Subgroups — #33999843

Order 18688.

10894 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624742	C₂	1	1	yes	maximal, characteristic, complemented
2	#22624711	C₂	1	15	yes	maximal, complemented
3	#22624710	C₂	1	15	yes	maximal, complemented
4	#22624702	C₂	1	15	yes	maximal, complemented
5	C₂×#17989390	C₂	1	15	yes	maximal, complemented
6	#22624643	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989424	—	2	6	no	—
9	#17989423	C₂²	1	15	yes	complemented
10	C₂×#14556637	C₂²	1	15	yes	complemented
11	H(5,𝔽₂)×D₇₃	—	2	10	no	—
12	#17989402	C₂²	1	20	yes	complemented
13	#17989398	C₂²	1	60	yes	complemented
14	#17989395	C₂²	1	60	yes	complemented
15	#17989390	C₂²	1	60	yes	complemented
16	#17989390	C₂²	1	120	yes	complemented
17	#17989385	C₂²	1	90	yes	complemented
18	C₂×#14556617	C₂²	1	45	yes	complemented
19	C₂×#14556616	C₂²	1	45	yes	complemented
20	C₂×#14556611	C₂²	1	45	yes	complemented
21	C₂²×#11924013	C₂²	1	15	yes	complemented
22	#17989354	C₂²	1	15	yes	complemented
23	C₂×#14556598	C₂²	1	15	yes	complemented
24	#585×C₇₃	—	2	6	no	—
25	#585×C₇₃	C₂²	1	1	yes	characteristic
26	#585×C₇₃	C₂²	1	15	yes	complemented
27	C₂×H(5,𝔽₂)×C₇₃	—	2	10	no	—
28	C₂×H(5,𝔽₂)×C₇₃	C₂²	1	15	yes	complemented
29	#14556639	—	2	15	no	—
30	#14556638	C₂³	1	20	yes	complemented
31	#14556637	—	2	60	no	—
32	#14556637	C₂³	1	60	yes	complemented
33	#14556636	—	2	45	no	—
34	C₂×#11924029	—	2	15	no	—
35	C₂×#11924028	C₂³	1	45	yes	complemented
36	#41×D₇₃	—	2	60	no	—
37	C₂²×D₂₉₂	C₂³	1	15	yes	complemented
38	C₂×D₄×D₇₃	—	2	45	no	—
39	#14556622	C₂³	1	60	yes	complemented
40	#14556621	C₂³	1	60	yes	complemented
41	#14556617	C₂³	1	180	yes	complemented
42	#14556616	C₂³	1	180	yes	complemented
43	#14556611	C₂³	1	180	yes	complemented
44	#14556611	C₂³	1	60	yes	complemented
45	C₂×#11924018	C₂³	1	15	yes	complemented
46	C₂×#11924016	C₂³	1	45	yes	complemented
47	C₂×#11924015	C₂³	1	45	yes	complemented
48	C₂×#11924013	C₂³	1	45	yes	complemented
49	C₂×#11924013	C₂³	1	90	yes	complemented
50	#14556598	C₂³	1	20	yes	complemented
51	#14556598	C₂³	1	60	yes	complemented
52	C₂×#11924012	C₂³	1	45	yes	complemented
53	C₂²×#10596831	C₂³	1	15	yes	complemented
54	2₋¹⁺⁴×C₇₃	—	2	15	no	—
55	2₋¹⁺⁴×C₇₃	D₄	1	6	yes	—
56	2₋¹⁺⁴×C₇₃	C₂³	1	20	yes	complemented
57	H(5,𝔽₂)×C₇₃	—	2	60	no	—
58	H(5,𝔽₂)×C₇₃	—	2	45	no	—
59	H(5,𝔽₂)×C₇₃	D₄	1	10	yes	—
60	H(5,𝔽₂)×C₇₃	C₂³	1	60	yes	complemented
61	C₂×#41×C₇₃	—	2	60	no	—
62	C₂×#41×C₇₃	—	2	15	no	—
63	C₂×#41×C₇₃	C₂³	1	15	yes	—
64	C₂×#41×C₇₃	C₂³	1	45	yes	complemented
65	C₂²×D₄×C₇₃	—	2	45	no	—
66	C₂²×D₄×C₇₃	C₂³	1	15	yes	complemented
67	C₂³×D₇₃	—	2	15	no	—
68	#11924029	—	2	60	no	—
69	#11924028	—	2	90	no	—
70	#11924028	C₂⁴	1	60	yes	—
71	#11924027	—	2	60	no	—
72	C₂×Dic₁₄₆	C₂⁴	1	15	yes	—
73	Q₈×D₇₃	—	2	20	no	—
74	C₂×D₂₉₂	—	2	45	no	—
75	C₂×D₂₉₂	C₂⁴	1	45	yes	—
76	C₂×C₂²⋊D₇₃	—	2	45	no	—
77	D₄×D₇₃	—	2	180	no	—
78	C₂×C₄×D₇₃	—	2	45	no	—
79	#11924018	C₂⁴	1	6	yes	—
80	#11924018	C₂⁴	1	30	yes	—
81	#11924016	C₂⁴	1	60	yes	—
82	#11924015	C₂⁴	1	60	yes	—
83	#11924013	C₂⁴	1	10	yes	—
84	#11924013	C₂⁴	1	90	yes	—
85	#11924012	C₂⁴	1	60	yes	—
86	C₂×#10596831	C₂⁴	1	15	yes	—
87	C₂×#10596831	C₂⁴	1	45	yes	—
88	C₂⁴×C₇₃	—	2	15	no	—
89	#41×C₇₃	—	2	60	no	—
90	#41×C₇₃	—	2	60	no	—
91	#41×C₇₃	—	2	90	no	—
92	#41×C₇₃	C₂×D₄	1	60	yes	—
93	#41×C₇₃	C₂⁴	1	60	yes	—
94	#41×C₇₃	C₂⁴	1	20	yes	—
95	C₂×Q₈×C₇₃	—	2	20	no	—
96	C₂×Q₈×C₇₃	C₂×D₄	1	15	yes	—
97	C₂×Q₈×C₇₃	C₂⁴	1	15	yes	—
98	C₂×D₄×C₇₃	—	2	45	no	—
99	C₂×D₄×C₇₃	—	2	180	no	—
100	C₂×D₄×C₇₃	—	2	45	no	—
101	C₂×D₄×C₇₃	C₂×D₄	1	45	yes	—
102	C₂×D₄×C₇₃	C₂⁴	1	45	yes	—
103	C₂²×C₄×C₇₃	—	2	45	no	—
104	C₂²×C₄×C₇₃	C₂⁴	1	15	yes	—
105	C₂³×C₇₃	—	2	45	no	—
106	C₂³×C₇₃	—	16	15	no	—
107	C₂³×C₇₃	C₂²×D₄	1	15	yes	—
108	C₂²×D₇₃	—	2	45	no	—
109	C₂²×D₇₃	—	16	15	no	—
110	Q₈×C₇₃	—	2	15	no	—
111	Q₈×C₇₃	C₂²×D₄	1	20	yes	—
112	Q₈×C₇₃	C₂⁵	1	6	yes	—
113	D₄×C₇₃	—	2	60	no	—
114	D₄×C₇₃	—	2	45	no	—
115	D₄×C₇₃	C₂²×D₄	1	60	yes	—
116	D₄×C₇₃	C₂⁵	1	10	yes	—
117	C₂×C₄×C₇₃	—	2	15	no	—
118	C₂×C₄×C₇₃	—	2	60	no	—
119	C₂×C₄×C₇₃	C₂²×D₄	1	45	yes	—
120	C₂×C₄×C₇₃	C₂⁵	1	15	yes	—
121	C₂²⋊D₇₃	—	2	60	no	—
122	C₂×Dic₇₃	—	2	15	no	—
123	D₂₉₂	—	2	45	no	—
124	D₂₉₂	C₂⁵	1	10	yes	—
125	C₄×D₇₃	—	2	60	no	—
126	Dic₁₄₆	—	2	15	no	—
127	Dic₁₄₆	C₂⁵	1	6	yes	—
128	#10596831	C₂⁵	1	1	yes	characteristic
129	#10596831	C₂⁵	1	15	yes	—
130	C₂²×C₇₃	—	2	10	no	—
131	C₂²×C₇₃	—	4	15	no	—
132	C₂²×C₇₃	—	8	45	no	—
133	C₂²×C₇₃	C₂³×D₄	1	15	yes	—
134	C₂×D₇₃	—	2	10	no	—
135	C₂×D₇₃	—	8	45	no	—
136	C₄×C₇₃	—	2	6	no	—
137	C₄×C₇₃	C₂³×D₄	1	15	yes	—
138	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
139	Dic₇₃	—	2	6	no	—
140	#63091	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	15	no	—
142	C₂×C₇₃	—	4	10	no	—
143	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
144	D₇₃	—	4	10	no	—
145	H(7,𝔽₂)	—	73	1	no	—
146	H(7,𝔽₂)	D₇₃	1	1	yes	characteristic, complemented
147	#3585	—	73	15	no	—
148	#3585	—	73	15	no	—
149	#3584	—	73	15	no	—
150	C₂×#576	—	73	15	no	—
151	#3573	—	73	1	no	—
152	C₇₃	#63091	1	1	yes	characteristic, complemented, Sylow
153	#585	—	2	6	no	—
154	#585	—	73	15	no	—
155	#585	—	146	6	no	—
156	#585	C₂×D₇₃	1	1	yes	characteristic
157	#585	C₂×D₇₃	1	15	yes	complemented
158	C₂×H(5,𝔽₂)	—	2	10	no	—
159	C₂×H(5,𝔽₂)	—	73	15	no	—
160	C₂×H(5,𝔽₂)	—	146	10	no	—
161	C₂×H(5,𝔽₂)	C₂×D₇₃	1	15	yes	complemented
162	#578	—	73	20	no	—
163	#577	—	73	60	no	—
164	#577	—	73	60	no	—
165	#576	—	73	60	no	—
166	#576	—	73	120	no	—
167	#575	—	73	90	no	—
168	C₂×AGL(1,ℤ/8)	—	73	45	no	—
169	C₂×AGL(1,ℤ/8)	—	73	45	no	—
170	C₂×#135	—	73	45	no	—
171	C₂²×D₈	—	73	15	no	—
172	#568	—	73	15	no	—
173	C₂×#131	—	73	15	no	—
174	2₋¹⁺⁴	—	2	15	no	—
175	2₋¹⁺⁴	—	73	20	no	—
176	2₋¹⁺⁴	—	146	15	no	—
177	2₋¹⁺⁴	C₂²⋊D₇₃	1	6	yes	—
178	2₋¹⁺⁴	C₂²×D₇₃	1	20	yes	complemented
179	H(5,𝔽₂)	—	2	60	no	—
180	H(5,𝔽₂)	—	2	45	no	—
181	H(5,𝔽₂)	—	73	60	no	—
182	H(5,𝔽₂)	—	146	60	no	—
183	H(5,𝔽₂)	—	146	45	no	—
184	H(5,𝔽₂)	C₂²⋊D₇₃	1	10	yes	—
185	H(5,𝔽₂)	C₂²×D₇₃	1	60	yes	complemented
186	C₂×#41	—	2	60	no	—
187	C₂×#41	—	2	15	no	—
188	C₂×#41	—	73	45	no	—
189	C₂×#41	—	146	60	no	—
190	C₂×#41	—	146	15	no	—
191	C₂×#41	C₂²×D₇₃	1	15	yes	—
192	C₂×#41	C₂²×D₇₃	1	45	yes	complemented
193	C₂²×D₄	—	2	45	no	—
194	C₂²×D₄	—	73	15	no	—
195	C₂²×D₄	—	146	45	no	—
196	C₂²×D₄	C₂²×D₇₃	1	15	yes	complemented
197	#137	—	73	60	no	—
198	#137	—	73	60	no	—
199	AGL(1,ℤ/8)	—	73	180	no	—
200	AGL(1,ℤ/8)	—	73	180	no	—
201	#135	—	73	60	no	—
202	#135	—	73	180	no	—
203	C₂×Q₁₆	—	73	15	no	—
204	C₂×SD₁₆	—	73	45	no	—
205	C₂×SD₁₆	—	73	45	no	—
206	C₂×D₈	—	73	45	no	—
207	C₂×D₈	—	73	90	no	—
208	#131	—	73	20	no	—
209	#131	—	73	60	no	—
210	C₂×M₁₆	—	73	45	no	—
211	C₂²×C₈	—	73	15	no	—
212	C₂⁴	—	2	15	no	—
213	C₂⁴	—	146	15	no	—
214	#41	—	2	60	no	—
215	#41	—	2	60	no	—
216	#41	—	2	90	no	—
217	#41	—	73	60	no	—
218	#41	—	146	60	no	—
219	#41	—	146	60	no	—
220	#41	—	146	90	no	—
221	#41	C₂×C₂²⋊D₇₃	1	60	yes	—
222	#41	C₂³×D₇₃	1	60	yes	—
223	#41	C₂³×D₇₃	1	20	yes	—
224	C₂×Q₈	—	2	20	no	—
225	C₂×Q₈	—	73	15	no	—
226	C₂×Q₈	—	146	20	no	—
227	C₂×Q₈	C₂×C₂²⋊D₇₃	1	15	yes	—
228	C₂×Q₈	C₂³×D₇₃	1	15	yes	—
229	C₂×D₄	—	2	45	no	—
230	C₂×D₄	—	2	180	no	—
231	C₂×D₄	—	2	45	no	—
232	C₂×D₄	—	73	45	no	—
233	C₂×D₄	—	146	45	no	—
234	C₂×D₄	—	146	180	no	—
235	C₂×D₄	—	146	45	no	—
236	C₂×D₄	C₂×C₂²⋊D₇₃	1	45	yes	—
237	C₂×D₄	C₂³×D₇₃	1	45	yes	—
238	C₂²×C₄	—	2	45	no	—
239	C₂²×C₄	—	146	45	no	—
240	C₂²×C₄	C₂³×D₇₃	1	15	yes	—
241	Q₁₆	—	73	6	no	—
242	Q₁₆	—	73	30	no	—
243	SD₁₆	—	73	60	no	—
244	SD₁₆	—	73	60	no	—
245	D₈	—	73	10	no	—
246	D₈	—	73	90	no	—
247	M₁₆	—	73	60	no	—
248	C₂×C₈	—	73	15	no	—
249	C₂×C₈	—	73	45	no	—
250	C₂³	—	2	45	no	—
251	C₂³	—	16	15	no	—
252	C₂³	—	1168	15	no	—
253	C₂³	—	146	45	no	—
254	C₂³	C₂²×C₂²⋊D₇₃	1	15	yes	—
255	Q₈	—	2	15	no	—
256	Q₈	—	73	6	no	—
257	Q₈	—	146	15	no	—
258	Q₈	C₂²×C₂²⋊D₇₃	1	20	yes	—
259	Q₈	C₂⁴×D₇₃	1	6	yes	—
260	D₄	—	2	60	no	—
261	D₄	—	2	45	no	—
262	D₄	—	73	10	no	—
263	D₄	—	146	60	no	—
264	D₄	—	146	45	no	—
265	D₄	C₂²×C₂²⋊D₇₃	1	60	yes	—
266	D₄	C₂⁴×D₇₃	1	10	yes	—
267	C₂×C₄	—	2	15	no	—
268	C₂×C₄	—	2	60	no	—
269	C₂×C₄	—	146	15	no	—
270	C₂×C₄	—	146	60	no	—
271	C₂×C₄	C₂²×C₂²⋊D₇₃	1	45	yes	—
272	C₂×C₄	C₂⁴×D₇₃	1	15	yes	—
273	C₈	—	73	1	no	—
274	C₈	—	73	15	no	—
275	C₂²	—	2	10	no	—
276	C₂²	—	4	15	no	—
277	C₂²	—	8	45	no	—
278	C₂²	—	584	45	no	—
279	C₂²	—	146	10	no	—
280	C₂²	C₂³×C₂²⋊D₇₃	1	15	yes	—
281	C₄	—	2	6	no	—
282	C₄	—	146	6	no	—
283	C₄	C₂³×C₂²⋊D₇₃	1	15	yes	—
284	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
285	C₂	—	2	15	no	—
286	C₂	—	4	10	no	—
287	C₂	—	292	10	no	—
288	C₂	C₂⁴×C₂²⋊D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×C₂²⋊D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₇₃×H(7,𝔽₂)	C₂
socle	C₂×C₇₃	C₂⁴×D₄
hypercenter	H(7,𝔽₂)	D₇₃
hypocenter	C₇₃	#63091
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ #585 ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #22624742	C₂	semidirect
2. #22624711	C₂	semidirect
3. #22624710	C₂	semidirect
4. #22624702	C₂	semidirect
5. C₂×#17989390	C₂	semidirect
6. #22624643	C₂	semidirect
7. C₇₃×H(7,𝔽₂)	C₂	semidirect
9. #17989423	C₂²	semidirect
10. C₂×#14556637	C₂²	semidirect
12. #17989402	C₂²	semidirect
13. #17989398	C₂²	semidirect
14. #17989395	C₂²	semidirect
15. #17989390	C₂²	semidirect
16. #17989390	C₂²	semidirect
17. #17989385	C₂²	semidirect
18. C₂×#14556617	C₂²	semidirect
19. C₂×#14556616	C₂²	semidirect
20. C₂×#14556611	C₂²	semidirect
21. C₂²×#11924013	C₂²	semidirect
22. #17989354	C₂²	semidirect
23. C₂×#14556598	C₂²	semidirect
26. #585×C₇₃	C₂²	semidirect
28. C₂×H(5,𝔽₂)×C₇₃	C₂²	semidirect
30. #14556638	C₂³	semidirect
32. #14556637	C₂³	semidirect
35. C₂×#11924028	C₂³	semidirect
37. C₂²×D₂₉₂	C₂³	semidirect
39. #14556622	C₂³	semidirect
40. #14556621	C₂³	semidirect
41. #14556617	C₂³	semidirect
42. #14556616	C₂³	semidirect
43. #14556611	C₂³	semidirect
44. #14556611	C₂³	semidirect
45. C₂×#11924018	C₂³	semidirect
46. C₂×#11924016	C₂³	semidirect
47. C₂×#11924015	C₂³	semidirect
48. C₂×#11924013	C₂³	semidirect
49. C₂×#11924013	C₂³	semidirect
50. #14556598	C₂³	semidirect
51. #14556598	C₂³	semidirect
52. C₂×#11924012	C₂³	semidirect
53. C₂²×#10596831	C₂³	semidirect
56. 2₋¹⁺⁴×C₇₃	C₂³	semidirect
60. H(5,𝔽₂)×C₇₃	C₂³	semidirect
64. C₂×#41×C₇₃	C₂³	semidirect
66. C₂²×D₄×C₇₃	C₂³	semidirect
146. H(7,𝔽₂)	D₇₃	semidirect
152. C₇₃	#63091	semidirect
157. #585	C₂×D₇₃	semidirect
161. C₂×H(5,𝔽₂)	C₂×D₇₃	semidirect
178. 2₋¹⁺⁴	C₂²×D₇₃	semidirect
185. H(5,𝔽₂)	C₂²×D₇₃	semidirect
192. C₂×#41	C₂²×D₇₃	semidirect
196. C₂²×D₄	C₂²×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22624742	2
#22624711	2
#22624710	2
#22624702	2
C₂×#17989390	2
#22624643	2
C₇₃×H(7,𝔽₂)	2
#63091	73