Subgroups — C₃₇×#28505 #22649758

Order 9472.

998 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3156	C₂	1	2	yes	maximal
2	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#473	C₂	1	2	yes	maximal, complemented
4	C₃₇×#3059	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₇×#443	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#2005	C₂	1	4	yes	maximal, complemented
7	C₃₇×#1979	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₃₇	C₂²	1	4	yes	—
10	C₂×#135×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂×#131×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#536	C₂²	1	4	yes	—
15	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
16	C₃₇×#497	—	2	4	no	—
17	C₃₇×#495	—	2	2	no	—
18	C₃₇×#494	—	2	2	no	—
19	C₃₇×#494	—	2	4	no	—
20	C₃₇×#473	—	2	8	no	—
21	C₃₇×#470	—	2	4	no	—
22	C₃₇×#467	—	2	4	no	—
23	C₃₇×#443	—	2	4	no	—
24	C₂×#108×C₃₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#427	C₂²	1	1	yes	characteristic
27	C₂×#104×C₃₇	C₂²	1	2	yes	—
28	C₃₇×#419	C₂²	1	4	yes	—
29	C₂×#101×C₃₇	C₂²	1	4	yes	—
30	C₃₇×#405	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
32	C₃₇×#339	C₂²	1	2	yes	—
33	C₂×#41×C₃₇	C₂³	1	2	yes	—
34	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
35	#137×C₃₇	—	2	8	no	—
36	#137×C₃₇	—	4	4	no	—
37	#135×C₃₇	—	2	2	no	—
38	#135×C₃₇	—	2	2	no	—
39	C₂×Q₁₆×C₃₇	—	2	4	no	—
40	C₂×Q₁₆×C₃₇	—	2	2	no	—
41	C₂×Q₁₆×C₃₇	—	2	2	no	—
42	C₂×Q₁₆×C₃₇	—	2	4	no	—
43	C₂×Q₁₆×C₃₇	—	2	4	no	—
44	C₂×Q₁₆×C₃₇	—	2	4	no	—
45	C₂×Q₁₆×C₃₇	—	2	2	no	—
46	C₂×Q₁₆×C₃₇	—	2	2	no	—
47	C₂×Q₁₆×C₃₇	—	2	4	no	—
48	C₂×Q₁₆×C₃₇	D₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₇	—	2	4	no	—
50	C₂×SD₁₆×C₃₇	D₄	1	2	yes	—
51	C₂×D₈×C₃₇	D₄	1	1	yes	characteristic
52	#131×C₃₇	—	2	4	no	—
53	#131×C₃₇	—	2	4	no	—
54	C₂×M₁₆×C₃₇	—	2	2	no	—
55	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
56	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₇	—	2	2	no	—
58	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₃₇	—	2	4	no	—
61	C₄⋊Q₈×C₃₇	—	2	4	no	—
62	C₄⋊Q₈×C₃₇	—	4	2	no	—
63	#124×C₃₇	—	2	4	no	—
64	#123×C₃₇	—	4	4	no	—
65	C₂²⋊Q₈×C₃₇	—	2	8	no	—
66	#117×C₃₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₇	—	4	2	no	—
68	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
69	#108×C₃₇	—	2	2	no	—
70	#108×C₃₇	D₄	1	4	yes	—
71	#107×C₃₇	D₄	1	2	yes	—
72	#106×C₃₇	D₄	1	2	yes	—
73	#104×C₃₇	—	2	4	no	—
74	#103×C₃₇	—	2	8	no	—
75	#101×C₃₇	—	2	8	no	—
76	#97×C₃₇	—	2	2	no	—
77	C₄×C₈×C₃₇	—	2	2	no	—
78	C₄×C₈×C₃₇	D₄	1	4	yes	—
79	#41×C₃₇	—	2	2	no	—
80	#41×C₃₇	—	2	2	no	—
81	#41×C₃₇	—	2	2	no	—
82	#41×C₃₇	—	2	2	no	—
83	C₂×Q₈×C₃₇	—	2	4	no	—
84	C₂×Q₈×C₃₇	—	2	4	no	—
85	C₂×Q₈×C₃₇	—	2	4	no	—
86	C₂×Q₈×C₃₇	—	2	4	no	—
87	C₂×Q₈×C₃₇	—	2	4	no	—
88	C₂×Q₈×C₃₇	—	2	4	no	—
89	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₃₇	—	2	2	no	—
91	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₇	—	2	1	no	—
93	C₂²×C₄×C₃₇	—	2	2	no	—
94	C₂²×C₄×C₃₇	—	2	4	no	—
95	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
96	Q₁₆×C₃₇	—	2	2	no	—
97	Q₁₆×C₃₇	—	4	8	no	—
98	Q₁₆×C₃₇	—	4	4	no	—
99	Q₁₆×C₃₇	—	4	4	no	—
100	Q₁₆×C₃₇	—	4	4	no	—
101	Q₁₆×C₃₇	—	4	4	no	—
102	SD₁₆×C₃₇	—	2	4	no	—
103	SD₁₆×C₃₇	—	4	8	no	—
104	D₈×C₃₇	—	2	2	no	—
105	M₁₆×C₃₇	—	2	2	no	—
106	M₁₆×C₃₇	—	2	2	no	—
107	M₁₆×C₃₇	—	2	2	no	—
108	M₁₆×C₃₇	—	4	4	no	—
109	M₁₆×C₃₇	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₃₇	—	2	2	no	—
111	C₂×C₈×C₃₇	—	2	2	no	—
112	C₂×C₈×C₃₇	—	4	4	no	—
113	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₃₇	#41	1	2	yes	—
121	C₄⋊C₄×C₃₇	—	4	8	no	—
122	C₄⋊C₄×C₃₇	—	4	8	no	—
123	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
124	C₂²⋊C₄×C₃₇	—	2	2	no	—
125	C₂²⋊C₄×C₃₇	—	4	8	no	—
126	C₄²×C₃₇	—	2	2	no	—
127	C₄²×C₃₇	—	2	1	no	—
128	C₄²×C₃₇	C₂×D₄	1	2	yes	—
129	C₂³×C₃₇	—	2	2	no	—
130	C₂³×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
131	Q₈×C₃₇	—	2	2	no	—
132	Q₈×C₃₇	—	2	2	no	—
133	Q₈×C₃₇	—	4	4	no	—
134	Q₈×C₃₇	—	4	4	no	—
135	Q₈×C₃₇	—	4	4	no	—
136	Q₈×C₃₇	—	4	4	no	—
137	D₄×C₃₇	—	2	2	no	—
138	D₄×C₃₇	—	2	2	no	—
139	D₄×C₃₇	—	4	2	no	—
140	D₄×C₃₇	—	4	2	no	—
141	C₂×C₄×C₃₇	—	2	4	no	—
142	C₂×C₄×C₃₇	—	2	2	no	—
143	C₂×C₄×C₃₇	—	2	4	no	—
144	C₂×C₄×C₃₇	—	2	2	no	—
145	C₂×C₄×C₃₇	—	4	2	no	—
146	C₂×C₄×C₃₇	—	4	2	no	—
147	C₂×C₄×C₃₇	—	4	4	no	—
148	C₂×C₄×C₃₇	—	4	4	no	—
149	C₂×C₄×C₃₇	—	4	4	no	—
150	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈×C₃₇	—	2	4	no	—
157	C₈×C₃₇	—	4	2	no	—
158	C₈×C₃₇	#121	1	2	yes	—
159	C₈×C₃₇	#121	1	2	yes	—
160	C₈×C₃₇	#127	1	2	yes	—
161	C₈×C₃₇	#127	1	2	yes	—
162	#28505	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₃₇	—	2	2	no	—
164	C₂²×C₃₇	—	4	4	no	—
165	C₂²×C₃₇	—	4	2	no	—
166	C₂²×C₃₇	#535	1	1	yes	characteristic
167	C₂²×C₃₇	D₄²	1	1	yes	characteristic
168	C₂²×C₃₇	#550	1	1	yes	characteristic
169	C₄×C₃₇	—	2	4	no	—
170	C₄×C₃₇	—	4	4	no	—
171	C₄×C₃₇	—	4	2	no	—
172	C₄×C₃₇	—	8	4	no	—
173	C₄×C₃₇	C₂×#121	1	1	yes	characteristic
174	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
175	C₄×C₃₇	D₄²	1	1	yes	characteristic
176	C₄×C₃₇	#550	1	1	yes	characteristic
177	#3156	C₂×C₃₇	1	2	yes	—
178	C₂×#494	C₂×C₃₇	1	1	yes	characteristic, complemented
179	C₂×#473	C₂×C₃₇	1	2	yes	complemented
180	#3059	C₂×C₃₇	1	2	yes	complemented
181	C₂×#443	C₂×C₃₇	1	1	yes	characteristic
182	#2005	C₂×C₃₇	1	4	yes	complemented
183	#1979	C₂×C₃₇	1	2	yes	complemented
184	#1772	C₂×C₃₇	1	1	yes	characteristic, complemented
185	C₂×C₃₇	—	2	1	no	—
186	C₂×C₃₇	—	2	1	no	—
187	C₂×C₃₇	—	8	2	no	—
188	C₂×C₃₇	#2541	1	1	yes	characteristic
189	C₂×C₃₇	#3293	1	2	yes	—
190	C₂×#137	C₂²×C₃₇	1	4	yes	—
191	C₂×#135	C₂²×C₃₇	1	1	yes	characteristic
192	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
193	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
194	C₂×#131	C₂²×C₃₇	1	2	yes	—
195	#536	C₂²×C₃₇	1	4	yes	—
196	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
197	#497	—	2	4	no	—
198	#495	—	2	2	no	—
199	#494	—	2	2	no	—
200	#494	—	2	4	no	—
201	#473	—	2	8	no	—
202	#470	—	2	4	no	—
203	#467	—	2	4	no	—
204	#443	—	2	4	no	—
205	C₂×#108	C₂²×C₃₇	1	1	yes	characteristic
206	C₂×#108	C₂²×C₃₇	1	1	yes	characteristic
207	#427	C₂²×C₃₇	1	1	yes	characteristic
208	C₂×#104	C₂²×C₃₇	1	2	yes	—
209	#419	C₂²×C₃₇	1	4	yes	—
210	C₂×#101	C₂²×C₃₇	1	4	yes	—
211	#405	C₂²×C₃₇	1	2	yes	—
212	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
213	#339	C₂²×C₃₇	1	2	yes	—
214	C₃₇	#28505	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#41	C₂³×C₃₇	1	2	yes	—
216	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
217	#137	—	2	8	no	—
218	#137	—	4	4	no	—
219	#135	—	2	2	no	—
220	#135	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×Q₁₆	—	2	4	no	—
226	C₂×Q₁₆	—	2	4	no	—
227	C₂×Q₁₆	—	2	2	no	—
228	C₂×Q₁₆	—	2	2	no	—
229	C₂×Q₁₆	—	2	4	no	—
230	C₂×Q₁₆	D₄×C₃₇	1	1	yes	characteristic
231	C₂×SD₁₆	—	2	4	no	—
232	C₂×SD₁₆	D₄×C₃₇	1	2	yes	—
233	C₂×D₈	D₄×C₃₇	1	1	yes	characteristic
234	#131	—	2	4	no	—
235	#131	—	2	4	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
238	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
242	C₄⋊Q₈	—	2	4	no	—
243	C₄⋊Q₈	—	2	4	no	—
244	C₄⋊Q₈	—	4	2	no	—
245	#124	—	2	4	no	—
246	#123	—	4	4	no	—
247	C₂²⋊Q₈	—	2	8	no	—
248	#117	C₂³×C₃₇	1	2	yes	—
249	C₂×C₄⋊C₄	—	4	2	no	—
250	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
251	#108	—	2	2	no	—
252	#108	D₄×C₃₇	1	4	yes	—
253	#107	D₄×C₃₇	1	2	yes	—
254	#106	D₄×C₃₇	1	2	yes	—
255	#104	—	2	4	no	—
256	#103	—	2	8	no	—
257	#101	—	2	8	no	—
258	#97	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	D₄×C₃₇	1	4	yes	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	C₂×Q₈	—	2	4	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	2	4	no	—
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	2	4	no	—
271	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	8	no	—
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	4	4	no	—
283	Q₁₆	—	4	4	no	—
284	SD₁₆	—	2	4	no	—
285	SD₁₆	—	4	8	no	—
286	D₈	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	4	4	no	—
291	M₁₆	C₂×D₄×C₃₇	1	4	yes	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	4	no	—
295	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
297	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
299	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
300	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
301	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
302	C₂×C₈	#41×C₃₇	1	2	yes	—
303	C₄⋊C₄	—	4	8	no	—
304	C₄⋊C₄	—	4	8	no	—
305	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	4	8	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	C₂×D₄×C₃₇	1	2	yes	—
311	C₂³	—	2	2	no	—
312	C₂³	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	4	4	no	—
316	Q₈	—	4	4	no	—
317	Q₈	—	4	4	no	—
318	Q₈	—	4	4	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
336	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
337	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
338	C₈	—	2	4	no	—
339	C₈	—	4	2	no	—
340	C₈	#121×C₃₇	1	2	yes	—
341	C₈	#121×C₃₇	1	2	yes	—
342	C₈	#127×C₃₇	1	2	yes	—
343	C₈	#127×C₃₇	1	2	yes	—
344	C₂²	—	2	2	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	2	no	—
347	C₂²	C₃₇×#535	1	1	yes	characteristic
348	C₂²	D₄²×C₃₇	1	1	yes	characteristic
349	C₂²	C₃₇×#550	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	4	no	—
354	C₄	C₂×#121×C₃₇	1	1	yes	characteristic
355	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
356	C₄	D₄²×C₃₇	1	1	yes	characteristic
357	C₄	C₃₇×#550	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	C₃₇×#2541	1	1	yes	characteristic
362	C₂	C₃₇×#3293	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#135×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#494	C₂	semidirect
3. C₂×C₃₇×#473	C₂	semidirect
4. C₃₇×#3059	C₂	semidirect
6. C₃₇×#2005	C₂	semidirect
7. C₃₇×#1979	C₂	semidirect
8. C₃₇×#1772	C₂	semidirect
162. #28505	C₃₇	direct (N × H)
178. C₂×#494	C₂×C₃₇	semidirect
179. C₂×#473	C₂×C₃₇	semidirect
180. #3059	C₂×C₃₇	semidirect
182. #2005	C₂×C₃₇	semidirect
183. #1979	C₂×C₃₇	semidirect
184. #1772	C₂×C₃₇	semidirect
214. C₃₇	#28505	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×#3156	2
C₂×C₃₇×#494	2
C₂×C₃₇×#473	2
C₃₇×#3059	2
C₂×C₃₇×#443	2
C₃₇×#2005	2
C₃₇×#1979	2
C₃₇×#1772	2
#28505	37

Subgroups — C37×#28505 #22649758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28505 #22649758