Subgroups — #33090274

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612582	C₂	1	2	yes	maximal
2	#22610803	C₂	1	2	yes	maximal
3	#22610670	C₂	1	2	yes	maximal
4	#22610609	C₂	1	4	yes	maximal
5	C₄×#14556523	C₂	1	1	yes	maximal, characteristic
6	#22609761	C₂	1	2	yes	maximal
7	#22609617	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556523	C₂²	1	2	yes	—
10	C₂×#14556515	C₂²	1	4	yes	—
11	C₂×#14556502	C₂²	1	1	yes	characteristic
12	C₂×#14556501	C₂²	1	2	yes	—
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	#17988494	C₂²	1	4	yes	—
15	#17988486	C₂²	1	4	yes	—
16	#17988473	—	2	4	no	—
17	#17988455	—	2	2	no	—
18	#17988453	—	2	2	no	—
19	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
20	#17988433	C₂²	1	1	yes	characteristic
21	#17988350	C₂²	1	2	yes	—
22	#17988349	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₃	C₄	1	8	yes	complemented
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#103×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×#10596831	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₇₃	—	2	4	no	—
31	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
34	#14556523	—	2	4	no	—
35	#14556515	—	2	8	no	—
36	#14556502	—	2	2	no	—
37	#14556501	—	2	4	no	—
38	C₈×Dic₇₃	—	2	4	no	—
39	#14556498	—	2	4	no	—
40	#14556498	—	2	4	no	—
41	#14556498	—	2	4	no	—
42	#14556498	—	2	2	no	—
43	#14556498	D₄	1	2	yes	—
44	#14556498	D₄	1	2	yes	—
45	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₇₃	—	2	4	no	—
47	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₇₃	—	2	4	no	—
51	C₄×Q₈×C₇₃	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₇₃	—	2	2	no	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	#107×C₇₃	C₂×C₄	1	4	yes	—
57	#103×C₇₃	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
59	C₂²×Dic₇₃	—	2	4	no	—
60	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	#41	1	4	yes	—
64	C₄⋊Dic₇₃	—	2	4	no	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	8	4	no	—
67	C₄×Dic₇₃	—	2	2	no	—
68	C₄×Dic₇₃	—	2	2	no	—
69	C₄×Dic₇₃	—	8	2	no	—
70	C₂×Q₈×C₇₃	—	2	2	no	—
71	C₂×Q₈×C₇₃	—	2	4	no	—
72	C₂×Q₈×C₇₃	—	2	4	no	—
73	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₇₃	—	2	2	no	—
75	C₂²×C₄×C₇₃	—	2	2	no	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	Q₁₆×C₇₃	—	2	8	no	—
80	C₂×C₈×C₇₃	—	2	2	no	—
81	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₇₃	#41	1	4	yes	—
84	C₄⋊C₄×C₇₃	—	2	8	no	—
85	C₄⋊C₄×C₇₃	—	2	2	no	—
86	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₇₃	—	2	8	no	—
88	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₇₃	#41	1	2	yes	—
90	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₇₃	—	2	4	no	—
92	Q₈×C₇₃	—	2	4	no	—
93	Q₈×C₇₃	—	4	4	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	4	4	no	—
99	C₂×C₄×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	—	4	2	no	—
101	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
105	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
107	C₂×Dic₇₃	—	2	4	no	—
108	C₂×Dic₇₃	—	2	4	no	—
109	C₂×Dic₇₃	—	2	4	no	—
110	C₂×Dic₇₃	—	4	4	no	—
111	C₂×Dic₇₃	—	4	4	no	—
112	C₂×Dic₇₃	—	4	4	no	—
113	C₈×C₇₃	—	2	2	no	—
114	C₈×C₇₃	#117	1	4	yes	—
115	#10596831	—	2	4	no	—
116	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂²×C₇₃	#532	1	2	yes	—
118	C₂²×C₇₃	#577	1	2	yes	—
119	C₂²×C₇₃	#578	1	2	yes	—
120	C₄×C₇₃	—	2	2	no	—
121	C₄×C₇₃	—	2	2	no	—
122	C₄×C₇₃	—	4	4	no	—
123	C₄×C₇₃	—	4	4	no	—
124	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
125	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
126	C₄×C₇₃	#529	1	2	yes	—
127	Dic₇₃	—	4	4	no	—
128	Dic₇₃	—	8	4	no	—
129	#21283	—	73	1	no	maximal, Sylow
130	C₂×C₇₃	#2293	1	1	yes	characteristic
131	C₂×C₇₃	#2950	1	1	yes	characteristic
132	C₂×C₇₃	#2952	1	1	yes	characteristic
133	C₂×C₇₃	#3178	1	4	yes	—
134	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
135	#2320	—	73	2	no	—
136	#1971	—	73	2	no	—
137	#1943	—	73	2	no	—
138	#1933	—	73	4	no	—
139	C₄×#106	—	73	1	no	—
140	#1763	—	73	2	no	—
141	#1726	—	73	1	no	—
142	C₇₃	#21283	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
145	C₄×Q₁₆	Dic₇₃	1	8	yes	complemented
146	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
147	C₂×#106	—	73	2	no	—
148	C₂×#103	—	73	4	no	—
149	C₂×#103	C₂×D₇₃	1	2	yes	—
150	C₂×#97	—	73	2	no	—
151	C₂×#97	—	73	1	no	—
152	C₂×C₄×C₈	—	73	2	no	—
153	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
154	#391	—	73	4	no	—
155	#389	—	73	4	no	—
156	#387	—	146	4	no	—
157	#383	—	146	2	no	—
158	#382	—	146	2	no	—
159	C₄×C₄⋊C₄	—	73	2	no	—
160	#376	—	73	1	no	—
161	#340	—	73	2	no	—
162	#340	—	73	4	no	—
163	C₂²×Q₈	C₂²×D₇₃	1	2	yes	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	C₂×Dic₇₃	1	4	yes	—
166	C₂²×C₈	—	73	2	no	—
167	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
169	C₄×Q₈	—	2	4	no	—
170	C₄×Q₈	C₂×Dic₇₃	1	8	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	73	4	no	—
173	C₂×C₄⋊C₄	—	146	4	no	—
174	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	73	2	no	—
177	C₂×C₄²	—	146	2	no	—
178	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
179	#107	C₂×Dic₇₃	1	4	yes	—
180	#106	—	146	4	no	—
181	#103	—	146	8	no	—
182	#103	C₂×Dic₇₃	1	8	yes	—
183	#97	—	146	4	no	—
184	#97	—	146	2	no	—
185	C₄×C₈	—	146	4	no	—
186	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
187	#95	—	73	2	no	—
188	#95	—	73	2	no	—
189	#95	—	146	4	no	—
190	#95	—	146	4	no	—
191	#95	—	146	4	no	—
192	#95	—	146	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	C₂²×Dic₇₃	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	73	4	no	—
200	C₂²×C₄	—	146	4	no	—
201	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
204	Q₁₆	—	2	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	73	4	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	—	146	2	no	—
209	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
210	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
211	C₂×C₈	#11924028	1	4	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	584	4	no	—
215	C₄⋊C₄	—	146	4	no	—
216	C₄⋊C₄	—	146	4	no	—
217	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
218	C₄²	—	2	8	no	—
219	C₄²	—	584	2	no	—
220	C₄²	—	146	2	no	—
221	C₄²	—	146	2	no	—
222	C₄²	C₂²×Dic₇₃	1	2	yes	—
223	C₄²	#11924027	1	2	yes	—
224	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	292	4	no	—
236	C₂×C₄	—	292	4	no	—
237	C₂×C₄	—	292	4	no	—
238	C₂×C₄	—	146	4	no	—
239	C₂×C₄	—	146	4	no	—
240	C₂×C₄	—	146	4	no	—
241	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
242	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
243	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
244	C₂×C₄	#41×D₇₃	1	2	yes	—
245	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
246	C₂×C₄	C₂×#11924028	1	2	yes	—
247	C₈	—	2	2	no	—
248	C₈	—	146	4	no	—
249	C₈	#14556550	1	4	yes	—
250	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
251	C₂²	#17989134	1	2	yes	—
252	C₂²	#17989394	1	2	yes	—
253	C₂²	#17989400	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₄	—	292	4	no	—
259	C₄	—	584	4	no	—
260	C₄	C₂×#14556550	1	1	yes	characteristic
261	C₄	#41×Dic₇₃	1	1	yes	characteristic
262	C₄	#17989119	1	2	yes	—
263	C₂	#22612396	1	1	yes	characteristic
264	C₂	#22618799	1	1	yes	characteristic
265	C₂	#22618815	1	1	yes	characteristic
266	C₂	#22620603	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	#21283
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₇₃	C₄	semidirect
142. C₇₃	#21283	semidirect
145. C₄×Q₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612582	2
#22610803	2
#22610670	2
#22610609	2
C₄×#14556523	2
#22609761	2
#22609617	2
C₂×C₄×Q₁₆×C₇₃	2
#21283	73