Subgroups — C₃₇×#34012 #22655265

Order 9472.

3158 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2285	C₂	1	6	yes	maximal, complemented
3	C₃₇×#2283	C₂	1	12	yes	maximal, complemented
4	C₃₇×#2280	C₂	1	6	yes	maximal
5	C₄×#117×C₃₇	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₄×C₃₇	C₂²	1	3	yes	—
7	C₂×#117×C₃₇	C₂²	1	12	yes	—
8	C₂²×C₄²×C₃₇	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₃₇	C₂²	1	6	yes	—
10	C₂²×C₄²×C₃₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₃₇	C₂²	1	3	yes	—
12	C₃₇×#384	C₂²	1	4	yes	complemented
13	C₃₇×#383	C₂²	1	8	yes	complemented
14	C₃₇×#382	C₂²	1	12	yes	complemented
15	C₃₇×#381	C₂²	1	12	yes	—
16	C₃₇×#380	C₂²	1	24	yes	—
17	C₃₇×#379	C₂²	1	4	yes	—
18	C₄×C₄⋊C₄×C₃₇	C₂²	1	24	yes	complemented
19	C₄×C₂²⋊C₄×C₃₇	C₂²	1	24	yes	—
20	C₃₇×#376	C₂²	1	12	yes	complemented
21	C₄³×C₃₇	C₂²	1	4	yes	complemented
22	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₃₇	—	2	24	no	—
24	C₂³×C₄×C₃₇	C₂×C₄	1	24	yes	—
25	C₂³×C₄×C₃₇	C₂³	1	6	yes	—
26	C₂³×C₄×C₃₇	C₂³	1	6	yes	—
27	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
29	#117×C₃₇	—	2	48	no	—
30	C₂×C₄⋊C₄×C₃₇	C₂³	1	24	yes	—
31	C₂×C₂²⋊C₄×C₃₇	C₂³	1	24	yes	—
32	C₂×C₄²×C₃₇	C₂×C₄	1	48	yes	—
33	C₂×C₄²×C₃₇	C₂×C₄	1	24	yes	—
34	C₂×C₄²×C₃₇	C₂×C₄	1	24	yes	—
35	C₂×C₄²×C₃₇	C₂³	1	24	yes	—
36	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
37	C₂×C₄²×C₃₇	C₂³	1	6	yes	—
38	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
39	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	#95×C₃₇	C₂³	1	48	yes	—
43	C₂⁴×C₃₇	—	2	12	no	—
44	C₂⁴×C₃₇	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₃₇	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₃₇	—	2	12	no	—
47	C₂²×C₄×C₃₇	—	2	96	no	—
48	C₂²×C₄×C₃₇	—	2	24	no	—
49	C₂²×C₄×C₃₇	—	2	24	no	—
50	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₃₇	C₂²×C₄	1	24	yes	—
52	C₂²×C₄×C₃₇	C₂²×C₄	1	24	yes	—
53	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₃₇	#41	1	24	yes	—
55	C₂²×C₄×C₃₇	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₃₇	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
59	C₄⋊C₄×C₃₇	—	2	48	no	—
60	C₂²⋊C₄×C₃₇	—	2	48	no	—
61	C₄²×C₃₇	—	2	48	no	—
62	C₄²×C₃₇	C₂²×C₄	1	8	yes	—
63	C₄²×C₃₇	C₂²×C₄	1	12	yes	—
64	C₄²×C₃₇	C₂²×C₄	1	12	yes	—
65	C₄²×C₃₇	C₂²×C₄	1	24	yes	—
66	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
68	C₂³×C₃₇	—	2	32	no	—
69	C₂³×C₃₇	—	2	24	no	—
70	C₂³×C₃₇	—	2	12	no	—
71	C₂³×C₃₇	C₂³×C₄	1	6	yes	—
72	C₂³×C₃₇	C₂×#41	1	12	yes	—
73	C₂³×C₃₇	C₂⁵	1	1	yes	characteristic
74	C₂×C₄×C₃₇	—	2	24	no	—
75	C₂×C₄×C₃₇	—	2	12	no	—
76	C₂×C₄×C₃₇	—	2	48	no	—
77	C₂×C₄×C₃₇	—	2	48	no	—
78	C₂×C₄×C₃₇	—	2	48	no	—
79	C₂×C₄×C₃₇	#117	1	48	yes	complemented
80	C₂×C₄×C₃₇	C₂³×C₄	1	12	yes	—
81	C₂×C₄×C₃₇	C₂³×C₄	1	6	yes	—
82	C₂×C₄×C₃₇	C₂³×C₄	1	6	yes	—
83	C₂×C₄×C₃₇	C₂×#41	1	12	yes	—
84	C₂×C₄×C₃₇	C₂×#41	1	12	yes	—
85	#34012	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₃₇	—	2	32	no	—
87	C₂²×C₃₇	—	2	12	no	—
88	C₂²×C₃₇	—	2	24	no	—
89	C₂²×C₃₇	C₂×#117	1	12	yes	—
90	C₂²×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
91	C₂²×C₃₇	C₂²×#41	1	6	yes	—
92	C₄×C₃₇	—	2	12	no	—
93	C₄×C₃₇	—	4	16	no	—
94	C₄×C₃₇	C₂×#117	1	12	yes	—
95	C₄×C₃₇	C₂×#117	1	12	yes	—
96	C₂³×C₄²	C₂×C₃₇	1	1	yes	characteristic
97	#2285	C₂×C₃₇	1	6	yes	complemented
98	#2283	C₂×C₃₇	1	12	yes	complemented
99	#2280	C₂×C₃₇	1	6	yes	—
100	C₄×#117	C₂×C₃₇	1	6	yes	complemented
101	C₂×C₃₇	—	2	12	no	—
102	C₂×C₃₇	C₂²×#117	1	3	yes	—
103	C₂×C₃₇	#3444	1	4	yes	—
104	C₂⁴×C₄	C₂²×C₃₇	1	3	yes	—
105	C₂×#117	C₂²×C₃₇	1	12	yes	—
106	C₂²×C₄²	C₄×C₃₇	1	16	yes	complemented
107	C₂²×C₄²	C₂²×C₃₇	1	3	yes	—
108	C₂²×C₄²	C₂²×C₃₇	1	6	yes	—
109	C₂²×C₄²	C₂²×C₃₇	1	3	yes	—
110	#384	C₂²×C₃₇	1	4	yes	complemented
111	#383	C₂²×C₃₇	1	8	yes	complemented
112	#382	C₂²×C₃₇	1	12	yes	complemented
113	#381	C₂²×C₃₇	1	12	yes	—
114	#380	C₂²×C₃₇	1	24	yes	—
115	#379	C₂²×C₃₇	1	4	yes	—
116	C₄×C₄⋊C₄	C₂²×C₃₇	1	24	yes	complemented
117	C₄×C₂²⋊C₄	C₂²×C₃₇	1	24	yes	—
118	#376	C₂²×C₃₇	1	12	yes	complemented
119	C₄³	C₂²×C₃₇	1	4	yes	complemented
120	C₃₇	#34012	1	1	yes	characteristic, direct product factor, Sylow
121	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
122	C₂³×C₄	—	2	24	no	—
123	C₂³×C₄	C₂×C₄×C₃₇	1	24	yes	—
124	C₂³×C₄	C₂³×C₃₇	1	6	yes	—
125	C₂³×C₄	C₂³×C₃₇	1	6	yes	—
126	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
127	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
128	#117	—	2	48	no	—
129	C₂×C₄⋊C₄	C₂³×C₃₇	1	24	yes	—
130	C₂×C₂²⋊C₄	C₂³×C₃₇	1	24	yes	—
131	C₂×C₄²	C₂×C₄×C₃₇	1	48	yes	—
132	C₂×C₄²	C₂×C₄×C₃₇	1	24	yes	—
133	C₂×C₄²	C₂×C₄×C₃₇	1	24	yes	—
134	C₂×C₄²	C₂³×C₃₇	1	24	yes	—
135	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
137	C₂×C₄²	C₂³×C₃₇	1	6	yes	—
138	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
139	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
140	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
141	#95	C₂³×C₃₇	1	48	yes	—
142	C₂⁴	—	2	12	no	—
143	C₂⁴	C₂²×C₄×C₃₇	1	4	yes	—
144	C₂⁴	C₂⁴×C₃₇	1	3	yes	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	2	96	no	—
147	C₂²×C₄	—	2	24	no	—
148	C₂²×C₄	—	2	24	no	—
149	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
150	C₂²×C₄	C₂²×C₄×C₃₇	1	24	yes	—
151	C₂²×C₄	C₂²×C₄×C₃₇	1	24	yes	—
152	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
153	C₂²×C₄	#41×C₃₇	1	24	yes	—
154	C₂²×C₄	C₂⁴×C₃₇	1	16	yes	—
155	C₂²×C₄	C₂⁴×C₃₇	1	6	yes	—
156	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
157	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
158	C₄⋊C₄	—	2	48	no	—
159	C₂²⋊C₄	—	2	48	no	—
160	C₄²	—	2	48	no	—
161	C₄²	C₂²×C₄×C₃₇	1	8	yes	—
162	C₄²	C₂²×C₄×C₃₇	1	12	yes	—
163	C₄²	C₂²×C₄×C₃₇	1	12	yes	—
164	C₄²	C₂²×C₄×C₃₇	1	24	yes	—
165	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
166	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
167	C₂³	—	2	32	no	—
168	C₂³	—	2	12	no	—
169	C₂³	—	2	24	no	—
170	C₂³	C₂³×C₄×C₃₇	1	6	yes	—
171	C₂³	C₂×#41×C₃₇	1	12	yes	—
172	C₂³	C₂⁵×C₃₇	1	1	yes	characteristic
173	C₂×C₄	—	2	12	no	—
174	C₂×C₄	—	2	24	no	—
175	C₂×C₄	—	2	48	no	—
176	C₂×C₄	—	2	48	no	—
177	C₂×C₄	—	2	48	no	—
178	C₂×C₄	#117×C₃₇	1	48	yes	complemented
179	C₂×C₄	C₂³×C₄×C₃₇	1	12	yes	—
180	C₂×C₄	C₂³×C₄×C₃₇	1	6	yes	—
181	C₂×C₄	C₂³×C₄×C₃₇	1	6	yes	—
182	C₂×C₄	C₂×#41×C₃₇	1	12	yes	—
183	C₂×C₄	C₂×#41×C₃₇	1	12	yes	—
184	C₂²	—	2	32	no	—
185	C₂²	—	2	12	no	—
186	C₂²	—	2	24	no	—
187	C₂²	C₂×#117×C₃₇	1	12	yes	—
188	C₂²	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
189	C₂²	C₂²×#41×C₃₇	1	6	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	4	16	no	—
192	C₄	C₂×#117×C₃₇	1	12	yes	—
193	C₄	C₂×#117×C₃₇	1	12	yes	—
194	C₂	—	2	12	no	—
195	C₂	C₂²×#117×C₃₇	1	3	yes	—
196	C₂	C₃₇×#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂⁵
prime radical	C₂⁵×C₃₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2285	C₂	semidirect
3. C₃₇×#2283	C₂	semidirect
5. C₄×#117×C₃₇	C₂	semidirect
8. C₂²×C₄²×C₃₇	C₄	semidirect
12. C₃₇×#384	C₂²	semidirect
13. C₃₇×#383	C₂²	semidirect
14. C₃₇×#382	C₂²	semidirect
18. C₄×C₄⋊C₄×C₃₇	C₂²	semidirect
20. C₃₇×#376	C₂²	semidirect
21. C₄³×C₃₇	C₂²	semidirect
79. C₂×C₄×C₃₇	#117	semidirect
85. #34012	C₃₇	direct (N × H)
97. #2285	C₂×C₃₇	semidirect
98. #2283	C₂×C₃₇	semidirect
100. C₄×#117	C₂×C₃₇	semidirect
106. C₂²×C₄²	C₄×C₃₇	semidirect
110. #384	C₂²×C₃₇	semidirect
111. #383	C₂²×C₃₇	semidirect
112. #382	C₂²×C₃₇	semidirect
116. C₄×C₄⋊C₄	C₂²×C₃₇	semidirect
118. #376	C₂²×C₃₇	semidirect
119. C₄³	C₂²×C₃₇	semidirect
120. C₃₇	#34012	direct (N × H)
178. C₂×C₄	#117×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₃₇	2
C₃₇×#2285	2
C₃₇×#2283	2
C₃₇×#2280	2
C₄×#117×C₃₇	2
#34012	37

Subgroups — C37×#34012 #22655265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#34012 #22655265