Subgroups — C₃₇×#3183 #17992001

Order 4736.

370 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#529	C₂	1	2	yes	maximal, complemented
2	C₃₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#488	C₂	1	2	yes	maximal, complemented
5	C₃₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#469	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#468	C₂	1	2	yes	maximal, complemented
8	C₃₇×#442	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₇×#440	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂×D₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	#126×C₃₇	—	2	2	no	—
18	#125×C₃₇	C₂²	1	2	yes	—
19	#124×C₃₇	C₂²	1	2	yes	—
20	#123×C₃₇	—	2	2	no	—
21	C₂²⋊Q₈×C₃₇	C₂²	1	2	yes	—
22	#121×C₃₇	C₂²	1	2	yes	—
23	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
24	C₄×D₄×C₃₇	—	2	2	no	—
25	C₄×D₄×C₃₇	C₂²	1	2	yes	—
26	#117×C₃₇	—	2	2	no	—
27	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
28	#107×C₃₇	C₂²	1	2	yes	—
29	#106×C₃₇	C₂²	1	1	yes	characteristic
30	#106×C₃₇	C₂²	1	1	yes	characteristic
31	#103×C₃₇	C₂²	1	2	yes	—
32	#103×C₃₇	C₂²	1	2	yes	—
33	#102×C₃₇	C₂²	1	2	yes	—
34	#102×C₃₇	C₂²	1	2	yes	—
35	#97×C₃₇	C₂²	1	1	yes	characteristic
36	#97×C₃₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
40	C₂×D₄×C₃₇	—	2	2	no	—
41	C₂×D₄×C₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₇	—	2	2	no	—
43	C₂²×C₄×C₃₇	D₄	1	2	yes	—
44	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
45	Q₁₆×C₃₇	—	2	2	no	—
46	SD₁₆×C₃₇	—	2	4	no	—
47	D₈×C₃₇	—	2	2	no	—
48	M₁₆×C₃₇	—	2	2	no	—
49	M₁₆×C₃₇	—	2	2	no	—
50	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₇	—	2	2	no	—
55	C₄⋊C₄×C₃₇	—	2	2	no	—
56	C₄⋊C₄×C₃₇	—	2	2	no	—
57	C₄⋊C₄×C₃₇	—	2	2	no	—
58	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
60	C₂²⋊C₄×C₃₇	—	2	2	no	—
61	C₂²⋊C₄×C₃₇	—	2	2	no	—
62	C₂²⋊C₄×C₃₇	—	2	2	no	—
63	C₂²⋊C₄×C₃₇	—	2	2	no	—
64	C₂²⋊C₄×C₃₇	—	2	2	no	—
65	C₄²×C₃₇	—	2	2	no	—
66	C₄²×C₃₇	D₄	1	2	yes	—
67	C₄²×C₃₇	C₂³	1	1	yes	characteristic
68	C₄²×C₃₇	C₂³	1	1	yes	characteristic
69	C₂³×C₃₇	—	2	2	no	—
70	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₃₇	—	2	2	no	—
72	Q₈×C₃₇	—	2	2	no	—
73	D₄×C₃₇	—	2	2	no	—
74	D₄×C₃₇	—	2	2	no	—
75	D₄×C₃₇	—	4	4	no	—
76	C₂×C₄×C₃₇	—	2	1	no	—
77	C₂×C₄×C₃₇	—	2	1	no	—
78	C₂×C₄×C₃₇	—	2	2	no	—
79	C₂×C₄×C₃₇	—	2	2	no	—
80	C₂×C₄×C₃₇	—	2	2	no	—
81	C₂×C₄×C₃₇	—	4	2	no	—
82	C₂×C₄×C₃₇	—	4	4	no	—
83	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
87	C₈×C₃₇	—	2	1	no	—
88	C₈×C₃₇	—	2	1	no	—
89	C₈×C₃₇	#41	1	2	yes	—
90	C₈×C₃₇	#41	1	2	yes	—
91	C₂²×C₃₇	—	2	1	no	—
92	C₂²×C₃₇	—	2	2	no	—
93	C₂²×C₃₇	—	4	2	no	—
94	C₂²×C₃₇	—	4	2	no	—
95	C₂²×C₃₇	—	4	2	no	—
96	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₃₇	—	2	1	no	—
98	C₄×C₃₇	—	2	2	no	—
99	C₄×C₃₇	—	2	2	no	—
100	C₄×C₃₇	—	4	2	no	—
101	C₄×C₃₇	—	4	2	no	—
102	C₄×C₃₇	—	4	2	no	—
103	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
104	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
105	#3183	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂×C₃₇	—	4	1	no	—
107	C₂×C₃₇	—	8	2	no	—
108	C₂×C₃₇	#532	1	1	yes	characteristic
109	C₂×C₃₇	#575	1	2	yes	—
110	#529	C₂×C₃₇	1	2	yes	complemented
111	#499	C₂×C₃₇	1	1	yes	characteristic, complemented
112	#495	C₂×C₃₇	1	1	yes	characteristic, complemented
113	#488	C₂×C₃₇	1	2	yes	complemented
114	#470	C₂×C₃₇	1	1	yes	characteristic, complemented
115	#469	C₂×C₃₇	1	1	yes	characteristic
116	#468	C₂×C₃₇	1	2	yes	complemented
117	#442	C₂×C₃₇	1	1	yes	characteristic, complemented
118	#441	C₂×C₃₇	1	1	yes	characteristic, complemented
119	#440	C₂×C₃₇	1	2	yes	complemented
120	C₄×M₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
121	C₃₇	#3183	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
123	C₂×SD₁₆	C₂²×C₃₇	1	2	yes	—
124	C₂×D₈	C₂²×C₃₇	1	1	yes	characteristic
125	C₂×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
126	C₂×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
127	#126	—	2	2	no	—
128	#125	C₂²×C₃₇	1	2	yes	—
129	#124	C₂²×C₃₇	1	2	yes	—
130	#123	—	2	2	no	—
131	C₂²⋊Q₈	C₂²×C₃₇	1	2	yes	—
132	#121	C₂²×C₃₇	1	2	yes	—
133	C₄×Q₈	C₂²×C₃₇	1	2	yes	—
134	C₄×D₄	—	2	2	no	—
135	C₄×D₄	C₂²×C₃₇	1	2	yes	—
136	#117	—	2	2	no	—
137	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
138	#107	C₂²×C₃₇	1	2	yes	—
139	#106	C₂²×C₃₇	1	1	yes	characteristic
140	#106	C₂²×C₃₇	1	1	yes	characteristic
141	#103	C₂²×C₃₇	1	2	yes	—
142	#103	C₂²×C₃₇	1	2	yes	—
143	#102	C₂²×C₃₇	1	2	yes	—
144	#102	C₂²×C₃₇	1	2	yes	—
145	#97	C₂²×C₃₇	1	1	yes	characteristic
146	#97	C₂²×C₃₇	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
149	C₂×Q₈	C₂³×C₃₇	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	C₂³×C₃₇	1	2	yes	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	D₄×C₃₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
155	Q₁₆	—	2	2	no	—
156	SD₁₆	—	2	4	no	—
157	D₈	—	2	2	no	—
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	C₂×C₈	C₂³×C₃₇	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₃₇	1	1	yes	characteristic
162	C₂×C₈	C₂³×C₃₇	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₃₇	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
169	C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	D₄×C₃₇	1	2	yes	—
177	C₄²	C₂³×C₃₇	1	1	yes	characteristic
178	C₄²	C₂³×C₃₇	1	1	yes	characteristic
179	C₂³	—	2	2	no	—
180	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
181	Q₈	—	2	2	no	—
182	Q₈	—	2	2	no	—
183	D₄	—	2	2	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	4	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
196	C₂×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	#41×C₃₇	1	2	yes	—
200	C₈	#41×C₃₇	1	2	yes	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×D₄×C₃₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₂×#41×C₃₇	1	1	yes	characteristic
214	C₄	C₂×#41×C₃₇	1	1	yes	characteristic
215	C₂	—	4	1	no	—
216	C₂	—	8	2	no	—
217	C₂	C₃₇×#532	1	1	yes	characteristic
218	C₂	C₃₇×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#529	C₂	semidirect
2. C₃₇×#499	C₂	semidirect
3. C₃₇×#495	C₂	semidirect
4. C₃₇×#488	C₂	semidirect
5. C₃₇×#470	C₂	semidirect
7. C₃₇×#468	C₂	semidirect
8. C₃₇×#442	C₂	semidirect
9. C₃₇×#441	C₂	semidirect
10. C₃₇×#440	C₂	semidirect
11. C₄×M₁₆×C₃₇	C₂	semidirect
105. #3183	C₃₇	direct (N × H)
110. #529	C₂×C₃₇	semidirect
111. #499	C₂×C₃₇	semidirect
112. #495	C₂×C₃₇	semidirect
113. #488	C₂×C₃₇	semidirect
114. #470	C₂×C₃₇	semidirect
116. #468	C₂×C₃₇	semidirect
117. #442	C₂×C₃₇	semidirect
118. #441	C₂×C₃₇	semidirect
119. #440	C₂×C₃₇	semidirect
120. C₄×M₁₆	C₂×C₃₇	semidirect
121. C₃₇	#3183	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×#529	2
C₃₇×#499	2
C₃₇×#495	2
C₃₇×#488	2
C₃₇×#470	2
C₃₇×#469	2
C₃₇×#468	2
C₃₇×#442	2
C₃₇×#441	2
C₃₇×#440	2
C₄×M₁₆×C₃₇	2
#3183	37

Subgroups — C37×#3183 #17992001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#3183 #17992001