Subgroups — C₂×C₃₇×#525 #17992253

Order 4736.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#525	C₂	1	16	yes	maximal, direct product factor
4	C₂×#123×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#121×C₃₇	C₂	1	2	yes	maximal, complemented
7	C₂×#120×C₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₃₇	C₂	1	4	yes	maximal, complemented
9	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#41×C₃₇	—	2	8	no	—
12	C₂×#41×C₃₇	C₂²	1	8	yes	complemented
13	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₄×C₃₇	C₂²	1	4	yes	—
15	C₂²×D₄×C₃₇	C₂²	1	2	yes	—
16	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₃₇	—	2	8	no	—
18	C₂³×C₄×C₃₇	C₂²	1	2	yes	—
19	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₃₇	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₃₇	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₃₇	C₂²	1	2	yes	complemented
24	#123×C₃₇	C₂²	1	16	yes	complemented
25	C₂²⋊Q₈×C₃₇	C₂²	1	16	yes	complemented
26	#121×C₃₇	C₂²	1	16	yes	complemented
27	#120×C₃₇	C₂²	1	16	yes	complemented
28	C₄×D₄×C₃₇	C₂²	1	32	yes	complemented
29	#117×C₃₇	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	complemented
31	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	complemented
32	C₂×C₂²⋊C₄×C₃₇	C₂²	1	4	yes	—
33	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₄×C₃₇	C₂²	1	4	yes	complemented
35	C₂×C₄²×C₃₇	C₂²	1	2	yes	complemented
36	C₂⁴×C₃₇	—	2	4	no	—
37	C₂⁴×C₃₇	—	2	8	no	—
38	C₂⁴×C₃₇	C₂³	1	2	yes	—
39	C₂⁴×C₃₇	C₂³	1	4	yes	—
40	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
41	C₂⁴×C₃₇	C₂³	1	2	yes	—
42	#41×C₃₇	—	2	32	no	—
43	C₂×Q₈×C₃₇	—	2	4	no	—
44	C₂×Q₈×C₃₇	C₂³	1	4	yes	complemented
45	C₂×D₄×C₃₇	—	2	8	no	—
46	C₂×D₄×C₃₇	—	2	16	no	—
47	C₂×D₄×C₃₇	—	2	4	no	—
48	C₂×D₄×C₃₇	C₂³	1	16	yes	—
49	C₂×D₄×C₃₇	C₂³	1	8	yes	—
50	C₂×D₄×C₃₇	C₂³	1	4	yes	complemented
51	C₂²×C₄×C₃₇	—	2	4	no	—
52	C₂²×C₄×C₃₇	—	2	8	no	—
53	C₂²×C₄×C₃₇	—	2	8	no	—
54	C₂²×C₄×C₃₇	—	2	8	no	—
55	C₂²×C₄×C₃₇	—	2	32	no	—
56	C₂²×C₄×C₃₇	D₄	1	8	yes	complemented
57	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
58	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₃₇	C₂³	1	8	yes	—
61	C₂²×C₄×C₃₇	C₂³	1	8	yes	—
62	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
63	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
65	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₂³	1	8	yes	complemented
67	C₄⋊C₄×C₃₇	C₂³	1	8	yes	complemented
68	C₄⋊C₄×C₃₇	C₂³	1	16	yes	complemented
69	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
70	C₂²⋊C₄×C₃₇	C₂³	1	16	yes	—
71	C₂²⋊C₄×C₃₇	C₂³	1	16	yes	complemented
72	C₄²×C₃₇	C₂³	1	8	yes	complemented
73	C₂³×C₃₇	—	2	32	no	—
74	C₂³×C₃₇	—	2	8	no	—
75	C₂³×C₃₇	—	2	4	no	—
76	C₂³×C₃₇	—	2	4	no	—
77	C₂³×C₃₇	—	2	4	no	—
78	C₂³×C₃₇	—	2	16	no	—
79	C₂³×C₃₇	—	2	4	no	—
80	C₂³×C₃₇	—	2	8	no	—
81	C₂³×C₃₇	C₂×D₄	1	4	yes	—
82	C₂³×C₃₇	#41	1	8	yes	complemented
83	C₂³×C₃₇	C₂⁴	1	4	yes	—
84	C₂³×C₃₇	C₂⁴	1	2	yes	—
85	C₂³×C₃₇	C₂⁴	1	4	yes	—
86	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
87	Q₈×C₃₇	—	2	8	no	—
88	D₄×C₃₇	—	2	16	no	—
89	D₄×C₃₇	—	2	32	no	—
90	D₄×C₃₇	—	2	8	no	—
91	C₂×C₄×C₃₇	—	2	8	no	—
92	C₂×C₄×C₃₇	—	2	4	no	—
93	C₂×C₄×C₃₇	—	2	4	no	—
94	C₂×C₄×C₃₇	—	2	8	no	—
95	C₂×C₄×C₃₇	—	2	16	no	—
96	C₂×C₄×C₃₇	—	2	16	no	—
97	C₂×C₄×C₃₇	—	2	16	no	—
98	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
99	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₃₇	C₂×D₄	1	16	yes	complemented
101	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
102	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
103	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
104	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
105	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
106	C₂²×C₃₇	—	2	32	no	—
107	C₂²×C₃₇	—	2	8	no	—
108	C₂²×C₃₇	—	2	16	no	—
109	C₂²×C₃₇	—	2	4	no	—
110	C₂²×C₃₇	—	2	4	no	—
111	C₂²×C₃₇	—	2	4	no	—
112	C₂²×C₃₇	—	2	4	no	—
113	C₂²×C₃₇	—	2	8	no	—
114	C₂²×C₃₇	—	4	8	no	—
115	C₂²×C₃₇	C₂²×D₄	1	4	yes	—
116	C₂²×C₃₇	C₂²×D₄	1	2	yes	—
117	C₂²×C₃₇	C₂×#41	1	4	yes	—
118	C₂²×C₃₇	C₂×#41	1	8	yes	complemented
119	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
120	C₄×C₃₇	—	2	4	no	—
121	C₄×C₃₇	—	2	4	no	—
122	C₄×C₃₇	—	2	8	no	—
123	C₄×C₃₇	C₂²×D₄	1	4	yes	—
124	C₄×C₃₇	C₂²×D₄	1	4	yes	—
125	C₂×#525	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₃₇	—	2	4	no	—
127	C₂×C₃₇	—	2	8	no	—
128	C₂×C₃₇	—	4	4	no	—
129	C₂×C₃₇	#525	1	4	yes	direct product factor
130	C₂×C₃₇	C₂³×D₄	1	1	yes	characteristic
131	C₂×C₃₇	C₂²×#41	1	2	yes	—
132	C₂²×#41	C₂×C₃₇	1	1	yes	characteristic, complemented
133	C₂⁴×C₄	C₂×C₃₇	1	1	yes	characteristic, complemented
134	#525	C₂×C₃₇	1	16	yes	direct product factor
135	C₂×#123	C₂×C₃₇	1	2	yes	complemented
136	C₂×C₂²⋊Q₈	C₂×C₃₇	1	2	yes	complemented
137	C₂×#121	C₂×C₃₇	1	2	yes	complemented
138	C₂×#120	C₂×C₃₇	1	2	yes	complemented
139	C₂×C₄×D₄	C₂×C₃₇	1	4	yes	complemented
140	C₂×#117	C₂×C₃₇	1	1	yes	characteristic, complemented
141	C₃₇	C₂×#525	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁵	C₂²×C₃₇	1	1	yes	characteristic
143	C₂×#41	—	2	8	no	—
144	C₂×#41	C₂²×C₃₇	1	8	yes	complemented
145	C₂²×Q₈	C₂²×C₃₇	1	1	yes	characteristic, complemented
146	C₂²×D₄	C₂²×C₃₇	1	4	yes	—
147	C₂²×D₄	C₂²×C₃₇	1	2	yes	—
148	C₂²×D₄	C₂²×C₃₇	1	1	yes	characteristic, complemented
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	C₂²×C₃₇	1	2	yes	—
151	C₂³×C₄	C₂²×C₃₇	1	2	yes	—
152	C₂³×C₄	C₂²×C₃₇	1	1	yes	characteristic
153	C₂³×C₄	C₂²×C₃₇	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₃₇	1	8	yes	complemented
155	C₂³×C₄	C₂²×C₃₇	1	2	yes	complemented
156	#123	C₂²×C₃₇	1	16	yes	complemented
157	C₂²⋊Q₈	C₂²×C₃₇	1	16	yes	complemented
158	#121	C₂²×C₃₇	1	16	yes	complemented
159	#120	C₂²×C₃₇	1	16	yes	complemented
160	C₄×D₄	C₂²×C₃₇	1	32	yes	complemented
161	#117	C₂²×C₃₇	1	8	yes	complemented
162	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	complemented
163	C₂×C₄⋊C₄	C₂²×C₃₇	1	4	yes	complemented
164	C₂×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₃₇	1	4	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₃₇	1	4	yes	complemented
167	C₂×C₄²	C₂²×C₃₇	1	2	yes	complemented
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	C₂³×C₃₇	1	2	yes	—
171	C₂⁴	C₂³×C₃₇	1	4	yes	—
172	C₂⁴	C₂³×C₃₇	1	1	yes	characteristic
173	C₂⁴	C₂³×C₃₇	1	2	yes	—
174	#41	—	2	32	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	C₂³×C₃₇	1	4	yes	complemented
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	16	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	C₂³×C₃₇	1	16	yes	—
181	C₂×D₄	C₂³×C₃₇	1	8	yes	—
182	C₂×D₄	C₂³×C₃₇	1	4	yes	complemented
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	32	no	—
188	C₂²×C₄	D₄×C₃₇	1	8	yes	complemented
189	C₂²×C₄	C₂³×C₃₇	1	4	yes	—
190	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
191	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
192	C₂²×C₄	C₂³×C₃₇	1	8	yes	—
193	C₂²×C₄	C₂³×C₃₇	1	8	yes	—
194	C₂²×C₄	C₂³×C₃₇	1	4	yes	—
195	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₃₇	1	4	yes	—
198	C₂²×C₄	C₂³×C₃₇	1	8	yes	complemented
199	C₄⋊C₄	C₂³×C₃₇	1	8	yes	complemented
200	C₄⋊C₄	C₂³×C₃₇	1	16	yes	complemented
201	C₂²⋊C₄	C₂³×C₃₇	1	8	yes	—
202	C₂²⋊C₄	C₂³×C₃₇	1	16	yes	—
203	C₂²⋊C₄	C₂³×C₃₇	1	16	yes	complemented
204	C₄²	C₂³×C₃₇	1	8	yes	complemented
205	C₂³	—	2	32	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	8	no	—
213	C₂³	C₂×D₄×C₃₇	1	4	yes	—
214	C₂³	#41×C₃₇	1	8	yes	complemented
215	C₂³	C₂⁴×C₃₇	1	4	yes	—
216	C₂³	C₂⁴×C₃₇	1	2	yes	—
217	C₂³	C₂⁴×C₃₇	1	4	yes	—
218	C₂³	C₂⁴×C₃₇	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	32	no	—
222	D₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	C₂×D₄×C₃₇	1	4	yes	—
231	C₂×C₄	C₂×D₄×C₃₇	1	4	yes	—
232	C₂×C₄	C₂×D₄×C₃₇	1	16	yes	complemented
233	C₂×C₄	C₂⁴×C₃₇	1	8	yes	—
234	C₂×C₄	C₂⁴×C₃₇	1	4	yes	—
235	C₂×C₄	C₂⁴×C₃₇	1	4	yes	—
236	C₂×C₄	C₂⁴×C₃₇	1	2	yes	—
237	C₂×C₄	C₂⁴×C₃₇	1	2	yes	—
238	C₂²	—	2	32	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	16	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	8	no	—
247	C₂²	C₂²×D₄×C₃₇	1	4	yes	—
248	C₂²	C₂²×D₄×C₃₇	1	2	yes	—
249	C₂²	C₂×#41×C₃₇	1	4	yes	—
250	C₂²	C₂×#41×C₃₇	1	8	yes	complemented
251	C₂²	C₂⁵×C₃₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	—	2	8	no	—
255	C₄	C₂²×D₄×C₃₇	1	4	yes	—
256	C₄	C₂²×D₄×C₃₇	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	2	8	no	—
259	C₂	—	4	4	no	—
260	C₂	C₃₇×#525	1	4	yes	direct product factor
261	C₂	C₂³×D₄×C₃₇	1	1	yes	characteristic
262	C₂	C₂²×#41×C₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₃₇	C₂	semidirect
2. C₂⁴×C₄×C₃₇	C₂	semidirect
3. C₃₇×#525	C₂	direct (N × H)
4. C₂×#123×C₃₇	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₃₇	C₂	semidirect
6. C₂×#121×C₃₇	C₂	semidirect
7. C₂×#120×C₃₇	C₂	semidirect
8. C₂×C₄×D₄×C₃₇	C₂	semidirect
9. C₂×#117×C₃₇	C₂	semidirect
12. C₂×#41×C₃₇	C₂²	semidirect
13. C₂²×Q₈×C₃₇	C₂²	semidirect
16. C₂²×D₄×C₃₇	C₂²	semidirect
22. C₂³×C₄×C₃₇	C₂²	semidirect
23. C₂³×C₄×C₃₇	C₂²	semidirect
24. #123×C₃₇	C₂²	semidirect
25. C₂²⋊Q₈×C₃₇	C₂²	semidirect
26. #121×C₃₇	C₂²	semidirect
27. #120×C₃₇	C₂²	semidirect
28. C₄×D₄×C₃₇	C₂²	semidirect
29. #117×C₃₇	C₂²	semidirect
30. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
31. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
34. C₂×C₂²⋊C₄×C₃₇	C₂²	semidirect
35. C₂×C₄²×C₃₇	C₂²	semidirect
44. C₂×Q₈×C₃₇	C₂³	semidirect
50. C₂×D₄×C₃₇	C₂³	semidirect
56. C₂²×C₄×C₃₇	D₄	semidirect
66. C₂²×C₄×C₃₇	C₂³	semidirect
67. C₄⋊C₄×C₃₇	C₂³	semidirect
68. C₄⋊C₄×C₃₇	C₂³	semidirect
71. C₂²⋊C₄×C₃₇	C₂³	semidirect
72. C₄²×C₃₇	C₂³	semidirect
82. C₂³×C₃₇	#41	semidirect
100. C₂×C₄×C₃₇	C₂×D₄	semidirect
118. C₂²×C₃₇	C₂×#41	semidirect
125. C₂×#525	C₃₇	direct (N × H)
129. C₂×C₃₇	#525	direct (N × H)
132. C₂²×#41	C₂×C₃₇	semidirect
133. C₂⁴×C₄	C₂×C₃₇	semidirect
134. #525	C₂×C₃₇	direct (N × H)
135. C₂×#123	C₂×C₃₇	semidirect
136. C₂×C₂²⋊Q₈	C₂×C₃₇	semidirect
137. C₂×#121	C₂×C₃₇	semidirect
138. C₂×#120	C₂×C₃₇	semidirect
139. C₂×C₄×D₄	C₂×C₃₇	semidirect
140. C₂×#117	C₂×C₃₇	semidirect
141. C₃₇	C₂×#525	direct (N × H)
144. C₂×#41	C₂²×C₃₇	semidirect
145. C₂²×Q₈	C₂²×C₃₇	semidirect
148. C₂²×D₄	C₂²×C₃₇	semidirect
154. C₂³×C₄	C₂²×C₃₇	semidirect
155. C₂³×C₄	C₂²×C₃₇	semidirect
156. #123	C₂²×C₃₇	semidirect
157. C₂²⋊Q₈	C₂²×C₃₇	semidirect
158. #121	C₂²×C₃₇	semidirect
159. #120	C₂²×C₃₇	semidirect
160. C₄×D₄	C₂²×C₃₇	semidirect
161. #117	C₂²×C₃₇	semidirect
162. C₂×C₄⋊C₄	C₂²×C₃₇	semidirect
163. C₂×C₄⋊C₄	C₂²×C₃₇	semidirect
166. C₂×C₂²⋊C₄	C₂²×C₃₇	semidirect
167. C₂×C₄²	C₂²×C₃₇	semidirect
176. C₂×Q₈	C₂³×C₃₇	semidirect
182. C₂×D₄	C₂³×C₃₇	semidirect
188. C₂²×C₄	D₄×C₃₇	semidirect
198. C₂²×C₄	C₂³×C₃₇	semidirect
199. C₄⋊C₄	C₂³×C₃₇	semidirect
200. C₄⋊C₄	C₂³×C₃₇	semidirect
203. C₂²⋊C₄	C₂³×C₃₇	semidirect
204. C₄²	C₂³×C₃₇	semidirect
214. C₂³	#41×C₃₇	semidirect
232. C₂×C₄	C₂×D₄×C₃₇	semidirect
250. C₂²	C₂×#41×C₃₇	semidirect
260. C₂	C₃₇×#525	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×#41×C₃₇	2
C₂⁴×C₄×C₃₇	2
C₃₇×#525	2
C₂×#123×C₃₇	2
C₂×C₂²⋊Q₈×C₃₇	2
C₂×#121×C₃₇	2
C₂×#120×C₃₇	2
C₂×C₄×D₄×C₃₇	2
C₂×#117×C₃₇	2
C₂×#525	37

Subgroups — C2×C37×#525 #17992253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#525 #17992253