Subgroups — C₃₇×#13236 #22634489

Order 9472.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#358	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#1969	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
7	C₂×Q₃₂×C₃₇	—	2	2	no	—
8	C₂×Q₃₂×C₃₇	—	2	2	no	—
9	C₂×Q₃₂×C₃₇	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₃₇	C₂²	1	2	yes	—
13	C₂×#103×C₃₇	—	2	2	no	—
14	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#391	—	2	2	no	—
16	C₃₇×#366	—	2	2	no	—
17	C₃₇×#358	—	2	4	no	—
18	C₄×C₁₆×C₃₇	—	2	2	no	—
19	C₂²×Q₈×C₃₇	—	2	2	no	—
20	C₂×Q₁₆×C₃₇	—	2	2	no	—
21	C₂×Q₁₆×C₃₇	—	2	2	no	—
22	C₂×Q₁₆×C₃₇	—	2	2	no	—
23	C₂×Q₁₆×C₃₇	—	2	2	no	—
24	C₂×Q₁₆×C₃₇	—	2	2	no	—
25	C₂×Q₁₆×C₃₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
30	Q₃₂×C₃₇	—	2	8	no	—
31	C₂×C₁₆×C₃₇	—	2	1	no	—
32	C₂×C₁₆×C₃₇	—	2	1	no	—
33	C₂×C₁₆×C₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₃₇	D₄	1	2	yes	—
35	C₂×C₁₆×C₃₇	D₄	1	2	yes	—
36	C₂×C₁₆×C₃₇	D₄	1	2	yes	—
37	#107×C₃₇	—	2	2	no	—
38	#107×C₃₇	—	2	2	no	—
39	#103×C₃₇	—	4	4	no	—
40	C₄×C₈×C₃₇	—	2	1	no	—
41	C₄×C₈×C₃₇	—	2	1	no	—
42	#95×C₃₇	—	4	4	no	—
43	C₂×Q₈×C₃₇	—	2	4	no	—
44	C₂×Q₈×C₃₇	—	4	2	no	—
45	C₂×Q₈×C₃₇	—	4	2	no	—
46	C₂×Q₈×C₃₇	—	4	2	no	—
47	C₂×Q₈×C₃₇	—	4	2	no	—
48	C₂×Q₈×C₃₇	—	4	2	no	—
49	C₂²×C₄×C₃₇	—	4	2	no	—
50	C₂²×C₄×C₃₇	—	4	2	no	—
51	C₂²×C₄×C₃₇	D₈	1	2	yes	—
52	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₃₇	—	2	4	no	—
54	Q₁₆×C₃₇	—	2	4	no	—
55	Q₁₆×C₃₇	—	4	2	no	—
56	Q₁₆×C₃₇	—	4	2	no	—
57	C₂×C₈×C₃₇	—	2	1	no	—
58	C₂×C₈×C₃₇	—	2	1	no	—
59	C₂×C₈×C₃₇	D₈	1	2	yes	—
60	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₇	#41	1	1	yes	characteristic
65	C₂×C₈×C₃₇	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₃₇	—	4	2	no	—
67	C₄⋊C₄×C₃₇	—	4	2	no	—
68	C₄²×C₃₇	—	2	1	no	—
69	C₄²×C₃₇	—	2	1	no	—
70	C₁₆×C₃₇	—	2	2	no	—
71	C₁₆×C₃₇	C₂²⋊C₄	1	4	yes	—
72	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₃₇	—	4	4	no	—
74	Q₈×C₃₇	—	4	4	no	—
75	Q₈×C₃₇	—	8	2	no	—
76	Q₈×C₃₇	—	8	2	no	—
77	C₂×C₄×C₃₇	—	2	2	no	—
78	C₂×C₄×C₃₇	—	2	2	no	—
79	C₂×C₄×C₃₇	—	4	4	no	—
80	C₂×C₄×C₃₇	—	8	2	no	—
81	C₂×C₄×C₃₇	—	8	2	no	—
82	C₂×C₄×C₃₇	—	8	2	no	—
83	C₂×C₄×C₃₇	—	8	4	no	—
84	C₂×C₄×C₃₇	Q₃₂	1	4	yes	complemented
85	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₃₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₃₇	#135	1	1	yes	characteristic
90	C₂×C₄×C₃₇	#135	1	1	yes	characteristic
91	C₈×C₃₇	—	2	2	no	—
92	C₈×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₈×C₃₇	C₄×D₄	1	1	yes	characteristic
94	C₈×C₃₇	#121	1	1	yes	characteristic
95	C₈×C₃₇	#124	1	1	yes	characteristic
96	#13236	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₃₇	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₃₇	#466	1	1	yes	characteristic
99	C₂²×C₃₇	#493	1	1	yes	characteristic
100	C₂²×C₃₇	C₂×Q₃₂	1	2	yes	—
101	C₂²×C₃₇	#508	1	2	yes	—
102	C₄×C₃₇	—	2	4	no	—
103	C₄×C₃₇	—	8	4	no	—
104	C₄×C₃₇	—	16	2	no	—
105	C₄×C₃₇	#390	1	1	yes	characteristic
106	C₄×C₃₇	C₄×D₈	1	1	yes	characteristic
107	C₄×C₃₇	#466	1	1	yes	characteristic
108	C₄×C₃₇	#495	1	1	yes	characteristic
109	C₂²×Q₃₂	C₂×C₃₇	1	1	yes	characteristic
110	C₂×#366	C₂×C₃₇	1	1	yes	characteristic
111	C₂×#358	C₂×C₃₇	1	2	yes	—
112	C₂×C₄×C₁₆	C₂×C₃₇	1	1	yes	characteristic
113	#1969	C₂×C₃₇	1	2	yes	—
114	C₂×C₃₇	#1970	1	1	yes	characteristic
115	C₂×C₃₇	C₄×Q₃₂	1	2	yes	—
116	C₂×C₃₇	#2216	1	2	yes	—
117	C₂×C₃₇	#2247	1	1	yes	characteristic
118	C₂×C₃₇	#2249	1	1	yes	characteristic
119	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
120	C₂×Q₃₂	—	2	2	no	—
121	C₂×Q₃₂	—	2	2	no	—
122	C₂×Q₃₂	C₄×C₃₇	1	4	yes	complemented
123	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
124	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
125	C₂×#107	C₂²×C₃₇	1	2	yes	—
126	C₂×#103	—	2	2	no	—
127	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
128	#391	—	2	2	no	—
129	#366	—	2	2	no	—
130	#358	—	2	4	no	—
131	C₄×C₁₆	—	2	2	no	—
132	C₃₇	#13236	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×Q₈	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	C₂×C₄×C₃₇	1	4	yes	—
140	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
144	Q₃₂	—	2	8	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	C₂×C₄×C₃₇	1	2	yes	—
148	C₂×C₁₆	D₄×C₃₇	1	2	yes	—
149	C₂×C₁₆	D₄×C₃₇	1	2	yes	—
150	C₂×C₁₆	D₄×C₃₇	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	2	2	no	—
153	#103	—	4	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	#95	—	4	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	D₈×C₃₇	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	4	2	no	—
170	Q₁₆	—	4	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	D₈×C₃₇	1	2	yes	—
174	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
178	C₂×C₈	#41×C₃₇	1	1	yes	characteristic
179	C₂×C₈	#41×C₃₇	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	C₂²⋊C₄×C₃₇	1	4	yes	—
186	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	4	4	no	—
189	Q₈	—	8	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	Q₃₂×C₃₇	1	4	yes	complemented
199	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
200	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
201	C₂×C₄	#127×C₃₇	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
203	C₂×C₄	#135×C₃₇	1	1	yes	characteristic
204	C₂×C₄	#135×C₃₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
207	C₈	C₄×D₄×C₃₇	1	1	yes	characteristic
208	C₈	#121×C₃₇	1	1	yes	characteristic
209	C₈	#124×C₃₇	1	1	yes	characteristic
210	C₂²	C₄×D₈×C₃₇	1	1	yes	characteristic
211	C₂²	C₃₇×#466	1	1	yes	characteristic
212	C₂²	C₃₇×#493	1	1	yes	characteristic
213	C₂²	C₂×Q₃₂×C₃₇	1	2	yes	—
214	C₂²	C₃₇×#508	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	8	4	no	—
217	C₄	—	16	2	no	—
218	C₄	C₃₇×#390	1	1	yes	characteristic
219	C₄	C₄×D₈×C₃₇	1	1	yes	characteristic
220	C₄	C₃₇×#466	1	1	yes	characteristic
221	C₄	C₃₇×#495	1	1	yes	characteristic
222	C₂	C₃₇×#1970	1	1	yes	characteristic
223	C₂	C₄×Q₃₂×C₃₇	1	2	yes	—
224	C₂	C₃₇×#2216	1	2	yes	—
225	C₂	C₃₇×#2247	1	1	yes	characteristic
226	C₂	C₃₇×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
9. C₂×Q₃₂×C₃₇	C₄	semidirect
84. C₂×C₄×C₃₇	Q₃₂	semidirect
96. #13236	C₃₇	direct (N × H)
122. C₂×Q₃₂	C₄×C₃₇	semidirect
132. C₃₇	#13236	direct (N × H)
198. C₂×C₄	Q₃₂×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₃₇	2
C₂×C₃₇×#366	2
C₂×C₃₇×#358	2
C₂×C₄×C₁₆×C₃₇	2
C₃₇×#1969	2
#13236	37

Subgroups — C37×#13236 #22634489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#13236 #22634489