Subgroups — C₇₃×#15253 #32924657

Order 18688.

1338 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₇₃	C₂	1	2	yes	maximal
2	C₇₃×#1446	C₂	1	4	yes	maximal
3	C₇₃×#1445	C₂	1	4	yes	maximal
4	C₇₃×#1434	C₂	1	4	yes	maximal
5	C₄×#95×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
10	#391×C₇₃	—	2	8	no	—
11	C₂×#95×C₇₃	C₂²	1	4	yes	—
12	C₂×#95×C₇₃	C₂²	1	4	yes	—
13	C₂×#95×C₇₃	C₂²	1	8	yes	—
14	C₂×#95×C₇₃	C₂²	1	4	yes	—
15	C₂×#95×C₇₃	C₂²	1	4	yes	—
16	C₂×#95×C₇₃	C₂²	1	4	yes	—
17	C₂²×Q₈×C₇₃	—	2	4	no	—
18	C₂²×Q₈×C₇₃	—	2	8	no	—
19	C₂²×Q₈×C₇₃	—	2	8	no	—
20	C₂²×Q₈×C₇₃	—	4	4	no	—
21	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
22	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
26	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
28	C₂×C₄²×C₇₃	—	4	2	no	—
29	C₂×C₄²×C₇₃	—	4	2	no	—
30	C₂×C₄²×C₇₃	—	4	2	no	—
31	C₂×C₄²×C₇₃	D₄	1	4	yes	—
32	C₂×C₄²×C₇₃	Q₈	1	8	yes	complemented
33	#95×C₇₃	—	2	8	no	—
34	#95×C₇₃	—	2	8	no	—
35	#95×C₇₃	—	2	8	no	—
36	#95×C₇₃	—	2	16	no	—
37	#95×C₇₃	—	4	4	no	—
38	#95×C₇₃	—	4	4	no	—
39	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₇₃	—	2	16	no	—
41	C₂×Q₈×C₇₃	—	4	32	no	—
42	C₂×Q₈×C₇₃	—	4	8	no	—
43	C₂×Q₈×C₇₃	—	4	8	no	—
44	C₂²×C₄×C₇₃	—	2	4	no	—
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	—	2	4	no	—
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	—	2	4	no	—
50	C₂²×C₄×C₇₃	—	2	8	no	—
51	C₂²×C₄×C₇₃	—	2	8	no	—
52	C₂²×C₄×C₇₃	—	2	4	no	—
53	C₂²×C₄×C₇₃	—	2	8	no	—
54	C₂²×C₄×C₇₃	—	2	4	no	—
55	C₂²×C₄×C₇₃	—	2	4	no	—
56	C₂²×C₄×C₇₃	—	2	8	no	—
57	C₂²×C₄×C₇₃	—	2	4	no	—
58	C₂²×C₄×C₇₃	—	2	8	no	—
59	C₂²×C₄×C₇₃	—	2	8	no	—
60	C₂²×C₄×C₇₃	—	2	4	no	—
61	C₂²×C₄×C₇₃	—	2	4	no	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂×Q₈	1	8	yes	—
65	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₃	#41	1	8	yes	—
67	C₂²×C₄×C₇₃	#41	1	8	yes	—
68	C₄⋊C₄×C₇₃	—	4	8	no	—
69	C₄²×C₇₃	—	4	4	no	—
70	C₄²×C₇₃	—	4	8	no	—
71	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
72	C₂³×C₇₃	C₂²×Q₈	1	2	yes	—
73	C₂³×C₇₃	C₂×#41	1	4	yes	—
74	C₂³×C₇₃	C₂×#41	1	4	yes	—
75	C₂³×C₇₃	H(5,𝔽₂)	1	2	yes	—
76	C₂³×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂³×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
78	Q₈×C₇₃	—	4	32	no	—
79	C₂×C₄×C₇₃	—	2	4	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	2	4	no	—
84	C₂×C₄×C₇₃	—	2	4	no	—
85	C₂×C₄×C₇₃	—	2	16	no	—
86	C₂×C₄×C₇₃	—	2	8	no	—
87	C₂×C₄×C₇₃	—	2	8	no	—
88	C₂×C₄×C₇₃	—	2	8	no	—
89	C₂×C₄×C₇₃	—	2	8	no	—
90	C₂×C₄×C₇₃	—	2	16	no	—
91	C₂×C₄×C₇₃	—	4	4	no	—
92	C₂×C₄×C₇₃	—	4	4	no	—
93	C₂×C₄×C₇₃	—	4	8	no	—
94	C₂×C₄×C₇₃	—	4	8	no	—
95	C₂×C₄×C₇₃	—	4	8	no	—
96	C₂×C₄×C₇₃	—	4	4	no	—
97	C₂×C₄×C₇₃	—	4	4	no	—
98	C₂×C₄×C₇₃	—	4	4	no	—
99	C₂×C₄×C₇₃	—	4	8	no	—
100	C₂×C₄×C₇₃	—	4	8	no	—
101	C₂×C₄×C₇₃	#124	1	8	yes	—
102	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
103	C₂²×C₇₃	#525	1	2	yes	—
104	C₂²×C₇₃	C₂×#124	1	2	yes	—
105	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₂²×C₇₃	#533	1	4	yes	—
107	C₂²×C₇₃	#533	1	4	yes	—
108	C₂²×C₇₃	#534	1	1	yes	characteristic
109	C₂²×C₇₃	#537	1	1	yes	characteristic
110	C₂²×C₇₃	#543	1	2	yes	—
111	C₂²×C₇₃	#544	1	2	yes	—
112	C₂²×C₇₃	#551	1	8	yes	—
113	C₂²×C₇₃	#556	1	4	yes	—
114	C₂²×C₇₃	#559	1	4	yes	—
115	C₄×C₇₃	—	2	8	no	—
116	C₄×C₇₃	—	2	4	no	—
117	C₄×C₇₃	—	4	8	no	—
118	C₄×C₇₃	—	4	8	no	—
119	C₄×C₇₃	—	8	4	no	—
120	#15253	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₇₃	#2399	1	1	yes	characteristic
122	C₂×C₇₃	#2561	1	2	yes	—
123	C₂×C₇₃	#2592	1	2	yes	—
124	C₂×C₇₃	#2605	1	4	yes	—
125	C₂×C₇₃	#2607	1	4	yes	—
126	C₂×C₇₃	#2659	1	1	yes	characteristic
127	C₂×C₇₃	#2862	1	1	yes	characteristic
128	C₂×#391	C₂×C₇₃	1	2	yes	—
129	#1446	C₂×C₇₃	1	4	yes	—
130	#1445	C₂×C₇₃	1	4	yes	—
131	#1434	C₂×C₇₃	1	4	yes	—
132	C₄×#95	C₂×C₇₃	1	1	yes	characteristic
133	C₇₃	#15253	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³×Q₈	C₂²×C₇₃	1	2	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
136	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
137	C₂²×C₄²	C₂²×C₇₃	1	2	yes	—
138	#391	—	2	8	no	—
139	C₂×#95	C₂²×C₇₃	1	4	yes	—
140	C₂×#95	C₂²×C₇₃	1	4	yes	—
141	C₂×#95	C₂²×C₇₃	1	8	yes	—
142	C₂×#95	C₂²×C₇₃	1	4	yes	—
143	C₂×#95	C₂²×C₇₃	1	4	yes	—
144	C₂×#95	C₂²×C₇₃	1	4	yes	—
145	C₂²×Q₈	—	2	4	no	—
146	C₂²×Q₈	—	2	8	no	—
147	C₂²×Q₈	—	2	8	no	—
148	C₂²×Q₈	—	4	4	no	—
149	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
150	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
151	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
152	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
153	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄²	—	4	2	no	—
157	C₂×C₄²	—	4	2	no	—
158	C₂×C₄²	—	4	2	no	—
159	C₂×C₄²	D₄×C₇₃	1	4	yes	—
160	C₂×C₄²	Q₈×C₇₃	1	8	yes	complemented
161	#95	—	2	8	no	—
162	#95	—	2	8	no	—
163	#95	—	2	8	no	—
164	#95	—	2	16	no	—
165	#95	—	4	4	no	—
166	#95	—	4	4	no	—
167	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
168	C₂×Q₈	—	2	16	no	—
169	C₂×Q₈	—	4	32	no	—
170	C₂×Q₈	—	4	8	no	—
171	C₂×Q₈	—	4	8	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
191	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
192	C₂²×C₄	C₂×Q₈×C₇₃	1	8	yes	—
193	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
194	C₂²×C₄	#41×C₇₃	1	8	yes	—
195	C₂²×C₄	#41×C₇₃	1	8	yes	—
196	C₄⋊C₄	—	4	8	no	—
197	C₄²	—	4	4	no	—
198	C₄²	—	4	8	no	—
199	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
200	C₂³	C₂²×Q₈×C₇₃	1	2	yes	—
201	C₂³	C₂×#41×C₇₃	1	4	yes	—
202	C₂³	C₂×#41×C₇₃	1	4	yes	—
203	C₂³	H(5,𝔽₂)×C₇₃	1	2	yes	—
204	C₂³	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
205	C₂³	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
206	Q₈	—	4	32	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	#124×C₇₃	1	8	yes	—
230	C₂×C₄	C₄⋊Q₈×C₇₃	1	4	yes	—
231	C₂²	#525×C₇₃	1	2	yes	—
232	C₂²	C₂×#124×C₇₃	1	2	yes	—
233	C₂²	C₂×C₄⋊Q₈×C₇₃	1	1	yes	characteristic
234	C₂²	#533×C₇₃	1	4	yes	—
235	C₂²	#533×C₇₃	1	4	yes	—
236	C₂²	#534×C₇₃	1	1	yes	characteristic
237	C₂²	#537×C₇₃	1	1	yes	characteristic
238	C₂²	#543×C₇₃	1	2	yes	—
239	C₂²	#544×C₇₃	1	2	yes	—
240	C₂²	#551×C₇₃	1	8	yes	—
241	C₂²	#556×C₇₃	1	4	yes	—
242	C₂²	#559×C₇₃	1	4	yes	—
243	C₄	—	2	8	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	—	4	8	no	—
247	C₄	—	8	4	no	—
248	C₂	C₇₃×#2399	1	1	yes	characteristic
249	C₂	C₇₃×#2561	1	2	yes	—
250	C₂	C₇₃×#2592	1	2	yes	—
251	C₂	C₇₃×#2605	1	4	yes	—
252	C₂	C₇₃×#2607	1	4	yes	—
253	C₂	C₇₃×#2659	1	1	yes	characteristic
254	C₂	C₇₃×#2862	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
32. C₂×C₄²×C₇₃	Q₈	semidirect
120. #15253	C₇₃	direct (N × H)
133. C₇₃	#15253	direct (N × H)
160. C₂×C₄²	Q₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#391×C₇₃	2
C₇₃×#1446	2
C₇₃×#1445	2
C₇₃×#1434	2
C₄×#95×C₇₃	2
#15253	73

Subgroups — C73×#15253 #32924657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#15253 #32924657