Subgroups — C₂×C₃₇×#3407 #22683876

Order 9472.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3407	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₇×#504	C₂	1	8	yes	maximal, complemented
4	C₂²×M₃₂×C₃₇	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₃₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₃₇	C₄	1	4	yes	—
7	C₂×#131×C₃₇	C₄	1	8	yes	—
8	C₂×#131×C₃₇	C₂²	1	4	yes	—
9	C₂×#131×C₃₇	C₂²	1	8	yes	—
10	C₂²×M₁₆×C₃₇	C₄	1	3	yes	—
11	C₂³×C₈×C₃₇	C₂²	1	3	yes	—
12	C₃₇×#504	C₂²	1	64	yes	complemented
13	C₂×M₃₂×C₃₇	C₂²	1	4	yes	—
14	C₂×M₃₂×C₃₇	C₂²	1	12	yes	complemented
15	C₂×M₃₂×C₃₇	C₂²	1	48	yes	complemented
16	C₂²×C₁₆×C₃₇	C₂²	1	12	yes	complemented
17	C₂×#41×C₃₇	C₈	1	4	yes	—
18	C₂×#41×C₃₇	C₈	1	8	yes	—
19	C₂×#41×C₃₇	C₂×C₄	1	4	yes	—
20	C₂×#41×C₃₇	C₂×C₄	1	8	yes	—
21	C₂²×Q₈×C₃₇	C₈	1	1	yes	characteristic
22	C₂²×D₄×C₃₇	C₈	1	3	yes	—
23	C₂³×C₄×C₃₇	C₂×C₄	1	3	yes	—
24	#131×C₃₇	C₂×C₄	1	16	yes	—
25	#131×C₃₇	C₂×C₄	1	32	yes	—
26	#131×C₃₇	C₂³	1	16	yes	—
27	C₂×M₁₆×C₃₇	C₂×C₄	1	12	yes	—
28	C₂×M₁₆×C₃₇	C₂×C₄	1	24	yes	—
29	C₂²×C₈×C₃₇	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₃₇	C₂×C₄	1	6	yes	—
31	C₂²×C₈×C₃₇	C₂×C₄	1	12	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	12	yes	—
33	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
35	M₃₂×C₃₇	C₂³	1	16	yes	—
36	M₃₂×C₃₇	C₂³	1	48	yes	complemented
37	C₂×C₁₆×C₃₇	C₂³	1	8	yes	—
38	C₂×C₁₆×C₃₇	C₂³	1	48	yes	complemented
39	C₂⁴×C₃₇	C₂×C₈	1	3	yes	—
40	#41×C₃₇	C₂×C₈	1	16	yes	—
41	#41×C₃₇	C₂×C₈	1	32	yes	—
42	#41×C₃₇	C₂²×C₄	1	16	yes	—
43	C₂×Q₈×C₃₇	C₂×C₈	1	8	yes	—
44	C₂×Q₈×C₃₇	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₃₇	C₂×C₈	1	12	yes	—
46	C₂×D₄×C₃₇	C₂×C₈	1	24	yes	—
47	C₂²×C₄×C₃₇	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₃₇	C₂×C₈	1	6	yes	—
49	C₂²×C₄×C₃₇	C₂×C₈	1	12	yes	—
50	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₃₇	C₂²×C₄	1	6	yes	—
53	M₁₆×C₃₇	C₂²×C₄	1	48	yes	—
54	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₃₇	C₂²×C₄	1	12	yes	—
57	C₂×C₈×C₃₇	C₂²×C₄	1	6	yes	—
58	C₂×C₈×C₃₇	C₂²×C₄	1	24	yes	—
59	C₂×C₈×C₃₇	C₂²×C₄	1	12	yes	—
60	C₂×C₈×C₃₇	C₂⁴	1	12	yes	—
61	C₂×C₈×C₃₇	C₂⁴	1	2	yes	—
62	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₃₇	C₂⁴	1	16	yes	—
64	C₂³×C₃₇	—	4	6	no	—
65	C₂³×C₃₇	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₃₇	C₂²×C₈	1	6	yes	—
67	C₂³×C₃₇	C₂²×C₈	1	12	yes	—
68	Q₈×C₃₇	C₂²×C₈	1	16	yes	—
69	D₄×C₃₇	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₃₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₃₇	C₂²×C₈	1	12	yes	—
73	C₂×C₄×C₃₇	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₃₇	C₂²×C₈	1	24	yes	—
75	C₂×C₄×C₃₇	C₂²×C₈	1	12	yes	—
76	C₂×C₄×C₃₇	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂³×C₄	1	12	yes	—
79	C₈×C₃₇	C₂³×C₄	1	2	yes	—
80	C₈×C₃₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₃₇	C₂³×C₄	1	12	yes	—
82	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
83	C₂×#3407	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₃₇	—	2	2	no	—
85	C₂²×C₃₇	—	2	12	no	—
86	C₂²×C₃₇	—	4	12	no	—
87	C₂²×C₃₇	C₂³×C₈	1	2	yes	—
88	C₂²×C₃₇	C₂³×C₈	1	12	yes	—
89	C₂²×C₃₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₃₇	C₂³×C₈	1	2	yes	—
91	C₄×C₃₇	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₃₇	C₂³×C₈	1	12	yes	—
93	C₄×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
94	C₂²×#131	C₂×C₃₇	1	1	yes	characteristic
95	#3407	C₂×C₃₇	1	16	yes	direct product factor
96	C₂×#504	C₂×C₃₇	1	8	yes	complemented
97	C₂²×M₃₂	C₂×C₃₇	1	6	yes	complemented
98	C₂×C₃₇	—	2	2	no	—
99	C₂×C₃₇	—	2	12	no	—
100	C₂×C₃₇	#3407	1	2	yes	direct product factor
101	C₂×C₃₇	C₂⁴×C₈	1	1	yes	characteristic
102	C₂²×#41	C₄×C₃₇	1	1	yes	characteristic
103	C₂×#131	C₄×C₃₇	1	4	yes	—
104	C₂×#131	C₄×C₃₇	1	8	yes	—
105	C₂×#131	C₂²×C₃₇	1	4	yes	—
106	C₂×#131	C₂²×C₃₇	1	8	yes	—
107	C₂²×M₁₆	C₄×C₃₇	1	3	yes	—
108	C₂³×C₈	C₂²×C₃₇	1	3	yes	—
109	#504	C₂²×C₃₇	1	64	yes	complemented
110	C₂×M₃₂	C₂²×C₃₇	1	4	yes	—
111	C₂×M₃₂	C₂²×C₃₇	1	12	yes	complemented
112	C₂×M₃₂	C₂²×C₃₇	1	48	yes	complemented
113	C₂²×C₁₆	C₂²×C₃₇	1	12	yes	complemented
114	C₃₇	C₂×#3407	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×#41	C₈×C₃₇	1	4	yes	—
116	C₂×#41	C₈×C₃₇	1	8	yes	—
117	C₂×#41	C₂×C₄×C₃₇	1	4	yes	—
118	C₂×#41	C₂×C₄×C₃₇	1	8	yes	—
119	C₂²×Q₈	C₈×C₃₇	1	1	yes	characteristic
120	C₂²×D₄	C₈×C₃₇	1	3	yes	—
121	C₂³×C₄	C₂×C₄×C₃₇	1	3	yes	—
122	#131	C₂×C₄×C₃₇	1	16	yes	—
123	#131	C₂×C₄×C₃₇	1	32	yes	—
124	#131	C₂³×C₃₇	1	16	yes	—
125	C₂×M₁₆	C₂×C₄×C₃₇	1	12	yes	—
126	C₂×M₁₆	C₂×C₄×C₃₇	1	24	yes	—
127	C₂²×C₈	C₂×C₄×C₃₇	1	3	yes	—
128	C₂²×C₈	C₂×C₄×C₃₇	1	6	yes	—
129	C₂²×C₈	C₂×C₄×C₃₇	1	12	yes	—
130	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
132	C₂²×C₈	C₂³×C₃₇	1	12	yes	—
133	M₃₂	C₂³×C₃₇	1	16	yes	—
134	M₃₂	C₂³×C₃₇	1	48	yes	complemented
135	C₂×C₁₆	C₂³×C₃₇	1	8	yes	—
136	C₂×C₁₆	C₂³×C₃₇	1	48	yes	complemented
137	C₂⁴	C₂×C₈×C₃₇	1	3	yes	—
138	#41	C₂×C₈×C₃₇	1	16	yes	—
139	#41	C₂×C₈×C₃₇	1	32	yes	—
140	#41	C₂²×C₄×C₃₇	1	16	yes	—
141	C₂×Q₈	C₂×C₈×C₃₇	1	8	yes	—
142	C₂×Q₈	C₂×C₈×C₃₇	1	4	yes	—
143	C₂×D₄	C₂×C₈×C₃₇	1	12	yes	—
144	C₂×D₄	C₂×C₈×C₃₇	1	24	yes	—
145	C₂²×C₄	C₂×C₈×C₃₇	1	3	yes	—
146	C₂²×C₄	C₂×C₈×C₃₇	1	6	yes	—
147	C₂²×C₄	C₂×C₈×C₃₇	1	12	yes	—
148	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₃₇	1	6	yes	—
150	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
151	M₁₆	C₂²×C₄×C₃₇	1	48	yes	—
152	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
153	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
154	C₂×C₈	C₂²×C₄×C₃₇	1	6	yes	—
155	C₂×C₈	C₂²×C₄×C₃₇	1	12	yes	—
156	C₂×C₈	C₂²×C₄×C₃₇	1	12	yes	—
157	C₂×C₈	C₂²×C₄×C₃₇	1	24	yes	—
158	C₂×C₈	C₂⁴×C₃₇	1	12	yes	—
159	C₂×C₈	C₂⁴×C₃₇	1	2	yes	—
160	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
161	C₁₆	C₂⁴×C₃₇	1	16	yes	—
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×C₈×C₃₇	1	1	yes	characteristic
164	C₂³	C₂²×C₈×C₃₇	1	6	yes	—
165	C₂³	C₂²×C₈×C₃₇	1	12	yes	—
166	Q₈	C₂²×C₈×C₃₇	1	16	yes	—
167	D₄	C₂²×C₈×C₃₇	1	48	yes	—
168	C₂×C₄	C₂²×C₈×C₃₇	1	2	yes	—
169	C₂×C₄	C₂²×C₈×C₃₇	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₈×C₃₇	1	12	yes	—
171	C₂×C₄	C₂²×C₈×C₃₇	1	6	yes	—
172	C₂×C₄	C₂²×C₈×C₃₇	1	24	yes	—
173	C₂×C₄	C₂²×C₈×C₃₇	1	12	yes	—
174	C₂×C₄	C₂³×C₄×C₃₇	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
176	C₂×C₄	C₂³×C₄×C₃₇	1	12	yes	—
177	C₈	C₂³×C₄×C₃₇	1	2	yes	—
178	C₈	C₂³×C₄×C₃₇	1	1	yes	characteristic
179	C₈	C₂³×C₄×C₃₇	1	12	yes	—
180	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×C₈×C₃₇	1	2	yes	—
185	C₂²	C₂³×C₈×C₃₇	1	12	yes	—
186	C₂²	C₂³×C₈×C₃₇	1	1	yes	characteristic
187	C₄	C₂³×C₈×C₃₇	1	1	yes	characteristic
188	C₄	C₂³×C₈×C₃₇	1	2	yes	—
189	C₄	C₂³×C₈×C₃₇	1	12	yes	—
190	C₄	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	C₃₇×#3407	1	2	yes	direct product factor
194	C₂	C₂⁴×C₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₇	C₂⁴
commutator	C₂	C₂⁴×C₈×C₃₇
Frattini, cosocle	C₈	C₂⁵×C₃₇
socle	C₂²×C₃₇	C₂³×C₈
prime radical	C₂²×#41×C₃₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#3407	C₂	direct (N × H)
3. C₂×C₃₇×#504	C₂	semidirect
4. C₂²×M₃₂×C₃₇	C₂	semidirect
12. C₃₇×#504	C₂²	semidirect
14. C₂×M₃₂×C₃₇	C₂²	semidirect
15. C₂×M₃₂×C₃₇	C₂²	semidirect
16. C₂²×C₁₆×C₃₇	C₂²	semidirect
36. M₃₂×C₃₇	C₂³	semidirect
38. C₂×C₁₆×C₃₇	C₂³	semidirect
83. C₂×#3407	C₃₇	direct (N × H)
95. #3407	C₂×C₃₇	direct (N × H)
96. C₂×#504	C₂×C₃₇	semidirect
97. C₂²×M₃₂	C₂×C₃₇	semidirect
100. C₂×C₃₇	#3407	direct (N × H)
109. #504	C₂²×C₃₇	semidirect
111. C₂×M₃₂	C₂²×C₃₇	semidirect
112. C₂×M₃₂	C₂²×C₃₇	semidirect
113. C₂²×C₁₆	C₂²×C₃₇	semidirect
114. C₃₇	C₂×#3407	direct (N × H)
134. M₃₂	C₂³×C₃₇	semidirect
136. C₂×C₁₆	C₂³×C₃₇	semidirect
193. C₂	C₃₇×#3407	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₃₇	2
C₃₇×#3407	2
C₂×C₃₇×#504	2
C₂²×M₃₂×C₃₇	2
C₂×#3407	37

Subgroups — C2×C37×#3407 #22683876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#3407 #22683876