Subgroups — C₃₇×#28514 #22649767

Order 9472.

1002 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#478	C₂	1	4	yes	maximal, complemented
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2092	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1957	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1957	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₃₇	—	2	2	no	—
13	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#478	—	2	16	no	—
15	C₂×#107×C₃₇	C₂²	1	4	yes	—
16	C₂×#106×C₃₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#102×C₃₇	C₂²	1	4	yes	—
19	C₂×#102×C₃₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#390	—	2	2	no	—
23	C₃₇×#390	C₂²	1	2	yes	—
24	C₃₇×#389	—	2	2	no	—
25	C₃₇×#389	—	2	2	no	—
26	C₃₇×#384	C₂²	1	1	yes	characteristic
27	C₃₇×#384	C₂²	1	1	yes	characteristic
28	C₃₇×#384	C₂²	1	1	yes	characteristic
29	C₃₇×#382	C₂²	1	2	yes	—
30	C₄³×C₃₇	C₂²	1	1	yes	characteristic
31	C₃₇×#336	C₂²	1	2	yes	—
32	C₂²×D₄×C₃₇	—	2	2	no	—
33	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₇	—	2	2	no	—
35	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	#127×C₃₇	—	2	2	no	—
38	#127×C₃₇	Q₈	1	4	yes	—
39	#125×C₃₇	—	2	8	no	—
40	C₄×D₄×C₃₇	—	2	8	no	—
41	C₄×D₄×C₃₇	—	4	8	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#107×C₃₇	—	2	8	no	—
58	#106×C₃₇	—	2	8	no	—
59	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
60	#102×C₃₇	—	2	8	no	—
61	#102×C₃₇	—	2	8	no	—
62	C₄×C₈×C₃₇	—	2	2	no	—
63	C₄×C₈×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×D₄×C₃₇	—	2	8	no	—
67	C₂×D₄×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	2	2	no	—
69	C₂×D₄×C₃₇	—	2	4	no	—
70	C₂×D₄×C₃₇	—	4	4	no	—
71	C₂×D₄×C₃₇	—	4	4	no	—
72	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	4	8	no	—
77	C₂²×C₄×C₃₇	—	4	4	no	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₃₇	—	2	2	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
89	C₄⋊C₄×C₃₇	—	2	4	no	—
90	C₄⋊C₄×C₃₇	—	2	4	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₄⋊C₄×C₃₇	—	4	4	no	—
97	C₂²⋊C₄×C₃₇	—	4	8	no	—
98	C₂²⋊C₄×C₃₇	—	4	4	no	—
99	C₄²×C₃₇	—	2	4	no	—
100	C₄²×C₃₇	—	2	4	no	—
101	C₄²×C₃₇	—	2	2	no	—
102	C₄²×C₃₇	C₂×D₄	1	4	yes	—
103	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₃₇	#41	1	2	yes	—
105	C₂³×C₃₇	—	2	4	no	—
106	C₂³×C₃₇	—	4	2	no	—
107	C₂³×C₃₇	—	4	2	no	—
108	C₂³×C₃₇	—	4	4	no	—
109	C₂³×C₃₇	—	4	4	no	—
110	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
111	D₄×C₃₇	—	2	8	no	—
112	D₄×C₃₇	—	4	4	no	—
113	D₄×C₃₇	—	4	16	no	—
114	C₂×C₄×C₃₇	—	2	2	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	—	4	4	no	—
125	C₂×C₄×C₃₇	—	8	8	no	—
126	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₃₇	#127	1	2	yes	—
128	C₂×C₄×C₃₇	#127	1	2	yes	—
129	C₂×C₄×C₃₇	#135	1	4	yes	—
130	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
131	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
135	C₈×C₃₇	—	2	4	no	—
136	C₈×C₃₇	—	2	4	no	—
137	#28514	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₃₇	—	4	8	no	—
139	C₂²×C₃₇	—	4	4	no	—
140	C₂²×C₃₇	—	8	2	no	—
141	C₂²×C₃₇	—	8	2	no	—
142	C₂²×C₃₇	—	8	4	no	—
143	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
145	C₂²×C₃₇	#536	1	1	yes	characteristic
146	C₂²×C₃₇	C₂×#135	1	2	yes	—
147	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	C₄×C₃₇	—	2	2	no	—
149	C₄×C₃₇	—	2	2	no	—
150	C₄×C₃₇	—	4	2	no	—
151	C₄×C₃₇	—	8	2	no	—
152	C₄×C₃₇	—	8	2	no	—
153	C₄×C₃₇	—	8	2	no	—
154	C₄×C₃₇	#478	1	8	yes	complemented
155	C₄×C₃₇	#547	1	2	yes	—
156	C₄×C₃₇	D₄×Q₈	1	2	yes	—
157	C₂×#478	C₂×C₃₇	1	4	yes	complemented
158	#2859	C₂×C₃₇	1	1	yes	characteristic, complemented
159	C₄×#127	C₂×C₃₇	1	1	yes	characteristic
160	#2092	C₂×C₃₇	1	4	yes	complemented
161	#1957	C₂×C₃₇	1	1	yes	characteristic
162	#1957	C₂×C₃₇	1	1	yes	characteristic
163	#1845	C₂×C₃₇	1	1	yes	characteristic, complemented
164	#1844	C₂×C₃₇	1	1	yes	characteristic, complemented
165	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
166	C₂×C₃₇	—	8	4	no	—
167	C₂×C₃₇	#2542	1	1	yes	characteristic
168	C₂×C₃₇	C₂×#478	1	2	yes	—
169	C₂×C₃₇	#3147	1	1	yes	characteristic
170	C₂×C₃₇	#3151	1	1	yes	characteristic
171	C₂×C₃₇	#3228	1	2	yes	—
172	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
173	C₂×#125	C₂²×C₃₇	1	2	yes	—
174	C₂×C₄×D₄	—	2	2	no	—
175	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
176	#478	—	2	16	no	—
177	C₂×#107	C₂²×C₃₇	1	4	yes	—
178	C₂×#106	C₂²×C₃₇	1	4	yes	—
179	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
180	C₂×#102	C₂²×C₃₇	1	4	yes	—
181	C₂×#102	C₂²×C₃₇	1	4	yes	—
182	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
184	#390	—	2	2	no	—
185	#390	C₂²×C₃₇	1	2	yes	—
186	#389	—	2	2	no	—
187	#389	—	2	2	no	—
188	#384	C₂²×C₃₇	1	1	yes	characteristic
189	#384	C₂²×C₃₇	1	1	yes	characteristic
190	#384	C₂²×C₃₇	1	1	yes	characteristic
191	#382	C₂²×C₃₇	1	2	yes	—
192	C₄³	C₂²×C₃₇	1	1	yes	characteristic
193	#336	C₂²×C₃₇	1	2	yes	—
194	C₃₇	#28514	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
197	C₂³×C₄	—	2	2	no	—
198	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
200	#127	—	2	2	no	—
201	#127	Q₈×C₃₇	1	4	yes	—
202	#125	—	2	8	no	—
203	C₄×D₄	—	2	8	no	—
204	C₄×D₄	—	4	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	D₄×C₃₇	1	2	yes	—
216	C₂×C₄²	D₄×C₃₇	1	2	yes	—
217	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
220	#107	—	2	8	no	—
221	#106	—	2	8	no	—
222	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
223	#102	—	2	8	no	—
224	#102	—	2	8	no	—
225	C₄×C₈	—	2	2	no	—
226	C₄×C₈	—	2	2	no	—
227	#95	—	2	4	no	—
228	C₂⁴	—	2	2	no	—
229	C₂×D₄	—	2	8	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
251	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	4	8	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	C₂×D₄×C₃₇	1	4	yes	—
266	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
267	C₄²	#41×C₃₇	1	2	yes	—
268	C₂³	—	2	4	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	4	no	—
272	C₂³	—	4	4	no	—
273	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
274	D₄	—	2	8	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	16	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	8	no	—
289	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
290	C₂×C₄	#127×C₃₇	1	2	yes	—
291	C₂×C₄	#127×C₃₇	1	2	yes	—
292	C₂×C₄	#135×C₃₇	1	4	yes	—
293	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
295	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
297	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	4	no	—
305	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
306	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
307	C₂²	C₃₇×#536	1	1	yes	characteristic
308	C₂²	C₂×#135×C₃₇	1	2	yes	—
309	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₃₇×#478	1	8	yes	complemented
317	C₄	C₃₇×#547	1	2	yes	—
318	C₄	D₄×Q₈×C₃₇	1	2	yes	—
319	C₂	—	8	4	no	—
320	C₂	C₃₇×#2542	1	1	yes	characteristic
321	C₂	C₂×C₃₇×#478	1	2	yes	—
322	C₂	C₃₇×#3147	1	1	yes	characteristic
323	C₂	C₃₇×#3151	1	1	yes	characteristic
324	C₂	C₃₇×#3228	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#478	C₂	semidirect
2. C₃₇×#2859	C₂	semidirect
4. C₃₇×#2092	C₂	semidirect
7. C₃₇×#1845	C₂	semidirect
8. C₃₇×#1844	C₂	semidirect
9. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
59. C₄⋊C₈×C₃₇	D₄	semidirect
137. #28514	C₃₇	direct (N × H)
154. C₄×C₃₇	#478	semidirect
157. C₂×#478	C₂×C₃₇	semidirect
158. #2859	C₂×C₃₇	semidirect
160. #2092	C₂×C₃₇	semidirect
163. #1845	C₂×C₃₇	semidirect
164. #1844	C₂×C₃₇	semidirect
165. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
194. C₃₇	#28514	direct (N × H)
222. C₄⋊C₈	D₄×C₃₇	semidirect
316. C₄	C₃₇×#478	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#478	2
C₃₇×#2859	2
C₄×#127×C₃₇	2
C₃₇×#2092	2
C₃₇×#1957	2
C₃₇×#1845	2
C₃₇×#1844	2
C₄×C₄⋊C₈×C₃₇	2
#28514	37

Subgroups — C37×#28514 #22649767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28514 #22649767