Subgroups — C₂²×C₃₇×#518 #22683898

Order 9472.

14590 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#518	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×D₄×C₃₇	C₂	1	8	yes	maximal, complemented
4	C₂²×#117×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂²⋊C₄×C₃₇	C₂	1	4	yes	maximal, complemented
6	C₂⁶×C₃₇	C₂²	1	2	yes	—
7	C₂³×D₄×C₃₇	C₄	1	32	yes	complemented
8	C₂³×D₄×C₃₇	C₂²	1	4	yes	—
9	C₂³×D₄×C₃₇	C₂²	1	24	yes	—
10	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₃₇	C₂²	1	8	yes	complemented
12	C₃₇×#518	C₂²	1	256	yes	direct product factor
13	C₂×C₄×D₄×C₃₇	C₂²	1	192	yes	complemented
14	C₂×#117×C₃₇	C₂²	1	48	yes	complemented
15	C₂²×C₄⋊C₄×C₃₇	C₂²	1	4	yes	complemented
16	C₂²×C₂²⋊C₄×C₃₇	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₃₇	C₂²	1	96	yes	complemented
18	C₂²×C₂²⋊C₄×C₃₇	C₂²	1	8	yes	complemented
19	C₂²×C₄²×C₃₇	C₂²	1	4	yes	complemented
20	C₂⁵×C₃₇	—	4	16	no	—
21	C₂⁵×C₃₇	C₂×C₄	1	32	yes	complemented
22	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
23	C₂⁵×C₃₇	C₂³	1	4	yes	—
24	C₂⁵×C₃₇	C₂³	1	24	yes	—
25	C₂²×D₄×C₃₇	C₂×C₄	1	64	yes	complemented
26	C₂²×D₄×C₃₇	C₂×C₄	1	384	yes	complemented
27	C₂²×D₄×C₃₇	C₂³	1	48	yes	—
28	C₂²×D₄×C₃₇	C₂³	1	64	yes	—
29	C₂³×C₄×C₃₇	—	4	32	no	—
30	C₂³×C₄×C₃₇	C₂×C₄	1	16	yes	complemented
31	C₂³×C₄×C₃₇	C₂³	1	8	yes	—
32	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
33	C₂³×C₄×C₃₇	C₂³	1	12	yes	—
34	C₂³×C₄×C₃₇	C₂³	1	96	yes	complemented
35	C₄×D₄×C₃₇	C₂³	1	512	yes	complemented
36	#117×C₃₇	C₂³	1	128	yes	complemented
37	C₂×C₄⋊C₄×C₃₇	C₂³	1	48	yes	complemented
38	C₂×C₂²⋊C₄×C₃₇	C₂³	1	48	yes	—
39	C₂×C₂²⋊C₄×C₃₇	C₂³	1	256	yes	complemented
40	C₂×C₂²⋊C₄×C₃₇	C₂³	1	96	yes	complemented
41	C₂×C₄²×C₃₇	C₂³	1	48	yes	complemented
42	C₂⁴×C₃₇	—	2	32	no	—
43	C₂⁴×C₃₇	—	2	8	no	—
44	C₂⁴×C₃₇	—	4	128	no	—
45	C₂⁴×C₃₇	—	4	96	no	—
46	C₂⁴×C₃₇	C₂²×C₄	1	8	yes	—
47	C₂⁴×C₃₇	C₂²×C₄	1	32	yes	—
48	C₂⁴×C₃₇	C₂²×C₄	1	192	yes	complemented
49	C₂⁴×C₃₇	C₂⁴	1	6	yes	—
50	C₂⁴×C₃₇	C₂⁴	1	24	yes	—
51	C₂⁴×C₃₇	C₂⁴	1	32	yes	—
52	C₂⁴×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₄×C₃₇	C₂²×C₄	1	384	yes	complemented
54	C₂×D₄×C₃₇	C₂²×C₄	1	512	yes	complemented
55	C₂×D₄×C₃₇	C₂⁴	1	64	yes	—
56	C₂²×C₄×C₃₇	—	2	32	no	—
57	C₂²×C₄×C₃₇	—	4	192	no	—
58	C₂²×C₄×C₃₇	C₂²×C₄	1	16	yes	—
59	C₂²×C₄×C₃₇	C₂²×C₄	1	96	yes	complemented
60	C₂²×C₄×C₃₇	C₂⁴	1	48	yes	—
61	C₂²×C₄×C₃₇	C₂⁴	1	12	yes	—
62	C₂²×C₄×C₃₇	C₂⁴	1	16	yes	—
63	C₂²×C₄×C₃₇	C₂⁴	1	128	yes	complemented
64	C₄⋊C₄×C₃₇	C₂⁴	1	64	yes	complemented
65	C₂²⋊C₄×C₃₇	C₂⁴	1	64	yes	—
66	C₂²⋊C₄×C₃₇	C₂⁴	1	128	yes	complemented
67	C₄²×C₃₇	C₂⁴	1	64	yes	complemented
68	C₂³×C₃₇	—	2	32	no	—
69	C₂³×C₃₇	—	2	128	no	—
70	C₂³×C₃₇	—	2	96	no	—
71	C₂³×C₃₇	—	2	24	no	—
72	C₂³×C₃₇	—	4	384	no	—
73	C₂³×C₃₇	—	4	64	no	—
74	C₂³×C₃₇	C₂³×C₄	1	24	yes	—
75	C₂³×C₃₇	C₂³×C₄	1	96	yes	—
76	C₂³×C₃₇	C₂³×C₄	1	4	yes	—
77	C₂³×C₃₇	C₂³×C₄	1	128	yes	complemented
78	C₂³×C₃₇	H(5,𝔽₂)	1	8	yes	—
79	C₂³×C₃₇	C₂⁵	1	4	yes	—
80	C₂³×C₃₇	C₂⁵	1	16	yes	—
81	C₂³×C₃₇	C₂⁵	1	3	yes	—
82	D₄×C₃₇	C₂³×C₄	1	256	yes	complemented
83	C₂×C₄×C₃₇	—	2	96	no	—
84	C₂×C₄×C₃₇	—	4	128	no	—
85	C₂×C₄×C₃₇	C₂³×C₄	1	48	yes	—
86	C₂×C₄×C₃₇	C₂³×C₄	1	64	yes	complemented
87	C₂×C₄×C₃₇	C₂⁵	1	32	yes	—
88	C₂×C₄×C₃₇	C₂⁵	1	8	yes	—
89	C₂²×#518	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₃₇	—	2	32	no	—
91	C₂²×C₃₇	—	2	8	no	—
92	C₂²×C₃₇	—	2	48	no	—
93	C₂²×C₃₇	—	2	192	no	—
94	C₂²×C₃₇	—	4	128	no	—
95	C₂²×C₃₇	#518	1	16	yes	direct product factor
96	C₂²×C₃₇	C₂⁴×C₄	1	8	yes	—
97	C₂²×C₃₇	C₂⁴×C₄	1	32	yes	—
98	C₂²×C₃₇	C₂⁴×C₄	1	6	yes	—
99	C₂²×C₃₇	C₂×H(5,𝔽₂)	1	12	yes	—
100	C₂²×C₃₇	C₂⁶	1	1	yes	characteristic
101	C₄×C₃₇	—	2	32	no	—
102	C₄×C₃₇	C₂⁴×C₄	1	16	yes	—
103	C₂⁴×D₄	C₂×C₃₇	1	1	yes	characteristic
104	C₂×#518	C₂×C₃₇	1	48	yes	direct product factor
105	C₂²×C₄×D₄	C₂×C₃₇	1	8	yes	complemented
106	C₂²×#117	C₂×C₃₇	1	2	yes	complemented
107	C₂³×C₂²⋊C₄	C₂×C₃₇	1	4	yes	complemented
108	C₂×C₃₇	—	2	8	no	—
109	C₂×C₃₇	—	2	32	no	—
110	C₂×C₃₇	C₂×#518	1	12	yes	direct product factor
111	C₂×C₃₇	C₂⁵×C₄	1	1	yes	characteristic
112	C₂×C₃₇	C₂²×H(5,𝔽₂)	1	2	yes	—
113	C₂⁶	C₂²×C₃₇	1	2	yes	—
114	C₂³×D₄	C₄×C₃₇	1	32	yes	complemented
115	C₂³×D₄	C₂²×C₃₇	1	4	yes	—
116	C₂³×D₄	C₂²×C₃₇	1	24	yes	—
117	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
118	C₂⁴×C₄	C₂²×C₃₇	1	8	yes	complemented
119	#518	C₂²×C₃₇	1	256	yes	direct product factor
120	C₂×C₄×D₄	C₂²×C₃₇	1	192	yes	complemented
121	C₂×#117	C₂²×C₃₇	1	48	yes	complemented
122	C₂²×C₄⋊C₄	C₂²×C₃₇	1	4	yes	complemented
123	C₂²×C₂²⋊C₄	C₂²×C₃₇	1	4	yes	—
124	C₂²×C₂²⋊C₄	C₂²×C₃₇	1	96	yes	complemented
125	C₂²×C₂²⋊C₄	C₂²×C₃₇	1	8	yes	complemented
126	C₂²×C₄²	C₂²×C₃₇	1	4	yes	complemented
127	C₃₇	C₂²×#518	1	1	yes	characteristic, direct product factor, Sylow
128	C₂⁵	—	4	16	no	—
129	C₂⁵	C₂×C₄×C₃₇	1	32	yes	complemented
130	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
131	C₂⁵	C₂³×C₃₇	1	4	yes	—
132	C₂⁵	C₂³×C₃₇	1	24	yes	—
133	C₂²×D₄	C₂×C₄×C₃₇	1	64	yes	complemented
134	C₂²×D₄	C₂×C₄×C₃₇	1	384	yes	complemented
135	C₂²×D₄	C₂³×C₃₇	1	48	yes	—
136	C₂²×D₄	C₂³×C₃₇	1	64	yes	—
137	C₂³×C₄	—	4	32	no	—
138	C₂³×C₄	C₂×C₄×C₃₇	1	16	yes	complemented
139	C₂³×C₄	C₂³×C₃₇	1	8	yes	—
140	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
141	C₂³×C₄	C₂³×C₃₇	1	12	yes	—
142	C₂³×C₄	C₂³×C₃₇	1	96	yes	complemented
143	C₄×D₄	C₂³×C₃₇	1	512	yes	complemented
144	#117	C₂³×C₃₇	1	128	yes	complemented
145	C₂×C₄⋊C₄	C₂³×C₃₇	1	48	yes	complemented
146	C₂×C₂²⋊C₄	C₂³×C₃₇	1	48	yes	—
147	C₂×C₂²⋊C₄	C₂³×C₃₇	1	256	yes	complemented
148	C₂×C₂²⋊C₄	C₂³×C₃₇	1	96	yes	complemented
149	C₂×C₄²	C₂³×C₃₇	1	48	yes	complemented
150	C₂⁴	—	2	32	no	—
151	C₂⁴	—	2	8	no	—
152	C₂⁴	—	4	128	no	—
153	C₂⁴	—	4	96	no	—
154	C₂⁴	C₂²×C₄×C₃₇	1	8	yes	—
155	C₂⁴	C₂²×C₄×C₃₇	1	32	yes	—
156	C₂⁴	C₂²×C₄×C₃₇	1	192	yes	complemented
157	C₂⁴	C₂⁴×C₃₇	1	6	yes	—
158	C₂⁴	C₂⁴×C₃₇	1	24	yes	—
159	C₂⁴	C₂⁴×C₃₇	1	32	yes	—
160	C₂⁴	C₂⁴×C₃₇	1	1	yes	characteristic
161	C₂×D₄	C₂²×C₄×C₃₇	1	384	yes	complemented
162	C₂×D₄	C₂²×C₄×C₃₇	1	512	yes	complemented
163	C₂×D₄	C₂⁴×C₃₇	1	64	yes	—
164	C₂²×C₄	—	2	32	no	—
165	C₂²×C₄	—	4	192	no	—
166	C₂²×C₄	C₂²×C₄×C₃₇	1	16	yes	—
167	C₂²×C₄	C₂²×C₄×C₃₇	1	96	yes	complemented
168	C₂²×C₄	C₂⁴×C₃₇	1	48	yes	—
169	C₂²×C₄	C₂⁴×C₃₇	1	12	yes	—
170	C₂²×C₄	C₂⁴×C₃₇	1	16	yes	—
171	C₂²×C₄	C₂⁴×C₃₇	1	128	yes	complemented
172	C₄⋊C₄	C₂⁴×C₃₇	1	64	yes	complemented
173	C₂²⋊C₄	C₂⁴×C₃₇	1	64	yes	—
174	C₂²⋊C₄	C₂⁴×C₃₇	1	128	yes	complemented
175	C₄²	C₂⁴×C₃₇	1	64	yes	complemented
176	C₂³	—	2	96	no	—
177	C₂³	—	2	24	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	128	no	—
180	C₂³	—	4	384	no	—
181	C₂³	—	4	64	no	—
182	C₂³	C₂³×C₄×C₃₇	1	24	yes	—
183	C₂³	C₂³×C₄×C₃₇	1	96	yes	—
184	C₂³	C₂³×C₄×C₃₇	1	4	yes	—
185	C₂³	C₂³×C₄×C₃₇	1	128	yes	complemented
186	C₂³	H(5,𝔽₂)×C₃₇	1	8	yes	—
187	C₂³	C₂⁵×C₃₇	1	4	yes	—
188	C₂³	C₂⁵×C₃₇	1	16	yes	—
189	C₂³	C₂⁵×C₃₇	1	3	yes	—
190	D₄	C₂³×C₄×C₃₇	1	256	yes	complemented
191	C₂×C₄	—	2	96	no	—
192	C₂×C₄	—	4	128	no	—
193	C₂×C₄	C₂³×C₄×C₃₇	1	48	yes	—
194	C₂×C₄	C₂³×C₄×C₃₇	1	64	yes	complemented
195	C₂×C₄	C₂⁵×C₃₇	1	32	yes	—
196	C₂×C₄	C₂⁵×C₃₇	1	8	yes	—
197	C₂²	—	2	48	no	—
198	C₂²	—	2	192	no	—
199	C₂²	—	2	32	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	128	no	—
202	C₂²	C₃₇×#518	1	16	yes	direct product factor
203	C₂²	C₂⁴×C₄×C₃₇	1	8	yes	—
204	C₂²	C₂⁴×C₄×C₃₇	1	32	yes	—
205	C₂²	C₂⁴×C₄×C₃₇	1	6	yes	—
206	C₂²	C₂×H(5,𝔽₂)×C₃₇	1	12	yes	—
207	C₂²	C₂⁶×C₃₇	1	1	yes	characteristic
208	C₄	—	2	32	no	—
209	C₄	C₂⁴×C₄×C₃₇	1	16	yes	—
210	C₂	—	2	8	no	—
211	C₂	—	2	32	no	—
212	C₂	C₂×C₃₇×#518	1	12	yes	direct product factor
213	C₂	C₂⁵×C₄×C₃₇	1	1	yes	characteristic
214	C₂	C₂²×H(5,𝔽₂)×C₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂⁴
commutator	C₂	C₂⁵×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁶×C₃₇
prime radical	C₂⁴×D₄×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#518	C₂	direct (N × H)
3. C₂²×C₄×D₄×C₃₇	C₂	semidirect
4. C₂²×#117×C₃₇	C₂	semidirect
5. C₂³×C₂²⋊C₄×C₃₇	C₂	semidirect
7. C₂³×D₄×C₃₇	C₄	semidirect
11. C₂⁴×C₄×C₃₇	C₂²	semidirect
12. C₃₇×#518	C₂²	direct (N × H)
13. C₂×C₄×D₄×C₃₇	C₂²	semidirect
14. C₂×#117×C₃₇	C₂²	semidirect
15. C₂²×C₄⋊C₄×C₃₇	C₂²	semidirect
17. C₂²×C₂²⋊C₄×C₃₇	C₂²	semidirect
18. C₂²×C₂²⋊C₄×C₃₇	C₂²	semidirect
19. C₂²×C₄²×C₃₇	C₂²	semidirect
21. C₂⁵×C₃₇	C₂×C₄	semidirect
25. C₂²×D₄×C₃₇	C₂×C₄	semidirect
26. C₂²×D₄×C₃₇	C₂×C₄	semidirect
30. C₂³×C₄×C₃₇	C₂×C₄	semidirect
34. C₂³×C₄×C₃₇	C₂³	semidirect
35. C₄×D₄×C₃₇	C₂³	semidirect
36. #117×C₃₇	C₂³	semidirect
37. C₂×C₄⋊C₄×C₃₇	C₂³	semidirect
39. C₂×C₂²⋊C₄×C₃₇	C₂³	semidirect
40. C₂×C₂²⋊C₄×C₃₇	C₂³	semidirect
41. C₂×C₄²×C₃₇	C₂³	semidirect
48. C₂⁴×C₃₇	C₂²×C₄	semidirect
53. C₂×D₄×C₃₇	C₂²×C₄	semidirect
54. C₂×D₄×C₃₇	C₂²×C₄	semidirect
59. C₂²×C₄×C₃₇	C₂²×C₄	semidirect
63. C₂²×C₄×C₃₇	C₂⁴	semidirect
64. C₄⋊C₄×C₃₇	C₂⁴	semidirect
66. C₂²⋊C₄×C₃₇	C₂⁴	semidirect
67. C₄²×C₃₇	C₂⁴	semidirect
77. C₂³×C₃₇	C₂³×C₄	semidirect
82. D₄×C₃₇	C₂³×C₄	semidirect
86. C₂×C₄×C₃₇	C₂³×C₄	semidirect
89. C₂²×#518	C₃₇	direct (N × H)
95. C₂²×C₃₇	#518	direct (N × H)
104. C₂×#518	C₂×C₃₇	direct (N × H)
105. C₂²×C₄×D₄	C₂×C₃₇	semidirect
106. C₂²×#117	C₂×C₃₇	semidirect
107. C₂³×C₂²⋊C₄	C₂×C₃₇	semidirect
110. C₂×C₃₇	C₂×#518	direct (N × H)
114. C₂³×D₄	C₄×C₃₇	semidirect
118. C₂⁴×C₄	C₂²×C₃₇	semidirect
119. #518	C₂²×C₃₇	direct (N × H)
120. C₂×C₄×D₄	C₂²×C₃₇	semidirect
121. C₂×#117	C₂²×C₃₇	semidirect
122. C₂²×C₄⋊C₄	C₂²×C₃₇	semidirect
124. C₂²×C₂²⋊C₄	C₂²×C₃₇	semidirect
125. C₂²×C₂²⋊C₄	C₂²×C₃₇	semidirect
126. C₂²×C₄²	C₂²×C₃₇	semidirect
127. C₃₇	C₂²×#518	direct (N × H)
129. C₂⁵	C₂×C₄×C₃₇	semidirect
133. C₂²×D₄	C₂×C₄×C₃₇	semidirect
134. C₂²×D₄	C₂×C₄×C₃₇	semidirect
138. C₂³×C₄	C₂×C₄×C₃₇	semidirect
142. C₂³×C₄	C₂³×C₃₇	semidirect
143. C₄×D₄	C₂³×C₃₇	semidirect
144. #117	C₂³×C₃₇	semidirect
145. C₂×C₄⋊C₄	C₂³×C₃₇	semidirect
147. C₂×C₂²⋊C₄	C₂³×C₃₇	semidirect
148. C₂×C₂²⋊C₄	C₂³×C₃₇	semidirect
149. C₂×C₄²	C₂³×C₃₇	semidirect
156. C₂⁴	C₂²×C₄×C₃₇	semidirect
161. C₂×D₄	C₂²×C₄×C₃₇	semidirect
162. C₂×D₄	C₂²×C₄×C₃₇	semidirect
167. C₂²×C₄	C₂²×C₄×C₃₇	semidirect
171. C₂²×C₄	C₂⁴×C₃₇	semidirect
172. C₄⋊C₄	C₂⁴×C₃₇	semidirect
174. C₂²⋊C₄	C₂⁴×C₃₇	semidirect
175. C₄²	C₂⁴×C₃₇	semidirect
185. C₂³	C₂³×C₄×C₃₇	semidirect
190. D₄	C₂³×C₄×C₃₇	semidirect
194. C₂×C₄	C₂³×C₄×C₃₇	semidirect
202. C₂²	C₃₇×#518	direct (N × H)
212. C₂	C₂×C₃₇×#518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₃₇	2
C₂×C₃₇×#518	2
C₂²×C₄×D₄×C₃₇	2
C₂²×#117×C₃₇	2
C₂³×C₂²⋊C₄×C₃₇	2
C₂²×#518	37

Subgroups — C22×C37×#518 #22683898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#518 #22683898