Subgroups — C₃₇×#28515 #22649768

Order 9472.

834 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#479	C₂	1	4	yes	maximal
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2097	C₂	1	4	yes	maximal
5	C₃₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1956	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1844	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#479	—	2	16	no	—
14	C₂×#107×C₃₇	C₂²	1	4	yes	—
15	C₂×#106×C₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#103×C₃₇	C₂²	1	4	yes	—
18	C₂×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#391	C₂²	1	2	yes	—
22	C₃₇×#389	—	2	2	no	—
23	C₃₇×#389	—	2	2	no	—
24	C₃₇×#389	—	2	2	no	—
25	C₃₇×#384	C₂²	1	1	yes	characteristic
26	C₃₇×#384	C₂²	1	1	yes	characteristic
27	C₃₇×#384	C₂²	1	1	yes	characteristic
28	C₃₇×#382	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
30	C₄³×C₃₇	C₂²	1	1	yes	characteristic
31	C₃₇×#336	C₂²	1	2	yes	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	Q₈	1	4	yes	—
37	#125×C₃₇	—	2	8	no	—
38	C₄×Q₈×C₃₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	2	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	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	C₄×C₈×C₃₇	—	2	2	no	—
61	C₄×C₈×C₃₇	—	2	2	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	4	no	—
66	C₂×Q₈×C₃₇	—	2	4	no	—
67	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₃₇	—	2	2	no	—
69	C₂²×C₄×C₃₇	—	2	2	no	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	2	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₃₇	—	2	8	no	—
84	C₄⋊C₄×C₃₇	—	2	4	no	—
85	C₄⋊C₄×C₃₇	—	2	4	no	—
86	C₄⋊C₄×C₃₇	—	2	4	no	—
87	C₄⋊C₄×C₃₇	—	4	4	no	—
88	C₄⋊C₄×C₃₇	—	4	4	no	—
89	C₄⋊C₄×C₃₇	—	4	4	no	—
90	C₄⋊C₄×C₃₇	—	4	4	no	—
91	C₄⋊C₄×C₃₇	—	4	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄²×C₃₇	—	2	4	no	—
94	C₄²×C₃₇	—	2	4	no	—
95	C₄²×C₃₇	—	2	2	no	—
96	C₄²×C₃₇	—	4	4	no	—
97	C₄²×C₃₇	C₂×D₄	1	4	yes	—
98	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
99	C₄²×C₃₇	#41	1	2	yes	—
100	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₃₇	—	2	8	no	—
102	Q₈×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	2	2	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	4	no	—
115	C₂×C₄×C₃₇	—	4	4	no	—
116	C₂×C₄×C₃₇	—	4	4	no	—
117	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₄×C₃₇	#127	1	2	yes	—
119	C₂×C₄×C₃₇	#127	1	2	yes	—
120	C₂×C₄×C₃₇	#135	1	4	yes	—
121	C₂×C₄×C₃₇	#137	1	4	yes	—
122	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₃₇	—	2	4	no	—
127	C₈×C₃₇	—	2	4	no	—
128	#28515	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
131	C₂²×C₃₇	#536	1	1	yes	characteristic
132	C₂²×C₃₇	C₂×#135	1	2	yes	—
133	C₂²×C₃₇	C₂×#137	1	2	yes	—
134	C₄×C₃₇	—	2	2	no	—
135	C₄×C₃₇	—	2	2	no	—
136	C₄×C₃₇	—	4	2	no	—
137	C₄×C₃₇	—	4	4	no	—
138	C₄×C₃₇	—	8	2	no	—
139	C₄×C₃₇	—	8	2	no	—
140	C₄×C₃₇	—	8	2	no	—
141	C₄×C₃₇	#479	1	8	yes	complemented
142	C₄×C₃₇	#547	1	2	yes	—
143	C₄×C₃₇	D₄×Q₈	1	2	yes	—
144	C₂×#479	C₂×C₃₇	1	4	yes	—
145	#2859	C₂×C₃₇	1	1	yes	characteristic
146	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
147	#2097	C₂×C₃₇	1	4	yes	—
148	#1956	C₂×C₃₇	1	1	yes	characteristic
149	#1956	C₂×C₃₇	1	1	yes	characteristic
150	#1845	C₂×C₃₇	1	1	yes	characteristic
151	#1844	C₂×C₃₇	1	1	yes	characteristic
152	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
153	C₂×C₃₇	#2542	1	1	yes	characteristic
154	C₂×C₃₇	C₂×#479	1	2	yes	—
155	C₂×C₃₇	#3147	1	1	yes	characteristic
156	C₂×C₃₇	#3152	1	1	yes	characteristic
157	C₂×C₃₇	#3231	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×#125	C₂²×C₃₇	1	2	yes	—
160	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
161	#479	—	2	16	no	—
162	C₂×#107	C₂²×C₃₇	1	4	yes	—
163	C₂×#106	C₂²×C₃₇	1	4	yes	—
164	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
165	C₂×#103	C₂²×C₃₇	1	4	yes	—
166	C₂×#103	C₂²×C₃₇	1	4	yes	—
167	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
169	#391	C₂²×C₃₇	1	2	yes	—
170	#389	—	2	2	no	—
171	#389	—	2	2	no	—
172	#389	—	2	2	no	—
173	#384	C₂²×C₃₇	1	1	yes	characteristic
174	#384	C₂²×C₃₇	1	1	yes	characteristic
175	#384	C₂²×C₃₇	1	1	yes	characteristic
176	#382	C₂²×C₃₇	1	2	yes	—
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₄³	C₂²×C₃₇	1	1	yes	characteristic
179	#336	C₂²×C₃₇	1	2	yes	—
180	C₃₇	#28515	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
182	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
183	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	Q₈×C₃₇	1	4	yes	—
186	#125	—	2	8	no	—
187	C₄×Q₈	—	2	8	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	D₄×C₃₇	1	2	yes	—
200	C₂×C₄²	D₄×C₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
204	#107	—	2	8	no	—
205	#106	—	2	8	no	—
206	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
207	#103	—	2	8	no	—
208	#103	—	2	8	no	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	2	no	—
211	#95	—	2	4	no	—
212	#95	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
231	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	4	no	—
246	C₄²	C₂×D₄×C₃₇	1	4	yes	—
247	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
248	C₄²	#41×C₃₇	1	2	yes	—
249	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	Q₈	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
267	C₂×C₄	#127×C₃₇	1	2	yes	—
268	C₂×C₄	#127×C₃₇	1	2	yes	—
269	C₂×C₄	#135×C₃₇	1	4	yes	—
270	C₂×C₄	#137×C₃₇	1	4	yes	—
271	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
272	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
274	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
278	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
279	C₂²	C₃₇×#536	1	1	yes	characteristic
280	C₂²	C₂×#135×C₃₇	1	2	yes	—
281	C₂²	C₂×#137×C₃₇	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	8	2	no	—
289	C₄	C₃₇×#479	1	8	yes	complemented
290	C₄	C₃₇×#547	1	2	yes	—
291	C₄	D₄×Q₈×C₃₇	1	2	yes	—
292	C₂	C₃₇×#2542	1	1	yes	characteristic
293	C₂	C₂×C₃₇×#479	1	2	yes	—
294	C₂	C₃₇×#3147	1	1	yes	characteristic
295	C₂	C₃₇×#3152	1	1	yes	characteristic
296	C₂	C₃₇×#3231	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
128. #28515	C₃₇	direct (N × H)
141. C₄×C₃₇	#479	semidirect
180. C₃₇	#28515	direct (N × H)
289. C₄	C₃₇×#479	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#479	2
C₃₇×#2859	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2097	2
C₃₇×#1956	2
C₃₇×#1845	2
C₃₇×#1844	2
C₄×C₄⋊C₈×C₃₇	2
#28515	37

Subgroups — C37×#28515 #22649768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28515 #22649768