Subgroups — C₃₇×#28522 #22649775

Order 9472.

838 conjugacy classes of subgroups.

The subgroup lattice has 308 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
2	C₂×C₃₇×#475	C₂	1	2	yes	maximal
3	C₃₇×#2862	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2097	C₂	1	4	yes	maximal
6	C₃₇×#1960	C₂	1	2	yes	maximal
7	C₃₇×#1847	C₂	1	2	yes	maximal
8	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#477	—	2	8	no	—
14	C₃₇×#475	—	2	8	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×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₃₇	C₂²	1	4	yes	—
20	C₂×#97×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#391	—	2	2	no	—
22	C₃₇×#391	—	2	2	no	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#389	—	2	2	no	—
25	C₃₇×#384	C₂²	1	1	yes	characteristic
26	C₃₇×#382	C₂²	1	2	yes	—
27	C₃₇×#382	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
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×Q₈×C₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₃₇	—	2	4	no	—
36	C₄⋊Q₈×C₃₇	—	2	4	no	—
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	Q₈	1	4	yes	complemented
39	C₄×Q₈×C₃₇	—	2	4	no	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	—	2	2	no	—
50	C₂×C₄²×C₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	#107×C₃₇	—	2	8	no	—
56	#106×C₃₇	—	2	8	no	—
57	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
58	#103×C₃₇	—	2	8	no	—
59	#103×C₃₇	—	2	8	no	—
60	#97×C₃₇	—	2	4	no	—
61	#95×C₃₇	—	2	4	no	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	C₂×Q₈×C₃₇	—	2	2	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	4	4	no	—
70	C₂×Q₈×C₃₇	—	4	2	no	—
71	C₂×Q₈×C₃₇	—	4	2	no	—
72	C₂×Q₈×C₃₇	—	4	4	no	—
73	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₃₇	—	2	4	no	—
84	C₂×C₈×C₃₇	—	2	4	no	—
85	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
86	C₄⋊C₄×C₃₇	—	2	8	no	—
87	C₄⋊C₄×C₃₇	—	2	4	no	—
88	C₄⋊C₄×C₃₇	—	2	4	no	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	4	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	2	no	—
94	C₄⋊C₄×C₃₇	—	4	2	no	—
95	C₄²×C₃₇	—	2	4	no	—
96	C₄²×C₃₇	—	2	2	no	—
97	C₄²×C₃₇	—	4	2	no	—
98	C₄²×C₃₇	—	4	4	no	—
99	C₄²×C₃₇	C₂×D₄	1	4	yes	—
100	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₃₇	#41	1	2	yes	—
102	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₃₇	—	2	4	no	—
104	Q₈×C₃₇	—	2	4	no	—
105	Q₈×C₃₇	—	4	2	no	—
106	Q₈×C₃₇	—	4	2	no	—
107	Q₈×C₃₇	—	8	4	no	—
108	Q₈×C₃₇	—	8	4	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	4	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	2	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	4	2	no	—
118	C₂×C₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	4	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₃₇	#127	1	2	yes	—
126	C₂×C₄×C₃₇	#127	1	2	yes	—
127	C₂×C₄×C₃₇	#137	1	4	yes	—
128	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₃₇	—	2	8	no	—
133	#28522	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
136	C₂²×C₃₇	#536	1	1	yes	characteristic
137	C₂²×C₃₇	C₂×#137	1	2	yes	—
138	C₂²×C₃₇	#575	1	2	yes	—
139	C₄×C₃₇	—	2	4	no	—
140	C₄×C₃₇	—	2	2	no	—
141	C₄×C₃₇	—	4	2	no	—
142	C₄×C₃₇	—	4	4	no	—
143	C₄×C₃₇	—	8	2	no	—
144	C₄×C₃₇	—	8	2	no	—
145	C₄×C₃₇	—	8	2	no	—
146	C₄×C₃₇	#497	1	4	yes	—
147	C₄×C₃₇	#547	1	2	yes	—
148	C₄×C₃₇	D₄×Q₈	1	2	yes	—
149	C₂×#477	C₂×C₃₇	1	2	yes	—
150	C₂×#475	C₂×C₃₇	1	2	yes	—
151	#2862	C₂×C₃₇	1	1	yes	characteristic
152	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
153	#2097	C₂×C₃₇	1	4	yes	—
154	#1960	C₂×C₃₇	1	2	yes	—
155	#1847	C₂×C₃₇	1	2	yes	—
156	#1767	C₂×C₃₇	1	1	yes	characteristic
157	C₂×C₃₇	#2542	1	1	yes	characteristic
158	C₂×C₃₇	#3078	1	2	yes	—
159	C₂×C₃₇	C₂×#497	1	1	yes	characteristic
160	C₂×C₃₇	#3153	1	1	yes	characteristic
161	C₂×C₃₇	#3237	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
163	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
166	#477	—	2	8	no	—
167	#475	—	2	8	no	—
168	C₂×#107	C₂²×C₃₇	1	4	yes	—
169	C₂×#106	C₂²×C₃₇	1	4	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
171	C₂×#103	C₂²×C₃₇	1	4	yes	—
172	C₂×#103	C₂²×C₃₇	1	4	yes	—
173	C₂×#97	C₂²×C₃₇	1	2	yes	—
174	#391	—	2	2	no	—
175	#391	—	2	2	no	—
176	#391	C₂²×C₃₇	1	2	yes	—
177	#389	—	2	2	no	—
178	#384	C₂²×C₃₇	1	1	yes	characteristic
179	#382	C₂²×C₃₇	1	2	yes	—
180	#382	C₂²×C₃₇	1	2	yes	—
181	C₄×C₄⋊C₄	—	2	2	no	—
182	C₄³	C₂²×C₃₇	1	1	yes	characteristic
183	#336	C₂²×C₃₇	1	2	yes	—
184	C₃₇	#28522	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	Q₈×C₃₇	1	4	yes	complemented
193	C₄×Q₈	—	2	4	no	—
194	C₄×Q₈	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₄×C₃₇	1	2	yes	—
205	C₂×C₄²	D₄×C₃₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
209	#107	—	2	8	no	—
210	#106	—	2	8	no	—
211	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
212	#103	—	2	8	no	—
213	#103	—	2	8	no	—
214	#97	—	2	4	no	—
215	#95	—	2	4	no	—
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
236	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₂×D₄×C₃₇	1	4	yes	—
254	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
255	C₄²	#41×C₃₇	1	2	yes	—
256	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	8	4	no	—
262	Q₈	—	8	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
279	C₂×C₄	#127×C₃₇	1	2	yes	—
280	C₂×C₄	#127×C₃₇	1	2	yes	—
281	C₂×C₄	#137×C₃₇	1	4	yes	—
282	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
283	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
285	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
288	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
289	C₂²	C₃₇×#536	1	1	yes	characteristic
290	C₂²	C₂×#137×C₃₇	1	2	yes	—
291	C₂²	C₃₇×#575	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₃₇×#497	1	4	yes	—
300	C₄	C₃₇×#547	1	2	yes	—
301	C₄	D₄×Q₈×C₃₇	1	2	yes	—
302	C₂	C₃₇×#2542	1	1	yes	characteristic
303	C₂	C₃₇×#3078	1	2	yes	—
304	C₂	C₂×C₃₇×#497	1	1	yes	characteristic
305	C₂	C₃₇×#3153	1	1	yes	characteristic
306	C₂	C₃₇×#3237	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
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
38. C₄⋊Q₈×C₃₇	Q₈	semidirect
133. #28522	C₃₇	direct (N × H)
184. C₃₇	#28522	direct (N × H)
192. C₄⋊Q₈	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×#475	2
C₃₇×#2862	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2097	2
C₃₇×#1960	2
C₃₇×#1847	2
C₃₇×#1767	2
#28522	37

Subgroups — C37×#28522 #22649775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28522 #22649775