Subgroups — C₃₇×#28734 #22649987

Order 9472.

666 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#3142	C₂	1	2	yes	maximal
3	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2079	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1998	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1816	C₂	1	4	yes	maximal, complemented
7	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₃₇	C₂²	1	2	yes	—
9	C₃₇×#532	C₂²	1	1	yes	characteristic
10	C₂×#125×C₃₇	C₂²	1	2	yes	—
11	C₃₇×#499	—	2	2	no	—
12	C₃₇×#499	—	2	2	no	—
13	C₃₇×#491	—	2	2	no	—
14	C₃₇×#490	—	2	2	no	—
15	C₃₇×#488	—	2	4	no	—
16	C₃₇×#436	—	2	2	no	—
17	C₈×D₄×C₃₇	—	2	2	no	—
18	C₂×#108×C₃₇	C₂²	1	2	yes	—
19	C₂×#107×C₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#423	C₂²	1	1	yes	characteristic
22	C₃₇×#417	C₂²	1	8	yes	—
23	C₃₇×#408	C₂²	1	2	yes	—
24	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#343	C₂²	1	8	yes	—
27	C₃₇×#341	C₂²	1	2	yes	—
28	C₂×#41×C₃₇	C₂³	1	2	yes	—
29	#131×C₃₇	—	2	8	no	—
30	C₂×M₁₆×C₃₇	—	2	2	no	—
31	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₇	—	2	2	no	—
34	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
37	#127×C₃₇	D₄	1	1	yes	characteristic
38	#125×C₃₇	—	2	2	no	—
39	#125×C₃₇	—	2	2	no	—
40	#125×C₃₇	—	2	4	no	—
41	#124×C₃₇	D₄	1	2	yes	—
42	#121×C₃₇	—	2	2	no	—
43	C₄×D₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#108×C₃₇	—	2	4	no	—
48	#107×C₃₇	—	2	2	no	—
49	#107×C₃₇	—	2	2	no	—
50	#106×C₃₇	—	2	4	no	—
51	C₄⋊C₈×C₃₇	—	2	2	no	—
52	#103×C₃₇	—	2	8	no	—
53	#102×C₃₇	—	2	8	no	—
54	C₂²⋊C₈×C₃₇	—	2	4	no	—
55	#97×C₃₇	—	2	2	no	—
56	C₄×C₈×C₃₇	—	2	2	no	—
57	C₄×C₈×C₃₇	D₄	1	2	yes	—
58	C₄×C₈×C₃₇	D₄	1	2	yes	—
59	#41×C₃₇	—	2	4	no	—
60	#41×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
62	C₂×D₄×C₃₇	—	2	2	no	—
63	C₂×D₄×C₃₇	—	2	2	no	—
64	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	—	2	4	no	—
67	C₂²×C₄×C₃₇	#41	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₃₇	—	2	4	no	—
70	M₁₆×C₃₇	—	2	4	no	—
71	M₁₆×C₃₇	—	2	4	no	—
72	M₁₆×C₃₇	—	4	4	no	—
73	C₂×C₈×C₃₇	—	2	2	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	—	2	2	no	—
76	C₂×C₈×C₃₇	—	4	4	no	—
77	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₃₇	#41	1	4	yes	—
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	—	2	4	no	—
83	C₄⋊C₄×C₃₇	—	2	4	no	—
84	C₄⋊C₄×C₃₇	—	4	4	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₂²⋊C₄×C₃₇	—	2	4	no	—
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₃₇	—	2	2	no	—
91	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
92	Q₈×C₃₇	—	2	4	no	—
93	D₄×C₃₇	—	2	4	no	—
94	D₄×C₃₇	—	4	4	no	—
95	D₄×C₃₇	—	4	4	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	4	2	no	—
98	C₂×C₄×C₃₇	—	4	2	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	8	4	no	—
102	C₂×C₄×C₃₇	#121	1	2	yes	—
103	C₂×C₄×C₃₇	#121	1	2	yes	—
104	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₃₇	—	2	2	no	—
109	C₈×C₃₇	—	4	2	no	—
110	C₈×C₃₇	—	4	4	no	—
111	C₈×C₃₇	#124	1	2	yes	—
112	C₈×C₃₇	#124	1	2	yes	—
113	#28734	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	—	4	4	no	—
117	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
118	C₂²×C₃₇	#556	1	2	yes	—
119	C₄×C₃₇	—	2	2	no	—
120	C₄×C₃₇	—	2	2	no	—
121	C₄×C₃₇	—	4	2	no	—
122	C₄×C₃₇	—	8	4	no	—
123	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
124	C₄×C₃₇	#536	1	1	yes	characteristic
125	C₄×C₃₇	#547	1	2	yes	—
126	C₂×#499	C₂×C₃₇	1	1	yes	characteristic, complemented
127	#3142	C₂×C₃₇	1	2	yes	—
128	#2931	C₂×C₃₇	1	1	yes	characteristic
129	#2079	C₂×C₃₇	1	4	yes	complemented
130	#1998	C₂×C₃₇	1	2	yes	complemented
131	#1816	C₂×C₃₇	1	4	yes	complemented
132	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
133	C₂×C₃₇	—	2	2	no	—
134	C₂×C₃₇	—	8	2	no	—
135	C₂×C₃₇	#2547	1	1	yes	characteristic
136	C₂×C₃₇	#3067	1	2	yes	—
137	C₂×#131	C₂²×C₃₇	1	2	yes	—
138	#532	C₂²×C₃₇	1	1	yes	characteristic
139	C₂×#125	C₂²×C₃₇	1	2	yes	—
140	#499	—	2	2	no	—
141	#499	—	2	2	no	—
142	#491	—	2	2	no	—
143	#490	—	2	2	no	—
144	#488	—	2	4	no	—
145	#436	—	2	2	no	—
146	C₈×D₄	—	2	2	no	—
147	C₂×#108	C₂²×C₃₇	1	2	yes	—
148	C₂×#107	C₂²×C₃₇	1	2	yes	—
149	C₂×#106	C₂²×C₃₇	1	2	yes	—
150	#423	C₂²×C₃₇	1	1	yes	characteristic
151	#417	C₂²×C₃₇	1	8	yes	—
152	#408	C₂²×C₃₇	1	2	yes	—
153	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
155	#343	C₂²×C₃₇	1	8	yes	—
156	#341	C₂²×C₃₇	1	2	yes	—
157	C₃₇	#28734	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#41	C₂³×C₃₇	1	2	yes	—
159	#131	—	2	8	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
162	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
166	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
167	#127	D₄×C₃₇	1	1	yes	characteristic
168	#125	—	2	2	no	—
169	#125	—	2	2	no	—
170	#125	—	2	4	no	—
171	#124	D₄×C₃₇	1	2	yes	—
172	#121	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
176	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
177	#108	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#106	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	#103	—	2	8	no	—
183	#102	—	2	8	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	D₄×C₃₇	1	2	yes	—
188	C₄×C₈	D₄×C₃₇	1	2	yes	—
189	#41	—	2	4	no	—
190	#41	—	2	4	no	—
191	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	#41×C₃₇	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
210	C₂×C₈	#41×C₃₇	1	4	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₃₇	1	2	yes	—
219	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
222	Q₈	—	2	4	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	#121×C₃₇	1	2	yes	—
233	C₂×C₄	#121×C₃₇	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
237	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	4	no	—
241	C₈	#124×C₃₇	1	2	yes	—
242	C₈	#124×C₃₇	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
247	C₂²	C₃₇×#556	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
253	C₄	C₃₇×#536	1	1	yes	characteristic
254	C₄	C₃₇×#547	1	2	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	8	2	no	—
257	C₂	C₃₇×#2547	1	1	yes	characteristic
258	C₂	C₃₇×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#499	C₂	semidirect
4. C₃₇×#2079	C₂	semidirect
5. C₃₇×#1998	C₂	semidirect
6. C₃₇×#1816	C₂	semidirect
7. C₄×#108×C₃₇	C₂	semidirect
113. #28734	C₃₇	direct (N × H)
126. C₂×#499	C₂×C₃₇	semidirect
129. #2079	C₂×C₃₇	semidirect
130. #1998	C₂×C₃₇	semidirect
131. #1816	C₂×C₃₇	semidirect
132. C₄×#108	C₂×C₃₇	semidirect
157. C₃₇	#28734	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₃₇×#3142	2
C₃₇×#2931	2
C₃₇×#2079	2
C₃₇×#1998	2
C₃₇×#1816	2
C₄×#108×C₃₇	2
#28734	37

Subgroups — C37×#28734 #22649987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28734 #22649987