Subgroups — C₃₇×#28532 #22649785

Order 9472.

822 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×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#475	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2568	C₂	1	2	yes	maximal
5	C₃₇×#2097	C₂	1	4	yes	maximal
6	C₃₇×#1969	C₂	1	2	yes	maximal
7	C₃₇×#1846	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#500	—	2	4	no	—
13	C₃₇×#475	—	2	8	no	—
14	C₄×Q₁₆×C₃₇	—	2	4	no	—
15	C₂×#107×C₃₇	C₂²	1	2	yes	—
16	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₃₇	C₂²	1	2	yes	—
18	C₂×#106×C₃₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₃₇	C₂²	1	2	yes	—
21	C₂×#103×C₃₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#400	—	2	4	no	—
25	C₃₇×#391	—	2	4	no	—
26	C₃₇×#391	C₂²	1	4	yes	—
27	C₃₇×#382	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	—	2	2	no	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂×Q₁₆×C₃₇	—	2	2	no	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	complemented
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₃₇	—	2	2	no	—
48	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#107×C₃₇	—	2	2	no	—
51	#107×C₃₇	—	2	4	no	—
52	#107×C₃₇	—	2	4	no	—
53	#106×C₃₇	—	2	8	no	—
54	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
55	#103×C₃₇	—	2	4	no	—
56	#103×C₃₇	—	2	8	no	—
57	C₄×C₈×C₃₇	—	2	4	no	—
58	C₄×C₈×C₃₇	D₄	1	4	yes	—
59	#95×C₃₇	—	2	8	no	—
60	#95×C₃₇	—	2	8	no	—
61	#95×C₃₇	D₄	1	2	yes	—
62	C₂×Q₈×C₃₇	—	2	2	no	—
63	C₂×Q₈×C₃₇	—	2	2	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	—	4	4	no	—
68	C₂×Q₈×C₃₇	—	4	8	no	—
69	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	2	4	no	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₃₇	—	2	8	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	1	no	—
80	C₂×C₈×C₃₇	—	2	1	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₃₇	#41	1	2	yes	—
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	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄²×C₃₇	—	2	2	no	—
95	C₄²×C₃₇	—	2	2	no	—
96	C₄²×C₃₇	—	4	4	no	—
97	C₄²×C₃₇	C₂×D₄	1	2	yes	—
98	C₄²×C₃₇	C₂×D₄	1	2	yes	—
99	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₃₇	—	2	4	no	—
101	Q₈×C₃₇	—	2	4	no	—
102	Q₈×C₃₇	—	4	2	no	—
103	Q₈×C₃₇	—	4	2	no	—
104	Q₈×C₃₇	—	8	8	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	—	2	4	no	—
108	C₂×C₄×C₃₇	—	2	4	no	—
109	C₂×C₄×C₃₇	—	2	4	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	4	no	—
114	C₂×C₄×C₃₇	—	4	8	no	—
115	C₂×C₄×C₃₇	—	4	4	no	—
116	C₂×C₄×C₃₇	#127	1	2	yes	—
117	C₂×C₄×C₃₇	#127	1	2	yes	—
118	C₂×C₄×C₃₇	#135	1	4	yes	—
119	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₈×C₃₇	—	2	2	no	—
126	C₈×C₃₇	—	2	4	no	—
127	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
128	#28532	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
130	C₂²×C₃₇	#547	1	1	yes	characteristic
131	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
132	C₂²×C₃₇	C₂×#135	1	2	yes	—
133	C₂²×C₃₇	#578	1	2	yes	—
134	C₄×C₃₇	—	2	2	no	—
135	C₄×C₃₇	—	4	4	no	—
136	C₄×C₃₇	—	4	4	no	—
137	C₄×C₃₇	—	8	4	no	—
138	C₄×C₃₇	—	8	2	no	—
139	C₄×C₃₇	#495	1	4	yes	—
140	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₄×C₃₇	#536	1	1	yes	characteristic
142	C₄×C₃₇	#547	1	1	yes	characteristic
143	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
144	C₂×#500	C₂×C₃₇	1	1	yes	characteristic
145	C₂×#475	C₂×C₃₇	1	2	yes	—
146	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
147	#2568	C₂×C₃₇	1	2	yes	—
148	#2097	C₂×C₃₇	1	4	yes	—
149	#1969	C₂×C₃₇	1	2	yes	—
150	#1846	C₂×C₃₇	1	2	yes	—
151	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
152	C₂×C₃₇	#2542	1	1	yes	characteristic
153	C₂×C₃₇	C₂×#495	1	1	yes	characteristic
154	C₂×C₃₇	#3156	1	1	yes	characteristic
155	C₂×C₃₇	#3342	1	2	yes	—
156	C₂×C₃₇	#3383	1	2	yes	—
157	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
159	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
160	#500	—	2	4	no	—
161	#475	—	2	8	no	—
162	C₄×Q₁₆	—	2	4	no	—
163	C₂×#107	C₂²×C₃₇	1	2	yes	—
164	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×#107	C₂²×C₃₇	1	2	yes	—
166	C₂×#106	C₂²×C₃₇	1	4	yes	—
167	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×#103	C₂²×C₃₇	1	4	yes	—
169	C₂×#103	C₂²×C₃₇	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
172	#400	—	2	4	no	—
173	#391	—	2	4	no	—
174	#391	C₂²×C₃₇	1	4	yes	—
175	#382	C₂²×C₃₇	1	4	yes	—
176	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
177	#336	C₂²×C₃₇	1	2	yes	—
178	C₃₇	#28532	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
184	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	4	no	—
188	C₄⋊Q₈	—	2	4	no	—
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
198	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	—	2	4	no	—
201	#107	—	2	4	no	—
202	#106	—	2	8	no	—
203	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
204	#103	—	2	4	no	—
205	#103	—	2	8	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	D₄×C₃₇	1	4	yes	—
208	#95	—	2	8	no	—
209	#95	—	2	8	no	—
210	#95	D₄×C₃₇	1	2	yes	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	8	no	—
218	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
233	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
234	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
235	C₂×C₈	#41×C₃₇	1	2	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	4	no	—
246	C₄²	C₂×D₄×C₃₇	1	2	yes	—
247	C₄²	C₂×D₄×C₃₇	1	2	yes	—
248	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	8	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	#127×C₃₇	1	2	yes	—
266	C₂×C₄	#127×C₃₇	1	2	yes	—
267	C₂×C₄	#135×C₃₇	1	4	yes	—
268	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
272	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
277	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
278	C₂²	C₃₇×#547	1	1	yes	characteristic
279	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
280	C₂²	C₂×#135×C₃₇	1	2	yes	—
281	C₂²	C₃₇×#578	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	4	no	—
286	C₄	—	8	2	no	—
287	C₄	C₃₇×#495	1	4	yes	—
288	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
289	C₄	C₃₇×#536	1	1	yes	characteristic
290	C₄	C₃₇×#547	1	1	yes	characteristic
291	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
292	C₂	C₃₇×#2542	1	1	yes	characteristic
293	C₂	C₂×C₃₇×#495	1	1	yes	characteristic
294	C₂	C₃₇×#3156	1	1	yes	characteristic
295	C₂	C₃₇×#3342	1	2	yes	—
296	C₂	C₃₇×#3383	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 ◁ C₄×C₄⋊C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂×Q₁₆×C₃₇	Q₈	semidirect
128. #28532	C₃₇	direct (N × H)
178. C₃₇	#28532	direct (N × H)
183. C₂×Q₁₆	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×#500	2
C₂×C₃₇×#475	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2568	2
C₃₇×#2097	2
C₃₇×#1969	2
C₃₇×#1846	2
C₈×C₄⋊C₄×C₃₇	2
#28532	37

Subgroups — C37×#28532 #22649785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28532 #22649785