Subgroups — C₃₇×#12979 #22634232

Order 9472.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#2955	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₇×#350	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#575	C₄	1	1	yes	characteristic
9	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
11	C₃₇×#504	—	2	2	no	—
12	C₃₇×#504	—	2	2	no	—
13	C₂×M₃₂×C₃₇	—	2	2	no	—
14	C₂×M₃₂×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	C₃₇×#444	—	2	2	no	—
19	C₃₇×#443	—	2	2	no	—
20	C₃₇×#430	C₄	1	1	yes	characteristic
21	C₃₇×#421	C₄	1	1	yes	characteristic
22	C₃₇×#421	C₄	1	1	yes	characteristic
23	C₃₇×#405	C₂²	1	1	yes	characteristic
24	C₃₇×#350	—	2	2	no	—
25	C₃₇×#350	—	2	2	no	—
26	C₃₇×C₂²⋊C₁₆	—	2	2	no	—
27	C₂×#41×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#41×C₃₇	C₂×C₄	1	1	yes	characteristic
29	#137×C₃₇	—	2	1	no	—
30	#137×C₃₇	C₈	1	2	yes	—
31	AGL(1,ℤ/8)×C₃₇	—	2	1	no	—
32	AGL(1,ℤ/8)×C₃₇	C₈	1	2	yes	—
33	#135×C₃₇	—	2	2	no	—
34	#131×C₃₇	—	2	1	no	—
35	#131×C₃₇	—	2	1	no	—
36	#131×C₃₇	—	2	1	no	—
37	#131×C₃₇	—	2	1	no	—
38	#131×C₃₇	—	2	1	no	—
39	#131×C₃₇	—	2	1	no	—
40	#131×C₃₇	D₄	1	2	yes	—
41	#131×C₃₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₃₇	—	2	1	no	—
43	C₂×M₁₆×C₃₇	—	2	1	no	—
44	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₇	—	2	1	no	—
48	C₂²×C₈×C₃₇	—	2	1	no	—
49	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
50	#117×C₃₇	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₃₇	—	2	4	no	—
52	M₃₂×C₃₇	—	2	1	no	—
53	M₃₂×C₃₇	—	2	1	no	—
54	M₃₂×C₃₇	—	2	1	no	—
55	M₃₂×C₃₇	—	2	1	no	—
56	M₃₂×C₃₇	—	2	1	no	—
57	M₃₂×C₃₇	D₄	1	2	yes	—
58	C₂×C₁₆×C₃₇	—	2	4	no	—
59	C₂×C₁₆×C₃₇	—	2	1	no	—
60	C₂×C₁₆×C₃₇	—	2	1	no	—
61	C₂×C₁₆×C₃₇	—	2	1	no	—
62	C₂×C₁₆×C₃₇	—	2	1	no	—
63	C₂×C₁₆×C₃₇	—	2	1	no	—
64	C₂×C₁₆×C₃₇	D₄	1	2	yes	—
65	#108×C₃₇	—	2	2	no	—
66	#104×C₃₇	—	2	2	no	—
67	#104×C₃₇	—	2	2	no	—
68	#97×C₃₇	—	2	1	no	—
69	C₄×C₈×C₃₇	—	2	1	no	—
70	#41×C₃₇	—	2	1	no	—
71	#41×C₃₇	—	2	1	no	—
72	#41×C₃₇	—	2	1	no	—
73	#41×C₃₇	—	2	1	no	—
74	#41×C₃₇	—	2	1	no	—
75	#41×C₃₇	—	2	1	no	—
76	#41×C₃₇	C₂²⋊C₄	1	2	yes	—
77	#41×C₃₇	C₂×C₈	1	2	yes	—
78	C₂×Q₈×C₃₇	C₂×C₈	1	1	yes	characteristic
79	C₂×Q₈×C₃₇	M₁₆	1	1	yes	characteristic
80	C₂×D₄×C₃₇	—	2	1	no	—
81	C₂×D₄×C₃₇	—	2	1	no	—
82	C₂×D₄×C₃₇	C₂×C₈	1	1	yes	characteristic
83	C₂×D₄×C₃₇	M₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₃₇	—	2	1	no	—
85	C₂²×C₄×C₃₇	—	2	1	no	—
86	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	Q₁₆×C₃₇	—	2	2	no	—
88	SD₁₆×C₃₇	—	2	2	no	—
89	SD₁₆×C₃₇	—	2	2	no	—
90	D₈×C₃₇	—	2	2	no	—
91	M₁₆×C₃₇	—	2	1	no	—
92	M₁₆×C₃₇	—	2	1	no	—
93	M₁₆×C₃₇	—	2	2	no	—
94	M₁₆×C₃₇	—	2	2	no	—
95	M₁₆×C₃₇	—	2	1	no	—
96	M₁₆×C₃₇	—	2	1	no	—
97	M₁₆×C₃₇	—	4	1	no	—
98	M₁₆×C₃₇	—	4	1	no	—
99	M₁₆×C₃₇	C₂²⋊C₄	1	2	yes	—
100	M₁₆×C₃₇	C₂×C₈	1	2	yes	—
101	C₂×C₈×C₃₇	—	2	1	no	—
102	C₂×C₈×C₃₇	—	2	1	no	—
103	C₂×C₈×C₃₇	—	2	2	no	—
104	C₂×C₈×C₃₇	—	2	2	no	—
105	C₂×C₈×C₃₇	—	2	1	no	—
106	C₂×C₈×C₃₇	—	4	1	no	—
107	C₂×C₈×C₃₇	—	4	1	no	—
108	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₃₇	#41	1	1	yes	characteristic
114	C₄⋊C₄×C₃₇	—	2	1	no	—
115	C₂²⋊C₄×C₃₇	—	2	1	no	—
116	C₄²×C₃₇	—	2	1	no	—
117	C₁₆×C₃₇	—	2	2	no	—
118	C₁₆×C₃₇	—	2	2	no	—
119	C₁₆×C₃₇	—	4	1	no	—
120	C₁₆×C₃₇	—	4	1	no	—
121	C₂³×C₃₇	—	2	1	no	—
122	C₂³×C₃₇	—	2	1	no	—
123	C₂³×C₃₇	#131	1	1	yes	characteristic
124	Q₈×C₃₇	—	2	1	no	—
125	Q₈×C₃₇	—	2	1	no	—
126	Q₈×C₃₇	—	2	1	no	—
127	Q₈×C₃₇	C₂²⋊C₈	1	2	yes	—
128	D₄×C₃₇	—	2	2	no	—
129	D₄×C₃₇	—	2	2	no	—
130	D₄×C₃₇	—	2	1	no	—
131	D₄×C₃₇	—	2	1	no	—
132	D₄×C₃₇	—	2	1	no	—
133	D₄×C₃₇	—	4	1	no	—
134	D₄×C₃₇	—	4	1	no	—
135	D₄×C₃₇	C₂²⋊C₈	1	2	yes	—
136	C₂×C₄×C₃₇	—	2	1	no	—
137	C₂×C₄×C₃₇	—	2	2	no	—
138	C₂×C₄×C₃₇	—	2	2	no	—
139	C₂×C₄×C₃₇	—	2	1	no	—
140	C₂×C₄×C₃₇	—	4	1	no	—
141	C₂×C₄×C₃₇	—	4	1	no	—
142	C₂×C₄×C₃₇	—	4	1	no	—
143	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₄×C₃₇	#131	1	1	yes	characteristic
149	C₈×C₃₇	—	2	2	no	—
150	C₈×C₃₇	—	2	2	no	—
151	C₈×C₃₇	—	4	1	no	—
152	C₈×C₃₇	#120	1	1	yes	characteristic
153	C₈×C₃₇	#121	1	1	yes	characteristic
154	C₈×C₃₇	#121	1	1	yes	characteristic
155	C₈×C₃₇	#123	1	1	yes	characteristic
156	#12979	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₃₇	—	2	2	no	—
158	C₂²×C₃₇	—	4	1	no	—
159	C₂²×C₃₇	—	4	2	no	—
160	C₂²×C₃₇	—	4	1	no	—
161	C₂²×C₃₇	—	8	1	no	—
162	C₂²×C₃₇	#408	1	1	yes	characteristic
163	C₂²×C₃₇	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₃₇	#435	1	1	yes	characteristic
165	C₄×C₃₇	—	2	2	no	—
166	C₄×C₃₇	—	4	1	no	—
167	C₄×C₃₇	—	8	1	no	—
168	C₄×C₃₇	#386	1	1	yes	characteristic
169	C₄×C₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₃₇	C₈×D₄	1	1	yes	characteristic
171	C₄×C₃₇	#435	1	1	yes	characteristic
172	#3407	C₂×C₃₇	1	1	yes	characteristic, complemented
173	#2955	C₂×C₃₇	1	1	yes	characteristic
174	#2117	C₂×C₃₇	1	1	yes	characteristic, complemented
175	C₂×#350	C₂×C₃₇	1	1	yes	characteristic, complemented
176	#2113	C₂×C₃₇	1	1	yes	characteristic, complemented
177	#1379	C₂×C₃₇	1	1	yes	characteristic, complemented
178	#1378	C₂×C₃₇	1	1	yes	characteristic, complemented
179	C₂×C₃₇	—	2	1	no	—
180	C₂×C₃₇	—	2	1	no	—
181	C₂×C₃₇	—	2	1	no	—
182	C₂×C₃₇	—	4	2	no	—
183	C₂×C₃₇	—	8	1	no	—
184	C₂×C₃₇	#1869	1	1	yes	characteristic
185	#575	C₄×C₃₇	1	1	yes	characteristic
186	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
188	#504	—	2	2	no	—
189	#504	—	2	2	no	—
190	C₂×M₃₂	—	2	2	no	—
191	C₂×M₃₂	C₂²×C₃₇	1	1	yes	characteristic
192	C₂×M₃₂	C₂²×C₃₇	1	1	yes	characteristic
193	C₂×M₃₂	C₂²×C₃₇	1	1	yes	characteristic
194	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
195	#444	—	2	2	no	—
196	#443	—	2	2	no	—
197	#430	C₄×C₃₇	1	1	yes	characteristic
198	#421	C₄×C₃₇	1	1	yes	characteristic
199	#421	C₄×C₃₇	1	1	yes	characteristic
200	#405	C₂²×C₃₇	1	1	yes	characteristic
201	#350	—	2	2	no	—
202	#350	—	2	2	no	—
203	C₂²⋊C₁₆	—	2	2	no	—
204	C₃₇	#12979	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×#41	C₂×C₄×C₃₇	1	1	yes	characteristic
206	C₂×#41	C₂×C₄×C₃₇	1	1	yes	characteristic
207	#137	—	2	1	no	—
208	#137	C₈×C₃₇	1	2	yes	—
209	AGL(1,ℤ/8)	—	2	1	no	—
210	AGL(1,ℤ/8)	C₈×C₃₇	1	2	yes	—
211	#135	—	2	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	2	1	no	—
218	#131	D₄×C₃₇	1	2	yes	—
219	#131	D₄×C₃₇	1	2	yes	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
223	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
224	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
228	#117	C₂×C₄×C₃₇	1	1	yes	characteristic
229	M₃₂	—	2	4	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	D₄×C₃₇	1	2	yes	—
236	C₂×C₁₆	—	2	4	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	2	1	no	—
242	C₂×C₁₆	D₄×C₃₇	1	2	yes	—
243	#108	—	2	2	no	—
244	#104	—	2	2	no	—
245	#104	—	2	2	no	—
246	#97	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	C₂²⋊C₄×C₃₇	1	2	yes	—
255	#41	C₂×C₈×C₃₇	1	2	yes	—
256	C₂×Q₈	C₂×C₈×C₃₇	1	1	yes	characteristic
257	C₂×Q₈	M₁₆×C₃₇	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×C₈×C₃₇	1	1	yes	characteristic
261	C₂×D₄	M₁₆×C₃₇	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
265	Q₁₆	—	2	2	no	—
266	SD₁₆	—	2	2	no	—
267	SD₁₆	—	2	2	no	—
268	D₈	—	2	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	C₂²⋊C₄×C₃₇	1	2	yes	—
278	M₁₆	C₂×C₈×C₃₇	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
290	C₂×C₈	#41×C₃₇	1	1	yes	characteristic
291	C₂×C₈	#41×C₃₇	1	1	yes	characteristic
292	C₄⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	4	1	no	—
298	C₁₆	—	4	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	#131×C₃₇	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	C₂²⋊C₈×C₃₇	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	C₂²⋊C₈×C₃₇	1	2	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
322	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
323	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₈×C₃₇	1	1	yes	characteristic
325	C₂×C₄	C₂×M₁₆×C₃₇	1	1	yes	characteristic
326	C₂×C₄	#131×C₃₇	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	4	1	no	—
330	C₈	#120×C₃₇	1	1	yes	characteristic
331	C₈	#121×C₃₇	1	1	yes	characteristic
332	C₈	#121×C₃₇	1	1	yes	characteristic
333	C₈	#123×C₃₇	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₃₇×#408	1	1	yes	characteristic
340	C₂²	C₈×D₄×C₃₇	1	1	yes	characteristic
341	C₂²	C₃₇×#435	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	1	no	—
345	C₄	C₃₇×#386	1	1	yes	characteristic
346	C₄	C₂×C₂²⋊C₈×C₃₇	1	1	yes	characteristic
347	C₄	C₈×D₄×C₃₇	1	1	yes	characteristic
348	C₄	C₃₇×#435	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	C₃₇×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₇	#120
commutator	C₂×C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₃₇
socle	C₂×C₃₇	#1869
prime radical	C₃₇×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3407	C₂	semidirect
3. C₃₇×#2117	C₂	semidirect
4. C₂×C₃₇×#350	C₂	semidirect
5. C₃₇×#2113	C₂	semidirect
6. C₃₇×#1379	C₂	semidirect
7. C₃₇×#1378	C₂	semidirect
156. #12979	C₃₇	direct (N × H)
172. #3407	C₂×C₃₇	semidirect
174. #2117	C₂×C₃₇	semidirect
175. C₂×#350	C₂×C₃₇	semidirect
176. #2113	C₂×C₃₇	semidirect
177. #1379	C₂×C₃₇	semidirect
178. #1378	C₂×C₃₇	semidirect
204. C₃₇	#12979	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×#3407	2
C₃₇×#2955	2
C₃₇×#2117	2
C₂×C₃₇×#350	2
C₃₇×#2113	2
C₃₇×#1379	2
C₃₇×#1378	2
#12979	37

Subgroups — C37×#12979 #22634232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#12979 #22634232