Subgroups — C₃₇×#28691 #22649944

Order 9472.

710 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#499	C₂	1	2	yes	maximal
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2085	C₂	1	4	yes	maximal
5	C₃₇×#1991	C₂	1	2	yes	maximal
6	C₃₇×#1948	C₂	1	2	yes	maximal
7	C₃₇×#1947	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₃₇	C₂²	1	2	yes	—
10	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
11	C₃₇×#499	—	2	8	no	—
12	C₂×#107×C₃₇	C₂²	1	4	yes	—
13	C₂×#106×C₃₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
17	C₃₇×#389	—	2	2	no	—
18	C₃₇×#389	—	2	2	no	—
19	C₃₇×#389	C₂²	1	2	yes	—
20	C₃₇×#387	—	2	2	no	—
21	C₃₇×#384	C₂²	1	2	yes	—
22	C₃₇×#382	C₂²	1	1	yes	characteristic
23	C₃₇×#382	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₄³×C₃₇	C₂²	1	1	yes	characteristic
27	C₃₇×#340	C₂²	1	4	yes	—
28	C₃₇×#340	C₂²	1	4	yes	—
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	#125×C₃₇	—	2	4	no	—
33	#125×C₃₇	—	2	2	no	—
34	#125×C₃₇	—	2	4	no	—
35	#125×C₃₇	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₇	—	2	2	no	—
45	C₂×C₄²×C₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₃₇	D₄	1	2	yes	—
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#107×C₃₇	—	2	8	no	—
51	#106×C₃₇	—	2	8	no	—
52	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
53	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
54	C₄×C₈×C₃₇	—	2	4	no	—
55	#95×C₃₇	—	2	2	no	—
56	#95×C₃₇	—	2	2	no	—
57	#95×C₃₇	—	2	2	no	—
58	#95×C₃₇	—	2	4	no	—
59	C₂²×C₄×C₃₇	—	2	2	no	—
60	C₂²×C₄×C₃₇	—	2	2	no	—
61	C₂²×C₄×C₃₇	—	2	4	no	—
62	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₃₇	—	2	2	no	—
68	C₂×C₈×C₃₇	—	2	2	no	—
69	C₂×C₈×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
73	C₄⋊C₄×C₃₇	—	2	8	no	—
74	C₄⋊C₄×C₃₇	—	2	2	no	—
75	C₄⋊C₄×C₃₇	—	2	4	no	—
76	C₄⋊C₄×C₃₇	—	2	4	no	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	4	4	no	—
80	C₄⋊C₄×C₃₇	—	4	4	no	—
81	C₄⋊C₄×C₃₇	—	4	8	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
84	C₄²×C₃₇	—	2	4	no	—
85	C₄²×C₃₇	—	2	1	no	—
86	C₄²×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	—	4	2	no	—
88	C₄²×C₃₇	—	4	4	no	—
89	C₄²×C₃₇	C₂×D₄	1	2	yes	—
90	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₃₇	#41	1	2	yes	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	2	no	—
101	C₂×C₄×C₃₇	—	4	2	no	—
102	C₂×C₄×C₃₇	—	4	2	no	—
103	C₂×C₄×C₃₇	—	4	4	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	4	no	—
109	C₂×C₄×C₃₇	#121	1	4	yes	—
110	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₃₇	#135	1	4	yes	—
113	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₃₇	—	2	4	no	—
120	C₈×C₃₇	—	2	4	no	—
121	#28691	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
123	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂²×C₃₇	#536	1	1	yes	characteristic
125	C₂²×C₃₇	C₂×#135	1	2	yes	—
126	C₂²×C₃₇	#575	1	2	yes	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	2	4	no	—
129	C₄×C₃₇	—	4	2	no	—
130	C₄×C₃₇	—	4	4	no	—
131	C₄×C₃₇	—	8	2	no	—
132	C₄×C₃₇	—	8	4	no	—
133	C₄×C₃₇	#499	1	4	yes	complemented
134	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
135	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
136	C₄×C₃₇	#557	1	1	yes	characteristic
137	C₄×C₃₇	#557	1	1	yes	characteristic
138	C₂×#499	C₂×C₃₇	1	2	yes	—
139	#2859	C₂×C₃₇	1	1	yes	characteristic
140	C₄×#125	C₂×C₃₇	1	1	yes	characteristic
141	#2085	C₂×C₃₇	1	4	yes	—
142	#1991	C₂×C₃₇	1	2	yes	—
143	#1948	C₂×C₃₇	1	2	yes	—
144	#1947	C₂×C₃₇	1	2	yes	—
145	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
146	C₂×C₃₇	#2546	1	1	yes	characteristic
147	C₂×C₃₇	#3035	1	2	yes	—
148	C₂×C₃₇	C₂×#499	1	1	yes	characteristic
149	C₂×C₃₇	#3166	1	1	yes	characteristic
150	C₂×C₃₇	#3233	1	2	yes	—
151	C₂×#125	C₂²×C₃₇	1	2	yes	—
152	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
153	#499	—	2	8	no	—
154	C₂×#107	C₂²×C₃₇	1	4	yes	—
155	C₂×#106	C₂²×C₃₇	1	4	yes	—
156	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
159	#389	—	2	2	no	—
160	#389	—	2	2	no	—
161	#389	C₂²×C₃₇	1	2	yes	—
162	#387	—	2	2	no	—
163	#384	C₂²×C₃₇	1	2	yes	—
164	#382	C₂²×C₃₇	1	1	yes	characteristic
165	#382	C₂²×C₃₇	1	2	yes	—
166	C₄×C₄⋊C₄	—	2	2	no	—
167	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
168	C₄³	C₂²×C₃₇	1	1	yes	characteristic
169	#340	C₂²×C₃₇	1	4	yes	—
170	#340	C₂²×C₃₇	1	4	yes	—
171	#336	C₂²×C₃₇	1	2	yes	—
172	C₃₇	#28691	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
175	#125	—	2	4	no	—
176	#125	—	2	2	no	—
177	#125	—	2	4	no	—
178	#125	D₄×C₃₇	1	4	yes	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₃₇	1	2	yes	—
189	C₂×C₄²	D₄×C₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
193	#107	—	2	8	no	—
194	#106	—	2	8	no	—
195	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
196	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
197	C₄×C₈	—	2	4	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
215	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	4	no	—
232	C₄²	C₂×D₄×C₃₇	1	2	yes	—
233	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
234	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
235	C₄²	#41×C₃₇	1	2	yes	—
236	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	#121×C₃₇	1	4	yes	—
253	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
254	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
255	C₂×C₄	#135×C₃₇	1	4	yes	—
256	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
260	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
261	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	4	no	—
264	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
265	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
266	C₂²	C₃₇×#536	1	1	yes	characteristic
267	C₂²	C₂×#135×C₃₇	1	2	yes	—
268	C₂²	C₃₇×#575	1	2	yes	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	8	2	no	—
274	C₄	—	8	4	no	—
275	C₄	C₃₇×#499	1	4	yes	complemented
276	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
277	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
278	C₄	C₃₇×#557	1	1	yes	characteristic
279	C₄	C₃₇×#557	1	1	yes	characteristic
280	C₂	C₃₇×#2546	1	1	yes	characteristic
281	C₂	C₃₇×#3035	1	2	yes	—
282	C₂	C₂×C₃₇×#499	1	1	yes	characteristic
283	C₂	C₃₇×#3166	1	1	yes	characteristic
284	C₂	C₃₇×#3233	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
121. #28691	C₃₇	direct (N × H)
133. C₄×C₃₇	#499	semidirect
172. C₃₇	#28691	direct (N × H)
275. C₄	C₃₇×#499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₃₇×#2859	2
C₄×#125×C₃₇	2
C₃₇×#2085	2
C₃₇×#1991	2
C₃₇×#1948	2
C₃₇×#1947	2
C₄×C₄⋊C₈×C₃₇	2
#28691	37

Subgroups — C37×#28691 #22649944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28691 #22649944