Subgroups — C₃₇×#19888 #22641141

Order 9472.

710 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×#3358	C₂	1	2	yes	maximal, complemented
2	C₃₇×#3278	C₂	1	2	yes	maximal, complemented
3	C₃₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#1685	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1682	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1649	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1621	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₃₇	—	2	2	no	—
11	C₃₇×#550	C₂²	1	2	yes	—
12	D₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#500	C₂²	1	1	yes	characteristic
14	C₃₇×#493	C₂²	1	1	yes	characteristic
15	C₃₇×#492	C₂²	1	2	yes	—
16	C₃₇×#486	—	2	2	no	—
17	C₃₇×#486	C₂²	1	2	yes	—
18	C₃₇×#474	—	2	2	no	—
19	C₃₇×#474	C₂²	1	2	yes	—
20	C₃₇×#468	—	2	2	no	—
21	C₃₇×#462	C₂²	1	2	yes	—
22	C₃₇×#460	C₂²	1	2	yes	—
23	C₃₇×#459	—	2	2	no	—
24	C₃₇×#459	C₂²	1	2	yes	—
25	C₃₇×#452	—	2	2	no	—
26	C₃₇×#448	—	2	2	no	—
27	C₈×Q₈×C₃₇	C₂²	1	2	yes	—
28	C₄×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₃₇	—	2	2	no	—
30	C₄×D₈×C₃₇	C₂²	1	1	yes	characteristic
31	C₃₇×#436	C₂²	1	2	yes	—
32	C₃₇×#335	C₂²	1	1	yes	characteristic
33	C₃₇×#335	C₂²	1	1	yes	characteristic
34	C₃₇×#332	C₂²	1	2	yes	—
35	C₃₇×#331	C₂²	1	2	yes	—
36	C₃₇×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₃₇×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₃₇	—	2	2	no	—
40	C₂²×Q₈×C₃₇	—	2	2	no	—
41	#137×C₃₇	—	4	8	no	—
42	C₂×Q₁₆×C₃₇	—	2	2	no	—
43	C₂×Q₁₆×C₃₇	D₄	1	1	yes	characteristic
44	C₂×SD₁₆×C₃₇	—	2	2	no	—
45	C₂×SD₁₆×C₃₇	—	2	2	no	—
46	C₂×SD₁₆×C₃₇	—	2	2	no	—
47	C₂×D₈×C₃₇	—	2	2	no	—
48	C₂×D₈×C₃₇	—	4	1	no	—
49	C₂×D₈×C₃₇	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₃₇	—	2	2	no	—
51	C₄⋊Q₈×C₃₇	—	2	2	no	—
52	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
53	#127×C₃₇	—	2	2	no	—
54	#127×C₃₇	C₂³	1	2	yes	—
55	C₂²⋊Q₈×C₃₇	—	2	2	no	—
56	C₂²⋊Q₈×C₃₇	—	4	2	no	—
57	#121×C₃₇	—	2	2	no	—
58	#121×C₃₇	—	4	2	no	—
59	C₄×Q₈×C₃₇	C₂³	1	2	yes	—
60	C₄×D₄×C₃₇	—	2	2	no	—
61	C₄×D₄×C₃₇	—	2	2	no	—
62	C₄×D₄×C₃₇	C₂³	1	2	yes	—
63	#107×C₃₇	—	2	2	no	—
64	#107×C₃₇	—	4	1	no	—
65	#107×C₃₇	D₄	1	1	yes	characteristic
66	#107×C₃₇	D₄	1	1	yes	characteristic
67	C₄⋊C₈×C₃₇	—	2	2	no	—
68	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
69	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
70	#103×C₃₇	—	2	2	no	—
71	#103×C₃₇	D₄	1	2	yes	—
72	#102×C₃₇	—	2	2	no	—
73	#102×C₃₇	—	2	2	no	—
74	#102×C₃₇	—	2	2	no	—
75	#102×C₃₇	—	2	2	no	—
76	#102×C₃₇	—	2	2	no	—
77	#102×C₃₇	D₄	1	2	yes	—
78	C₂²⋊C₈×C₃₇	—	2	2	no	—
79	C₄×C₈×C₃₇	—	2	2	no	—
80	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
83	#41×C₃₇	—	4	4	no	—
84	#41×C₃₇	—	8	2	no	—
85	C₂×Q₈×C₃₇	—	2	2	no	—
86	C₂×Q₈×C₃₇	—	4	2	no	—
87	C₂×Q₈×C₃₇	—	4	4	no	—
88	C₂×Q₈×C₃₇	—	8	2	no	—
89	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₃₇	—	2	2	no	—
91	C₂×D₄×C₃₇	—	2	2	no	—
92	C₂×D₄×C₃₇	—	2	2	no	—
93	C₂×D₄×C₃₇	—	4	2	no	—
94	C₂×D₄×C₃₇	—	4	2	no	—
95	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₇	—	2	2	no	—
97	C₂²×C₄×C₃₇	—	4	2	no	—
98	C₂²×C₄×C₃₇	—	4	2	no	—
99	Q₁₆×C₃₇	—	4	4	no	—
100	Q₁₆×C₃₇	D₈	1	4	yes	complemented
101	SD₁₆×C₃₇	—	4	4	no	—
102	SD₁₆×C₃₇	—	4	4	no	—
103	SD₁₆×C₃₇	—	8	2	no	—
104	D₈×C₃₇	—	4	1	no	—
105	D₈×C₃₇	—	4	4	no	—
106	D₈×C₃₇	—	8	2	no	—
107	M₁₆×C₃₇	—	4	4	no	—
108	C₂×C₈×C₃₇	—	2	1	no	—
109	C₂×C₈×C₃₇	—	2	2	no	—
110	C₂×C₈×C₃₇	—	2	2	no	—
111	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₃₇	—	2	2	no	—
116	C₄⋊C₄×C₃₇	—	2	2	no	—
117	C₄⋊C₄×C₃₇	—	2	2	no	—
118	C₄⋊C₄×C₃₇	—	4	2	no	—
119	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄×C₃₇	—	2	2	no	—
122	C₂²⋊C₄×C₃₇	—	4	2	no	—
123	C₂²⋊C₄×C₃₇	—	4	2	no	—
124	C₄²×C₃₇	—	2	2	no	—
125	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
126	C₂³×C₃₇	—	2	2	no	—
127	C₂³×C₃₇	—	4	2	no	—
128	Q₈×C₃₇	—	2	2	no	—
129	Q₈×C₃₇	—	4	2	no	—
130	Q₈×C₃₇	—	4	2	no	—
131	Q₈×C₃₇	—	8	4	no	—
132	Q₈×C₃₇	C₂×D₈	1	4	yes	—
133	D₄×C₃₇	—	2	2	no	—
134	D₄×C₃₇	—	2	2	no	—
135	D₄×C₃₇	—	4	2	no	—
136	D₄×C₃₇	—	4	2	no	—
137	D₄×C₃₇	—	4	2	no	—
138	D₄×C₃₇	—	4	2	no	—
139	D₄×C₃₇	—	8	4	no	—
140	D₄×C₃₇	—	8	2	no	—
141	D₄×C₃₇	—	16	2	no	—
142	C₂×C₄×C₃₇	—	2	2	no	—
143	C₂×C₄×C₃₇	—	2	2	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	2	no	—
148	C₂×C₄×C₃₇	—	8	4	no	—
149	C₂×C₄×C₃₇	—	8	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	H(5,𝔽₂)	1	1	yes	characteristic
153	C₈×C₃₇	—	2	2	no	—
154	C₈×C₃₇	—	2	1	no	—
155	C₈×C₃₇	—	2	1	no	—
156	C₈×C₃₇	—	2	1	no	—
157	C₈×C₃₇	—	4	2	no	—
158	C₈×C₃₇	—	4	2	no	—
159	C₈×C₃₇	C₂×D₈	1	2	yes	—
160	#19888	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂²×C₃₇	—	4	2	no	—
162	C₂²×C₃₇	—	4	2	no	—
163	C₂²×C₃₇	—	4	2	no	—
164	C₂²×C₃₇	—	8	2	no	—
165	C₂²×C₃₇	—	8	2	no	—
166	C₂²×C₃₇	—	8	2	no	—
167	C₂²×C₃₇	D₄²	1	1	yes	characteristic
168	C₄×C₃₇	—	2	4	no	—
169	C₄×C₃₇	—	2	1	no	—
170	C₄×C₃₇	—	4	2	no	—
171	C₄×C₃₇	—	4	2	no	—
172	C₄×C₃₇	—	8	2	no	—
173	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₃₇	#575	1	1	yes	characteristic
175	C₄×C₃₇	#576	1	1	yes	characteristic
176	C₄×C₃₇	#578	1	1	yes	characteristic
177	#3358	C₂×C₃₇	1	2	yes	complemented
178	#3278	C₂×C₃₇	1	2	yes	complemented
179	#1706	C₂×C₃₇	1	1	yes	characteristic, complemented
180	#1685	C₂×C₃₇	1	2	yes	complemented
181	#1682	C₂×C₃₇	1	2	yes	complemented
182	#1649	C₂×C₃₇	1	2	yes	complemented
183	#1621	C₂×C₃₇	1	2	yes	complemented
184	#1589	C₂×C₃₇	1	1	yes	characteristic, complemented
185	C₈×Q₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
186	C₂×C₃₇	—	8	2	no	—
187	C₂×C₃₇	—	16	2	no	—
188	C₂×C₃₇	#3277	1	1	yes	characteristic
189	C₂×C₃₇	D₄×D₈	1	1	yes	characteristic
190	C₂×C₃₇	#3291	1	1	yes	characteristic
191	C₂×#137	—	2	2	no	—
192	#550	C₂²×C₃₇	1	2	yes	—
193	D₄×Q₈	C₂²×C₃₇	1	2	yes	—
194	#500	C₂²×C₃₇	1	1	yes	characteristic
195	#493	C₂²×C₃₇	1	1	yes	characteristic
196	#492	C₂²×C₃₇	1	2	yes	—
197	#486	—	2	2	no	—
198	#486	C₂²×C₃₇	1	2	yes	—
199	#474	—	2	2	no	—
200	#474	C₂²×C₃₇	1	2	yes	—
201	#468	—	2	2	no	—
202	#462	C₂²×C₃₇	1	2	yes	—
203	#460	C₂²×C₃₇	1	2	yes	—
204	#459	—	2	2	no	—
205	#459	C₂²×C₃₇	1	2	yes	—
206	#452	—	2	2	no	—
207	#448	—	2	2	no	—
208	C₈×Q₈	C₂²×C₃₇	1	2	yes	—
209	C₄×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
210	C₄×D₈	—	2	2	no	—
211	C₄×D₈	C₂²×C₃₇	1	1	yes	characteristic
212	#436	C₂²×C₃₇	1	2	yes	—
213	#335	C₂²×C₃₇	1	1	yes	characteristic
214	#335	C₂²×C₃₇	1	1	yes	characteristic
215	#332	C₂²×C₃₇	1	2	yes	—
216	#331	C₂²×C₃₇	1	2	yes	—
217	Q₈⋊C₈	C₂²×C₃₇	1	2	yes	—
218	D₄⋊C₈	C₂²×C₃₇	1	2	yes	—
219	C₈²	C₂²×C₃₇	1	1	yes	characteristic
220	C₃₇	#19888	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#41	—	2	2	no	—
222	C₂²×Q₈	—	2	2	no	—
223	#137	—	4	8	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	D₄×C₃₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	2	2	no	—
229	C₂×D₈	—	2	2	no	—
230	C₂×D₈	—	4	1	no	—
231	C₂×D₈	D₄×C₃₇	1	1	yes	characteristic
232	C₂×M₁₆	—	2	2	no	—
233	C₄⋊Q₈	—	2	2	no	—
234	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
235	#127	—	2	2	no	—
236	#127	C₂³×C₃₇	1	2	yes	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₂²⋊Q₈	—	4	2	no	—
239	#121	—	2	2	no	—
240	#121	—	4	2	no	—
241	C₄×Q₈	C₂³×C₃₇	1	2	yes	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	C₂³×C₃₇	1	2	yes	—
245	#107	—	2	2	no	—
246	#107	—	4	1	no	—
247	#107	D₄×C₃₇	1	1	yes	characteristic
248	#107	D₄×C₃₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
251	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
252	#103	—	2	2	no	—
253	#103	D₄×C₃₇	1	2	yes	—
254	#102	—	2	2	no	—
255	#102	—	2	2	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	—	2	2	no	—
259	#102	D₄×C₃₇	1	2	yes	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
265	#41	—	4	4	no	—
266	#41	—	8	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	8	2	no	—
271	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	D₈×C₃₇	1	4	yes	complemented
283	SD₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	8	2	no	—
286	D₈	—	4	1	no	—
287	D₈	—	4	4	no	—
288	D₈	—	8	2	no	—
289	M₁₆	—	4	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
302	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	—	4	2	no	—
310	Q₈	—	2	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	4	no	—
314	Q₈	C₂×D₈×C₃₇	1	4	yes	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	4	no	—
322	D₄	—	8	2	no	—
323	D₄	—	16	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	—	8	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	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	C₂×D₈×C₃₇	1	2	yes	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	D₄²×C₃₇	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
355	C₄	C₃₇×#575	1	1	yes	characteristic
356	C₄	C₃₇×#576	1	1	yes	characteristic
357	C₄	C₃₇×#578	1	1	yes	characteristic
358	C₂	—	8	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₃₇×#3277	1	1	yes	characteristic
361	C₂	D₄×D₈×C₃₇	1	1	yes	characteristic
362	C₂	C₃₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3358	C₂	semidirect
2. C₃₇×#3278	C₂	semidirect
3. C₃₇×#1706	C₂	semidirect
4. C₃₇×#1685	C₂	semidirect
5. C₃₇×#1682	C₂	semidirect
6. C₃₇×#1649	C₂	semidirect
7. C₃₇×#1621	C₂	semidirect
8. C₃₇×#1589	C₂	semidirect
9. C₈×Q₁₆×C₃₇	C₂	semidirect
100. Q₁₆×C₃₇	D₈	semidirect
160. #19888	C₃₇	direct (N × H)
177. #3358	C₂×C₃₇	semidirect
178. #3278	C₂×C₃₇	semidirect
179. #1706	C₂×C₃₇	semidirect
180. #1685	C₂×C₃₇	semidirect
181. #1682	C₂×C₃₇	semidirect
182. #1649	C₂×C₃₇	semidirect
183. #1621	C₂×C₃₇	semidirect
184. #1589	C₂×C₃₇	semidirect
185. C₈×Q₁₆	C₂×C₃₇	semidirect
220. C₃₇	#19888	direct (N × H)
282. Q₁₆	D₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3358	2
C₃₇×#3278	2
C₃₇×#1706	2
C₃₇×#1685	2
C₃₇×#1682	2
C₃₇×#1649	2
C₃₇×#1621	2
C₃₇×#1589	2
C₈×Q₁₆×C₃₇	2
#19888	37

Subgroups — C37×#19888 #22641141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#19888 #22641141