Subgroups — #33999851

Order 18688.

10796 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624744	C₂	1	1	yes	maximal, characteristic, complemented
2	#22624717	C₂	1	15	yes	maximal, complemented
3	#22624711	C₂	1	15	yes	maximal, complemented
4	#22624702	C₂	1	15	yes	maximal, complemented
5	C₂×#17989398	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	#17989423	C₂²	1	15	yes	complemented
9	#17989422	—	2	10	no	—
10	C₂×#14556638	C₂²	1	15	yes	complemented
11	2₋¹⁺⁴×D₇₃	—	2	6	no	—
12	#17989402	C₂²	1	60	yes	complemented
13	#17989398	C₂²	1	60	yes	complemented
14	#17989398	C₂²	1	120	yes	complemented
15	#17989395	C₂²	1	20	yes	complemented
16	#17989390	C₂²	1	60	yes	complemented
17	#17989385	C₂²	1	90	yes	complemented
18	C₂×#14556621	C₂²	1	45	yes	complemented
19	C₂×#14556617	C₂²	1	45	yes	complemented
20	C₂×#14556611	C₂²	1	45	yes	complemented
21	C₂²×#11924016	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	—	2	60	no	—
31	#14556638	C₂³	1	60	yes	complemented
32	#14556637	C₂³	1	20	yes	complemented
33	#14556636	—	2	45	no	—
34	C₂×#11924028	C₂³	1	45	yes	complemented
35	C₂×#11924027	—	2	45	no	—
36	#41×D₇₃	—	2	60	no	—
37	C₂²×Dic₁₄₆	C₂³	1	15	yes	complemented
38	C₂×Q₈×D₇₃	—	2	15	no	—
39	#14556622	C₂³	1	60	yes	complemented
40	#14556621	C₂³	1	180	yes	complemented
41	#14556617	C₂³	1	180	yes	complemented
42	#14556616	C₂³	1	60	yes	complemented
43	#14556611	C₂³	1	180	yes	complemented
44	#14556611	C₂³	1	60	yes	complemented
45	C₂×#11924018	C₂³	1	45	yes	complemented
46	C₂×#11924016	C₂³	1	90	yes	complemented
47	C₂×#11924016	C₂³	1	45	yes	complemented
48	C₂×#11924015	C₂³	1	15	yes	complemented
49	C₂×#11924013	C₂³	1	45	yes	complemented
50	#14556598	C₂³	1	60	yes	complemented
51	#14556598	C₂³	1	20	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	—
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	—
61	C₂×#41×C₇₃	—	2	60	no	—
62	C₂×#41×C₇₃	—	2	15	no	—
63	C₂×#41×C₇₃	C₂³	1	45	yes	—
64	C₂×#41×C₇₃	C₂³	1	15	yes	—
65	C₂²×D₄×C₇₃	—	2	45	no	—
66	C₂²×D₄×C₇₃	C₂³	1	15	yes	—
67	#11924029	—	2	20	no	—
68	#11924028	—	2	90	no	—
69	#11924028	C₂⁴	1	60	yes	—
70	#11924027	—	2	180	no	—
71	C₂×Dic₁₄₆	—	2	45	no	—
72	C₂×Dic₁₄₆	C₂⁴	1	45	yes	—
73	Q₈×D₇₃	—	2	60	no	—
74	C₂×D₂₉₂	C₂⁴	1	15	yes	—
75	C₂×C₂²⋊D₇₃	—	2	45	no	—
76	D₄×D₇₃	—	2	60	no	—
77	C₂²×Dic₇₃	—	2	15	no	—
78	C₂×C₄×D₇₃	—	2	45	no	—
79	#11924018	C₂⁴	1	60	yes	—
80	#11924016	C₂⁴	1	10	yes	—
81	#11924016	C₂⁴	1	90	yes	—
82	#11924015	C₂⁴	1	6	yes	—
83	#11924015	C₂⁴	1	30	yes	—
84	#11924013	C₂⁴	1	60	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	15	no	—
109	Q₈×C₇₃	—	2	15	no	—
110	Q₈×C₇₃	C₂²×D₄	1	20	yes	—
111	Q₈×C₇₃	C₂⁵	1	6	yes	—
112	D₄×C₇₃	—	2	60	no	—
113	D₄×C₇₃	—	2	45	no	—
114	D₄×C₇₃	C₂²×D₄	1	60	yes	—
115	D₄×C₇₃	C₂⁵	1	10	yes	—
116	C₂×C₄×C₇₃	—	2	15	no	—
117	C₂×C₄×C₇₃	—	2	60	no	—
118	C₂×C₄×C₇₃	C₂²×D₄	1	45	yes	—
119	C₂×C₄×C₇₃	C₂⁵	1	15	yes	—
120	C₂²⋊D₇₃	—	2	60	no	—
121	C₂×Dic₇₃	—	2	45	no	—
122	D₂₉₂	—	2	15	no	—
123	D₂₉₂	C₂⁵	1	6	yes	—
124	C₄×D₇₃	—	2	60	no	—
125	Dic₁₄₆	—	2	45	no	—
126	Dic₁₄₆	C₂⁵	1	10	yes	—
127	#10596831	C₂⁵	1	1	yes	characteristic
128	#10596831	C₂⁵	1	15	yes	—
129	C₂²×C₇₃	—	2	10	no	—
130	C₂²×C₇₃	—	4	15	no	—
131	C₂²×C₇₃	—	8	45	no	—
132	C₂²×C₇₃	C₂³×D₄	1	15	yes	—
133	C₂×D₇₃	—	2	6	no	—
134	C₂×D₇₃	—	8	15	no	—
135	C₄×C₇₃	—	2	6	no	—
136	C₄×C₇₃	C₂³×D₄	1	15	yes	—
137	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
138	Dic₇₃	—	2	10	no	—
139	#63092	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	—	2	15	no	—
141	C₂×C₇₃	—	4	10	no	—
142	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
143	D₇₃	—	4	6	no	—
144	2₋¹⁺⁶	—	73	1	no	—
145	H(7,𝔽₂)	D₇₃	1	1	yes	characteristic, complemented
146	#3586	—	73	15	no	—
147	#3585	—	73	15	no	—
148	#3584	—	73	15	no	—
149	C₂×#577	—	73	15	no	—
150	#3573	—	73	1	no	—
151	C₇₃	#63092	1	1	yes	characteristic, complemented, Sylow
152	#585	—	2	6	no	—
153	#585	—	73	15	no	—
154	#585	—	146	10	no	—
155	#585	C₂×D₇₃	1	1	yes	characteristic
156	#585	C₂×D₇₃	1	15	yes	complemented
157	C₂×2₋¹⁺⁴	—	73	15	no	—
158	C₂×2₋¹⁺⁴	—	146	6	no	—
159	C₂×H(5,𝔽₂)	—	2	10	no	—
160	C₂×H(5,𝔽₂)	C₂×D₇₃	1	15	yes	complemented
161	#578	—	73	60	no	—
162	#577	—	73	20	no	—
163	#577	—	73	60	no	—
164	#577	—	73	120	no	—
165	#576	—	73	60	no	—
166	#575	—	73	90	no	—
167	C₂×#137	—	73	45	no	—
168	C₂×AGL(1,ℤ/8)	—	73	45	no	—
169	C₂×#135	—	73	45	no	—
170	C₂²×SD₁₆	—	73	15	no	—
171	#568	—	73	15	no	—
172	C₂×#131	—	73	15	no	—
173	2₋¹⁺⁴	—	2	15	no	—
174	2₋¹⁺⁴	—	73	60	no	—
175	2₋¹⁺⁴	—	146	60	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	—
179	H(5,𝔽₂)	—	2	60	no	—
180	H(5,𝔽₂)	—	2	45	no	—
181	H(5,𝔽₂)	—	73	20	no	—
182	H(5,𝔽₂)	—	146	45	no	—
183	H(5,𝔽₂)	C₂²⋊D₇₃	1	10	yes	—
184	H(5,𝔽₂)	C₂²×D₇₃	1	60	yes	—
185	C₂×#41	—	2	60	no	—
186	C₂×#41	—	2	15	no	—
187	C₂×#41	—	73	45	no	—
188	C₂×#41	—	146	60	no	—
189	C₂×#41	—	146	45	no	—
190	C₂×#41	C₂²×D₇₃	1	45	yes	—
191	C₂×#41	C₂²×D₇₃	1	15	yes	—
192	C₂²×Q₈	—	73	15	no	—
193	C₂²×Q₈	—	146	15	no	—
194	C₂²×D₄	—	2	45	no	—
195	C₂²×D₄	C₂²×D₇₃	1	15	yes	—
196	#137	—	73	180	no	—
197	#137	—	73	60	no	—
198	AGL(1,ℤ/8)	—	73	60	no	—
199	AGL(1,ℤ/8)	—	73	180	no	—
200	#135	—	73	60	no	—
201	#135	—	73	180	no	—
202	C₂×Q₁₆	—	73	45	no	—
203	C₂×SD₁₆	—	73	15	no	—
204	C₂×SD₁₆	—	73	45	no	—
205	C₂×SD₁₆	—	73	90	no	—
206	C₂×D₈	—	73	45	no	—
207	#131	—	73	20	no	—
208	#131	—	73	60	no	—
209	C₂×M₁₆	—	73	45	no	—
210	C₂²×C₈	—	73	15	no	—
211	C₂⁴	—	2	15	no	—
212	#41	—	2	60	no	—
213	#41	—	2	60	no	—
214	#41	—	2	90	no	—
215	#41	—	73	60	no	—
216	#41	—	146	20	no	—
217	#41	—	146	180	no	—
218	#41	—	146	90	no	—
219	#41	C₂×C₂²⋊D₇₃	1	60	yes	—
220	#41	C₂³×D₇₃	1	60	yes	—
221	#41	C₂³×D₇₃	1	20	yes	—
222	C₂×Q₈	—	2	20	no	—
223	C₂×Q₈	—	73	45	no	—
224	C₂×Q₈	—	146	60	no	—
225	C₂×Q₈	—	146	45	no	—
226	C₂×Q₈	C₂×C₂²⋊D₇₃	1	15	yes	—
227	C₂×Q₈	C₂³×D₇₃	1	15	yes	—
228	C₂×D₄	—	2	45	no	—
229	C₂×D₄	—	2	180	no	—
230	C₂×D₄	—	2	45	no	—
231	C₂×D₄	—	73	15	no	—
232	C₂×D₄	—	146	45	no	—
233	C₂×D₄	—	146	60	no	—
234	C₂×D₄	C₂×C₂²⋊D₇₃	1	45	yes	—
235	C₂×D₄	C₂³×D₇₃	1	45	yes	—
236	C₂²×C₄	—	2	45	no	—
237	C₂²×C₄	—	146	15	no	—
238	C₂²×C₄	—	146	45	no	—
239	C₂²×C₄	C₂³×D₇₃	1	15	yes	—
240	Q₁₆	—	73	60	no	—
241	SD₁₆	—	73	6	no	—
242	SD₁₆	—	73	30	no	—
243	SD₁₆	—	73	10	no	—
244	SD₁₆	—	73	90	no	—
245	D₈	—	73	60	no	—
246	M₁₆	—	73	60	no	—
247	C₂×C₈	—	73	15	no	—
248	C₂×C₈	—	73	45	no	—
249	C₂³	—	2	45	no	—
250	C₂³	—	16	15	no	—
251	C₂³	—	146	15	no	—
252	C₂³	C₂²×C₂²⋊D₇₃	1	15	yes	—
253	Q₈	—	2	15	no	—
254	Q₈	—	73	10	no	—
255	Q₈	—	146	45	no	—
256	Q₈	C₂²×C₂²⋊D₇₃	1	20	yes	—
257	Q₈	C₂⁴×D₇₃	1	6	yes	—
258	D₄	—	2	60	no	—
259	D₄	—	2	45	no	—
260	D₄	—	73	6	no	—
261	D₄	—	146	60	no	—
262	D₄	—	146	15	no	—
263	D₄	C₂²×C₂²⋊D₇₃	1	60	yes	—
264	D₄	C₂⁴×D₇₃	1	10	yes	—
265	C₂×C₄	—	2	15	no	—
266	C₂×C₄	—	2	60	no	—
267	C₂×C₄	—	146	45	no	—
268	C₂×C₄	—	146	60	no	—
269	C₂×C₄	C₂²×C₂²⋊D₇₃	1	45	yes	—
270	C₂×C₄	C₂⁴×D₇₃	1	15	yes	—
271	C₈	—	73	1	no	—
272	C₈	—	73	15	no	—
273	C₂²	—	2	10	no	—
274	C₂²	—	4	15	no	—
275	C₂²	—	8	45	no	—
276	C₂²	—	584	15	no	—
277	C₂²	—	146	6	no	—
278	C₂²	C₂³×C₂²⋊D₇₃	1	15	yes	—
279	C₄	—	2	6	no	—
280	C₄	—	146	10	no	—
281	C₄	C₂³×C₂²⋊D₇₃	1	15	yes	—
282	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
283	C₂	—	2	15	no	—
284	C₂	—	4	10	no	—
285	C₂	—	292	6	no	—
286	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	#63092
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. #22624744	C₂	semidirect
2. #22624717	C₂	semidirect
3. #22624711	C₂	semidirect
4. #22624702	C₂	semidirect
5. C₂×#17989398	C₂	semidirect
6. #22624643	C₂	semidirect
7. C₇₃×H(7,𝔽₂)	C₂	semidirect
8. #17989423	C₂²	semidirect
10. C₂×#14556638	C₂²	semidirect
12. #17989402	C₂²	semidirect
13. #17989398	C₂²	semidirect
14. #17989398	C₂²	semidirect
15. #17989395	C₂²	semidirect
16. #17989390	C₂²	semidirect
17. #17989385	C₂²	semidirect
18. C₂×#14556621	C₂²	semidirect
19. C₂×#14556617	C₂²	semidirect
20. C₂×#14556611	C₂²	semidirect
21. C₂²×#11924016	C₂²	semidirect
22. #17989354	C₂²	semidirect
23. C₂×#14556598	C₂²	semidirect
26. #585×C₇₃	C₂²	semidirect
28. C₂×H(5,𝔽₂)×C₇₃	C₂²	semidirect
31. #14556638	C₂³	semidirect
32. #14556637	C₂³	semidirect
34. C₂×#11924028	C₂³	semidirect
37. C₂²×Dic₁₄₆	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×#11924016	C₂³	semidirect
48. C₂×#11924015	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
145. H(7,𝔽₂)	D₇₃	semidirect
151. C₇₃	#63092	semidirect
156. #585	C₂×D₇₃	semidirect
160. C₂×H(5,𝔽₂)	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22624744	2
#22624717	2
#22624711	2
#22624702	2
C₂×#17989398	2
#22624643	2
C₇₃×H(7,𝔽₂)	2
#63092	73