Subgroups — C₃₇×#28495 #22649748

Order 9472.

1022 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×#3153	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#471	C₂	1	4	yes	maximal, complemented
4	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2020	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2018	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1967	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₃₇	C₂²	1	4	yes	—
10	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#532	C₂²	1	1	yes	characteristic
16	C₂×#124×C₃₇	C₂²	1	2	yes	—
17	C₃₇×#497	—	2	2	no	—
18	C₃₇×#496	—	2	2	no	—
19	C₃₇×#495	—	2	4	no	—
20	C₃₇×#495	—	2	2	no	—
21	C₃₇×#495	—	2	2	no	—
22	C₃₇×#471	—	2	16	no	—
23	C₃₇×#436	—	2	2	no	—
24	C₈×D₄×C₃₇	—	2	2	no	—
25	C₂×#108×C₃₇	C₂²	1	2	yes	—
26	C₃₇×#423	C₂²	1	1	yes	characteristic
27	C₂×#101×C₃₇	C₂²	1	4	yes	—
28	C₂×#100×C₃₇	C₂²	1	4	yes	—
29	C₃₇×#408	C₂²	1	2	yes	—
30	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
32	C₃₇×#341	C₂²	1	2	yes	—
33	C₂×#41×C₃₇	C₂³	1	2	yes	—
34	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
35	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
36	#137×C₃₇	—	2	8	no	—
37	#137×C₃₇	—	4	4	no	—
38	AGL(1,ℤ/8)×C₃₇	—	2	8	no	—
39	AGL(1,ℤ/8)×C₃₇	—	4	4	no	—
40	C₂×Q₁₆×C₃₇	—	2	4	no	—
41	C₂×Q₁₆×C₃₇	—	2	2	no	—
42	C₂×Q₁₆×C₃₇	—	4	2	no	—
43	C₂×SD₁₆×C₃₇	—	2	4	no	—
44	C₂×SD₁₆×C₃₇	—	2	4	no	—
45	C₂×SD₁₆×C₃₇	—	2	8	no	—
46	C₂×SD₁₆×C₃₇	—	2	2	no	—
47	C₂×SD₁₆×C₃₇	—	2	2	no	—
48	C₂×D₈×C₃₇	—	2	4	no	—
49	C₂×D₈×C₃₇	—	2	2	no	—
50	C₂×D₈×C₃₇	—	4	2	no	—
51	#131×C₃₇	—	2	8	no	—
52	C₂×M₁₆×C₃₇	—	2	2	no	—
53	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
54	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₃₇	—	2	2	no	—
56	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
59	#127×C₃₇	D₄	1	1	yes	characteristic
60	#124×C₃₇	—	2	2	no	—
61	#124×C₃₇	—	2	4	no	—
62	#124×C₃₇	—	2	2	no	—
63	#124×C₃₇	D₄	1	2	yes	—
64	#121×C₃₇	—	2	2	no	—
65	C₄×D₄×C₃₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
67	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
68	#108×C₃₇	D₄	1	8	yes	complemented
69	C₄⋊C₈×C₃₇	—	2	2	no	—
70	#101×C₃₇	—	2	8	no	—
71	#100×C₃₇	—	2	8	no	—
72	C₂²⋊C₈×C₃₇	—	2	4	no	—
73	#97×C₃₇	—	2	2	no	—
74	C₄×C₈×C₃₇	—	2	2	no	—
75	C₄×C₈×C₃₇	D₄	1	2	yes	—
76	C₄×C₈×C₃₇	D₄	1	2	yes	—
77	C₂⁴×C₃₇	—	2	2	no	—
78	#41×C₃₇	—	2	4	no	—
79	#41×C₃₇	—	2	4	no	—
80	C₂×Q₈×C₃₇	—	2	4	no	—
81	C₂×Q₈×C₃₇	—	2	2	no	—
82	C₂×Q₈×C₃₇	—	2	2	no	—
83	C₂×Q₈×C₃₇	—	2	4	no	—
84	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₃₇	—	2	2	no	—
86	C₂×D₄×C₃₇	—	2	2	no	—
87	C₂×D₄×C₃₇	—	2	4	no	—
88	C₂×D₄×C₃₇	—	2	2	no	—
89	C₂×D₄×C₃₇	—	2	2	no	—
90	C₂×D₄×C₃₇	—	2	4	no	—
91	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₇	—	2	2	no	—
93	C₂²×C₄×C₃₇	—	2	2	no	—
94	C₂²×C₄×C₃₇	#41	1	2	yes	—
95	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
96	Q₁₆×C₃₇	—	4	8	no	—
97	Q₁₆×C₃₇	—	4	4	no	—
98	SD₁₆×C₃₇	—	4	8	no	—
99	SD₁₆×C₃₇	—	4	8	no	—
100	SD₁₆×C₃₇	—	4	4	no	—
101	SD₁₆×C₃₇	—	4	4	no	—
102	D₈×C₃₇	—	4	8	no	—
103	D₈×C₃₇	—	4	4	no	—
104	M₁₆×C₃₇	—	2	4	no	—
105	M₁₆×C₃₇	—	2	4	no	—
106	M₁₆×C₃₇	—	4	4	no	—
107	M₁₆×C₃₇	C₂×D₄	1	8	yes	—
108	C₂×C₈×C₃₇	—	2	4	no	—
109	C₂×C₈×C₃₇	—	2	2	no	—
110	C₂×C₈×C₃₇	—	2	2	no	—
111	C₂×C₈×C₃₇	—	4	4	no	—
112	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
116	C₄⋊C₄×C₃₇	—	2	2	no	—
117	C₂²⋊C₄×C₃₇	—	2	4	no	—
118	C₂²⋊C₄×C₃₇	—	4	4	no	—
119	C₂²⋊C₄×C₃₇	—	4	4	no	—
120	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₃₇	C₂×D₄	1	2	yes	—
122	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
123	C₂³×C₃₇	—	2	2	no	—
124	C₂³×C₃₇	—	4	2	no	—
125	C₂³×C₃₇	—	4	4	no	—
126	C₂³×C₃₇	—	8	4	no	—
127	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
128	Q₈×C₃₇	—	2	4	no	—
129	Q₈×C₃₇	—	4	2	no	—
130	Q₈×C₃₇	—	4	2	no	—
131	Q₈×C₃₇	—	4	4	no	—
132	D₄×C₃₇	—	2	4	no	—
133	D₄×C₃₇	—	4	4	no	—
134	D₄×C₃₇	—	4	4	no	—
135	D₄×C₃₇	—	4	2	no	—
136	D₄×C₃₇	—	4	2	no	—
137	D₄×C₃₇	—	4	4	no	—
138	C₂×C₄×C₃₇	—	2	2	no	—
139	C₂×C₄×C₃₇	—	4	2	no	—
140	C₂×C₄×C₃₇	—	4	2	no	—
141	C₂×C₄×C₃₇	—	4	2	no	—
142	C₂×C₄×C₃₇	—	4	2	no	—
143	C₂×C₄×C₃₇	—	4	4	no	—
144	C₂×C₄×C₃₇	#121	1	2	yes	—
145	C₂×C₄×C₃₇	#121	1	2	yes	—
146	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
150	C₈×C₃₇	—	2	2	no	—
151	C₈×C₃₇	—	2	8	no	—
152	C₈×C₃₇	—	4	2	no	—
153	C₈×C₃₇	#127	1	2	yes	—
154	C₈×C₃₇	#127	1	2	yes	—
155	#28495	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₃₇	—	2	2	no	—
157	C₂²×C₃₇	—	4	2	no	—
158	C₂²×C₃₇	—	4	4	no	—
159	C₂²×C₃₇	—	8	2	no	—
160	C₂²×C₃₇	—	8	4	no	—
161	C₂²×C₃₇	—	16	4	no	—
162	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
163	C₂²×C₃₇	#550	1	2	yes	—
164	C₄×C₃₇	—	2	2	no	—
165	C₄×C₃₇	—	2	2	no	—
166	C₄×C₃₇	—	4	2	no	—
167	C₄×C₃₇	—	8	2	no	—
168	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
169	C₄×C₃₇	#535	1	1	yes	characteristic
170	C₄×C₃₇	D₄²	1	2	yes	—
171	#3153	C₂×C₃₇	1	2	yes	complemented
172	C₂×#495	C₂×C₃₇	1	1	yes	characteristic, complemented
173	C₂×#471	C₂×C₃₇	1	4	yes	complemented
174	#2931	C₂×C₃₇	1	1	yes	characteristic, complemented
175	#2020	C₂×C₃₇	1	2	yes	complemented
176	#2018	C₂×C₃₇	1	2	yes	complemented
177	#1967	C₂×C₃₇	1	2	yes	complemented
178	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
179	C₂×C₃₇	—	2	2	no	—
180	C₂×C₃₇	—	8	2	no	—
181	C₂×C₃₇	—	16	2	no	—
182	C₂×C₃₇	#2541	1	1	yes	characteristic
183	C₂×C₃₇	#3067	1	2	yes	—
184	C₂×#137	C₂²×C₃₇	1	4	yes	—
185	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	4	yes	—
186	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
187	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
188	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
189	C₂×#131	C₂²×C₃₇	1	2	yes	—
190	#532	C₂²×C₃₇	1	1	yes	characteristic
191	C₂×#124	C₂²×C₃₇	1	2	yes	—
192	#497	—	2	2	no	—
193	#496	—	2	2	no	—
194	#495	—	2	4	no	—
195	#495	—	2	2	no	—
196	#495	—	2	2	no	—
197	#471	—	2	16	no	—
198	#436	—	2	2	no	—
199	C₈×D₄	—	2	2	no	—
200	C₂×#108	C₂²×C₃₇	1	2	yes	—
201	#423	C₂²×C₃₇	1	1	yes	characteristic
202	C₂×#101	C₂²×C₃₇	1	4	yes	—
203	C₂×#100	C₂²×C₃₇	1	4	yes	—
204	#408	C₂²×C₃₇	1	2	yes	—
205	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
206	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
207	#341	C₂²×C₃₇	1	2	yes	—
208	C₃₇	#28495	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×#41	C₂³×C₃₇	1	2	yes	—
210	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
211	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
212	#137	—	2	8	no	—
213	#137	—	4	4	no	—
214	AGL(1,ℤ/8)	—	2	8	no	—
215	AGL(1,ℤ/8)	—	4	4	no	—
216	C₂×Q₁₆	—	2	4	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×Q₁₆	—	4	2	no	—
219	C₂×SD₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×SD₁₆	—	2	8	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×D₈	—	2	4	no	—
225	C₂×D₈	—	2	2	no	—
226	C₂×D₈	—	4	2	no	—
227	#131	—	2	8	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
230	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
234	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
235	#127	D₄×C₃₇	1	1	yes	characteristic
236	#124	—	2	2	no	—
237	#124	—	2	4	no	—
238	#124	—	2	2	no	—
239	#124	D₄×C₃₇	1	2	yes	—
240	#121	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₂×C₂²⋊C₄	—	2	4	no	—
243	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
244	#108	D₄×C₃₇	1	8	yes	complemented
245	C₄⋊C₈	—	2	2	no	—
246	#101	—	2	8	no	—
247	#100	—	2	8	no	—
248	C₂²⋊C₈	—	2	4	no	—
249	#97	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	D₄×C₃₇	1	2	yes	—
252	C₄×C₈	D₄×C₃₇	1	2	yes	—
253	C₂⁴	—	2	2	no	—
254	#41	—	2	4	no	—
255	#41	—	2	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	#41×C₃₇	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
272	Q₁₆	—	4	8	no	—
273	Q₁₆	—	4	4	no	—
274	SD₁₆	—	4	8	no	—
275	SD₁₆	—	4	8	no	—
276	SD₁₆	—	4	4	no	—
277	SD₁₆	—	4	4	no	—
278	D₈	—	4	8	no	—
279	D₈	—	4	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	2	4	no	—
282	M₁₆	—	4	4	no	—
283	M₁₆	C₂×D₄×C₃₇	1	8	yes	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	4	no	—
288	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	—	4	4	no	—
296	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₃₇	1	2	yes	—
298	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
299	C₂³	—	2	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
304	Q₈	—	2	4	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	4	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	#121×C₃₇	1	2	yes	—
321	C₂×C₄	#121×C₃₇	1	2	yes	—
322	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
324	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
325	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
326	C₈	—	2	2	no	—
327	C₈	—	2	8	no	—
328	C₈	—	4	2	no	—
329	C₈	#127×C₃₇	1	2	yes	—
330	C₈	#127×C₃₇	1	2	yes	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	16	4	no	—
337	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
338	C₂²	C₃₇×#550	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
344	C₄	C₃₇×#535	1	1	yes	characteristic
345	C₄	D₄²×C₃₇	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	8	2	no	—
348	C₂	—	16	2	no	—
349	C₂	C₃₇×#2541	1	1	yes	characteristic
350	C₂	C₃₇×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
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
1. C₃₇×#3153	C₂	semidirect
2. C₂×C₃₇×#495	C₂	semidirect
3. C₂×C₃₇×#471	C₂	semidirect
4. C₃₇×#2931	C₂	semidirect
5. C₃₇×#2020	C₂	semidirect
6. C₃₇×#2018	C₂	semidirect
7. C₃₇×#1967	C₂	semidirect
8. C₄×#108×C₃₇	C₂	semidirect
68. #108×C₃₇	D₄	semidirect
155. #28495	C₃₇	direct (N × H)
171. #3153	C₂×C₃₇	semidirect
172. C₂×#495	C₂×C₃₇	semidirect
173. C₂×#471	C₂×C₃₇	semidirect
174. #2931	C₂×C₃₇	semidirect
175. #2020	C₂×C₃₇	semidirect
176. #2018	C₂×C₃₇	semidirect
177. #1967	C₂×C₃₇	semidirect
178. C₄×#108	C₂×C₃₇	semidirect
208. C₃₇	#28495	direct (N × H)
244. #108	D₄×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3153	2
C₂×C₃₇×#495	2
C₂×C₃₇×#471	2
C₃₇×#2931	2
C₃₇×#2020	2
C₃₇×#2018	2
C₃₇×#1967	2
C₄×#108×C₃₇	2
#28495	37

Subgroups — C37×#28495 #22649748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28495 #22649748