Subgroups — C₂×C₃₇×#3417 #22683887

Order 9472.

2022 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#578	C₂	1	2	yes	maximal, complemented
2	C₃₇×#3417	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₇×#510	C₂	1	6	yes	maximal, complemented
4	C₂×C₃₇×#508	C₂	1	3	yes	maximal, complemented
5	C₂²×Q₃₂×C₃₇	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×2₋¹⁺⁴×C₃₇	—	2	2	no	—
8	C₃₇×#578	—	2	8	no	—
9	C₃₇×#578	C₂²	1	16	yes	complemented
10	C₂×#137×C₃₇	—	2	6	no	—
11	C₂×#135×C₃₇	C₂²	1	6	yes	—
12	C₂²×Q₁₆×C₃₇	C₂²	1	6	yes	—
13	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#510	C₂²	1	48	yes	complemented
15	C₃₇×#508	C₂²	1	24	yes	complemented
16	C₂×Q₃₂×C₃₇	C₂²	1	3	yes	—
17	C₂×Q₃₂×C₃₇	C₂²	1	6	yes	—
18	C₂×Q₃₂×C₃₇	C₂²	1	24	yes	complemented
19	C₂×SD₃₂×C₃₇	C₂²	1	6	yes	—
20	C₂×D₁₆×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#504	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₃₇	C₂²	1	3	yes	—
23	C₂²×C₁₆×C₃₇	C₂²	1	3	yes	—
24	2₋¹⁺⁴×C₃₇	—	2	16	no	—
25	2₋¹⁺⁴×C₃₇	—	4	8	no	—
26	C₂×#41×C₃₇	—	2	6	no	—
27	C₂×#41×C₃₇	—	4	6	no	—
28	C₂×#41×C₃₇	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₃₇	—	2	6	no	—
30	C₂²×Q₈×C₃₇	—	4	2	no	—
31	#137×C₃₇	—	2	48	no	—
32	#135×C₃₇	—	2	12	no	—
33	#135×C₃₇	C₂³	1	24	yes	—
34	C₂×Q₁₆×C₃₇	—	2	6	no	—
35	C₂×Q₁₆×C₃₇	—	2	12	no	—
36	C₂×Q₁₆×C₃₇	C₂³	1	6	yes	—
37	C₂×Q₁₆×C₃₇	C₂³	1	24	yes	—
38	C₂×SD₁₆×C₃₇	—	2	6	no	—
39	C₂×D₈×C₃₇	C₂³	1	2	yes	—
40	#131×C₃₇	D₄	1	4	yes	—
41	#131×C₃₇	C₂³	1	4	yes	—
42	C₂×M₁₆×C₃₇	D₄	1	3	yes	—
43	C₂²×C₈×C₃₇	C₂³	1	3	yes	—
44	Q₃₂×C₃₇	C₂³	1	12	yes	—
45	Q₃₂×C₃₇	C₂³	1	24	yes	—
46	SD₃₂×C₃₇	C₂³	1	24	yes	—
47	D₁₆×C₃₇	C₂³	1	4	yes	—
48	M₃₂×C₃₇	C₂³	1	12	yes	—
49	C₂×C₁₆×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₃₇	C₂³	1	3	yes	—
51	C₂×C₁₆×C₃₇	C₂³	1	12	yes	—
52	#41×C₃₇	—	2	24	no	—
53	#41×C₃₇	—	4	48	no	—
54	#41×C₃₇	—	4	12	no	—
55	#41×C₃₇	D₈	1	4	yes	—
56	#41×C₃₇	C₂×D₄	1	4	yes	—
57	C₂×Q₈×C₃₇	—	2	6	no	—
58	C₂×Q₈×C₃₇	—	2	24	no	—
59	C₂×Q₈×C₃₇	—	4	16	no	—
60	C₂×Q₈×C₃₇	—	4	6	no	—
61	C₂×Q₈×C₃₇	—	4	12	no	—
62	C₂×Q₈×C₃₇	D₈	1	1	yes	characteristic
63	C₂×D₄×C₃₇	—	2	2	no	—
64	C₂×D₄×C₃₇	—	4	6	no	—
65	C₂×D₄×C₃₇	D₈	1	3	yes	—
66	C₂²×C₄×C₃₇	—	4	6	no	—
67	C₂²×C₄×C₃₇	—	4	6	no	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
69	Q₁₆×C₃₇	—	2	6	no	—
70	Q₁₆×C₃₇	—	2	24	no	—
71	Q₁₆×C₃₇	C₂⁴	1	12	yes	—
72	SD₁₆×C₃₇	—	2	24	no	—
73	D₈×C₃₇	—	2	2	no	—
74	D₈×C₃₇	C₂⁴	1	4	yes	—
75	M₁₆×C₃₇	C₂×D₄	1	12	yes	—
76	C₂×C₈×C₃₇	C₂×D₄	1	3	yes	—
77	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
78	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₃₇	C₂⁴	1	6	yes	—
80	C₁₆×C₃₇	C₂⁴	1	2	yes	—
81	C₁₆×C₃₇	C₂⁴	1	6	yes	—
82	C₂³×C₃₇	—	4	2	no	—
83	C₂³×C₃₇	C₂×D₈	1	3	yes	—
84	Q₈×C₃₇	—	2	12	no	—
85	Q₈×C₃₇	—	4	24	no	—
86	Q₈×C₃₇	—	4	6	no	—
87	Q₈×C₃₇	C₂×D₈	1	4	yes	—
88	D₄×C₃₇	—	2	4	no	—
89	D₄×C₃₇	—	4	24	no	—
90	D₄×C₃₇	—	4	2	no	—
91	D₄×C₃₇	C₂×D₈	1	12	yes	—
92	C₂×C₄×C₃₇	—	4	24	no	—
93	C₂×C₄×C₃₇	—	4	6	no	—
94	C₂×C₄×C₃₇	—	4	24	no	—
95	C₂×C₄×C₃₇	C₂×D₈	1	6	yes	—
96	C₂×C₄×C₃₇	C₂×D₈	1	3	yes	—
97	C₂×C₄×C₃₇	C₂²×D₄	1	6	yes	—
98	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₃₇	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₃₇	C₂²×D₄	1	6	yes	—
101	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
102	C₂×#3417	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₃₇	—	2	6	no	—
104	C₂²×C₃₇	—	4	4	no	—
105	C₂²×C₃₇	—	8	4	no	—
106	C₂²×C₃₇	C₂²×D₈	1	6	yes	—
107	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
108	C₄×C₃₇	—	4	12	no	—
109	C₄×C₃₇	C₂²×D₈	1	6	yes	—
110	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
112	C₂×#578	C₂×C₃₇	1	2	yes	complemented
113	#3417	C₂×C₃₇	1	16	yes	direct product factor
114	C₂×#510	C₂×C₃₇	1	6	yes	complemented
115	C₂×#508	C₂×C₃₇	1	3	yes	complemented
116	C₂²×Q₃₂	C₂×C₃₇	1	3	yes	complemented
117	C₂×#504	C₂×C₃₇	1	1	yes	characteristic, complemented
118	C₂×C₃₇	—	2	6	no	—
119	C₂×C₃₇	—	8	4	no	—
120	C₂×C₃₇	#3417	1	2	yes	direct product factor
121	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
122	C₂×2₋¹⁺⁴	—	2	2	no	—
123	#578	—	2	8	no	—
124	#578	C₂²×C₃₇	1	16	yes	complemented
125	C₂×#137	—	2	6	no	—
126	C₂×#135	C₂²×C₃₇	1	6	yes	—
127	C₂²×Q₁₆	C₂²×C₃₇	1	6	yes	—
128	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
129	#510	C₂²×C₃₇	1	48	yes	complemented
130	#508	C₂²×C₃₇	1	24	yes	complemented
131	C₂×Q₃₂	C₂²×C₃₇	1	3	yes	—
132	C₂×Q₃₂	C₂²×C₃₇	1	6	yes	—
133	C₂×Q₃₂	C₂²×C₃₇	1	24	yes	complemented
134	C₂×SD₃₂	C₂²×C₃₇	1	6	yes	—
135	C₂×D₁₆	C₂²×C₃₇	1	1	yes	characteristic
136	#504	C₂²×C₃₇	1	8	yes	complemented
137	C₂×M₃₂	C₂²×C₃₇	1	3	yes	—
138	C₂²×C₁₆	C₂²×C₃₇	1	3	yes	—
139	C₃₇	C₂×#3417	1	1	yes	characteristic, direct product factor, Sylow
140	2₋¹⁺⁴	—	2	16	no	—
141	2₋¹⁺⁴	—	4	8	no	—
142	C₂×#41	—	2	6	no	—
143	C₂×#41	—	4	6	no	—
144	C₂×#41	D₄×C₃₇	1	1	yes	characteristic
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	—	4	2	no	—
147	#137	—	2	48	no	—
148	#135	—	2	12	no	—
149	#135	C₂³×C₃₇	1	24	yes	—
150	C₂×Q₁₆	—	2	6	no	—
151	C₂×Q₁₆	—	2	12	no	—
152	C₂×Q₁₆	C₂³×C₃₇	1	6	yes	—
153	C₂×Q₁₆	C₂³×C₃₇	1	24	yes	—
154	C₂×SD₁₆	—	2	6	no	—
155	C₂×D₈	C₂³×C₃₇	1	2	yes	—
156	#131	D₄×C₃₇	1	4	yes	—
157	#131	C₂³×C₃₇	1	4	yes	—
158	C₂×M₁₆	D₄×C₃₇	1	3	yes	—
159	C₂²×C₈	C₂³×C₃₇	1	3	yes	—
160	Q₃₂	C₂³×C₃₇	1	12	yes	—
161	Q₃₂	C₂³×C₃₇	1	24	yes	—
162	SD₃₂	C₂³×C₃₇	1	24	yes	—
163	D₁₆	C₂³×C₃₇	1	4	yes	—
164	M₃₂	C₂³×C₃₇	1	12	yes	—
165	C₂×C₁₆	C₂³×C₃₇	1	1	yes	characteristic
166	C₂×C₁₆	C₂³×C₃₇	1	3	yes	—
167	C₂×C₁₆	C₂³×C₃₇	1	12	yes	—
168	#41	—	2	24	no	—
169	#41	—	4	48	no	—
170	#41	—	4	12	no	—
171	#41	D₈×C₃₇	1	4	yes	—
172	#41	C₂×D₄×C₃₇	1	4	yes	—
173	C₂×Q₈	—	2	6	no	—
174	C₂×Q₈	—	2	24	no	—
175	C₂×Q₈	—	4	16	no	—
176	C₂×Q₈	—	4	6	no	—
177	C₂×Q₈	—	4	12	no	—
178	C₂×Q₈	D₈×C₃₇	1	1	yes	characteristic
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	6	no	—
181	C₂×D₄	D₈×C₃₇	1	3	yes	—
182	C₂²×C₄	—	4	6	no	—
183	C₂²×C₄	—	4	6	no	—
184	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
185	Q₁₆	—	2	6	no	—
186	Q₁₆	—	2	24	no	—
187	Q₁₆	C₂⁴×C₃₇	1	12	yes	—
188	SD₁₆	—	2	24	no	—
189	D₈	—	2	2	no	—
190	D₈	C₂⁴×C₃₇	1	4	yes	—
191	M₁₆	C₂×D₄×C₃₇	1	12	yes	—
192	C₂×C₈	C₂×D₄×C₃₇	1	3	yes	—
193	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
194	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
195	C₂×C₈	C₂⁴×C₃₇	1	6	yes	—
196	C₁₆	C₂⁴×C₃₇	1	2	yes	—
197	C₁₆	C₂⁴×C₃₇	1	6	yes	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×D₈×C₃₇	1	3	yes	—
200	Q₈	—	2	12	no	—
201	Q₈	—	4	24	no	—
202	Q₈	—	4	6	no	—
203	Q₈	C₂×D₈×C₃₇	1	4	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	4	24	no	—
206	D₄	—	4	2	no	—
207	D₄	C₂×D₈×C₃₇	1	12	yes	—
208	C₂×C₄	—	4	6	no	—
209	C₂×C₄	—	4	24	no	—
210	C₂×C₄	—	4	24	no	—
211	C₂×C₄	C₂×D₈×C₃₇	1	3	yes	—
212	C₂×C₄	C₂×D₈×C₃₇	1	6	yes	—
213	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₃₇	1	6	yes	—
215	C₈	C₂²×D₄×C₃₇	1	1	yes	characteristic
216	C₈	C₂²×D₄×C₃₇	1	6	yes	—
217	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
218	C₂²	—	2	6	no	—
219	C₂²	—	4	4	no	—
220	C₂²	—	8	4	no	—
221	C₂²	C₂²×D₈×C₃₇	1	6	yes	—
222	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
223	C₄	—	4	12	no	—
224	C₄	C₂²×D₈×C₃₇	1	6	yes	—
225	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
226	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
227	C₂	—	2	6	no	—
228	C₂	—	8	4	no	—
229	C₂	C₃₇×#3417	1	2	yes	direct product factor
230	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×#41×C₃₇ ◁ C₂×#131×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#578	C₂	semidirect
2. C₃₇×#3417	C₂	direct (N × H)
3. C₂×C₃₇×#510	C₂	semidirect
4. C₂×C₃₇×#508	C₂	semidirect
5. C₂²×Q₃₂×C₃₇	C₂	semidirect
6. C₂×C₃₇×#504	C₂	semidirect
9. C₃₇×#578	C₂²	semidirect
14. C₃₇×#510	C₂²	semidirect
15. C₃₇×#508	C₂²	semidirect
18. C₂×Q₃₂×C₃₇	C₂²	semidirect
21. C₃₇×#504	C₂²	semidirect
102. C₂×#3417	C₃₇	direct (N × H)
112. C₂×#578	C₂×C₃₇	semidirect
113. #3417	C₂×C₃₇	direct (N × H)
114. C₂×#510	C₂×C₃₇	semidirect
115. C₂×#508	C₂×C₃₇	semidirect
116. C₂²×Q₃₂	C₂×C₃₇	semidirect
117. C₂×#504	C₂×C₃₇	semidirect
120. C₂×C₃₇	#3417	direct (N × H)
124. #578	C₂²×C₃₇	semidirect
129. #510	C₂²×C₃₇	semidirect
130. #508	C₂²×C₃₇	semidirect
133. C₂×Q₃₂	C₂²×C₃₇	semidirect
136. #504	C₂²×C₃₇	semidirect
139. C₃₇	C₂×#3417	direct (N × H)
229. C₂	C₃₇×#3417	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×#578	2
C₃₇×#3417	2
C₂×C₃₇×#510	2
C₂×C₃₇×#508	2
C₂²×Q₃₂×C₃₇	2
C₂×C₃₇×#504	2
C₂×#3417	37

Subgroups — C2×C37×#3417 #22683887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#3417 #22683887