Subgroups — #18007988

Order 4736.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558157	C₂	1	4	yes	maximal, complemented
2	#14558146	C₂	1	2	yes	maximal, complemented
3	#14558139	C₂	1	2	yes	maximal, complemented
4	#14558132	C₂	1	4	yes	maximal, complemented
5	#14558123	C₂	1	4	yes	maximal, complemented
6	#14558083	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558070	C₂	1	8	yes	maximal, complemented
8	#14558045	C₂	1	2	yes	maximal, complemented
9	#14558001	C₂	1	2	yes	maximal, complemented
10	C₂×#11924493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	2	yes	—
13	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
14	#11924536	C₂²	1	1	yes	characteristic, complemented
15	#11924533	C₂²	1	1	yes	characteristic, complemented
16	#11924531	C₂²	1	8	yes	—
17	#11924531	C₂²	1	8	yes	complemented
18	#11924527	C₂²	1	8	yes	complemented
19	#11924525	C₂²	1	8	yes	—
20	#11924524	C₂²	1	2	yes	complemented
21	#11924521	C₂²	1	8	yes	—
22	#11924521	C₂²	1	8	yes	complemented
23	#11924520	C₂²	1	4	yes	—
24	#11924516	C₂²	1	4	yes	—
25	#11924512	C₂²	1	8	yes	complemented
26	#11924511	C₂²	1	4	yes	—
27	#11924511	C₂²	1	8	yes	—
28	#11924510	C₂²	1	4	yes	—
29	#11924506	C₂²	1	4	yes	—
30	#11924501	C₂²	1	4	yes	complemented
31	C₄×D₁₄₈	C₂²	1	8	yes	—
32	C₄×D₁₄₈	C₂²	1	4	yes	complemented
33	#11924498	C₂²	1	4	yes	complemented
34	#11924496	C₂²	1	8	yes	—
35	#11924493	C₂²	1	4	yes	—
36	#11924493	C₂²	1	2	yes	—
37	#11924493	C₂²	1	2	yes	—
38	#11924493	C₂²	1	4	yes	complemented
39	C₂×#10596936	C₂²	1	2	yes	—
40	C₂×#10596938	C₂²	1	2	yes	—
41	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
42	C₄²×D₃₇	C₂²	1	4	yes	complemented
43	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	complemented
44	#125×C₃₇	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈×C₃₇	C₂²	1	4	yes	—
46	C₄×Q₈×C₃₇	C₂²	1	4	yes	complemented
47	#117×C₃₇	C₂²	1	2	yes	—
48	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
49	C₂³×D₃₇	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
51	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₃₇	C₂³	1	4	yes	—
54	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₃₇	C₂³	1	2	yes	—
56	#10596962	—	2	8	no	—
57	C₂×D₁₄₈	C₂³	1	2	yes	—
58	C₂×D₁₄₈	C₂³	1	8	yes	—
59	C₂×D₁₄₈	C₂³	1	4	yes	—
60	C₂×C₄×D₃₇	—	2	4	no	—
61	C₂×C₄×D₃₇	C₂³	1	4	yes	—
62	C₂×C₄×D₃₇	C₂³	1	2	yes	—
63	C₂×C₄×D₃₇	C₂³	1	2	yes	—
64	C₂×C₄×D₃₇	C₂³	1	8	yes	—
65	C₂×Dic₇₄	C₂³	1	2	yes	—
66	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
67	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
68	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₃₇	C₂³	1	4	yes	—
70	C₄²×C₃₇	C₂³	1	4	yes	—
71	C₄²×C₃₇	C₂³	1	2	yes	—
72	C₄²×C₃₇	C₂³	1	1	yes	characteristic
73	C₄²×C₃₇	C₂³	1	1	yes	characteristic
74	#10596938	C₂³	1	4	yes	—
75	#10596938	C₂³	1	4	yes	—
76	#10596938	C₂³	1	8	yes	—
77	#10596938	C₂³	1	8	yes	—
78	#10596938	C₂³	1	8	yes	—
79	#10596938	C₂³	1	8	yes	—
80	C₄⋊Dic₃₇	C₂³	1	8	yes	—
81	C₄⋊Dic₃₇	C₂³	1	4	yes	—
82	#10596936	C₂³	1	4	yes	—
83	#10596936	C₂³	1	4	yes	—
84	#10596936	C₂³	1	8	yes	—
85	C₄×Dic₃₇	C₂³	1	2	yes	—
86	C₄×Dic₃₇	C₂³	1	8	yes	—
87	C₄×Dic₃₇	C₂³	1	2	yes	—
88	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
89	C₂²×D₃₇	—	2	2	no	—
90	C₂²×D₃₇	—	4	2	no	—
91	C₂²×D₃₇	C₂⁴	1	2	yes	—
92	C₂²×D₃₇	C₂⁴	1	4	yes	—
93	Q₈×C₃₇	—	4	4	no	—
94	C₂×C₄×C₃₇	—	4	2	no	—
95	C₂×C₄×C₃₇	#41	1	4	yes	—
96	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
98	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₃₇	—	2	8	no	—
102	C₂×Dic₃₇	—	2	2	no	—
103	C₂×Dic₃₇	—	4	2	no	—
104	C₂×Dic₃₇	C₂⁴	1	4	yes	—
105	C₂×Dic₃₇	C₂⁴	1	4	yes	—
106	C₂×Dic₃₇	C₂⁴	1	2	yes	—
107	D₁₄₈	—	2	4	no	—
108	D₁₄₈	—	4	8	no	—
109	D₁₄₈	—	4	4	no	—
110	C₄×D₃₇	—	2	4	no	—
111	C₄×D₃₇	—	2	4	no	—
112	C₄×D₃₇	—	2	8	no	—
113	C₄×D₃₇	—	4	8	no	—
114	Dic₇₄	—	2	4	no	—
115	C₂²×C₃₇	—	2	2	no	—
116	C₂²×C₃₇	C₂×#41	1	2	yes	—
117	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₃₇	—	2	2	no	—
119	C₂×D₃₇	—	2	4	no	—
120	C₂×D₃₇	—	2	8	no	—
121	C₂×D₃₇	—	2	4	no	—
122	C₂×D₃₇	—	4	4	no	—
123	C₄×C₃₇	—	2	8	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	C₂×#41	1	2	yes	—
127	C₄×C₃₇	C₂×#41	1	2	yes	—
128	Dic₃₇	—	2	4	no	—
129	Dic₃₇	—	2	2	no	—
130	Dic₃₇	—	2	4	no	—
131	#3455	—	37	1	no	maximal, Sylow
132	C₂×C₃₇	—	2	2	no	—
133	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
134	C₂×C₃₇	#585	1	2	yes	—
135	D₃₇	—	4	4	no	—
136	D₃₇	—	4	2	no	—
137	#542	—	37	4	no	—
138	#541	—	37	2	no	—
139	#540	—	37	2	no	—
140	#539	—	37	4	no	—
141	#538	—	37	4	no	—
142	#533	D₃₇	1	1	yes	characteristic, complemented
143	#532	—	37	1	no	—
144	#529	—	37	8	no	—
145	#525	—	37	2	no	—
146	C₄×#41	—	37	2	no	—
147	C₂×#117	—	37	1	no	—
148	C₃₇	#3455	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	37	2	no	—
150	C₂³×C₄	—	37	1	no	—
151	C₄⋊Q₈	—	37	1	no	—
152	C₄⋊Q₈	C₂×D₃₇	1	2	yes	complemented
153	#127	—	37	1	no	—
154	#126	—	37	8	no	—
155	#126	—	37	8	no	—
156	#125	—	37	8	no	—
157	#125	C₂×D₃₇	1	2	yes	complemented
158	#124	—	37	8	no	—
159	#124	—	37	2	no	—
160	#123	—	37	8	no	—
161	#123	—	37	8	no	—
162	#123	—	37	4	no	—
163	C₂²⋊Q₈	—	37	8	no	—
164	C₂²⋊Q₈	—	37	4	no	—
165	C₂²⋊Q₈	C₂×D₃₇	1	4	yes	—
166	#121	—	37	4	no	—
167	#121	—	37	4	no	—
168	#121	—	37	8	no	—
169	#120	—	37	4	no	—
170	C₄×Q₈	—	37	4	no	—
171	C₄×Q₈	C₂×D₃₇	1	4	yes	complemented
172	C₄×D₄	—	37	4	no	—
173	C₄×D₄	—	37	8	no	—
174	C₄×D₄	—	37	8	no	—
175	C₄×D₄	—	37	4	no	—
176	#117	—	37	4	no	—
177	#117	—	37	2	no	—
178	#117	—	37	2	no	—
179	#117	—	37	4	no	—
180	#117	C₂×D₃₇	1	2	yes	—
181	C₂×C₄⋊C₄	—	37	2	no	—
182	C₂×C₂²⋊C₄	—	37	2	no	—
183	C₂×C₄²	—	37	1	no	—
184	C₂×C₄²	—	37	4	no	—
185	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
186	C₂⁴	—	37	1	no	—
187	#41	—	74	8	no	—
188	C₂×Q₈	—	37	2	no	—
189	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
190	C₂×D₄	—	37	4	no	—
191	C₂×D₄	—	37	2	no	—
192	C₂×D₄	—	37	8	no	—
193	C₂×D₄	—	37	4	no	—
194	C₂²×C₄	—	37	1	no	—
195	C₂²×C₄	—	37	2	no	—
196	C₂²×C₄	—	37	2	no	—
197	C₂²×C₄	—	37	4	no	—
198	C₂²×C₄	—	37	2	no	—
199	C₂²×C₄	—	37	8	no	—
200	C₂²×C₄	—	74	4	no	—
201	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
203	C₄⋊C₄	—	37	4	no	—
204	C₄⋊C₄	—	37	4	no	—
205	C₄⋊C₄	—	37	8	no	—
206	C₄⋊C₄	—	37	4	no	—
207	C₄⋊C₄	—	37	8	no	—
208	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
209	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
210	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
211	C₂²⋊C₄	—	37	4	no	—
212	C₂²⋊C₄	—	37	4	no	—
213	C₂²⋊C₄	—	37	8	no	—
214	C₂²⋊C₄	—	37	8	no	—
215	C₂²⋊C₄	—	37	8	no	—
216	C₂²⋊C₄	—	37	8	no	—
217	C₂²⋊C₄	C₂²×D₃₇	1	4	yes	—
218	C₄²	—	37	2	no	—
219	C₄²	—	37	8	no	—
220	C₄²	—	37	2	no	—
221	C₄²	C₂²×D₃₇	1	1	yes	characteristic
222	C₄²	C₂²×D₃₇	1	1	yes	characteristic
223	C₄²	C₂²×D₃₇	1	4	yes	—
224	C₄²	C₂²×D₃₇	1	2	yes	—
225	C₂³	—	37	2	no	—
226	C₂³	—	37	4	no	—
227	C₂³	—	74	2	no	—
228	C₂³	—	148	2	no	—
229	C₂³	C₂³×D₃₇	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	74	4	no	—
232	D₄	—	74	8	no	—
233	D₄	—	74	4	no	—
234	D₄	—	148	4	no	—
235	D₄	—	148	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	37	4	no	—
238	C₂×C₄	—	37	4	no	—
239	C₂×C₄	—	37	2	no	—
240	C₂×C₄	—	74	2	no	—
241	C₂×C₄	—	74	4	no	—
242	C₂×C₄	—	74	4	no	—
243	C₂×C₄	—	74	8	no	—
244	C₂×C₄	—	148	8	no	—
245	C₂×C₄	—	148	2	no	—
246	C₂×C₄	#10596966	1	4	yes	—
247	C₂×C₄	C₂³×D₃₇	1	8	yes	—
248	C₂×C₄	C₂³×D₃₇	1	2	yes	—
249	C₂×C₄	C₂³×D₃₇	1	2	yes	—
250	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	74	2	no	—
254	C₂²	—	74	4	no	—
255	C₂²	—	74	8	no	—
256	C₂²	—	74	4	no	—
257	C₂²	—	148	4	no	—
258	C₂²	C₂×#10596966	1	2	yes	—
259	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	74	2	no	—
264	C₄	—	74	4	no	—
265	C₄	—	74	4	no	—
266	C₄	C₂×#10596966	1	2	yes	—
267	C₄	C₂×#10596966	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	148	2	no	—
270	C₂	—	148	4	no	—
271	C₂	C₂²×#10596966	1	1	yes	characteristic
272	C₂	#14558369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₃₇×#533	C₂
hypercenter	#533	D₃₇
hypocenter	C₃₇	#3455
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14558157	C₂	semidirect
2. #14558146	C₂	semidirect
3. #14558139	C₂	semidirect
4. #14558132	C₂	semidirect
5. #14558123	C₂	semidirect
6. #14558083	C₂	semidirect
7. #14558070	C₂	semidirect
8. #14558045	C₂	semidirect
9. #14558001	C₂	semidirect
10. C₂×#11924493	C₂	semidirect
11. C₃₇×#533	C₂	semidirect
14. #11924536	C₂²	semidirect
15. #11924533	C₂²	semidirect
17. #11924531	C₂²	semidirect
18. #11924527	C₂²	semidirect
20. #11924524	C₂²	semidirect
22. #11924521	C₂²	semidirect
25. #11924512	C₂²	semidirect
30. #11924501	C₂²	semidirect
32. C₄×D₁₄₈	C₂²	semidirect
33. #11924498	C₂²	semidirect
38. #11924493	C₂²	semidirect
42. C₄²×D₃₇	C₂²	semidirect
43. C₄⋊Q₈×C₃₇	C₂²	semidirect
44. #125×C₃₇	C₂²	semidirect
46. C₄×Q₈×C₃₇	C₂²	semidirect
142. #533	D₃₇	semidirect
148. C₃₇	#3455	semidirect
152. C₄⋊Q₈	C₂×D₃₇	semidirect
157. #125	C₂×D₃₇	semidirect
171. C₄×Q₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558157	2
#14558146	2
#14558139	2
#14558132	2
#14558123	2
#14558083	2
#14558070	2
#14558045	2
#14558001	2
C₂×#11924493	2
C₃₇×#533	2
#3455	37