Subgroups — C₃₇×#28635 #22649888

Order 9472.

806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3151	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2051	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1965	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1907	C₂	1	4	yes	maximal, complemented
9	C₃₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	4	yes	—
11	C₂×#131×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#532	C₂²	1	1	yes	characteristic
13	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#496	—	2	2	no	—
16	C₃₇×#493	—	2	1	no	—
17	C₃₇×#492	—	2	1	no	—
18	C₃₇×#491	—	2	1	no	—
19	C₃₇×#490	—	2	1	no	—
20	C₃₇×#490	—	2	4	no	—
21	C₃₇×#488	—	2	2	no	—
22	C₃₇×#435	—	2	4	no	—
23	C₂×#108×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#423	C₂²	1	1	yes	characteristic
25	C₃₇×#417	C₂²	1	4	yes	—
26	C₂×#102×C₃₇	C₂²	1	2	yes	—
27	C₂×#102×C₃₇	C₂²	1	2	yes	—
28	C₂×#100×C₃₇	C₂²	1	4	yes	—
29	C₃₇×#408	C₂²	1	2	yes	—
30	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
31	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
33	C₃₇×#343	C₂²	1	4	yes	—
34	C₃₇×#341	C₂²	1	2	yes	—
35	C₂×#41×C₃₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₃₇	—	2	2	no	—
37	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
38	AGL(1,ℤ/8)×C₃₇	—	2	8	no	—
39	AGL(1,ℤ/8)×C₃₇	—	4	4	no	—
40	C₂×SD₁₆×C₃₇	—	2	4	no	—
41	C₂×D₈×C₃₇	—	2	4	no	—
42	#131×C₃₇	—	4	4	no	—
43	C₂×M₁₆×C₃₇	—	2	2	no	—
44	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₇	—	2	2	no	—
48	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
51	#127×C₃₇	—	2	2	no	—
52	#127×C₃₇	—	2	1	no	—
53	#127×C₃₇	—	2	1	no	—
54	#127×C₃₇	D₄	1	1	yes	characteristic
55	#125×C₃₇	—	2	1	no	—
56	#125×C₃₇	—	2	2	no	—
57	#125×C₃₇	—	2	1	no	—
58	#124×C₃₇	D₄	1	2	yes	—
59	#121×C₃₇	—	2	2	no	—
60	C₄×D₄×C₃₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
64	#108×C₃₇	—	2	4	no	—
65	C₄⋊C₈×C₃₇	—	2	2	no	—
66	#103×C₃₇	—	2	4	no	—
67	#102×C₃₇	—	2	4	no	—
68	#102×C₃₇	—	2	4	no	—
69	#102×C₃₇	—	2	4	no	—
70	#100×C₃₇	—	4	4	no	—
71	C₂²⋊C₈×C₃₇	—	2	4	no	—
72	#97×C₃₇	—	2	1	no	—
73	#97×C₃₇	—	2	1	no	—
74	#97×C₃₇	D₄	1	4	yes	—
75	C₄×C₈×C₃₇	—	2	1	no	—
76	C₄×C₈×C₃₇	—	2	1	no	—
77	C₂⁴×C₃₇	—	2	2	no	—
78	#41×C₃₇	—	2	4	no	—
79	#41×C₃₇	—	2	4	no	—
80	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₃₇	—	2	2	no	—
82	C₂×D₄×C₃₇	—	2	2	no	—
83	C₂×D₄×C₃₇	—	2	2	no	—
84	C₂×D₄×C₃₇	—	2	2	no	—
85	C₂×D₄×C₃₇	—	2	4	no	—
86	C₂×D₄×C₃₇	—	2	4	no	—
87	C₂×D₄×C₃₇	—	4	2	no	—
88	C₂×D₄×C₃₇	—	4	2	no	—
89	C₂×D₄×C₃₇	—	8	4	no	—
90	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₃₇	—	2	2	no	—
92	C₂²×C₄×C₃₇	—	2	2	no	—
93	C₂²×C₄×C₃₇	#41	1	2	yes	—
94	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
95	SD₁₆×C₃₇	—	4	8	no	—
96	D₈×C₃₇	—	4	8	no	—
97	M₁₆×C₃₇	—	2	2	no	—
98	M₁₆×C₃₇	—	2	2	no	—
99	M₁₆×C₃₇	—	2	2	no	—
100	M₁₆×C₃₇	—	2	4	no	—
101	M₁₆×C₃₇	—	4	4	no	—
102	M₁₆×C₃₇	#41	1	4	yes	—
103	C₂×C₈×C₃₇	—	2	2	no	—
104	C₂×C₈×C₃₇	—	2	2	no	—
105	C₂×C₈×C₃₇	—	2	2	no	—
106	C₂×C₈×C₃₇	—	2	2	no	—
107	C₂×C₈×C₃₇	—	2	2	no	—
108	C₂×C₈×C₃₇	—	4	4	no	—
109	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₃₇	—	2	2	no	—
112	C₄⋊C₄×C₃₇	—	2	2	no	—
113	C₄⋊C₄×C₃₇	—	2	2	no	—
114	C₄⋊C₄×C₃₇	—	4	2	no	—
115	C₄⋊C₄×C₃₇	—	4	2	no	—
116	C₂²⋊C₄×C₃₇	—	2	4	no	—
117	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₇	C₂×D₄	1	2	yes	—
120	C₂³×C₃₇	—	2	2	no	—
121	C₂³×C₃₇	—	4	2	no	—
122	C₂³×C₃₇	—	4	4	no	—
123	C₂³×C₃₇	—	8	4	no	—
124	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
125	Q₈×C₃₇	—	2	4	no	—
126	D₄×C₃₇	—	2	4	no	—
127	D₄×C₃₇	—	4	4	no	—
128	D₄×C₃₇	—	4	2	no	—
129	D₄×C₃₇	—	4	2	no	—
130	D₄×C₃₇	—	4	4	no	—
131	D₄×C₃₇	—	8	2	no	—
132	D₄×C₃₇	—	8	4	no	—
133	D₄×C₃₇	—	8	4	no	—
134	C₂×C₄×C₃₇	—	2	2	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	—	4	2	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	4	2	no	—
139	C₂×C₄×C₃₇	—	8	2	no	—
140	C₂×C₄×C₃₇	#121	1	2	yes	—
141	C₂×C₄×C₃₇	#121	1	2	yes	—
142	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₃₇	—	2	2	no	—
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	—	2	4	no	—
149	C₈×C₃₇	—	4	2	no	—
150	C₈×C₃₇	—	4	2	no	—
151	#28635	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₃₇	—	2	2	no	—
153	C₂²×C₃₇	—	4	2	no	—
154	C₂²×C₃₇	—	4	4	no	—
155	C₂²×C₃₇	—	8	2	no	—
156	C₂²×C₃₇	—	8	4	no	—
157	C₂²×C₃₇	—	16	4	no	—
158	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
159	C₂²×C₃₇	#559	1	2	yes	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	2	2	no	—
162	C₄×C₃₇	—	4	2	no	—
163	C₄×C₃₇	—	8	2	no	—
164	C₄×C₃₇	#532	1	1	yes	characteristic
165	C₄×C₃₇	#535	1	1	yes	characteristic
166	C₄×C₃₇	#546	1	2	yes	—
167	#3151	C₂×C₃₇	1	1	yes	characteristic
168	#3142	C₂×C₃₇	1	1	yes	characteristic, complemented
169	C₂×#490	C₂×C₃₇	1	1	yes	characteristic, complemented
170	#2932	C₂×C₃₇	1	1	yes	characteristic, complemented
171	#2051	C₂×C₃₇	1	4	yes	complemented
172	#1998	C₂×C₃₇	1	1	yes	characteristic, complemented
173	#1965	C₂×C₃₇	1	1	yes	characteristic, complemented
174	#1907	C₂×C₃₇	1	4	yes	complemented
175	#1778	C₂×C₃₇	1	1	yes	characteristic, complemented
176	C₂×C₃₇	—	2	2	no	—
177	C₂×C₃₇	—	8	2	no	—
178	C₂×C₃₇	—	16	2	no	—
179	C₂×C₃₇	#2544	1	1	yes	characteristic
180	C₂×C₃₇	#3068	1	2	yes	—
181	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	4	yes	—
182	C₂×#131	C₂²×C₃₇	1	2	yes	—
183	#532	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
186	#496	—	2	2	no	—
187	#493	—	2	1	no	—
188	#492	—	2	1	no	—
189	#491	—	2	1	no	—
190	#490	—	2	1	no	—
191	#490	—	2	4	no	—
192	#488	—	2	2	no	—
193	#435	—	2	4	no	—
194	C₂×#108	C₂²×C₃₇	1	2	yes	—
195	#423	C₂²×C₃₇	1	1	yes	characteristic
196	#417	C₂²×C₃₇	1	4	yes	—
197	C₂×#102	C₂²×C₃₇	1	2	yes	—
198	C₂×#102	C₂²×C₃₇	1	2	yes	—
199	C₂×#100	C₂²×C₃₇	1	4	yes	—
200	#408	C₂²×C₃₇	1	2	yes	—
201	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
202	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
203	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
204	#343	C₂²×C₃₇	1	4	yes	—
205	#341	C₂²×C₃₇	1	2	yes	—
206	C₃₇	#28635	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×#41	C₂³×C₃₇	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
210	AGL(1,ℤ/8)	—	2	8	no	—
211	AGL(1,ℤ/8)	—	4	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	4	no	—
214	#131	—	4	4	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
217	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
218	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
222	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
223	#127	—	2	2	no	—
224	#127	—	2	1	no	—
225	#127	—	2	1	no	—
226	#127	D₄×C₃₇	1	1	yes	characteristic
227	#125	—	2	1	no	—
228	#125	—	2	2	no	—
229	#125	—	2	1	no	—
230	#124	D₄×C₃₇	1	2	yes	—
231	#121	—	2	2	no	—
232	C₄×D₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
236	#108	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	#100	—	4	4	no	—
243	C₂²⋊C₈	—	2	4	no	—
244	#97	—	2	1	no	—
245	#97	—	2	1	no	—
246	#97	D₄×C₃₇	1	4	yes	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₂⁴	—	2	2	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	4	no	—
262	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	#41×C₃₇	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
267	SD₁₆	—	4	8	no	—
268	D₈	—	4	8	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	4	no	—
274	M₁₆	#41×C₃₇	1	4	yes	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₃₇	1	2	yes	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	4	no	—
296	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	#121×C₃₇	1	2	yes	—
313	C₂×C₄	#121×C₃₇	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
317	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
330	C₂²	C₃₇×#559	1	2	yes	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₃₇×#532	1	1	yes	characteristic
336	C₄	C₃₇×#535	1	1	yes	characteristic
337	C₄	C₃₇×#546	1	2	yes	—
338	C₂	—	2	2	no	—
339	C₂	—	8	2	no	—
340	C₂	—	16	2	no	—
341	C₂	C₃₇×#2544	1	1	yes	characteristic
342	C₂	C₃₇×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#3151	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#3142	C₂	semidirect
3. C₂×C₃₇×#490	C₂	semidirect
4. C₃₇×#2932	C₂	semidirect
5. C₃₇×#2051	C₂	semidirect
6. C₃₇×#1998	C₂	semidirect
7. C₃₇×#1965	C₂	semidirect
8. C₃₇×#1907	C₂	semidirect
9. C₃₇×#1778	C₂	semidirect
151. #28635	C₃₇	direct (N × H)
168. #3142	C₂×C₃₇	semidirect
169. C₂×#490	C₂×C₃₇	semidirect
170. #2932	C₂×C₃₇	semidirect
171. #2051	C₂×C₃₇	semidirect
172. #1998	C₂×C₃₇	semidirect
173. #1965	C₂×C₃₇	semidirect
174. #1907	C₂×C₃₇	semidirect
175. #1778	C₂×C₃₇	semidirect
206. C₃₇	#28635	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×#3151	2
C₃₇×#3142	2
C₂×C₃₇×#490	2
C₃₇×#2932	2
C₃₇×#2051	2
C₃₇×#1998	2
C₃₇×#1965	2
C₃₇×#1907	2
C₃₇×#1778	2
#28635	37

Subgroups — C37×#28635 #22649888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28635 #22649888