Subgroups — C₃₇×#28521 #22649774

Order 9472.

1006 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#476	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#474	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2092	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1961	C₂	1	2	yes	maximal
7	C₃₇×#1847	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₇	—	2	2	no	—
13	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#476	—	2	8	no	—
15	C₃₇×#474	—	2	8	no	—
16	C₂×#107×C₃₇	C₂²	1	4	yes	—
17	C₂×#106×C₃₇	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#102×C₃₇	C₂²	1	4	yes	—
20	C₂×#102×C₃₇	C₂²	1	4	yes	—
21	C₂×#97×C₃₇	C₂²	1	2	yes	—
22	C₃₇×#391	—	2	2	no	—
23	C₃₇×#391	—	2	2	no	—
24	C₃₇×#390	—	2	2	no	—
25	C₃₇×#390	C₂²	1	2	yes	—
26	C₃₇×#384	C₂²	1	1	yes	characteristic
27	C₃₇×#382	C₂²	1	2	yes	—
28	C₃₇×#382	C₂²	1	2	yes	—
29	C₄³×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	—	2	1	no	—
32	C₂²×Q₈×C₃₇	—	2	1	no	—
33	C₂²×D₄×C₃₇	—	2	2	no	—
34	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₃₇	—	2	2	no	—
36	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₃₇	—	2	4	no	—
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	#127×C₃₇	—	2	2	no	—
40	#127×C₃₇	Q₈	1	4	yes	complemented
41	C₄×D₄×C₃₇	—	2	4	no	—
42	C₄×D₄×C₃₇	—	2	4	no	—
43	C₄×D₄×C₃₇	—	4	8	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	1	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	#97×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	4	4	no	—
67	C₂×Q₈×C₃₇	—	4	2	no	—
68	C₂×Q₈×C₃₇	—	4	2	no	—
69	C₂×Q₈×C₃₇	—	4	4	no	—
70	C₂×D₄×C₃₇	—	2	8	no	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	—	2	2	no	—
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	2	no	—
76	C₂×D₄×C₃₇	—	4	4	no	—
77	C₂×D₄×C₃₇	—	4	4	no	—
78	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	—	2	2	no	—
82	C₂²×C₄×C₃₇	—	4	4	no	—
83	C₂²×C₄×C₃₇	—	4	8	no	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	—	2	4	no	—
91	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	4	4	no	—
97	C₄⋊C₄×C₃₇	—	4	2	no	—
98	C₄⋊C₄×C₃₇	—	4	2	no	—
99	C₂²⋊C₄×C₃₇	—	4	8	no	—
100	C₂²⋊C₄×C₃₇	—	4	4	no	—
101	C₄²×C₃₇	—	2	4	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	—	4	2	no	—
104	C₄²×C₃₇	C₂×D₄	1	4	yes	—
105	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
106	C₄²×C₃₇	#41	1	2	yes	—
107	C₂³×C₃₇	—	2	4	no	—
108	C₂³×C₃₇	—	4	2	no	—
109	C₂³×C₃₇	—	4	2	no	—
110	C₂³×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	—	4	4	no	—
112	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₃₇	—	8	4	no	—
114	Q₈×C₃₇	—	8	4	no	—
115	D₄×C₃₇	—	2	4	no	—
116	D₄×C₃₇	—	2	4	no	—
117	D₄×C₃₇	—	4	16	no	—
118	D₄×C₃₇	—	4	2	no	—
119	D₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	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	—	4	2	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	4	no	—
130	C₂×C₄×C₃₇	—	4	4	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	—	8	8	no	—
133	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
134	C₂×C₄×C₃₇	#127	1	2	yes	—
135	C₂×C₄×C₃₇	#127	1	2	yes	—
136	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
141	C₈×C₃₇	—	2	8	no	—
142	#28521	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₃₇	—	4	8	no	—
144	C₂²×C₃₇	—	4	4	no	—
145	C₂²×C₃₇	—	8	2	no	—
146	C₂²×C₃₇	—	8	2	no	—
147	C₂²×C₃₇	—	8	4	no	—
148	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
150	C₂²×C₃₇	#536	1	1	yes	characteristic
151	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₃₇	#575	1	2	yes	—
153	C₄×C₃₇	—	2	4	no	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	—	8	2	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	—	8	2	no	—
159	C₄×C₃₇	#496	1	4	yes	—
160	C₄×C₃₇	#547	1	2	yes	—
161	C₄×C₃₇	D₄×Q₈	1	2	yes	—
162	C₂×#476	C₂×C₃₇	1	2	yes	complemented
163	C₂×#474	C₂×C₃₇	1	2	yes	complemented
164	#2862	C₂×C₃₇	1	1	yes	characteristic, complemented
165	C₄×#127	C₂×C₃₇	1	1	yes	characteristic
166	#2092	C₂×C₃₇	1	4	yes	complemented
167	#1961	C₂×C₃₇	1	2	yes	—
168	#1847	C₂×C₃₇	1	2	yes	complemented
169	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
170	C₂×C₃₇	—	8	4	no	—
171	C₂×C₃₇	#2542	1	1	yes	characteristic
172	C₂×C₃₇	#3077	1	2	yes	—
173	C₂×C₃₇	C₂×#496	1	1	yes	characteristic
174	C₂×C₃₇	#3153	1	1	yes	characteristic
175	C₂×C₃₇	#3235	1	2	yes	—
176	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×C₄×D₄	—	2	2	no	—
180	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
181	#476	—	2	8	no	—
182	#474	—	2	8	no	—
183	C₂×#107	C₂²×C₃₇	1	4	yes	—
184	C₂×#106	C₂²×C₃₇	1	4	yes	—
185	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
186	C₂×#102	C₂²×C₃₇	1	4	yes	—
187	C₂×#102	C₂²×C₃₇	1	4	yes	—
188	C₂×#97	C₂²×C₃₇	1	2	yes	—
189	#391	—	2	2	no	—
190	#391	—	2	2	no	—
191	#390	—	2	2	no	—
192	#390	C₂²×C₃₇	1	2	yes	—
193	#384	C₂²×C₃₇	1	1	yes	characteristic
194	#382	C₂²×C₃₇	1	2	yes	—
195	#382	C₂²×C₃₇	1	2	yes	—
196	C₄³	C₂²×C₃₇	1	1	yes	characteristic
197	#336	C₂²×C₃₇	1	2	yes	—
198	C₃₇	#28521	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
203	C₂³×C₄	—	2	2	no	—
204	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
205	C₄⋊Q₈	—	2	4	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	#127	—	2	2	no	—
208	#127	Q₈×C₃₇	1	4	yes	complemented
209	C₄×D₄	—	2	4	no	—
210	C₄×D₄	—	2	4	no	—
211	C₄×D₄	—	4	8	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₄²	D₄×C₃₇	1	2	yes	—
221	C₂×C₄²	D₄×C₃₇	1	2	yes	—
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
225	#107	—	2	8	no	—
226	#106	—	2	8	no	—
227	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
228	#102	—	2	8	no	—
229	#102	—	2	8	no	—
230	#97	—	2	4	no	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	C₂⁴	—	2	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×D₄	—	2	8	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×D₄×C₃₇	1	4	yes	—
273	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
274	C₄²	#41×C₃₇	1	2	yes	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
281	Q₈	—	8	4	no	—
282	Q₈	—	8	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	4	16	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	8	no	—
301	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
302	C₂×C₄	#127×C₃₇	1	2	yes	—
303	C₂×C₄	#127×C₃₇	1	2	yes	—
304	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
305	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
306	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
308	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
309	C₈	—	2	8	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	4	no	—
315	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
316	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
317	C₂²	C₃₇×#536	1	1	yes	characteristic
318	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
319	C₂²	C₃₇×#575	1	2	yes	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₃₇×#496	1	4	yes	—
327	C₄	C₃₇×#547	1	2	yes	—
328	C₄	D₄×Q₈×C₃₇	1	2	yes	—
329	C₂	—	8	4	no	—
330	C₂	C₃₇×#2542	1	1	yes	characteristic
331	C₂	C₃₇×#3077	1	2	yes	—
332	C₂	C₂×C₃₇×#496	1	1	yes	characteristic
333	C₂	C₃₇×#3153	1	1	yes	characteristic
334	C₂	C₃₇×#3235	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×#476	C₂	semidirect
2. C₂×C₃₇×#474	C₂	semidirect
3. C₃₇×#2862	C₂	semidirect
5. C₃₇×#2092	C₂	semidirect
7. C₃₇×#1847	C₂	semidirect
8. C₃₇×#1767	C₂	semidirect
40. #127×C₃₇	Q₈	semidirect
59. C₄⋊C₈×C₃₇	D₄	semidirect
142. #28521	C₃₇	direct (N × H)
162. C₂×#476	C₂×C₃₇	semidirect
163. C₂×#474	C₂×C₃₇	semidirect
164. #2862	C₂×C₃₇	semidirect
166. #2092	C₂×C₃₇	semidirect
168. #1847	C₂×C₃₇	semidirect
169. #1767	C₂×C₃₇	semidirect
198. C₃₇	#28521	direct (N × H)
208. #127	Q₈×C₃₇	semidirect
227. C₄⋊C₈	D₄×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×#476	2
C₂×C₃₇×#474	2
C₃₇×#2862	2
C₄×#127×C₃₇	2
C₃₇×#2092	2
C₃₇×#1961	2
C₃₇×#1847	2
C₃₇×#1767	2
#28521	37

Subgroups — C37×#28521 #22649774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28521 #22649774