Subgroups — C₃₇×#21960 #22643213

Order 9472.

818 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2283	C₂	1	2	yes	maximal
3	C₃₇×#1949	C₂	1	8	yes	maximal, complemented
4	C₃₇×#1808	C₂	1	2	yes	maximal
5	C₃₇×#1807	C₂	1	2	yes	maximal
6	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×#107×C₃₇	C₂²	1	4	yes	—
10	C₂×#106×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
13	C₃₇×#383	—	2	2	no	—
14	C₃₇×#382	C₂²	1	4	yes	—
15	C₃₇×#381	—	2	2	no	—
16	C₃₇×#380	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
18	C₄×C₂²⋊C₄×C₃₇	—	2	2	no	—
19	C₃₇×#376	—	2	2	no	—
20	C₃₇×#340	C₂²	1	4	yes	—
21	C₃₇×#340	C₂²	1	4	yes	—
22	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₃₇	—	4	2	no	—
26	C₂²×C₈×C₃₇	—	4	2	no	—
27	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
32	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
33	C₂×C₄²×C₃₇	—	2	4	no	—
34	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₃₇	D₄	1	2	yes	—
36	C₂×C₄²×C₃₇	Q₈	1	4	yes	—
37	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
38	#107×C₃₇	—	4	4	no	—
39	#106×C₃₇	—	4	4	no	—
40	C₄⋊C₈×C₃₇	C₂×C₄	1	16	yes	—
41	#95×C₃₇	—	2	4	no	—
42	#95×C₃₇	—	2	4	no	—
43	#95×C₃₇	—	2	4	no	—
44	#95×C₃₇	—	2	4	no	—
45	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₇	—	2	4	no	—
47	C₂²×C₄×C₃₇	—	2	2	no	—
48	C₂²×C₄×C₃₇	—	2	2	no	—
49	C₂²×C₄×C₃₇	—	2	8	no	—
50	C₂²×C₄×C₃₇	—	4	2	no	—
51	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₃₇	—	2	4	no	—
60	C₂×C₈×C₃₇	—	2	4	no	—
61	C₂×C₈×C₃₇	—	4	4	no	—
62	C₂×C₈×C₃₇	—	4	4	no	—
63	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
65	C₄⋊C₄×C₃₇	—	2	8	no	—
66	C₄⋊C₄×C₃₇	—	2	8	no	—
67	C₂²⋊C₄×C₃₇	—	4	8	no	—
68	C₄²×C₃₇	—	4	8	no	—
69	C₄²×C₃₇	C₄⋊C₄	1	8	yes	—
70	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
71	C₄²×C₃₇	C₂×Q₈	1	4	yes	—
72	C₂³×C₃₇	—	2	2	no	—
73	C₂³×C₃₇	—	2	2	no	—
74	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₃₇	#135	1	4	yes	—
76	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₃₇	—	2	2	no	—
78	C₂×C₄×C₃₇	—	2	4	no	—
79	C₂×C₄×C₃₇	—	2	4	no	—
80	C₂×C₄×C₃₇	—	2	8	no	—
81	C₂×C₄×C₃₇	—	2	2	no	—
82	C₂×C₄×C₃₇	—	2	2	no	—
83	C₂×C₄×C₃₇	—	2	4	no	—
84	C₂×C₄×C₃₇	—	4	4	no	—
85	C₂×C₄×C₃₇	—	4	8	no	—
86	C₂×C₄×C₃₇	—	4	8	no	—
87	C₂×C₄×C₃₇	—	4	8	no	—
88	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
94	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
95	C₈×C₃₇	—	2	4	no	—
96	C₈×C₃₇	—	2	4	no	—
97	#21960	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₃₇	—	2	2	no	—
99	C₂²×C₃₇	—	2	2	no	—
100	C₂²×C₃₇	—	2	8	no	—
101	C₂²×C₃₇	#427	1	4	yes	—
102	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃₇	#537	1	2	yes	—
104	C₂²×C₃₇	C₂×#135	1	2	yes	—
105	C₂²×C₃₇	#575	1	2	yes	—
106	C₄×C₃₇	—	2	2	no	—
107	C₄×C₃₇	—	2	4	no	—
108	C₄×C₃₇	—	2	4	no	—
109	C₄×C₃₇	—	8	8	no	—
110	C₄×C₃₇	#518	1	1	yes	characteristic
111	C₄×C₃₇	#519	1	1	yes	characteristic
112	C₄×C₃₇	#544	1	2	yes	—
113	C₂²×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
114	#2283	C₂×C₃₇	1	2	yes	—
115	#1949	C₂×C₃₇	1	8	yes	complemented
116	#1808	C₂×C₃₇	1	2	yes	—
117	#1807	C₂×C₃₇	1	2	yes	—
118	C₂×C₃₇	—	2	4	no	—
119	C₂×C₃₇	#2316	1	1	yes	characteristic
120	C₂×C₃₇	C₂×#427	1	1	yes	characteristic
121	C₂×C₃₇	#2911	1	1	yes	characteristic
122	C₂×C₃₇	#3250	1	4	yes	—
123	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
126	C₂×#107	C₂²×C₃₇	1	4	yes	—
127	C₂×#106	C₂²×C₃₇	1	4	yes	—
128	C₂×C₄⋊C₈	C₄×C₃₇	1	8	yes	complemented
129	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
130	#383	—	2	2	no	—
131	#382	C₂²×C₃₇	1	4	yes	—
132	#381	—	2	2	no	—
133	#380	C₂²×C₃₇	1	4	yes	—
134	C₄×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
135	C₄×C₂²⋊C₄	—	2	2	no	—
136	#376	—	2	2	no	—
137	#340	C₂²×C₃₇	1	4	yes	—
138	#340	C₂²×C₃₇	1	4	yes	—
139	C₃₇	#21960	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
141	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	—	4	2	no	—
145	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
146	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
147	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
148	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂³×C₃₇	1	8	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₄²	—	2	4	no	—
152	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
153	C₂×C₄²	D₄×C₃₇	1	2	yes	—
154	C₂×C₄²	Q₈×C₃₇	1	4	yes	—
155	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
156	#107	—	4	4	no	—
157	#106	—	4	4	no	—
158	C₄⋊C₈	C₂×C₄×C₃₇	1	16	yes	—
159	#95	—	2	4	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
170	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
176	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₄²	—	4	8	no	—
187	C₄²	C₄⋊C₄×C₃₇	1	8	yes	—
188	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
189	C₄²	C₂×Q₈×C₃₇	1	4	yes	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
193	C₂³	#135×C₃₇	1	4	yes	—
194	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	4	yes	—
209	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
211	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
212	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	C₃₇×#427	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
220	C₂²	C₃₇×#537	1	2	yes	—
221	C₂²	C₂×#135×C₃₇	1	2	yes	—
222	C₂²	C₃₇×#575	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	8	8	no	—
227	C₄	C₃₇×#518	1	1	yes	characteristic
228	C₄	C₃₇×#519	1	1	yes	characteristic
229	C₄	C₃₇×#544	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₃₇×#2316	1	1	yes	characteristic
232	C₂	C₂×C₃₇×#427	1	1	yes	characteristic
233	C₂	C₃₇×#2911	1	1	yes	characteristic
234	C₂	C₃₇×#3250	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂⁴×C₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂²×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1949	C₂	semidirect
11. C₂×C₄⋊C₈×C₃₇	C₄	semidirect
97. #21960	C₃₇	direct (N × H)
115. #1949	C₂×C₃₇	semidirect
128. C₂×C₄⋊C₈	C₄×C₃₇	semidirect
139. C₃₇	#21960	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×C₄⋊C₈×C₃₇	2
C₃₇×#2283	2
C₃₇×#1949	2
C₃₇×#1808	2
C₃₇×#1807	2
#21960	37

Subgroups — C37×#21960 #22643213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#21960 #22643213