Subgroups — C₃₇×#33716 #22654969

Order 9472.

826 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#508	C₂	1	4	yes	maximal, complemented
2	C₃₇×#3147	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2251	C₂	1	4	yes	maximal, complemented
4	C₃₇×#2250	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₃₇	—	2	2	no	—
7	C₂×#135×C₃₇	C₂²	1	4	yes	—
8	C₃₇×#532	—	2	2	no	—
9	C₃₇×#508	—	2	16	no	—
10	C₂×Q₃₂×C₃₇	C₂²	1	4	yes	—
11	C₂×SD₃₂×C₃₇	C₂²	1	8	yes	—
12	C₂×D₁₆×C₃₇	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#495	—	2	2	no	—
15	C₃₇×#495	C₂²	1	4	yes	—
16	C₃₇×#494	C₂²	1	2	yes	—
17	C₃₇×#493	C₂²	1	2	yes	—
18	C₃₇×#492	—	2	2	no	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#346	C₂²	1	4	yes	—
21	C₂×#41×C₃₇	—	2	4	no	—
22	#135×C₃₇	—	2	16	no	—
23	#135×C₃₇	—	4	8	no	—
24	C₂×Q₁₆×C₃₇	—	2	2	no	—
25	C₂×Q₁₆×C₃₇	C₂³	1	4	yes	—
26	C₂×SD₁₆×C₃₇	—	2	4	no	—
27	C₂×SD₁₆×C₃₇	—	2	4	no	—
28	C₂×D₈×C₃₇	—	2	2	no	—
29	C₂×D₈×C₃₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₃₇	—	2	2	no	—
33	#127×C₃₇	—	2	2	no	—
34	#124×C₃₇	—	2	4	no	—
35	#121×C₃₇	—	4	4	no	—
36	C₄×D₄×C₃₇	—	4	4	no	—
37	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
38	Q₃₂×C₃₇	—	2	8	no	—
39	SD₃₂×C₃₇	—	2	16	no	—
40	D₁₆×C₃₇	—	2	8	no	—
41	C₂×C₁₆×C₃₇	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₃₇	C₂³	1	4	yes	—
43	C₄×C₈×C₃₇	D₄	1	2	yes	—
44	C₄×C₈×C₃₇	C₂³	1	2	yes	—
45	#41×C₃₇	—	4	8	no	—
46	#41×C₃₇	—	4	8	no	—
47	C₂×Q₈×C₃₇	—	2	4	no	—
48	C₂×D₄×C₃₇	—	2	4	no	—
49	C₂×D₄×C₃₇	—	4	4	no	—
50	C₂×D₄×C₃₇	—	4	4	no	—
51	C₂²×C₄×C₃₇	—	4	4	no	—
52	C₂²×C₄×C₃₇	D₈	1	2	yes	—
53	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₃₇	—	2	8	no	—
55	Q₁₆×C₃₇	—	4	4	no	—
56	SD₁₆×C₃₇	—	4	8	no	—
57	SD₁₆×C₃₇	—	4	8	no	—
58	D₈×C₃₇	—	2	8	no	—
59	D₈×C₃₇	—	4	4	no	—
60	C₂×C₈×C₃₇	D₈	1	4	yes	—
61	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₃₇	—	4	4	no	—
66	C₂²⋊C₄×C₃₇	—	4	8	no	—
67	C₄²×C₃₇	D₈	1	2	yes	—
68	C₄²×C₃₇	C₂×D₄	1	2	yes	—
69	C₁₆×C₃₇	C₂×D₄	1	8	yes	—
70	C₂³×C₃₇	—	4	4	no	—
71	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
72	Q₈×C₃₇	—	4	4	no	—
73	Q₈×C₃₇	—	4	4	no	—
74	D₄×C₃₇	—	4	4	no	—
75	D₄×C₃₇	—	4	4	no	—
76	D₄×C₃₇	—	8	8	no	—
77	D₄×C₃₇	—	8	8	no	—
78	C₂×C₄×C₃₇	—	4	2	no	—
79	C₂×C₄×C₃₇	—	4	4	no	—
80	C₂×C₄×C₃₇	—	8	8	no	—
81	C₂×C₄×C₃₇	#127	1	2	yes	—
82	C₂×C₄×C₃₇	C₂×D₈	1	4	yes	—
83	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₃₇	#127	1	2	yes	—
87	C₈×C₃₇	C₂×D₈	1	4	yes	—
88	C₈×C₃₇	C₂²×D₄	1	2	yes	—
89	#33716	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₃₇	—	2	2	no	—
91	C₂²×C₃₇	—	8	4	no	—
92	C₂²×C₃₇	—	8	8	no	—
93	C₂²×C₃₇	#493	1	2	yes	—
94	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₃₇	—	2	4	no	—
96	C₄×C₃₇	—	8	4	no	—
97	C₄×C₃₇	#493	1	2	yes	—
98	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
99	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
100	C₂×#508	C₂×C₃₇	1	4	yes	complemented
101	#3147	C₂×C₃₇	1	2	yes	complemented
102	#2251	C₂×C₃₇	1	4	yes	complemented
103	#2250	C₂×C₃₇	1	4	yes	complemented
104	C₂×#346	C₂×C₃₇	1	1	yes	characteristic, complemented
105	C₂×C₃₇	—	2	2	no	—
106	C₂×C₃₇	—	16	4	no	—
107	C₂×C₃₇	C₂×#493	1	1	yes	characteristic
108	C₂×C₃₇	#3414	1	2	yes	—
109	C₂×#135	—	2	2	no	—
110	C₂×#135	C₂²×C₃₇	1	4	yes	—
111	#532	—	2	2	no	—
112	#508	—	2	16	no	—
113	C₂×Q₃₂	C₂²×C₃₇	1	4	yes	—
114	C₂×SD₃₂	C₂²×C₃₇	1	8	yes	—
115	C₂×D₁₆	C₂²×C₃₇	1	4	yes	—
116	C₂²×C₁₆	C₂²×C₃₇	1	2	yes	—
117	#495	—	2	2	no	—
118	#495	C₂²×C₃₇	1	4	yes	—
119	#494	C₂²×C₃₇	1	2	yes	—
120	#493	C₂²×C₃₇	1	2	yes	—
121	#492	—	2	2	no	—
122	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
123	#346	C₂²×C₃₇	1	4	yes	—
124	C₃₇	#33716	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×#41	—	2	4	no	—
126	#135	—	2	16	no	—
127	#135	—	4	8	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	C₂³×C₃₇	1	4	yes	—
130	C₂×SD₁₆	—	2	4	no	—
131	C₂×SD₁₆	—	2	4	no	—
132	C₂×D₈	—	2	2	no	—
133	C₂×D₈	C₂³×C₃₇	1	4	yes	—
134	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#127	—	2	2	no	—
138	#124	—	2	4	no	—
139	#121	—	4	4	no	—
140	C₄×D₄	—	4	4	no	—
141	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
142	Q₃₂	—	2	8	no	—
143	SD₃₂	—	2	16	no	—
144	D₁₆	—	2	8	no	—
145	C₂×C₁₆	D₄×C₃₇	1	8	yes	complemented
146	C₂×C₁₆	C₂³×C₃₇	1	4	yes	—
147	C₄×C₈	D₄×C₃₇	1	2	yes	—
148	C₄×C₈	C₂³×C₃₇	1	2	yes	—
149	#41	—	4	8	no	—
150	#41	—	4	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	D₈×C₃₇	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
158	Q₁₆	—	2	8	no	—
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	8	no	—
161	SD₁₆	—	4	8	no	—
162	D₈	—	2	8	no	—
163	D₈	—	4	4	no	—
164	C₂×C₈	D₈×C₃₇	1	4	yes	—
165	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
168	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	4	8	no	—
171	C₄²	D₈×C₃₇	1	2	yes	—
172	C₄²	C₂×D₄×C₃₇	1	2	yes	—
173	C₁₆	C₂×D₄×C₃₇	1	8	yes	—
174	C₂³	—	4	4	no	—
175	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
176	Q₈	—	4	4	no	—
177	Q₈	—	4	4	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	8	no	—
181	D₄	—	8	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	8	no	—
185	C₂×C₄	#127×C₃₇	1	2	yes	—
186	C₂×C₄	C₂×D₈×C₃₇	1	4	yes	—
187	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
190	C₈	#127×C₃₇	1	2	yes	—
191	C₈	C₂×D₈×C₃₇	1	4	yes	—
192	C₈	C₂²×D₄×C₃₇	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	8	4	no	—
195	C₂²	—	8	8	no	—
196	C₂²	C₃₇×#493	1	2	yes	—
197	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	C₃₇×#493	1	2	yes	—
201	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
202	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	C₂×C₃₇×#493	1	1	yes	characteristic
206	C₂	C₃₇×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#508	C₂	semidirect
2. C₃₇×#3147	C₂	semidirect
3. C₃₇×#2251	C₂	semidirect
4. C₃₇×#2250	C₂	semidirect
5. C₂×C₃₇×#346	C₂	semidirect
41. C₂×C₁₆×C₃₇	D₄	semidirect
89. #33716	C₃₇	direct (N × H)
100. C₂×#508	C₂×C₃₇	semidirect
101. #3147	C₂×C₃₇	semidirect
102. #2251	C₂×C₃₇	semidirect
103. #2250	C₂×C₃₇	semidirect
104. C₂×#346	C₂×C₃₇	semidirect
124. C₃₇	#33716	direct (N × H)
145. C₂×C₁₆	D₄×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#508	2
C₃₇×#3147	2
C₃₇×#2251	2
C₃₇×#2250	2
C₂×C₃₇×#346	2
#33716	37

Subgroups — C37×#33716 #22654969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33716 #22654969