Subgroups — C₂×C₃₇×#2112 #22654584

Order 9472.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₃₇	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2112	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₇×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₃₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₇	—	2	8	no	—
7	C₂³×C₈×C₃₇	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₃₇	C₄	1	4	yes	—
9	C₂³×C₈×C₃₇	C₄	1	2	yes	—
10	C₂³×C₈×C₃₇	C₂²	1	4	yes	—
11	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₃₇	—	2	8	no	—
14	C₂×M₃₂×C₃₇	C₂²	1	8	yes	complemented
15	C₂²×C₁₆×C₃₇	C₂²	1	4	yes	complemented
16	C₃₇×C₂²⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂⁵×C₃₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₃₇	—	2	4	no	—
19	C₂³×C₄×C₃₇	—	2	4	no	—
20	C₂³×C₄×C₃₇	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₃₇	C₈	1	4	yes	—
22	C₂³×C₄×C₃₇	C₈	1	2	yes	—
23	C₂³×C₄×C₃₇	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₇	—	2	4	no	—
27	C₂²×C₈×C₃₇	—	2	16	no	—
28	C₂²×C₈×C₃₇	—	2	8	no	—
29	C₂²×C₈×C₃₇	—	2	8	no	—
30	C₂²×C₈×C₃₇	—	2	16	no	—
31	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₃₇	D₄	1	8	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
39	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	M₃₂×C₃₇	—	2	16	no	—
42	C₂×C₁₆×C₃₇	—	2	8	no	—
43	C₂×C₁₆×C₃₇	C₂³	1	8	yes	complemented
44	C₂⁴×C₃₇	—	2	4	no	—
45	C₂⁴×C₃₇	—	2	8	no	—
46	C₂⁴×C₃₇	C₂×C₈	1	4	yes	—
47	C₂⁴×C₃₇	C₂×C₈	1	1	yes	characteristic
48	C₂⁴×C₃₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₃₇	—	2	4	no	—
50	C₂²×C₄×C₃₇	—	2	16	no	—
51	C₂²×C₄×C₃₇	—	2	8	no	—
52	C₂²×C₄×C₃₇	—	2	8	no	—
53	C₂²×C₄×C₃₇	—	2	4	no	—
54	C₂²×C₄×C₃₇	—	2	8	no	—
55	C₂²×C₄×C₃₇	—	2	4	no	—
56	C₂²×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₃₇	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₃₇	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₃₇	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₃₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	M₁₆	1	4	yes	—
64	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₃₇	—	2	8	no	—
68	C₂×C₈×C₃₇	—	2	4	no	—
69	C₂×C₈×C₃₇	—	2	16	no	—
70	C₂×C₈×C₃₇	—	2	8	no	—
71	C₂×C₈×C₃₇	—	2	8	no	—
72	C₂×C₈×C₃₇	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₃₇	C₂²⋊C₄	1	8	yes	—
74	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
79	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₁₆×C₃₇	—	2	8	no	—
81	C₂³×C₃₇	—	2	32	no	—
82	C₂³×C₃₇	—	2	8	no	—
83	C₂³×C₃₇	—	2	4	no	—
84	C₂³×C₃₇	—	2	4	no	—
85	C₂³×C₃₇	—	2	4	no	—
86	C₂³×C₃₇	—	2	16	no	—
87	C₂³×C₃₇	C₂²⋊C₈	1	4	yes	—
88	C₂³×C₃₇	M₃₂	1	8	yes	complemented
89	C₂³×C₃₇	C₂²×C₈	1	2	yes	—
90	C₂³×C₃₇	C₂²×C₈	1	4	yes	—
91	C₂³×C₃₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₃₇	—	2	8	no	—
93	C₂×C₄×C₃₇	—	2	4	no	—
94	C₂×C₄×C₃₇	—	2	4	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	2	8	no	—
97	C₂×C₄×C₃₇	—	2	16	no	—
98	C₂×C₄×C₃₇	C₂²⋊C₈	1	8	yes	—
99	C₂×C₄×C₃₇	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₃₇	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₃₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₃₇	—	2	4	no	—
109	C₈×C₃₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₈×C₃₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₈×C₃₇	C₂²×D₄	1	2	yes	—
112	C₂×#2112	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₃₇	—	2	32	no	—
114	C₂²×C₃₇	—	2	4	no	—
115	C₂²×C₃₇	—	2	8	no	—
116	C₂²×C₃₇	—	2	16	no	—
117	C₂²×C₃₇	—	2	4	no	—
118	C₂²×C₃₇	—	2	4	no	—
119	C₂²×C₃₇	C₂×C₂²⋊C₈	1	4	yes	—
120	C₂²×C₃₇	C₂×C₂²⋊C₈	1	2	yes	—
121	C₂²×C₃₇	C₂×M₃₂	1	4	yes	—
122	C₂²×C₃₇	C₂×M₃₂	1	8	yes	complemented
123	C₂²×C₃₇	C₂³×C₈	1	1	yes	characteristic
124	C₄×C₃₇	—	2	4	no	—
125	C₄×C₃₇	C₂×C₂²⋊C₈	1	2	yes	—
126	C₄×C₃₇	C₂×C₂²⋊C₈	1	4	yes	—
127	C₄×C₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
129	C₂⁴×C₈	C₂×C₃₇	1	1	yes	characteristic
130	C₂²×M₃₂	C₂×C₃₇	1	2	yes	complemented
131	#2112	C₂×C₃₇	1	8	yes	direct product factor
132	C₂×C₂²⋊C₁₆	C₂×C₃₇	1	4	yes	complemented
133	C₂×C₃₇	—	2	4	no	—
134	C₂×C₃₇	—	2	8	no	—
135	C₂×C₃₇	#2112	1	4	yes	direct product factor
136	C₂×C₃₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₃₇	C₂²×M₃₂	1	2	yes	—
138	C₂⁴×C₄	C₄×C₃₇	1	1	yes	characteristic
139	C₂³×C₈	—	2	8	no	—
140	C₂³×C₈	C₄×C₃₇	1	1	yes	characteristic
141	C₂³×C₈	C₄×C₃₇	1	2	yes	—
142	C₂³×C₈	C₄×C₃₇	1	4	yes	—
143	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
144	C₂³×C₈	C₂²×C₃₇	1	4	yes	—
145	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
146	C₂×M₃₂	—	2	8	no	—
147	C₂×M₃₂	C₂²×C₃₇	1	8	yes	complemented
148	C₂²×C₁₆	C₂²×C₃₇	1	4	yes	complemented
149	C₂²⋊C₁₆	C₂²×C₃₇	1	16	yes	complemented
150	C₃₇	C₂×#2112	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁵	C₈×C₃₇	1	1	yes	characteristic
152	C₂³×C₄	—	2	4	no	—
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	C₈×C₃₇	1	1	yes	characteristic
155	C₂³×C₄	C₈×C₃₇	1	4	yes	—
156	C₂³×C₄	C₈×C₃₇	1	2	yes	—
157	C₂³×C₄	C₂×C₄×C₃₇	1	4	yes	—
158	C₂³×C₄	C₂×C₄×C₃₇	1	2	yes	—
159	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	—	2	16	no	—
162	C₂²×C₈	—	2	8	no	—
163	C₂²×C₈	—	2	8	no	—
164	C₂²×C₈	—	2	16	no	—
165	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
167	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
168	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	—
169	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
171	C₂²×C₈	D₄×C₃₇	1	8	yes	—
172	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
173	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
175	M₃₂	—	2	16	no	—
176	C₂×C₁₆	—	2	8	no	—
177	C₂×C₁₆	C₂³×C₃₇	1	8	yes	complemented
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	8	no	—
180	C₂⁴	C₂×C₈×C₃₇	1	4	yes	—
181	C₂⁴	C₂×C₈×C₃₇	1	1	yes	characteristic
182	C₂⁴	C₂×C₈×C₃₇	1	2	yes	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	C₂²⋊C₄×C₃₇	1	4	yes	—
191	C₂²×C₄	C₂×C₈×C₃₇	1	4	yes	—
192	C₂²×C₄	C₂×C₈×C₃₇	1	2	yes	—
193	C₂²×C₄	C₂×C₈×C₃₇	1	4	yes	—
194	C₂²×C₄	C₂×C₈×C₃₇	1	8	yes	—
195	C₂²×C₄	C₂×C₈×C₃₇	1	2	yes	—
196	C₂²×C₄	C₂×C₈×C₃₇	1	1	yes	characteristic
197	C₂²×C₄	M₁₆×C₃₇	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	16	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²⋊C₄×C₃₇	1	8	yes	—
207	C₂×C₈	C₂²⋊C₄×C₃₇	1	4	yes	—
208	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
212	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
213	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
214	C₁₆	—	2	8	no	—
215	C₂³	—	2	32	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	C₂²⋊C₈×C₃₇	1	4	yes	—
222	C₂³	M₃₂×C₃₇	1	8	yes	complemented
223	C₂³	C₂²×C₈×C₃₇	1	2	yes	—
224	C₂³	C₂²×C₈×C₃₇	1	4	yes	—
225	C₂³	C₂²×C₈×C₃₇	1	1	yes	characteristic
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₂²⋊C₈×C₃₇	1	8	yes	—
233	C₂×C₄	C₂²⋊C₈×C₃₇	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	2	yes	—
236	C₂×C₄	C₂²×C₈×C₃₇	1	4	yes	—
237	C₂×C₄	C₂²×C₈×C₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₃₇	1	2	yes	—
239	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
240	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	C₂×C₂²⋊C₄×C₃₇	1	2	yes	—
244	C₈	C₂×C₂²⋊C₄×C₃₇	1	4	yes	—
245	C₈	C₂²×D₄×C₃₇	1	2	yes	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	2	16	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×C₂²⋊C₈×C₃₇	1	4	yes	—
253	C₂²	C₂×C₂²⋊C₈×C₃₇	1	2	yes	—
254	C₂²	C₂×M₃₂×C₃₇	1	4	yes	—
255	C₂²	C₂×M₃₂×C₃₇	1	8	yes	complemented
256	C₂²	C₂³×C₈×C₃₇	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₂²⋊C₈×C₃₇	1	2	yes	—
259	C₄	C₂×C₂²⋊C₈×C₃₇	1	4	yes	—
260	C₄	C₂²×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
261	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
262	C₂	—	2	4	no	—
263	C₂	—	2	8	no	—
264	C₂	C₃₇×#2112	1	4	yes	direct product factor
265	C₂	C₂²×C₂²⋊C₈×C₃₇	1	1	yes	characteristic
266	C₂	C₂²×M₃₂×C₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₃₇	C₂³
commutator	C₂²	C₂³×C₈×C₃₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂²×C₈
prime radical	C₂⁵×C₃₇	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₃₇	C₂	semidirect
3. C₃₇×#2112	C₂	direct (N × H)
4. C₂×C₃₇×C₂²⋊C₁₆	C₂	semidirect
14. C₂×M₃₂×C₃₇	C₂²	semidirect
15. C₂²×C₁₆×C₃₇	C₂²	semidirect
16. C₃₇×C₂²⋊C₁₆	C₂²	semidirect
43. C₂×C₁₆×C₃₇	C₂³	semidirect
88. C₂³×C₃₇	M₃₂	semidirect
112. C₂×#2112	C₃₇	direct (N × H)
122. C₂²×C₃₇	C₂×M₃₂	semidirect
130. C₂²×M₃₂	C₂×C₃₇	semidirect
131. #2112	C₂×C₃₇	direct (N × H)
132. C₂×C₂²⋊C₁₆	C₂×C₃₇	semidirect
135. C₂×C₃₇	#2112	direct (N × H)
147. C₂×M₃₂	C₂²×C₃₇	semidirect
148. C₂²×C₁₆	C₂²×C₃₇	semidirect
149. C₂²⋊C₁₆	C₂²×C₃₇	semidirect
150. C₃₇	C₂×#2112	direct (N × H)
177. C₂×C₁₆	C₂³×C₃₇	semidirect
222. C₂³	M₃₂×C₃₇	semidirect
255. C₂²	C₂×M₃₂×C₃₇	semidirect
264. C₂	C₃₇×#2112	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₃₇	2
C₂²×M₃₂×C₃₇	2
C₃₇×#2112	2
C₂×C₃₇×C₂²⋊C₁₆	2
C₂×#2112	37

Subgroups — C2×C37×#2112 #22654584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#2112 #22654584