Subgroups — C₂×C₃₇×#2240 #22654947

Order 9472.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 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, complemented
2	C₂×C₃₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2240	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₇×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₃₇	—	2	1	no	—
7	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₃₇	—	2	1	no	—
9	C₂×#127×C₃₇	—	2	1	no	—
10	C₂²×C₁₆×C₃₇	C₂²	1	2	yes	—
11	C₃₇×#500	—	2	2	no	—
12	C₃₇×#500	C₂²	1	4	yes	complemented
13	C₃₇×#493	—	2	2	no	—
14	C₃₇×#493	C₂²	1	4	yes	—
15	C₂×#107×C₃₇	—	2	1	no	—
16	C₂×#107×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₃₇×#357	C₂²	1	16	yes	complemented
19	C₄×C₁₆×C₃₇	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₃₇	—	4	1	no	—
21	C₂²×D₄×C₃₇	—	2	2	no	—
22	C₂²×D₄×C₃₇	—	4	1	no	—
23	C₂²×D₄×C₃₇	—	4	1	no	—
24	C₂×D₈×C₃₇	—	2	4	no	—
25	C₂×D₈×C₃₇	—	2	2	no	—
26	C₂×D₈×C₃₇	—	2	8	no	—
27	C₂×D₈×C₃₇	—	4	4	no	—
28	C₂×D₈×C₃₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₃₇	—	2	4	no	—
32	C₄⋊Q₈×C₃₇	—	4	2	no	—
33	#127×C₃₇	—	2	4	no	—
34	#127×C₃₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₇	—	4	1	no	—
37	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₃₇	—	2	4	no	—
39	C₂×C₁₆×C₃₇	C₂³	1	4	yes	—
40	#107×C₃₇	—	2	4	no	—
41	#107×C₃₇	—	2	2	no	—
42	#107×C₃₇	C₂³	1	4	yes	—
43	C₄×C₈×C₃₇	D₄	1	2	yes	—
44	C₄×C₈×C₃₇	C₂³	1	2	yes	—
45	C₂⁴×C₃₇	—	4	2	no	—
46	C₂×Q₈×C₃₇	—	4	4	no	—
47	C₂×Q₈×C₃₇	—	8	4	no	—
48	C₂×D₄×C₃₇	—	2	4	no	—
49	C₂×D₄×C₃₇	—	4	4	no	—
50	C₂×D₄×C₃₇	—	4	4	no	—
51	C₂×D₄×C₃₇	—	4	4	no	—
52	C₂×D₄×C₃₇	—	4	2	no	—
53	C₂×D₄×C₃₇	—	4	4	no	—
54	C₂×D₄×C₃₇	—	8	4	no	—
55	C₂×D₄×C₃₇	—	8	4	no	—
56	C₂²×C₄×C₃₇	—	4	2	no	—
57	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	D₈×C₃₇	—	2	8	no	—
61	D₈×C₃₇	—	4	4	no	—
62	D₈×C₃₇	—	4	8	no	—
63	C₂×C₈×C₃₇	SD₁₆	1	4	yes	—
64	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₃₇	#41	1	4	yes	—
67	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₃₇	—	2	4	no	—
69	C₄⋊C₄×C₃₇	—	4	2	no	—
70	C₄⋊C₄×C₃₇	—	4	4	no	—
71	C₄²×C₃₇	D₈	1	2	yes	—
72	C₄²×C₃₇	C₂×D₄	1	2	yes	—
73	C₁₆×C₃₇	—	2	4	no	—
74	C₂³×C₃₇	—	4	4	no	—
75	C₂³×C₃₇	—	8	4	no	—
76	C₂³×C₃₇	—	8	4	no	—
77	C₂³×C₃₇	—	8	4	no	—
78	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₃₇	—	8	8	no	—
80	D₄×C₃₇	—	4	4	no	—
81	D₄×C₃₇	—	4	4	no	—
82	D₄×C₃₇	—	8	4	no	—
83	D₄×C₃₇	—	8	8	no	—
84	D₄×C₃₇	—	8	8	no	—
85	C₂×C₄×C₃₇	—	4	4	no	—
86	C₂×C₄×C₃₇	—	8	4	no	—
87	C₂×C₄×C₃₇	D₁₆	1	4	yes	—
88	C₂×C₄×C₃₇	SD₃₂	1	4	yes	—
89	C₂×C₄×C₃₇	#124	1	2	yes	—
90	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₃₇	#124	1	2	yes	—
96	C₈×C₃₇	C₂×SD₁₆	1	4	yes	—
97	C₈×C₃₇	C₂×#41	1	2	yes	—
98	C₂×#2240	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₃₇	—	8	4	no	—
100	C₂²×C₃₇	—	8	4	no	—
101	C₂²×C₃₇	—	8	4	no	—
102	C₂²×C₃₇	—	16	8	no	—
103	C₂²×C₃₇	#486	1	2	yes	—
104	C₂²×C₃₇	C₂×D₁₆	1	2	yes	—
105	C₂²×C₃₇	C₂×SD₃₂	1	2	yes	—
106	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₃₇	—	8	4	no	—
108	C₄×C₃₇	#486	1	2	yes	—
109	C₄×C₃₇	C₂×D₁₆	1	4	yes	—
110	C₄×C₃₇	C₂×SD₃₂	1	4	yes	—
111	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
112	C₄×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
113	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
114	C₂×#493	C₂×C₃₇	1	1	yes	characteristic
115	#2240	C₂×C₃₇	1	8	yes	direct product factor
116	C₂×#357	C₂×C₃₇	1	4	yes	complemented
117	C₂×C₄×C₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
118	C₂×C₃₇	—	16	4	no	—
119	C₂×C₃₇	#2240	1	4	yes	direct product factor
120	C₂×C₃₇	C₂×#486	1	1	yes	characteristic
121	C₂×C₃₇	C₂²×D₁₆	1	1	yes	characteristic
122	C₂×C₃₇	C₂²×SD₃₂	1	1	yes	characteristic
123	C₂²×D₈	—	2	1	no	—
124	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
125	C₂×C₄⋊Q₈	—	2	1	no	—
126	C₂×#127	—	2	1	no	—
127	C₂²×C₁₆	C₂²×C₃₇	1	2	yes	—
128	#500	—	2	2	no	—
129	#500	C₂²×C₃₇	1	4	yes	complemented
130	#493	—	2	2	no	—
131	#493	C₂²×C₃₇	1	4	yes	—
132	C₂×#107	—	2	1	no	—
133	C₂×#107	C₂²×C₃₇	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
135	#357	C₂²×C₃₇	1	16	yes	complemented
136	C₄×C₁₆	C₂²×C₃₇	1	4	yes	complemented
137	C₃₇	C₂×#2240	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₈	—	4	1	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	—	4	1	no	—
141	C₂²×D₄	—	4	1	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	8	no	—
145	C₂×D₈	—	4	4	no	—
146	C₂×D₈	C₂³×C₃₇	1	4	yes	—
147	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	C₂³×C₃₇	1	4	yes	—
158	#107	—	2	4	no	—
159	#107	—	2	2	no	—
160	#107	C₂³×C₃₇	1	4	yes	—
161	C₄×C₈	D₄×C₃₇	1	2	yes	—
162	C₄×C₈	C₂³×C₃₇	1	2	yes	—
163	C₂⁴	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	4	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	8	4	no	—
173	C₂×D₄	—	8	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
176	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
178	D₈	—	2	8	no	—
179	D₈	—	4	4	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	SD₁₆×C₃₇	1	4	yes	—
182	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
184	C₂×C₈	#41×C₃₇	1	4	yes	—
185	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	D₈×C₃₇	1	2	yes	—
190	C₄²	C₂×D₄×C₃₇	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	8	4	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	8	no	—
202	D₄	—	8	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	D₁₆×C₃₇	1	4	yes	—
206	C₂×C₄	SD₃₂×C₃₇	1	4	yes	—
207	C₂×C₄	#124×C₃₇	1	2	yes	—
208	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
210	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
212	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
213	C₈	#124×C₃₇	1	2	yes	—
214	C₈	C₂×SD₁₆×C₃₇	1	4	yes	—
215	C₈	C₂×#41×C₃₇	1	2	yes	—
216	C₂²	—	8	4	no	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	8	no	—
220	C₂²	C₃₇×#486	1	2	yes	—
221	C₂²	C₂×D₁₆×C₃₇	1	2	yes	—
222	C₂²	C₂×SD₃₂×C₃₇	1	2	yes	—
223	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	C₃₇×#486	1	2	yes	—
226	C₄	C₂×D₁₆×C₃₇	1	4	yes	—
227	C₄	C₂×SD₃₂×C₃₇	1	4	yes	—
228	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
229	C₄	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	C₃₇×#2240	1	4	yes	direct product factor
232	C₂	C₂×C₃₇×#486	1	1	yes	characteristic
233	C₂	C₂²×D₁₆×C₃₇	1	1	yes	characteristic
234	C₂	C₂²×SD₃₂×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
prime radical	C₂×C₃₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#500	C₂	semidirect
3. C₃₇×#2240	C₂	direct (N × H)
4. C₂×C₃₇×#357	C₂	semidirect
5. C₂×C₄×C₁₆×C₃₇	C₂	semidirect
12. C₃₇×#500	C₂²	semidirect
18. C₃₇×#357	C₂²	semidirect
19. C₄×C₁₆×C₃₇	C₂²	semidirect
98. C₂×#2240	C₃₇	direct (N × H)
113. C₂×#500	C₂×C₃₇	semidirect
115. #2240	C₂×C₃₇	direct (N × H)
116. C₂×#357	C₂×C₃₇	semidirect
117. C₂×C₄×C₁₆	C₂×C₃₇	semidirect
119. C₂×C₃₇	#2240	direct (N × H)
129. #500	C₂²×C₃₇	semidirect
135. #357	C₂²×C₃₇	semidirect
136. C₄×C₁₆	C₂²×C₃₇	semidirect
137. C₃₇	C₂×#2240	direct (N × H)
231. C₂	C₃₇×#2240	direct (N × H)

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×#493	2
C₃₇×#2240	2
C₂×C₃₇×#357	2
C₂×C₄×C₁₆×C₃₇	2
C₂×#2240	37

Subgroups — C2×C37×#2240 #22654947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#2240 #22654947