2	yes	—
141	C₂²×C₃₇	#568	1	2	yes	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	2	4	no	—
144	C₄×C₃₇	—	2	4	no	—
145	C₄×C₃₇	—	2	4	no	—
146	C₄×C₃₇	—	4	4	no	—
147	C₄×C₃₇	C₄×M₁₆	1	4	yes	—
148	C₄×C₃₇	#518	1	2	yes	—
149	C₄×C₃₇	#520	1	2	yes	—
150	C₂×#436	C₂×C₃₇	1	2	yes	—
151	C₄²×D₄	C₂×C₃₇	1	1	yes	characteristic
152	#1770	C₂×C₃₇	1	2	yes	complemented
153	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
154	#1753	C₂×C₃₇	1	4	yes	complemented
155	#1749	C₂×C₃₇	1	2	yes	complemented
156	#1744	C₂×C₃₇	1	1	yes	characteristic, complemented
157	#1728	C₂×C₃₇	1	2	yes	complemented
158	C₂×C₃₇	—	4	4	no	—
159	C₂×C₃₇	#2275	1	1	yes	characteristic
160	C₂×C₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
161	C₂×C₃₇	#2875	1	1	yes	characteristic
162	C₂×C₃₇	#2975	1	2	yes	—
163	C₂×C₃₇	#2989	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₃₇	1	4	yes	—
165	C₂×C₄×D₄	C₄×C₃₇	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂²×C₃₇	1	2	yes	—
168	#436	C₄×C₃₇	1	16	yes	complemented
169	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
170	C₂×C₂²⋊C₈	C₂²×C₃₇	1	4	yes	—
171	C₂×#97	C₂²×C₃₇	1	4	yes	—
172	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
174	C₄×C₄⋊C₄	C₄×C₃₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
176	C₄×C₂²⋊C₄	C₄×C₃₇	1	4	yes	—
177	C₄×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
178	C₄³	C₂²×C₃₇	1	1	yes	characteristic
179	#336	C₂²×C₃₇	1	1	yes	characteristic
180	#336	C₂²×C₃₇	1	2	yes	—
181	#336	C₂²×C₃₇	1	4	yes	—
182	#336	C₂²×C₃₇	1	1	yes	characteristic
183	#336	C₂²×C₃₇	1	2	yes	—
184	C₃₇	#20568	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×D₄	C₂×C₄×C₃₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₃₇	1	4	yes	—
187	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	—	4	4	no	—
190	C₂×M₁₆	C₂×C₄×C₃₇	1	16	yes	complemented
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	C₄×D₄	—	2	8	no	—
195	C₄×D₄	C₂×C₄×C₃₇	1	8	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
200	C₂×C₄²	—	4	4	no	—
201	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
203	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	complemented
209	C₂²⋊C₈	C₂×C₄×C₃₇	1	16	yes	complemented
210	#97	—	4	4	no	—
211	C₄×C₈	—	4	2	no	—
212	C₄×C₈	C₂×C₄×C₃₇	1	8	yes	complemented
213	#95	C₂×C₄×C₃₇	1	2	yes	—
214	#95	C₂×C₄×C₃₇	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₃₇	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	C₄²×C₃₇	1	4	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₇	1	8	yes	—
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	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
232	M₁₆	—	4	16	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄²×C₃₇	1	4	yes	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	C₄²×C₃₇	1	8	yes	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	4	no	—
251	C₄²	—	4	2	no	—
252	C₄²	M₁₆×C₃₇	1	8	yes	complemented
253	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄²×C₃₇	1	4	yes	—
259	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₃₇	1	4	yes	—
276	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
277	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
278	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
279	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₃₇	1	2	yes	—
281	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
282	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
283	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	2	8	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
293	C₂²	C₃₇×#518	1	1	yes	characteristic
294	C₂²	C₃₇×#520	1	1	yes	characteristic
295	C₂²	C₂²×M₁₆×C₃₇	1	2	yes	—
296	C₂²	C₃₇×#568	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	4	no	—
302	C₄	C₄×M₁₆×C₃₇	1	4	yes	—
303	C₄	C₃₇×#518	1	2	yes	—
304	C₄	C₃₇×#520	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	C₃₇×#2275	1	1	yes	characteristic
307	C₂	C₂×C₄×M₁₆×C₃₇	1	1	yes	characteristic
308	C₂	C₃₇×#2875	1	1	yes	characteristic
309	C₂	C₃₇×#2975	1	2	yes	—
310	C₂	C₃₇×#2989	1	2	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	C₂²×D₄×C₃₇	C₂×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
3. C₃₇×#1770	C₂	semidirect
4. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
5. C₃₇×#1753	C₂	semidirect
6. C₃₇×#1749	C₂	semidirect
7. C₃₇×#1744	C₂	semidirect
8. C₃₇×#1728	C₂	semidirect
13. C₃₇×#436	C₄	semidirect
34. C₂×M₁₆×C₃₇	C₂×C₄	semidirect
52. C₄⋊C₈×C₃₇	C₂×C₄	semidirect
53. C₂²⋊C₈×C₃₇	C₂×C₄	semidirect
56. C₄×C₈×C₃₇	C₂×C₄	semidirect
96. C₄²×C₃₇	M₁₆	semidirect
131. #20568	C₃₇	direct (N × H)
152. #1770	C₂×C₃₇	semidirect
153. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
154. #1753	C₂×C₃₇	semidirect
155. #1749	C₂×C₃₇	semidirect
156. #1744	C₂×C₃₇	semidirect
157. #1728	C₂×C₃₇	semidirect
168. #436	C₄×C₃₇	semidirect
184. C₃₇	#20568	direct (N × H)
190. C₂×M₁₆	C₂×C₄×C₃₇	semidirect
208. C₄⋊C₈	C₂×C₄×C₃₇	semidirect
209. C₂²⋊C₈	C₂×C₄×C₃₇	semidirect
212. C₄×C₈	C₂×C₄×C₃₇	semidirect
252. 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×#436	2
C₄²×D₄×C₃₇	2
C₃₇×#1770	2
C₄×C₄⋊C₈×C₃₇	2
C₃₇×#1753	2
C₃₇×#1749	2
C₃₇×#1744	2
C₃₇×#1728	2
#20568	37

Subgroups — C37×#20568 #22641821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#20568 #22641821