Subgroups — C₃₇×#29791 #22651044

Order 9472.

738 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#469	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2090	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2063	C₂	1	4	yes	maximal, complemented
6	C₃₇×#2045	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1900	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₃₇	C₂²	1	2	yes	—
13	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#482	—	2	4	no	—
15	C₃₇×#481	—	2	2	no	—
16	C₃₇×#480	—	2	2	no	—
17	C₃₇×#469	—	2	2	no	—
18	C₃₇×#469	—	2	2	no	—
19	C₂×#108×C₃₇	C₂²	1	2	yes	—
20	C₃₇×#428	C₂²	1	4	yes	—
21	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#424	—	2	2	no	—
23	C₃₇×#424	C₂²	1	2	yes	—
24	C₃₇×#424	C₂²	1	2	yes	—
25	C₃₇×#423	—	2	2	no	—
26	C₃₇×#418	C₂²	1	4	yes	—
27	C₂×#102×C₃₇	C₂²	1	2	yes	—
28	C₂×#100×C₃₇	C₂²	1	4	yes	—
29	C₃₇×#407	C₂²	1	2	yes	—
30	C₃₇×#343	C₂²	1	4	yes	—
31	C₃₇×#341	C₂²	1	2	yes	—
32	C₂²×D₄×C₃₇	—	2	2	no	—
33	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₃₇	—	2	2	no	—
36	C₂×D₈×C₃₇	—	4	2	no	—
37	C₂×M₁₆×C₃₇	—	2	1	no	—
38	C₂×M₁₆×C₃₇	—	4	2	no	—
39	C₂×M₁₆×C₃₇	—	4	2	no	—
40	C₂×M₁₆×C₃₇	D₄	1	2	yes	—
41	C₂×M₁₆×C₃₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
44	C₂²×C₈×C₃₇	—	2	1	no	—
45	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
47	#121×C₃₇	—	2	2	no	—
48	#121×C₃₇	—	2	2	no	—
49	#121×C₃₇	—	2	2	no	—
50	#121×C₃₇	—	2	2	no	—
51	#117×C₃₇	—	2	1	no	—
52	#117×C₃₇	—	2	1	no	—
53	#117×C₃₇	D₄	1	2	yes	—
54	#117×C₃₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
59	C₂×C₄²×C₃₇	—	2	1	no	—
60	#108×C₃₇	—	4	2	no	—
61	#107×C₃₇	—	2	4	no	—
62	#106×C₃₇	—	2	2	no	—
63	#106×C₃₇	—	2	4	no	—
64	C₄⋊C₈×C₃₇	—	2	4	no	—
65	C₄⋊C₈×C₃₇	—	2	4	no	—
66	#102×C₃₇	—	2	4	no	—
67	#102×C₃₇	—	2	4	no	—
68	#102×C₃₇	—	2	4	no	—
69	#100×C₃₇	—	4	4	no	—
70	C₂²⋊C₈×C₃₇	—	2	2	no	—
71	C₂²⋊C₈×C₃₇	—	2	2	no	—
72	C₂⁴×C₃₇	—	2	2	no	—
73	C₂⁴×C₃₇	#41	1	1	yes	characteristic
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	2	no	—
76	C₂×D₄×C₃₇	—	2	4	no	—
77	C₂×D₄×C₃₇	—	2	4	no	—
78	C₂×D₄×C₃₇	—	4	2	no	—
79	C₂×D₄×C₃₇	—	4	2	no	—
80	C₂×D₄×C₃₇	—	8	4	no	—
81	C₂²×C₄×C₃₇	—	2	2	no	—
82	C₂²×C₄×C₃₇	—	2	2	no	—
83	C₂²×C₄×C₃₇	—	2	2	no	—
84	C₂²×C₄×C₃₇	—	2	2	no	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
90	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
91	D₈×C₃₇	—	4	4	no	—
92	M₁₆×C₃₇	—	2	4	no	—
93	M₁₆×C₃₇	—	2	4	no	—
94	M₁₆×C₃₇	—	2	4	no	—
95	M₁₆×C₃₇	—	4	2	no	—
96	M₁₆×C₃₇	—	4	2	no	—
97	M₁₆×C₃₇	—	4	2	no	—
98	C₂×C₈×C₃₇	—	2	2	no	—
99	C₂×C₈×C₃₇	—	2	2	no	—
100	C₂×C₈×C₃₇	—	2	4	no	—
101	C₂×C₈×C₃₇	—	2	2	no	—
102	C₂×C₈×C₃₇	—	4	2	no	—
103	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
105	C₄⋊C₄×C₃₇	—	2	2	no	—
106	C₄⋊C₄×C₃₇	—	2	2	no	—
107	C₄⋊C₄×C₃₇	—	2	2	no	—
108	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₃₇	—	2	2	no	—
110	C₂²⋊C₄×C₃₇	—	2	2	no	—
111	C₂²⋊C₄×C₃₇	—	4	2	no	—
112	C₂²⋊C₄×C₃₇	—	4	2	no	—
113	C₄²×C₃₇	—	2	2	no	—
114	C₄²×C₃₇	—	2	2	no	—
115	C₂³×C₃₇	—	2	2	no	—
116	C₂³×C₃₇	—	4	2	no	—
117	C₂³×C₃₇	—	4	2	no	—
118	C₂³×C₃₇	—	4	4	no	—
119	C₂³×C₃₇	—	8	2	no	—
120	C₂³×C₃₇	—	8	2	no	—
121	C₂³×C₃₇	#121	1	1	yes	characteristic
122	C₂³×C₃₇	#121	1	1	yes	characteristic
123	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
124	D₄×C₃₇	—	4	2	no	—
125	D₄×C₃₇	—	4	2	no	—
126	D₄×C₃₇	—	4	4	no	—
127	D₄×C₃₇	—	8	4	no	—
128	D₄×C₃₇	—	8	4	no	—
129	C₂×C₄×C₃₇	—	2	2	no	—
130	C₂×C₄×C₃₇	—	2	1	no	—
131	C₂×C₄×C₃₇	—	2	1	no	—
132	C₂×C₄×C₃₇	—	2	1	no	—
133	C₂×C₄×C₃₇	—	2	1	no	—
134	C₂×C₄×C₃₇	—	2	2	no	—
135	C₂×C₄×C₃₇	—	2	2	no	—
136	C₂×C₄×C₃₇	—	2	4	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	8	2	no	—
139	C₂×C₄×C₃₇	—	8	2	no	—
140	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
145	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
146	C₈×C₃₇	—	2	2	no	—
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	—	4	2	no	—
149	C₈×C₃₇	—	4	4	no	—
150	#29791	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₃₇	—	2	2	no	—
152	C₂²×C₃₇	—	2	1	no	—
153	C₂²×C₃₇	—	2	1	no	—
154	C₂²×C₃₇	—	2	2	no	—
155	C₂²×C₃₇	—	2	2	no	—
156	C₂²×C₃₇	—	8	2	no	—
157	C₂²×C₃₇	—	8	2	no	—
158	C₂²×C₃₇	—	8	2	no	—
159	C₂²×C₃₇	—	8	2	no	—
160	C₂²×C₃₇	—	16	4	no	—
161	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
162	C₂²×C₃₇	#539	1	2	yes	—
163	C₄×C₃₇	—	2	1	no	—
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	4	4	no	—
166	C₄×C₃₇	—	8	2	no	—
167	C₄×C₃₇	#534	1	1	yes	characteristic
168	C₄×C₃₇	#536	1	1	yes	characteristic
169	C₄×C₃₇	#537	1	2	yes	—
170	#3094	C₂×C₃₇	1	2	yes	complemented
171	C₂×#469	C₂×C₃₇	1	1	yes	characteristic
172	#2905	C₂×C₃₇	1	1	yes	characteristic, complemented
173	#2090	C₂×C₃₇	1	2	yes	complemented
174	#2063	C₂×C₃₇	1	4	yes	complemented
175	#2045	C₂×C₃₇	1	2	yes	complemented
176	#1900	C₂×C₃₇	1	2	yes	complemented
177	#1811	C₂×C₃₇	1	1	yes	characteristic, complemented
178	C₂×C₃₇	—	2	2	no	—
179	C₂×C₃₇	—	4	1	no	—
180	C₂×C₃₇	—	4	1	no	—
181	C₂×C₃₇	—	16	2	no	—
182	C₂×C₃₇	#2618	1	1	yes	characteristic
183	C₂×C₃₇	#3068	1	2	yes	—
184	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
185	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
186	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#121	C₂²×C₃₇	1	2	yes	—
188	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
189	#482	—	2	4	no	—
190	#481	—	2	2	no	—
191	#480	—	2	2	no	—
192	#469	—	2	2	no	—
193	#469	—	2	2	no	—
194	C₂×#108	C₂²×C₃₇	1	2	yes	—
195	#428	C₂²×C₃₇	1	4	yes	—
196	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
197	#424	—	2	2	no	—
198	#424	C₂²×C₃₇	1	2	yes	—
199	#424	C₂²×C₃₇	1	2	yes	—
200	#423	—	2	2	no	—
201	#418	C₂²×C₃₇	1	4	yes	—
202	C₂×#102	C₂²×C₃₇	1	2	yes	—
203	C₂×#100	C₂²×C₃₇	1	4	yes	—
204	#407	C₂²×C₃₇	1	2	yes	—
205	#343	C₂²×C₃₇	1	4	yes	—
206	#341	C₂²×C₃₇	1	2	yes	—
207	C₃₇	#29791	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
210	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂×M₁₆	D₄×C₃₇	1	2	yes	—
217	C₂×M₁₆	D₄×C₃₇	1	2	yes	—
218	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
219	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
220	C₂²×C₈	—	2	1	no	—
221	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#117	—	2	1	no	—
228	#117	—	2	1	no	—
229	#117	D₄×C₃₇	1	2	yes	—
230	#117	C₂³×C₃₇	1	2	yes	—
231	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₄²	—	2	1	no	—
236	#108	—	4	2	no	—
237	#107	—	2	4	no	—
238	#106	—	2	2	no	—
239	#106	—	2	4	no	—
240	C₄⋊C₈	—	2	4	no	—
241	C₄⋊C₈	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#100	—	4	4	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂⁴	—	2	2	no	—
249	C₂⁴	#41×C₃₇	1	1	yes	characteristic
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	8	4	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
265	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
266	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
267	D₈	—	4	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	2	4	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	4	2	no	—
273	M₁₆	—	4	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	#121×C₃₇	1	1	yes	characteristic
298	C₂³	#121×C₃₇	1	1	yes	characteristic
299	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
317	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
320	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
321	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	4	no	—
336	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
337	C₂²	C₃₇×#539	1	2	yes	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	2	no	—
342	C₄	C₃₇×#534	1	1	yes	characteristic
343	C₄	C₃₇×#536	1	1	yes	characteristic
344	C₄	C₃₇×#537	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	4	1	no	—
348	C₂	—	16	2	no	—
349	C₂	C₃₇×#2618	1	1	yes	characteristic
350	C₂	C₃₇×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×C₃₇×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3094	C₂	semidirect
3. C₃₇×#2905	C₂	semidirect
4. C₃₇×#2090	C₂	semidirect
5. C₃₇×#2063	C₂	semidirect
6. C₃₇×#2045	C₂	semidirect
7. C₃₇×#1900	C₂	semidirect
8. C₃₇×#1811	C₂	semidirect
150. #29791	C₃₇	direct (N × H)
170. #3094	C₂×C₃₇	semidirect
172. #2905	C₂×C₃₇	semidirect
173. #2090	C₂×C₃₇	semidirect
174. #2063	C₂×C₃₇	semidirect
175. #2045	C₂×C₃₇	semidirect
176. #1900	C₂×C₃₇	semidirect
177. #1811	C₂×C₃₇	semidirect
207. C₃₇	#29791	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#3094	2
C₂×C₃₇×#469	2
C₃₇×#2905	2
C₃₇×#2090	2
C₃₇×#2063	2
C₃₇×#2045	2
C₃₇×#1900	2
C₃₇×#1811	2
#29791	37

Subgroups — C37×#29791 #22651044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29791 #22651044