Subgroups — C₃₇×#17815 #22639068

Order 9472.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2920	C₂	1	2	yes	maximal
2	C₂×C₃₇×#431	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1572	C₂	1	8	yes	maximal, complemented
4	C₃₇×#1567	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1566	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
7	C₃₇×#432	—	2	2	no	—
8	C₃₇×#432	C₂²	1	4	yes	—
9	C₃₇×#431	—	2	2	no	—
10	C₃₇×#431	C₄	1	4	yes	—
11	C₃₇×#431	C₂²	1	4	yes	—
12	C₃₇×#431	C₂²	1	4	yes	—
13	C₃₇×#423	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₃₇	C₄	1	2	yes	—
15	C₃₇×#407	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₃₇	C₄	1	2	yes	—
17	C₄×M₁₆×C₃₇	—	2	2	no	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#335	C₂²	1	8	yes	complemented
21	C₃₇×#334	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
25	C₂×M₁₆×C₃₇	—	2	4	no	—
26	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₇	D₄	1	2	yes	—
32	C₂×C₄²×C₃₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
36	C₄⋊C₈×C₃₇	—	2	4	no	—
37	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
38	C₄⋊C₈×C₃₇	C₂³	1	8	yes	—
39	C₂²⋊C₈×C₃₇	—	2	4	no	—
40	C₂²⋊C₈×C₃₇	—	2	4	no	—
41	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
42	#97×C₃₇	—	2	4	no	—
43	C₄×C₈×C₃₇	—	2	4	no	—
44	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₃₇	C₂³	1	2	yes	—
47	C₄×C₈×C₃₇	C₂³	1	2	yes	—
48	C₂⁴×C₃₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	—	2	4	no	—
50	C₂²×C₄×C₃₇	—	4	2	no	—
51	C₂²×C₄×C₃₇	—	4	2	no	—
52	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
64	M₁₆×C₃₇	—	4	8	no	—
65	C₂×C₈×C₃₇	—	2	8	no	—
66	C₂×C₈×C₃₇	—	4	2	no	—
67	C₂×C₈×C₃₇	—	4	2	no	—
68	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₃₇	C₂×D₄	1	2	yes	—
76	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₃₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₂³×C₃₇	—	2	2	no	—
81	C₂³×C₃₇	—	4	1	no	—
82	C₂³×C₃₇	—	4	1	no	—
83	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₇	—	2	2	no	—
86	C₂×C₄×C₃₇	—	2	2	no	—
87	C₂×C₄×C₃₇	—	2	4	no	—
88	C₂×C₄×C₃₇	—	2	4	no	—
89	C₂×C₄×C₃₇	—	2	4	no	—
90	C₂×C₄×C₃₇	—	2	4	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	4	1	no	—
93	C₂×C₄×C₃₇	—	4	2	no	—
94	C₂×C₄×C₃₇	—	4	1	no	—
95	C₂×C₄×C₃₇	—	4	2	no	—
96	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	#135	1	4	yes	—
103	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₃₇	—	2	4	no	—
107	C₈×C₃₇	—	2	4	no	—
108	C₈×C₃₇	—	4	8	no	—
109	#17815	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₃₇	—	2	1	no	—
111	C₂²×C₃₇	—	2	1	no	—
112	C₂²×C₃₇	—	2	2	no	—
113	C₂²×C₃₇	—	2	2	no	—
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	4	4	no	—
116	C₂²×C₃₇	#427	1	2	yes	—
117	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₃₇	—	2	1	no	—
119	C₄×C₃₇	—	2	2	no	—
120	C₄×C₃₇	—	2	2	no	—
121	C₄×C₃₇	—	2	2	no	—
122	C₄×C₃₇	—	2	1	no	—
123	C₄×C₃₇	#427	1	2	yes	—
124	C₄×C₃₇	#430	1	2	yes	—
125	C₄×C₃₇	#568	1	2	yes	—
126	C₄×C₃₇	C₂×#135	1	2	yes	—
127	#2920	C₂×C₃₇	1	2	yes	—
128	C₂×#431	C₂×C₃₇	1	1	yes	characteristic
129	#1572	C₂×C₃₇	1	8	yes	complemented
130	#1567	C₂×C₃₇	1	2	yes	complemented
131	#1566	C₂×C₃₇	1	2	yes	complemented
132	C₂×C₃₇	—	2	2	no	—
133	C₂×C₃₇	—	4	2	no	—
134	C₂×C₃₇	#2905	1	1	yes	characteristic
135	C₂×C₃₇	C₂×#427	1	1	yes	characteristic
136	C₂×C₃₇	C₂×#430	1	1	yes	characteristic
137	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
138	#432	—	2	2	no	—
139	#432	C₂²×C₃₇	1	4	yes	—
140	#431	—	2	2	no	—
141	#431	C₄×C₃₇	1	4	yes	—
142	#431	C₂²×C₃₇	1	4	yes	—
143	#431	C₂²×C₃₇	1	4	yes	—
144	#423	C₂²×C₃₇	1	4	yes	—
145	C₂×C₄⋊C₈	C₄×C₃₇	1	2	yes	—
146	#407	—	2	2	no	—
147	C₂×C₂²⋊C₈	C₄×C₃₇	1	2	yes	—
148	C₄×M₁₆	—	2	2	no	—
149	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
151	#335	C₂²×C₃₇	1	8	yes	complemented
152	#334	C₂²×C₃₇	1	8	yes	complemented
153	C₃₇	#17815	1	1	yes	characteristic, direct product factor, Sylow
154	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
155	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
156	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
162	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₃₇	1	2	yes	—
164	C₂×C₄²	Q₈×C₃₇	1	1	yes	characteristic
165	C₂×C₄²	Q₈×C₃₇	1	2	yes	—
166	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
170	C₄⋊C₈	C₂³×C₃₇	1	8	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
178	C₄×C₈	C₂³×C₃₇	1	2	yes	—
179	C₄×C₈	C₂³×C₃₇	1	2	yes	—
180	C₂⁴	C₄⋊C₄×C₃₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
185	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
189	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
196	M₁₆	—	4	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
201	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
202	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
203	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
204	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
205	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
206	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
207	C₄²	C₂×D₄×C₃₇	1	2	yes	—
208	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
210	C₄²	C₂×Q₈×C₃₇	1	1	yes	characteristic
211	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
216	C₂³	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
234	C₂×C₄	#135×C₃₇	1	4	yes	—
235	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	4	8	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	C₃₇×#427	1	2	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	C₃₇×#427	1	2	yes	—
255	C₄	C₃₇×#430	1	2	yes	—
256	C₄	C₃₇×#568	1	2	yes	—
257	C₄	C₂×#135×C₃₇	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₃₇×#2905	1	1	yes	characteristic
261	C₂	C₂×C₃₇×#427	1	1	yes	characteristic
262	C₂	C₂×C₃₇×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1572	C₂	semidirect
4. C₃₇×#1567	C₂	semidirect
5. C₃₇×#1566	C₂	semidirect
20. C₃₇×#335	C₂²	semidirect
21. C₃₇×#334	C₂²	semidirect
109. #17815	C₃₇	direct (N × H)
129. #1572	C₂×C₃₇	semidirect
130. #1567	C₂×C₃₇	semidirect
131. #1566	C₂×C₃₇	semidirect
151. #335	C₂²×C₃₇	semidirect
152. #334	C₂²×C₃₇	semidirect
153. C₃₇	#17815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2920	2
C₂×C₃₇×#431	2
C₃₇×#1572	2
C₃₇×#1567	2
C₃₇×#1566	2
#17815	37

Subgroups — C37×#17815 #22639068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#17815 #22639068