Subgroups — C₂×C₃₇×#1591 #22639138

Order 9472.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#446	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1591	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₇×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₈²×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
9	C₃₇×#446	—	2	4	no	—
10	C₃₇×#446	C₂²	1	8	yes	—
11	C₄×Q₁₆×C₃₇	C₄	1	4	yes	—
12	C₄×Q₁₆×C₃₇	C₂²	1	4	yes	—
13	C₂×#107×C₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₃₇	—	2	2	no	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₃₇	C₄	1	2	yes	—
17	C₂×#97×C₃₇	—	2	2	no	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#335	C₂²	1	4	yes	—
22	C₃₇×Q₈⋊C₈	C₂²	1	8	yes	—
23	C₈²×C₃₇	C₂²	1	4	yes	—
24	C₂²×Q₈×C₃₇	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₃₇	—	2	4	no	—
26	C₂×Q₁₆×C₃₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₃₇	—	2	1	no	—
28	C₂²×C₈×C₃₇	—	2	2	no	—
29	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₃₇	—	2	2	no	—
34	C₄×Q₈×C₃₇	—	2	2	no	—
35	C₄×Q₈×C₃₇	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₃₇	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₇	—	2	2	no	—
40	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
41	#107×C₃₇	C₂×C₄	1	4	yes	—
42	C₄⋊C₈×C₃₇	—	2	8	no	—
43	C₄⋊C₈×C₃₇	—	2	2	no	—
44	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
45	#103×C₃₇	C₂×C₄	1	8	yes	—
46	#97×C₃₇	—	2	8	no	—
47	C₄×C₈×C₃₇	—	2	1	no	—
48	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₃₇	D₄	1	2	yes	—
50	C₄×C₈×C₃₇	C₂³	1	2	yes	—
51	C₄×C₈×C₃₇	C₂³	1	2	yes	—
52	C₄×C₈×C₃₇	C₂³	1	2	yes	—
53	C₂×Q₈×C₃₇	—	2	2	no	—
54	C₂×Q₈×C₃₇	—	2	4	no	—
55	C₂×Q₈×C₃₇	—	2	4	no	—
56	C₂×Q₈×C₃₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₃₇	—	2	2	no	—
58	C₂²×C₄×C₃₇	—	2	2	no	—
59	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
62	Q₁₆×C₃₇	—	2	8	no	—
63	C₂×C₈×C₃₇	—	2	4	no	—
64	C₂×C₈×C₃₇	—	2	2	no	—
65	C₂×C₈×C₃₇	—	2	2	no	—
66	C₂×C₈×C₃₇	—	2	2	no	—
67	C₂×C₈×C₃₇	—	2	4	no	—
68	C₂×C₈×C₃₇	—	4	4	no	—
69	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
70	C₂×C₈×C₃₇	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
75	C₄⋊C₄×C₃₇	—	2	4	no	—
76	C₄⋊C₄×C₃₇	—	2	2	no	—
77	C₄⋊C₄×C₃₇	—	2	4	no	—
78	C₄⋊C₄×C₃₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₃₇	—	2	4	no	—
80	C₄²×C₃₇	—	2	4	no	—
81	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₇	#41	1	1	yes	characteristic
84	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
85	C₂³×C₃₇	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₃₇	—	2	4	no	—
87	Q₈×C₃₇	—	2	4	no	—
88	Q₈×C₃₇	—	4	4	no	—
89	C₂×C₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	—	2	4	no	—
91	C₂×C₄×C₃₇	—	2	4	no	—
92	C₂×C₄×C₃₇	—	4	4	no	—
93	C₂×C₄×C₃₇	—	4	4	no	—
94	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₃₇	#131	1	2	yes	—
99	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₃₇	#135	1	2	yes	—
101	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
104	C₈×C₃₇	—	2	4	no	—
105	C₈×C₃₇	—	2	2	no	—
106	C₈×C₃₇	—	2	2	no	—
107	C₈×C₃₇	—	4	4	no	—
108	C₈×C₃₇	C₂×M₁₆	1	4	yes	—
109	C₈×C₃₇	C₂×Q₁₆	1	4	yes	complemented
110	C₂×#1591	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₇	#436	1	2	yes	—
112	C₂²×C₃₇	C₄×Q₁₆	1	2	yes	—
113	C₂²×C₃₇	#443	1	2	yes	—
114	C₂²×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₃₇	—	2	1	no	—
116	C₄×C₃₇	—	2	1	no	—
117	C₄×C₃₇	—	4	4	no	—
118	C₄×C₃₇	—	4	4	no	—
119	C₄×C₃₇	#436	1	2	yes	—
120	C₄×C₃₇	C₄×Q₁₆	1	2	yes	—
121	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₄×C₃₇	C₂×#131	1	1	yes	characteristic
123	C₄×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₃₇	C₂×#135	1	1	yes	characteristic
125	C₂×#446	C₂×C₃₇	1	2	yes	—
126	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
127	#1591	C₂×C₃₇	1	8	yes	direct product factor
128	C₂×#335	C₂×C₃₇	1	1	yes	characteristic
129	C₂×Q₈⋊C₈	C₂×C₃₇	1	2	yes	—
130	C₂×C₈²	C₂×C₃₇	1	1	yes	characteristic
131	C₂×C₃₇	#1591	1	4	yes	direct product factor
132	C₂×C₃₇	C₂×#436	1	1	yes	characteristic
133	C₂×C₃₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
134	C₂×C₃₇	C₂×#443	1	1	yes	characteristic
135	C₂²×Q₁₆	C₄×C₃₇	1	1	yes	characteristic
136	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
137	#446	—	2	4	no	—
138	#446	C₂²×C₃₇	1	8	yes	—
139	C₄×Q₁₆	C₄×C₃₇	1	4	yes	—
140	C₄×Q₁₆	C₂²×C₃₇	1	4	yes	—
141	C₂×#107	C₄×C₃₇	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	2	2	no	—
143	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
144	C₂×#103	C₄×C₃₇	1	2	yes	—
145	C₂×#97	—	2	2	no	—
146	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
149	#335	C₂²×C₃₇	1	4	yes	—
150	Q₈⋊C₈	C₂²×C₃₇	1	8	yes	—
151	C₈²	C₂²×C₃₇	1	4	yes	—
152	C₃₇	C₂×#1591	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×Q₈	C₂×C₄×C₃₇	1	2	yes	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	C₂×C₄×C₃₇	1	4	yes	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	C₂×C₄×C₃₇	1	4	yes	—
165	C₄×Q₈	C₂³×C₃₇	1	4	yes	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
170	#107	C₂×C₄×C₃₇	1	4	yes	—
171	C₄⋊C₈	—	2	8	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
174	#103	C₂×C₄×C₃₇	1	8	yes	—
175	#97	—	2	8	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
178	C₄×C₈	D₄×C₃₇	1	2	yes	—
179	C₄×C₈	C₂³×C₃₇	1	2	yes	—
180	C₄×C₈	C₂³×C₃₇	1	2	yes	—
181	C₄×C₈	C₂³×C₃₇	1	2	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	C₂²×C₄×C₃₇	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
190	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
191	Q₁₆	—	2	8	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
199	C₂×C₈	Q₁₆×C₃₇	1	4	yes	complemented
200	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
201	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₃₇	1	4	yes	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
212	C₄²	#41×C₃₇	1	1	yes	characteristic
213	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
214	C₂³	C₄×D₄×C₃₇	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
226	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
227	C₂×C₄	#131×C₃₇	1	2	yes	—
228	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
229	C₂×C₄	#135×C₃₇	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	4	4	no	—
237	C₈	C₂×M₁₆×C₃₇	1	4	yes	—
238	C₈	C₂×Q₁₆×C₃₇	1	4	yes	complemented
239	C₂²	C₃₇×#436	1	2	yes	—
240	C₂²	C₄×Q₁₆×C₃₇	1	2	yes	—
241	C₂²	C₃₇×#443	1	2	yes	—
242	C₂²	C₂×C₄×D₄×C₃₇	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	4	4	no	—
247	C₄	C₃₇×#436	1	2	yes	—
248	C₄	C₄×Q₁₆×C₃₇	1	2	yes	—
249	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
250	C₄	C₂×#131×C₃₇	1	1	yes	characteristic
251	C₄	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
252	C₄	C₂×#135×C₃₇	1	1	yes	characteristic
253	C₂	C₃₇×#1591	1	4	yes	direct product factor
254	C₂	C₂×C₃₇×#436	1	1	yes	characteristic
255	C₂	C₂×C₄×Q₁₆×C₃₇	1	1	yes	characteristic
256	C₂	C₂×C₃₇×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle, prime radical	C₂³×C₃₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1591	C₂	direct (N × H)
70. C₂×C₈×C₃₇	Q₁₆	semidirect
109. C₈×C₃₇	C₂×Q₁₆	semidirect
110. C₂×#1591	C₃₇	direct (N × H)
127. #1591	C₂×C₃₇	direct (N × H)
131. C₂×C₃₇	#1591	direct (N × H)
152. C₃₇	C₂×#1591	direct (N × H)
199. C₂×C₈	Q₁₆×C₃₇	semidirect
238. C₈	C₂×Q₁₆×C₃₇	semidirect
253. C₂	C₃₇×#1591	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×#446	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#1591	2
C₂×C₃₇×#335	2
C₂×C₃₇×Q₈⋊C₈	2
C₂×C₈²×C₃₇	2
C₂×#1591	37

Subgroups — C2×C37×#1591 #22639138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1591 #22639138