Subgroups — C₃₇×#28519 #22649772

Order 9472.

942 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#477	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#476	C₂	1	2	yes	maximal
3	C₃₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#124×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2096	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2093	C₂	1	2	yes	maximal
7	C₃₇×#1959	C₂	1	2	yes	maximal
8	C₃₇×#1847	C₂	1	2	yes	maximal, complemented
9	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₃₇	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₂×#124×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#477	—	2	8	no	—
16	C₃₇×#476	—	2	8	no	—
17	C₂×#107×C₃₇	C₂²	1	4	yes	—
18	C₂×#106×C₃₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
20	C₂×#103×C₃₇	C₂²	1	4	yes	—
21	C₂×#102×C₃₇	C₂²	1	4	yes	—
22	C₂×#97×C₃₇	C₂²	1	2	yes	—
23	C₃₇×#391	C₂²	1	1	yes	characteristic
24	C₃₇×#390	C₂²	1	1	yes	characteristic
25	C₃₇×#388	—	2	2	no	—
26	C₃₇×#384	—	2	2	no	—
27	C₃₇×#384	C₂²	1	2	yes	—
28	C₃₇×#384	C₂²	1	2	yes	—
29	C₃₇×#382	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₃₇	—	2	2	no	—
31	C₄³×C₃₇	C₂²	1	1	yes	characteristic
32	C₃₇×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₇	—	2	2	no	—
34	C₂²×Q₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×D₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₃₇	—	2	1	no	—
37	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	—	4	8	no	—
41	#124×C₃₇	—	2	2	no	—
42	#124×C₃₇	Q₈	1	4	yes	complemented
43	C₄×Q₈×C₃₇	—	2	4	no	—
44	C₄×D₄×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
55	C₂×C₄²×C₃₇	—	2	1	no	—
56	C₂×C₄²×C₃₇	D₄	1	2	yes	—
57	C₂×C₄²×C₃₇	D₄	1	2	yes	—
58	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
61	#107×C₃₇	—	2	8	no	—
62	#106×C₃₇	—	2	8	no	—
63	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
64	#103×C₃₇	—	2	8	no	—
65	#102×C₃₇	—	2	8	no	—
66	#97×C₃₇	—	2	4	no	—
67	#95×C₃₇	—	2	2	no	—
68	#95×C₃₇	—	2	2	no	—
69	C₂⁴×C₃₇	—	2	1	no	—
70	C₂×Q₈×C₃₇	—	2	1	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	2	2	no	—
73	C₂×Q₈×C₃₇	—	4	8	no	—
74	C₂×Q₈×C₃₇	—	4	4	no	—
75	C₂×Q₈×C₃₇	C₂×Q₈	1	2	yes	—
76	C₂×D₄×C₃₇	—	2	2	no	—
77	C₂×D₄×C₃₇	—	2	1	no	—
78	C₂×D₄×C₃₇	—	2	2	no	—
79	C₂×D₄×C₃₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₃₇	—	2	1	no	—
81	C₂²×C₄×C₃₇	—	2	1	no	—
82	C₂²×C₄×C₃₇	—	2	1	no	—
83	C₂²×C₄×C₃₇	—	2	4	no	—
84	C₂²×C₄×C₃₇	—	4	2	no	—
85	C₂²×C₄×C₃₇	—	4	4	no	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₂×C₈×C₃₇	—	2	4	no	—
93	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	2	4	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	—	4	8	no	—
99	C₄⋊C₄×C₃₇	—	4	2	no	—
100	C₄⋊C₄×C₃₇	—	4	4	no	—
101	C₄⋊C₄×C₃₇	—	4	8	no	—
102	C₂²⋊C₄×C₃₇	—	2	8	no	—
103	C₂²⋊C₄×C₃₇	—	4	2	no	—
104	C₂²⋊C₄×C₃₇	—	4	4	no	—
105	C₄²×C₃₇	—	2	4	no	—
106	C₄²×C₃₇	—	2	4	no	—
107	C₄²×C₃₇	—	2	2	no	—
108	C₄²×C₃₇	—	4	2	no	—
109	C₄²×C₃₇	C₂×D₄	1	4	yes	—
110	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
111	C₄²×C₃₇	#41	1	2	yes	—
112	C₂³×C₃₇	—	2	2	no	—
113	C₂³×C₃₇	—	4	1	no	—
114	C₂³×C₃₇	—	4	1	no	—
115	C₂³×C₃₇	—	4	2	no	—
116	C₂³×C₃₇	—	4	2	no	—
117	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₃₇	—	2	4	no	—
119	Q₈×C₃₇	—	4	2	no	—
120	Q₈×C₃₇	—	8	8	no	—
121	D₄×C₃₇	—	2	4	no	—
122	D₄×C₃₇	—	4	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	4	1	no	—
130	C₂×C₄×C₃₇	—	4	1	no	—
131	C₂×C₄×C₃₇	—	4	1	no	—
132	C₂×C₄×C₃₇	—	4	1	no	—
133	C₂×C₄×C₃₇	—	4	4	no	—
134	C₂×C₄×C₃₇	—	4	2	no	—
135	C₂×C₄×C₃₇	—	4	8	no	—
136	C₂×C₄×C₃₇	—	4	2	no	—
137	C₂×C₄×C₃₇	—	8	4	no	—
138	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
139	C₂×C₄×C₃₇	#127	1	2	yes	—
140	C₂×C₄×C₃₇	#127	1	2	yes	—
141	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
142	C₂×C₄×C₃₇	#137	1	4	yes	—
143	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
147	C₈×C₃₇	—	2	8	no	—
148	#28519	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₃₇	—	4	4	no	—
150	C₂²×C₃₇	—	4	2	no	—
151	C₂²×C₃₇	—	8	1	no	—
152	C₂²×C₃₇	—	8	1	no	—
153	C₂²×C₃₇	—	8	2	no	—
154	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
156	C₂²×C₃₇	#536	1	1	yes	characteristic
157	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	C₂²×C₃₇	C₂×#137	1	2	yes	—
159	C₄×C₃₇	—	2	4	no	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	2	4	no	—
162	C₄×C₃₇	—	4	2	no	—
163	C₄×C₃₇	—	8	1	no	—
164	C₄×C₃₇	—	8	1	no	—
165	C₄×C₃₇	—	8	4	no	—
166	C₄×C₃₇	#497	1	4	yes	—
167	C₄×C₃₇	#547	1	2	yes	—
168	C₄×C₃₇	D₄×Q₈	1	2	yes	—
169	C₂×#477	C₂×C₃₇	1	2	yes	complemented
170	C₂×#476	C₂×C₃₇	1	2	yes	—
171	#2868	C₂×C₃₇	1	1	yes	characteristic, complemented
172	C₄×#124	C₂×C₃₇	1	1	yes	characteristic
173	#2096	C₂×C₃₇	1	2	yes	complemented
174	#2093	C₂×C₃₇	1	2	yes	—
175	#1959	C₂×C₃₇	1	2	yes	—
176	#1847	C₂×C₃₇	1	2	yes	complemented
177	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
178	C₂×C₃₇	—	8	2	no	—
179	C₂×C₃₇	#2542	1	1	yes	characteristic
180	C₂×C₃₇	#3079	1	2	yes	—
181	C₂×C₃₇	C₂×#497	1	1	yes	characteristic
182	C₂×C₃₇	#3151	1	1	yes	characteristic
183	C₂×C₃₇	#3238	1	2	yes	—
184	C₂×C₄⋊Q₈	—	2	2	no	—
185	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
186	C₂×#124	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
188	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
189	#477	—	2	8	no	—
190	#476	—	2	8	no	—
191	C₂×#107	C₂²×C₃₇	1	4	yes	—
192	C₂×#106	C₂²×C₃₇	1	4	yes	—
193	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
194	C₂×#103	C₂²×C₃₇	1	4	yes	—
195	C₂×#102	C₂²×C₃₇	1	4	yes	—
196	C₂×#97	C₂²×C₃₇	1	2	yes	—
197	#391	C₂²×C₃₇	1	1	yes	characteristic
198	#390	C₂²×C₃₇	1	1	yes	characteristic
199	#388	—	2	2	no	—
200	#384	—	2	2	no	—
201	#384	C₂²×C₃₇	1	2	yes	—
202	#384	C₂²×C₃₇	1	2	yes	—
203	#382	C₂²×C₃₇	1	1	yes	characteristic
204	C₄×C₂²⋊C₄	—	2	2	no	—
205	C₄³	C₂²×C₃₇	1	1	yes	characteristic
206	#336	C₂²×C₃₇	1	2	yes	—
207	C₃₇	#28519	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	C₂³×C₃₇	1	1	yes	characteristic
210	C₂²×D₄	C₂³×C₃₇	1	1	yes	characteristic
211	C₂³×C₄	—	2	1	no	—
212	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
213	C₄⋊Q₈	—	2	4	no	—
214	C₄⋊Q₈	—	2	4	no	—
215	C₄⋊Q₈	—	4	8	no	—
216	#124	—	2	2	no	—
217	#124	Q₈×C₃₇	1	4	yes	complemented
218	C₄×Q₈	—	2	4	no	—
219	C₄×D₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	D₄×C₃₇	1	2	yes	—
232	C₂×C₄²	D₄×C₃₇	1	2	yes	—
233	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
236	#107	—	2	8	no	—
237	#106	—	2	8	no	—
238	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
239	#103	—	2	8	no	—
240	#102	—	2	8	no	—
241	#97	—	2	4	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₂⁴	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	8	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	C₂×Q₈×C₃₇	1	2	yes	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	C₂×Q₈×C₃₇	1	2	yes	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	8	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	8	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₃₇	1	4	yes	—
285	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
286	C₄²	#41×C₃₇	1	2	yes	—
287	C₂³	—	2	2	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	8	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	8	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
314	C₂×C₄	#127×C₃₇	1	2	yes	—
315	C₂×C₄	#127×C₃₇	1	2	yes	—
316	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
317	C₂×C₄	#137×C₃₇	1	4	yes	—
318	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
319	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
321	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
322	C₈	—	2	8	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
329	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
330	C₂²	C₃₇×#536	1	1	yes	characteristic
331	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
332	C₂²	C₂×#137×C₃₇	1	2	yes	—
333	C₄	—	2	2	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	4	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	4	no	—
339	C₄	—	8	1	no	—
340	C₄	C₃₇×#497	1	4	yes	—
341	C₄	C₃₇×#547	1	2	yes	—
342	C₄	D₄×Q₈×C₃₇	1	2	yes	—
343	C₂	—	8	2	no	—
344	C₂	C₃₇×#2542	1	1	yes	characteristic
345	C₂	C₃₇×#3079	1	2	yes	—
346	C₂	C₂×C₃₇×#497	1	1	yes	characteristic
347	C₂	C₃₇×#3151	1	1	yes	characteristic
348	C₂	C₃₇×#3238	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂³
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×#477	C₂	semidirect
3. C₃₇×#2868	C₂	semidirect
5. C₃₇×#2096	C₂	semidirect
8. C₃₇×#1847	C₂	semidirect
9. C₃₇×#1767	C₂	semidirect
42. #124×C₃₇	Q₈	semidirect
148. #28519	C₃₇	direct (N × H)
169. C₂×#477	C₂×C₃₇	semidirect
171. #2868	C₂×C₃₇	semidirect
173. #2096	C₂×C₃₇	semidirect
176. #1847	C₂×C₃₇	semidirect
177. #1767	C₂×C₃₇	semidirect
207. C₃₇	#28519	direct (N × H)
217. #124	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#477	2
C₂×C₃₇×#476	2
C₃₇×#2868	2
C₄×#124×C₃₇	2
C₃₇×#2096	2
C₃₇×#2093	2
C₃₇×#1959	2
C₃₇×#1847	2
C₃₇×#1767	2
#28519	37

Subgroups — C37×#28519 #22649772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28519 #22649772