Subgroups — #18007802

Order 4736.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924582	C₂	1	1	yes	maximal, characteristic
2	#14558015	C₂	1	2	yes	maximal, complemented
3	#14558014	C₂	1	3	yes	maximal, complemented
4	#14558007	C₂	1	3	yes	maximal, complemented
5	#14558005	C₂	1	6	yes	maximal, complemented
6	#41×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#10596966	C₂	1	2	yes	maximal, complemented
8	#14558001	C₂	1	3	yes	maximal, complemented
9	C₂×#11924498	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₃₇	C₂	1	6	yes	maximal, complemented
11	C₃₇×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	#11924582	C₄	1	16	yes	complemented
13	C₂×#10596966	C₂²	1	2	yes	—
14	C₂×#10596962	C₂²	1	3	yes	—
15	C₂²×D₁₄₈	C₂²	1	3	yes	—
16	C₂×D₄×D₃₇	C₂²	1	6	yes	—
17	C₂²×C₄×D₃₇	C₂²	1	6	yes	complemented
18	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
20	#11924501	C₂²	1	1	yes	characteristic, complemented
21	#11924501	C₂²	1	3	yes	complemented
22	C₄×D₁₄₈	C₂²	1	6	yes	complemented
23	C₄×D₁₄₈	C₂²	1	6	yes	complemented
24	D₄×Dic₃₇	C₂²	1	3	yes	complemented
25	D₄×Dic₃₇	C₂²	1	3	yes	complemented
26	#11924498	C₂²	1	6	yes	complemented
27	#11924498	C₂²	1	3	yes	complemented
28	#11924498	C₂²	1	6	yes	complemented
29	#11924498	C₂²	1	6	yes	complemented
30	#11924498	C₂²	1	3	yes	complemented
31	C₄×C₂²⋊D₃₇	C₂²	1	12	yes	complemented
32	#11924496	C₂²	1	6	yes	complemented
33	#11924496	C₂²	1	6	yes	complemented
34	#11924495	C₂²	1	3	yes	complemented
35	#11924494	C₂²	1	6	yes	complemented
36	#11924493	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₃₇	C₂²	1	6	yes	complemented
38	C₂×#10596938	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₃₇	C₂²	1	12	yes	complemented
40	C₂×C₄×Dic₃₇	C₂²	1	3	yes	complemented
41	C₄²×D₃₇	C₂²	1	6	yes	complemented
42	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₃₇	C₂²	1	2	yes	complemented
44	C₄×D₄×C₃₇	C₂²	1	6	yes	complemented
45	#117×C₃₇	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₃₇	C₂²	1	3	yes	complemented
47	C₂³×D₃₇	C₂³	1	6	yes	—
48	#41×C₃₇	C₂×C₄	1	8	yes	complemented
49	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₇	C₂³	1	3	yes	—
51	C₂²×C₄×C₃₇	C₂³	1	6	yes	—
52	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
53	C₂×C₂²⋊D₃₇	C₂³	1	6	yes	—
54	C₂²×Dic₃₇	C₂³	1	3	yes	—
55	C₂²×Dic₃₇	C₂³	1	3	yes	—
56	#10596966	C₂×C₄	1	16	yes	complemented
57	D₄×D₃₇	C₂×C₄	1	48	yes	complemented
58	#10596962	C₂×C₄	1	24	yes	complemented
59	C₂×D₁₄₈	C₂×C₄	1	24	yes	complemented
60	C₂×D₁₄₈	C₂³	1	3	yes	—
61	C₂×D₁₄₈	C₂³	1	6	yes	—
62	C₂×C₄×D₃₇	—	4	12	no	—
63	C₂×C₄×D₃₇	C₂³	1	6	yes	—
64	C₂×C₄×D₃₇	C₂³	1	6	yes	—
65	C₂×C₄×D₃₇	C₂³	1	12	yes	—
66	C₂×C₄×D₃₇	C₂³	1	6	yes	—
67	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
69	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
70	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₃₇	C₂³	1	6	yes	—
72	C₄²×C₃₇	C₂³	1	6	yes	—
73	C₂²⋊Dic₃₇	C₂³	1	6	yes	—
74	#10596938	C₂³	1	6	yes	—
75	#10596938	C₂³	1	12	yes	—
76	#10596938	C₂³	1	6	yes	—
77	C₄⋊Dic₃₇	C₂³	1	3	yes	—
78	C₄⋊Dic₃₇	C₂³	1	3	yes	—
79	#10596936	C₂³	1	6	yes	—
80	#10596936	C₂³	1	2	yes	—
81	C₄×Dic₃₇	C₂³	1	2	yes	—
82	C₄×Dic₃₇	C₂³	1	3	yes	—
83	C₄×Dic₃₇	C₂³	1	6	yes	—
84	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₃₇	C₂³	1	3	yes	—
86	C₄×Dic₃₇	C₂³	1	3	yes	—
87	C₂³×C₃₇	C₂⁴	1	3	yes	—
88	C₂²×D₃₇	—	4	6	no	—
89	C₂²×D₃₇	—	4	6	no	—
90	C₂²×D₃₇	C₂²×C₄	1	24	yes	complemented
91	C₂²×D₃₇	C₂⁴	1	6	yes	—
92	Q₈×C₃₇	C₂²×C₄	1	4	yes	complemented
93	D₄×C₃₇	C₂²×C₄	1	12	yes	complemented
94	C₂×C₄×C₃₇	—	4	6	no	—
95	C₂×C₄×C₃₇	C₂²×C₄	1	12	yes	complemented
96	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
98	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
100	C₂²⋊D₃₇	C₂²×C₄	1	24	yes	complemented
101	C₂×Dic₃₇	—	2	6	no	—
102	C₂×Dic₃₇	—	4	3	no	—
103	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₃₇	C₂⁴	1	3	yes	—
105	C₂×Dic₃₇	C₂⁴	1	3	yes	—
106	C₂×Dic₃₇	C₂⁴	1	2	yes	—
107	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
108	D₁₄₈	C₂²×C₄	1	12	yes	complemented
109	D₁₄₈	C₂²×C₄	1	24	yes	complemented
110	C₄×D₃₇	—	2	12	no	—
111	C₄×D₃₇	—	4	12	no	—
112	C₄×D₃₇	—	4	6	no	—
113	C₄×D₃₇	C₂²×C₄	1	24	yes	complemented
114	Dic₇₄	C₂²×C₄	1	4	yes	complemented
115	C₂²×C₃₇	—	2	3	no	—
116	C₂²×C₃₇	—	2	3	no	—
117	C₂²×C₃₇	C₂³×C₄	1	6	yes	—
118	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
119	C₂×D₃₇	—	2	6	no	—
120	C₂×D₃₇	—	2	6	no	—
121	C₂×D₃₇	—	4	24	no	—
122	C₂×D₃₇	C₂³×C₄	1	12	yes	—
123	C₄×C₃₇	—	2	6	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	C₂³×C₄	1	2	yes	—
126	C₄×C₃₇	C₂³×C₄	1	6	yes	—
127	Dic₃₇	—	2	3	no	—
128	Dic₃₇	—	2	3	no	—
129	Dic₃₇	—	2	1	no	—
130	Dic₃₇	C₂³×C₄	1	4	yes	—
131	Dic₃₇	H(5,𝔽₂)	1	2	yes	complemented
132	C₄×H(5,𝔽₂)	—	37	1	no	maximal, Sylow
133	C₂×C₃₇	—	2	6	no	—
134	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
135	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₃₇	#585	1	1	yes	characteristic
137	D₃₇	—	2	12	no	—
138	C₂×H(5,𝔽₂)	—	37	1	no	—
139	#520	—	37	2	no	—
140	#520	—	37	3	no	—
141	#520	D₃₇	1	1	yes	characteristic, complemented
142	#518	—	37	6	no	—
143	#518	—	37	3	no	—
144	C₄×#41	—	37	1	no	—
145	C₄×#41	—	37	2	no	—
146	C₄×#41	—	37	3	no	—
147	C₂×C₄×D₄	—	37	6	no	—
148	C₂×C₄×D₄	—	37	3	no	—
149	C₃₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
150	H(5,𝔽₂)	—	37	16	no	—
151	C₂×#41	—	37	2	no	—
152	C₂×#41	—	37	3	no	—
153	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
154	C₂²×D₄	—	37	6	no	—
155	C₂²×D₄	—	37	3	no	—
156	C₂³×C₄	—	37	6	no	—
157	C₄×Q₈	—	37	1	no	—
158	C₄×Q₈	—	37	1	no	—
159	C₄×Q₈	—	37	1	no	—
160	C₄×Q₈	—	37	3	no	—
161	C₄×Q₈	C₂×D₃₇	1	2	yes	complemented
162	C₄×D₄	—	37	3	no	—
163	C₄×D₄	—	37	3	no	—
164	C₄×D₄	—	37	12	no	—
165	C₄×D₄	—	37	6	no	—
166	C₄×D₄	—	37	3	no	—
167	C₄×D₄	—	37	6	no	—
168	C₄×D₄	—	37	6	no	—
169	C₄×D₄	—	37	3	no	—
170	C₄×D₄	—	37	6	no	—
171	C₄×D₄	—	37	6	no	—
172	C₄×D₄	—	37	6	no	—
173	C₄×D₄	—	37	6	no	—
174	C₄×D₄	C₂×D₃₇	1	6	yes	complemented
175	#117	—	37	3	no	—
176	#117	—	37	6	no	—
177	#117	—	37	6	no	—
178	#117	C₂×D₃₇	1	3	yes	complemented
179	C₂×C₄⋊C₄	—	37	6	no	—
180	C₂×C₄⋊C₄	C₂×D₃₇	1	3	yes	complemented
181	C₂×C₂²⋊C₄	—	37	6	no	—
182	C₂×C₂²⋊C₄	—	37	12	no	—
183	C₂×C₄²	—	37	3	no	—
184	C₂×C₄²	—	37	6	no	—
185	C₂⁴	—	37	6	no	—
186	#41	—	37	16	no	—
187	#41	—	37	24	no	—
188	#41	C₄×D₃₇	1	8	yes	complemented
189	C₂×Q₈	—	37	1	no	—
190	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
191	C₂×D₄	—	37	6	no	—
192	C₂×D₄	—	37	48	no	—
193	C₂×D₄	—	37	24	no	—
194	C₂×D₄	—	37	3	no	—
195	C₂×D₄	—	37	6	no	—
196	C₂×D₄	C₂²×D₃₇	1	3	yes	—
197	C₂²×C₄	—	37	6	no	—
198	C₂²×C₄	—	37	3	no	—
199	C₂²×C₄	—	37	3	no	—
200	C₂²×C₄	—	37	6	no	—
201	C₂²×C₄	—	37	6	no	—
202	C₂²×C₄	—	37	12	no	—
203	C₂²×C₄	—	148	12	no	—
204	C₂²×C₄	C₂²×D₃₇	1	6	yes	—
205	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
206	C₄⋊C₄	—	37	6	no	—
207	C₄⋊C₄	—	37	2	no	—
208	C₄⋊C₄	—	37	3	no	—
209	C₄⋊C₄	—	37	3	no	—
210	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
211	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
212	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
213	C₂²⋊C₄	—	37	6	no	—
214	C₂²⋊C₄	—	37	12	no	—
215	C₂²⋊C₄	—	37	6	no	—
216	C₂²⋊C₄	—	37	6	no	—
217	C₂²⋊C₄	C₂²×D₃₇	1	6	yes	—
218	C₄²	—	37	2	no	—
219	C₄²	—	37	1	no	—
220	C₄²	—	37	3	no	—
221	C₄²	—	37	6	no	—
222	C₄²	—	37	3	no	—
223	C₄²	—	37	3	no	—
224	C₄²	C₂²×D₃₇	1	6	yes	—
225	C₂³	—	37	6	no	—
226	C₂³	—	37	24	no	—
227	C₂³	—	148	6	no	—
228	C₂³	—	148	6	no	—
229	C₂³	C₂³×D₃₇	1	3	yes	—
230	Q₈	—	37	4	no	—
231	Q₈	C₂×C₄×D₃₇	1	4	yes	complemented
232	D₄	—	37	24	no	—
233	D₄	—	37	24	no	—
234	D₄	—	37	12	no	—
235	D₄	C₂×C₄×D₃₇	1	12	yes	complemented
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	37	1	no	—
238	C₂×C₄	—	37	2	no	—
239	C₂×C₄	—	37	24	no	—
240	C₂×C₄	—	37	3	no	—
241	C₂×C₄	—	37	1	no	—
242	C₂×C₄	—	37	3	no	—
243	C₂×C₄	—	74	12	no	—
244	C₂×C₄	—	74	6	no	—
245	C₂×C₄	—	148	12	no	—
246	C₂×C₄	—	148	6	no	—
247	C₂×C₄	—	148	3	no	—
248	C₂×C₄	C₂×C₄×D₃₇	1	12	yes	complemented
249	C₂×C₄	C₂³×D₃₇	1	2	yes	—
250	C₂×C₄	C₂³×D₃₇	1	6	yes	—
251	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₃₇	1	3	yes	—
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	37	12	no	—
256	C₂²	—	74	6	no	—
257	C₂²	—	74	6	no	—
258	C₂²	—	148	24	no	—
259	C₂²	C₂²×C₄×D₃₇	1	6	yes	—
260	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	—	37	4	no	—
264	C₄	—	37	2	no	—
265	C₄	—	74	3	no	—
266	C₄	—	74	3	no	—
267	C₄	—	74	1	no	—
268	C₄	C₂²×C₄×D₃₇	1	6	yes	—
269	C₄	C₂²×C₄×D₃₇	1	2	yes	—
270	C₂	—	2	6	no	—
271	C₂	—	74	12	no	—
272	C₂	C₂³×C₄×D₃₇	1	1	yes	characteristic
273	C₂	H(5,𝔽₂)×D₃₇	1	1	yes	characteristic
274	C₂	#14558370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator	C₂×C₃₇	C₂⁴×C₄
Fitting	C₃₇×#520	C₂
socle, cosocle	C₂²×C₃₇	C₂⁵
prime radical	C₂×#11924582	C₂
hypercenter	#520	D₃₇
hypocenter	C₃₇	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14558015	C₂	semidirect
3. #14558014	C₂	semidirect
4. #14558007	C₂	semidirect
5. #14558005	C₂	semidirect
6. #41×Dic₃₇	C₂	semidirect
7. C₄×#10596966	C₂	semidirect
8. #14558001	C₂	semidirect
9. C₂×#11924498	C₂	semidirect
10. C₄×D₄×D₃₇	C₂	semidirect
11. C₃₇×#520	C₂	semidirect
12. #11924582	C₄	semidirect
17. C₂²×C₄×D₃₇	C₂²	semidirect
18. Q₈×Dic₃₇	C₂²	semidirect
19. Q₈×Dic₃₇	C₂²	semidirect
20. #11924501	C₂²	semidirect
21. #11924501	C₂²	semidirect
22. C₄×D₁₄₈	C₂²	semidirect
23. C₄×D₁₄₈	C₂²	semidirect
24. D₄×Dic₃₇	C₂²	semidirect
25. D₄×Dic₃₇	C₂²	semidirect
26. #11924498	C₂²	semidirect
27. #11924498	C₂²	semidirect
28. #11924498	C₂²	semidirect
29. #11924498	C₂²	semidirect
30. #11924498	C₂²	semidirect
31. C₄×C₂²⋊D₃₇	C₂²	semidirect
32. #11924496	C₂²	semidirect
33. #11924496	C₂²	semidirect
34. #11924495	C₂²	semidirect
35. #11924494	C₂²	semidirect
36. #11924493	C₂²	semidirect
37. C₄⋊C₄×D₃₇	C₂²	semidirect
38. C₂×#10596938	C₂²	semidirect
39. C₂²⋊C₄×D₃₇	C₂²	semidirect
40. C₂×C₄×Dic₃₇	C₂²	semidirect
41. C₄²×D₃₇	C₂²	semidirect
43. C₄×Q₈×C₃₇	C₂²	semidirect
44. C₄×D₄×C₃₇	C₂²	semidirect
45. #117×C₃₇	C₂²	semidirect
46. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
48. #41×C₃₇	C₂×C₄	semidirect
56. #10596966	C₂×C₄	semidirect
57. D₄×D₃₇	C₂×C₄	semidirect
58. #10596962	C₂×C₄	semidirect
59. C₂×D₁₄₈	C₂×C₄	semidirect
90. C₂²×D₃₇	C₂²×C₄	semidirect
92. Q₈×C₃₇	C₂²×C₄	semidirect
93. D₄×C₃₇	C₂²×C₄	semidirect
95. C₂×C₄×C₃₇	C₂²×C₄	semidirect
100. C₂²⋊D₃₇	C₂²×C₄	semidirect
108. D₁₄₈	C₂²×C₄	semidirect
109. D₁₄₈	C₂²×C₄	semidirect
113. C₄×D₃₇	C₂²×C₄	semidirect
114. Dic₇₄	C₂²×C₄	semidirect
131. Dic₃₇	H(5,𝔽₂)	semidirect
141. #520	D₃₇	semidirect
149. C₃₇	C₄×H(5,𝔽₂)	semidirect
161. C₄×Q₈	C₂×D₃₇	semidirect
174. C₄×D₄	C₂×D₃₇	semidirect
178. #117	C₂×D₃₇	semidirect
180. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
188. #41	C₄×D₃₇	semidirect
231. Q₈	C₂×C₄×D₃₇	semidirect
235. D₄	C₂×C₄×D₃₇	semidirect
248. C₂×C₄	C₂×C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924582	2
#14558015	2
#14558014	2
#14558007	2
#14558005	2
#41×Dic₃₇	2
C₄×#10596966	2
#14558001	2
C₂×#11924498	2
C₄×D₄×D₃₇	2
C₃₇×#520	2
C₄×H(5,𝔽₂)	37