Subgroups — C₂²×C₃₇×#509 #22683882

Order 9472.

2334 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#509	C₂	1	24	yes	maximal, direct product factor
4	C₂²×SD₃₂×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×D₁₆×C₃₇	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₃₇	—	2	1	no	—
8	C₂³×D₄×C₃₇	—	2	1	no	—
9	C₂×#135×C₃₇	—	2	6	no	—
10	C₂×#135×C₃₇	C₂²	1	12	yes	complemented
11	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²×SD₁₆×C₃₇	—	2	1	no	—
13	C₂²×D₈×C₃₇	—	2	1	no	—
14	C₂²×D₈×C₃₇	—	2	6	no	—
15	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₇	C₂²	1	12	yes	complemented
17	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₃₇×#509	C₂²	1	64	yes	direct product factor
20	C₂×SD₃₂×C₃₇	C₂²	1	24	yes	complemented
21	C₂×D₁₆×C₃₇	C₂²	1	24	yes	complemented
22	C₂×M₃₂×C₃₇	C₂²	1	12	yes	complemented
23	C₂²×C₁₆×C₃₇	C₂²	1	2	yes	complemented
24	C₂⁵×C₃₇	—	4	1	no	—
25	C₂×#41×C₃₇	—	2	12	no	—
26	C₂×#41×C₃₇	—	4	6	no	—
27	C₂²×Q₈×C₃₇	—	2	1	no	—
28	C₂²×D₄×C₃₇	—	2	2	no	—
29	C₂²×D₄×C₃₇	—	2	12	no	—
30	C₂²×D₄×C₃₇	—	2	1	no	—
31	C₂²×D₄×C₃₇	—	4	1	no	—
32	C₂²×D₄×C₃₇	—	4	6	no	—
33	C₂²×D₄×C₃₇	—	4	1	no	—
34	C₂³×C₄×C₃₇	—	4	1	no	—
35	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
36	#135×C₃₇	—	2	24	no	—
37	#135×C₃₇	C₂³	1	16	yes	complemented
38	C₂×Q₁₆×C₃₇	—	2	3	no	—
39	C₂×Q₁₆×C₃₇	C₂³	1	6	yes	complemented
40	C₂×SD₁₆×C₃₇	—	2	12	no	—
41	C₂×D₈×C₃₇	—	2	6	no	—
42	C₂×D₈×C₃₇	—	2	12	no	—
43	C₂×D₈×C₃₇	—	2	24	no	—
44	C₂×D₈×C₃₇	—	2	3	no	—
45	C₂×D₈×C₃₇	C₂³	1	12	yes	—
46	C₂×D₈×C₃₇	C₂³	1	16	yes	complemented
47	C₂×D₈×C₃₇	C₂³	1	6	yes	complemented
48	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₇	D₄	1	6	yes	—
50	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
52	SD₃₂×C₃₇	C₂³	1	32	yes	complemented
53	D₁₆×C₃₇	C₂³	1	32	yes	complemented
54	M₃₂×C₃₇	C₂³	1	16	yes	complemented
55	C₂×C₁₆×C₃₇	C₂³	1	12	yes	complemented
56	C₂⁴×C₃₇	—	4	12	no	—
57	C₂⁴×C₃₇	—	4	1	no	—
58	C₂⁴×C₃₇	—	4	2	no	—
59	C₂⁴×C₃₇	—	8	8	no	—
60	C₂⁴×C₃₇	D₈	1	1	yes	characteristic
61	#41×C₃₇	—	2	16	no	—
62	#41×C₃₇	—	4	24	no	—
63	C₂×Q₈×C₃₇	—	2	6	no	—
64	C₂×Q₈×C₃₇	—	4	3	no	—
65	C₂×D₄×C₃₇	—	2	12	no	—
66	C₂×D₄×C₃₇	—	2	16	no	—
67	C₂×D₄×C₃₇	—	2	6	no	—
68	C₂×D₄×C₃₇	—	4	12	no	—
69	C₂×D₄×C₃₇	—	4	12	no	—
70	C₂×D₄×C₃₇	—	4	6	no	—
71	C₂×D₄×C₃₇	—	4	24	no	—
72	C₂×D₄×C₃₇	—	4	3	no	—
73	C₂²×C₄×C₃₇	—	4	1	no	—
74	C₂²×C₄×C₃₇	—	4	12	no	—
75	C₂²×C₄×C₃₇	D₈	1	6	yes	—
76	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₃₇	—	2	6	no	—
80	Q₁₆×C₃₇	C₂⁴	1	4	yes	complemented
81	SD₁₆×C₃₇	—	2	16	no	—
82	D₈×C₃₇	—	2	12	no	—
83	D₈×C₃₇	—	2	16	no	—
84	D₈×C₃₇	—	2	6	no	—
85	D₈×C₃₇	C₂⁴	1	8	yes	—
86	D₈×C₃₇	C₂⁴	1	4	yes	complemented
87	C₂×C₈×C₃₇	C₂×D₄	1	3	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₃₇	C₂×D₄	1	12	yes	—
90	C₂×C₈×C₃₇	C₂⁴	1	3	yes	—
91	C₂×C₈×C₃₇	C₂⁴	1	4	yes	—
92	C₁₆×C₃₇	C₂⁴	1	8	yes	complemented
93	C₂³×C₃₇	—	2	4	no	—
94	C₂³×C₃₇	—	4	6	no	—
95	C₂³×C₃₇	—	4	16	no	—
96	C₂³×C₃₇	—	4	12	no	—
97	C₂³×C₃₇	—	8	48	no	—
98	C₂³×C₃₇	—	8	4	no	—
99	C₂³×C₃₇	—	8	8	no	—
100	C₂³×C₃₇	C₂×D₈	1	6	yes	—
101	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
102	Q₈×C₃₇	—	2	4	no	—
103	Q₈×C₃₇	—	4	6	no	—
104	D₄×C₃₇	—	2	8	no	—
105	D₄×C₃₇	—	2	4	no	—
106	D₄×C₃₇	—	4	16	no	—
107	D₄×C₃₇	—	4	16	no	—
108	D₄×C₃₇	—	4	12	no	—
109	D₄×C₃₇	—	4	6	no	—
110	C₂×C₄×C₃₇	—	4	6	no	—
111	C₂×C₄×C₃₇	—	4	16	no	—
112	C₂×C₄×C₃₇	C₂×D₈	1	6	yes	—
113	C₂×C₄×C₃₇	C₂×D₈	1	3	yes	—
114	C₂×C₄×C₃₇	C₂×D₈	1	12	yes	—
115	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
117	C₈×C₃₇	C₂²×D₄	1	3	yes	—
118	C₈×C₃₇	C₂²×D₄	1	4	yes	—
119	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
120	C₂²×#509	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	—	2	12	no	—
122	C₂²×C₃₇	—	4	4	no	—
123	C₂²×C₃₇	—	4	8	no	—
124	C₂²×C₃₇	—	8	32	no	—
125	C₂²×C₃₇	—	8	12	no	—
126	C₂²×C₃₇	—	8	24	no	—
127	C₂²×C₃₇	#509	1	4	yes	direct product factor
128	C₂²×C₃₇	C₂²×D₈	1	4	yes	—
129	C₂²×C₃₇	C₂²×D₈	1	3	yes	—
130	C₄×C₃₇	—	4	4	no	—
131	C₄×C₃₇	C₂²×D₈	1	4	yes	—
132	C₄×C₃₇	C₂²×D₈	1	3	yes	—
133	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
134	C₂²×#135	C₂×C₃₇	1	1	yes	characteristic, complemented
135	C₂³×D₈	C₂×C₃₇	1	1	yes	characteristic, complemented
136	C₂×#509	C₂×C₃₇	1	24	yes	direct product factor
137	C₂²×SD₃₂	C₂×C₃₇	1	2	yes	complemented
138	C₂²×D₁₆	C₂×C₃₇	1	2	yes	complemented
139	C₂²×M₃₂	C₂×C₃₇	1	1	yes	characteristic, complemented
140	C₂×C₃₇	—	2	4	no	—
141	C₂×C₃₇	—	8	4	no	—
142	C₂×C₃₇	—	8	8	no	—
143	C₂×C₃₇	C₂×#509	1	6	yes	direct product factor
144	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
145	C₂²×#41	—	2	1	no	—
146	C₂³×D₄	—	2	1	no	—
147	C₂×#135	—	2	6	no	—
148	C₂×#135	C₂²×C₃₇	1	12	yes	complemented
149	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic, complemented
150	C₂²×SD₁₆	—	2	1	no	—
151	C₂²×D₈	—	2	1	no	—
152	C₂²×D₈	—	2	6	no	—
153	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
154	C₂²×D₈	C₂²×C₃₇	1	12	yes	complemented
155	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic, complemented
156	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
157	#509	C₂²×C₃₇	1	64	yes	direct product factor
158	C₂×SD₃₂	C₂²×C₃₇	1	24	yes	complemented
159	C₂×D₁₆	C₂²×C₃₇	1	24	yes	complemented
160	C₂×M₃₂	C₂²×C₃₇	1	12	yes	complemented
161	C₂²×C₁₆	C₂²×C₃₇	1	2	yes	complemented
162	C₃₇	C₂²×#509	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	—	4	1	no	—
164	C₂×#41	—	2	12	no	—
165	C₂×#41	—	4	6	no	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	12	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂²×D₄	—	4	6	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
175	#135	—	2	24	no	—
176	#135	C₂³×C₃₇	1	16	yes	complemented
177	C₂×Q₁₆	—	2	3	no	—
178	C₂×Q₁₆	C₂³×C₃₇	1	6	yes	complemented
179	C₂×SD₁₆	—	2	12	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂×D₈	—	2	12	no	—
182	C₂×D₈	—	2	24	no	—
183	C₂×D₈	—	2	3	no	—
184	C₂×D₈	C₂³×C₃₇	1	12	yes	—
185	C₂×D₈	C₂³×C₃₇	1	16	yes	complemented
186	C₂×D₈	C₂³×C₃₇	1	6	yes	complemented
187	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₃₇	1	6	yes	—
189	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
191	SD₃₂	C₂³×C₃₇	1	32	yes	complemented
192	D₁₆	C₂³×C₃₇	1	32	yes	complemented
193	M₃₂	C₂³×C₃₇	1	16	yes	complemented
194	C₂×C₁₆	C₂³×C₃₇	1	12	yes	complemented
195	C₂⁴	—	4	1	no	—
196	C₂⁴	—	4	12	no	—
197	C₂⁴	—	4	2	no	—
198	C₂⁴	—	8	8	no	—
199	C₂⁴	D₈×C₃₇	1	1	yes	characteristic
200	#41	—	2	16	no	—
201	#41	—	4	24	no	—
202	C₂×Q₈	—	2	6	no	—
203	C₂×Q₈	—	4	3	no	—
204	C₂×D₄	—	2	12	no	—
205	C₂×D₄	—	2	16	no	—
206	C₂×D₄	—	2	6	no	—
207	C₂×D₄	—	4	12	no	—
208	C₂×D₄	—	4	12	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	24	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	12	no	—
214	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
215	C₂²×C₄	D₈×C₃₇	1	6	yes	—
216	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
218	Q₁₆	—	2	6	no	—
219	Q₁₆	C₂⁴×C₃₇	1	4	yes	complemented
220	SD₁₆	—	2	16	no	—
221	D₈	—	2	12	no	—
222	D₈	—	2	16	no	—
223	D₈	—	2	6	no	—
224	D₈	C₂⁴×C₃₇	1	8	yes	—
225	D₈	C₂⁴×C₃₇	1	4	yes	complemented
226	C₂×C₈	C₂×D₄×C₃₇	1	3	yes	—
227	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
228	C₂×C₈	C₂×D₄×C₃₇	1	12	yes	—
229	C₂×C₈	C₂⁴×C₃₇	1	3	yes	—
230	C₂×C₈	C₂⁴×C₃₇	1	4	yes	—
231	C₁₆	C₂⁴×C₃₇	1	8	yes	complemented
232	C₂³	—	2	4	no	—
233	C₂³	—	4	6	no	—
234	C₂³	—	4	16	no	—
235	C₂³	—	4	12	no	—
236	C₂³	—	8	48	no	—
237	C₂³	—	8	8	no	—
238	C₂³	—	8	4	no	—
239	C₂³	C₂×D₈×C₃₇	1	6	yes	—
240	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	6	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	4	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	16	no	—
247	D₄	—	4	12	no	—
248	D₄	—	4	6	no	—
249	C₂×C₄	—	4	6	no	—
250	C₂×C₄	—	4	16	no	—
251	C₂×C₄	C₂×D₈×C₃₇	1	6	yes	—
252	C₂×C₄	C₂×D₈×C₃₇	1	3	yes	—
253	C₂×C₄	C₂×D₈×C₃₇	1	12	yes	—
254	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
256	C₈	C₂²×D₄×C₃₇	1	3	yes	—
257	C₈	C₂²×D₄×C₃₇	1	4	yes	—
258	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
259	C₂²	—	2	12	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	32	no	—
263	C₂²	—	8	24	no	—
264	C₂²	—	8	12	no	—
265	C₂²	C₃₇×#509	1	4	yes	direct product factor
266	C₂²	C₂²×D₈×C₃₇	1	4	yes	—
267	C₂²	C₂²×D₈×C₃₇	1	3	yes	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×D₈×C₃₇	1	4	yes	—
270	C₄	C₂²×D₈×C₃₇	1	3	yes	—
271	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	8	no	—
274	C₂	—	8	4	no	—
275	C₂	C₂×C₃₇×#509	1	6	yes	direct product factor
276	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₃₇	C₂	semidirect
2. C₂³×D₈×C₃₇	C₂	semidirect
3. C₂×C₃₇×#509	C₂	direct (N × H)
4. C₂²×SD₃₂×C₃₇	C₂	semidirect
5. C₂²×D₁₆×C₃₇	C₂	semidirect
6. C₂²×M₃₂×C₃₇	C₂	semidirect
10. C₂×#135×C₃₇	C₂²	semidirect
11. C₂²×Q₁₆×C₃₇	C₂²	semidirect
16. C₂²×D₈×C₃₇	C₂²	semidirect
17. C₂²×D₈×C₃₇	C₂²	semidirect
19. C₃₇×#509	C₂²	direct (N × H)
20. C₂×SD₃₂×C₃₇	C₂²	semidirect
21. C₂×D₁₆×C₃₇	C₂²	semidirect
22. C₂×M₃₂×C₃₇	C₂²	semidirect
23. C₂²×C₁₆×C₃₇	C₂²	semidirect
37. #135×C₃₇	C₂³	semidirect
39. C₂×Q₁₆×C₃₇	C₂³	semidirect
46. C₂×D₈×C₃₇	C₂³	semidirect
47. C₂×D₈×C₃₇	C₂³	semidirect
52. SD₃₂×C₃₇	C₂³	semidirect
53. D₁₆×C₃₇	C₂³	semidirect
54. M₃₂×C₃₇	C₂³	semidirect
55. C₂×C₁₆×C₃₇	C₂³	semidirect
80. Q₁₆×C₃₇	C₂⁴	semidirect
86. D₈×C₃₇	C₂⁴	semidirect
92. C₁₆×C₃₇	C₂⁴	semidirect
120. C₂²×#509	C₃₇	direct (N × H)
127. C₂²×C₃₇	#509	direct (N × H)
134. C₂²×#135	C₂×C₃₇	semidirect
135. C₂³×D₈	C₂×C₃₇	semidirect
136. C₂×#509	C₂×C₃₇	direct (N × H)
137. C₂²×SD₃₂	C₂×C₃₇	semidirect
138. C₂²×D₁₆	C₂×C₃₇	semidirect
139. C₂²×M₃₂	C₂×C₃₇	semidirect
143. C₂×C₃₇	C₂×#509	direct (N × H)
148. C₂×#135	C₂²×C₃₇	semidirect
149. C₂²×Q₁₆	C₂²×C₃₇	semidirect
154. C₂²×D₈	C₂²×C₃₇	semidirect
155. C₂²×D₈	C₂²×C₃₇	semidirect
157. #509	C₂²×C₃₇	direct (N × H)
158. C₂×SD₃₂	C₂²×C₃₇	semidirect
159. C₂×D₁₆	C₂²×C₃₇	semidirect
160. C₂×M₃₂	C₂²×C₃₇	semidirect
161. C₂²×C₁₆	C₂²×C₃₇	semidirect
162. C₃₇	C₂²×#509	direct (N × H)
176. #135	C₂³×C₃₇	semidirect
178. C₂×Q₁₆	C₂³×C₃₇	semidirect
185. C₂×D₈	C₂³×C₃₇	semidirect
186. C₂×D₈	C₂³×C₃₇	semidirect
191. SD₃₂	C₂³×C₃₇	semidirect
192. D₁₆	C₂³×C₃₇	semidirect
193. M₃₂	C₂³×C₃₇	semidirect
194. C₂×C₁₆	C₂³×C₃₇	semidirect
219. Q₁₆	C₂⁴×C₃₇	semidirect
225. D₈	C₂⁴×C₃₇	semidirect
231. C₁₆	C₂⁴×C₃₇	semidirect
265. C₂²	C₃₇×#509	direct (N × H)
275. C₂	C₂×C₃₇×#509	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×#135×C₃₇	2
C₂³×D₈×C₃₇	2
C₂×C₃₇×#509	2
C₂²×SD₃₂×C₃₇	2
C₂²×D₁₆×C₃₇	2
C₂²×M₃₂×C₃₇	2
C₂²×#509	37

Subgroups — C22×C37×#509 #22683882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#509 #22683882