Subgroups — C₇₃×#20010 #32929414

Order 18688.

566 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3395	C₂	1	2	yes	maximal
2	C₇₃×#3383	C₂	1	2	yes	maximal
3	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1674	C₂	1	2	yes	maximal
5	C₇₃×#1670	C₂	1	2	yes	maximal
6	C₇₃×#1648	C₂	1	4	yes	maximal
7	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#557×C₇₃	C₂²	1	4	yes	—
10	#500×C₇₃	C₂²	1	2	yes	—
11	#500×C₇₃	C₂²	1	2	yes	—
12	#499×C₇₃	—	2	2	no	—
13	#494×C₇₃	—	2	2	no	—
14	#494×C₇₃	C₂²	1	2	yes	—
15	#479×C₇₃	—	2	4	no	—
16	#475×C₇₃	C₂²	1	4	yes	—
17	#462×C₇₃	—	2	4	no	—
18	#462×C₇₃	C₂²	1	4	yes	—
19	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
20	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₇₃	—	2	2	no	—
22	C₄×Q₁₆×C₇₃	—	2	2	no	—
23	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#335×C₇₃	C₂²	1	1	yes	characteristic
26	#335×C₇₃	C₂²	1	1	yes	characteristic
27	#332×C₇₃	C₂²	1	4	yes	—
28	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
29	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
30	C₈²×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₇₃	—	2	4	no	—
32	C₂×Q₁₆×C₇₃	—	2	2	no	—
33	C₂×Q₁₆×C₇₃	—	2	2	no	—
34	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₇₃	—	2	4	no	—
37	C₄⋊Q₈×C₇₃	C₂³	1	4	yes	—
38	#125×C₇₃	—	2	4	no	—
39	#125×C₇₃	—	4	4	no	—
40	C₄×Q₈×C₇₃	—	2	4	no	—
41	C₄×Q₈×C₇₃	—	2	4	no	—
42	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
43	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
44	#107×C₇₃	—	2	2	no	—
45	#107×C₇₃	—	2	2	no	—
46	#107×C₇₃	—	4	2	no	—
47	#107×C₇₃	D₄	1	1	yes	characteristic
48	#107×C₇₃	D₄	1	1	yes	characteristic
49	#106×C₇₃	—	2	4	no	—
50	C₄⋊C₈×C₇₃	—	2	2	no	—
51	C₄⋊C₈×C₇₃	—	2	2	no	—
52	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
53	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
54	#103×C₇₃	—	2	4	no	—
55	#103×C₇₃	—	2	4	no	—
56	#103×C₇₃	D₄	1	2	yes	—
57	#103×C₇₃	D₄	1	2	yes	—
58	C₄×C₈×C₇₃	—	2	2	no	—
59	C₄×C₈×C₇₃	—	2	2	no	—
60	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₇₃	—	2	4	no	—
64	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
65	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
66	Q₁₆×C₇₃	—	2	2	no	—
67	Q₁₆×C₇₃	—	4	4	no	—
68	Q₁₆×C₇₃	—	8	2	no	—
69	Q₁₆×C₇₃	—	8	4	no	—
70	Q₁₆×C₇₃	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₇₃	—	2	1	no	—
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₇₃	—	2	2	no	—
79	C₄⋊C₄×C₇₃	—	2	2	no	—
80	C₄⋊C₄×C₇₃	—	2	4	no	—
81	C₄⋊C₄×C₇₃	—	2	4	no	—
82	C₄⋊C₄×C₇₃	—	4	4	no	—
83	C₄⋊C₄×C₇₃	—	4	4	no	—
84	C₄⋊C₄×C₇₃	—	4	4	no	—
85	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
87	C₄²×C₇₃	—	2	2	no	—
88	C₄²×C₇₃	—	2	2	no	—
89	C₄²×C₇₃	—	4	4	no	—
90	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
91	Q₈×C₇₃	—	2	2	no	—
92	Q₈×C₇₃	—	2	2	no	—
93	Q₈×C₇₃	—	2	2	no	—
94	Q₈×C₇₃	—	4	4	no	—
95	Q₈×C₇₃	—	4	4	no	—
96	Q₈×C₇₃	C₂×Q₁₆	1	4	yes	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	4	4	no	—
102	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
105	C₈×C₇₃	—	2	2	no	—
106	C₈×C₇₃	—	2	1	no	—
107	C₈×C₇₃	—	2	1	no	—
108	C₈×C₇₃	—	4	2	no	—
109	C₈×C₇₃	—	4	2	no	—
110	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
111	C₈×C₇₃	#135	1	2	yes	—
112	C₂²×C₇₃	D₄²	1	1	yes	characteristic
113	C₄×C₇₃	—	2	4	no	—
114	C₄×C₇₃	—	2	1	no	—
115	C₄×C₇₃	—	4	2	no	—
116	C₄×C₇₃	—	4	2	no	—
117	C₄×C₇₃	—	4	2	no	—
118	C₄×C₇₃	—	8	4	no	—
119	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
120	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
121	C₄×C₇₃	#576	1	1	yes	characteristic
122	C₄×C₇₃	#578	1	1	yes	characteristic
123	#20010	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
125	C₂×C₇₃	#3287	1	1	yes	characteristic
126	C₂×C₇₃	#3293	1	1	yes	characteristic
127	#3395	C₂×C₇₃	1	2	yes	—
128	#3383	C₂×C₇₃	1	2	yes	—
129	#1713	C₂×C₇₃	1	1	yes	characteristic
130	#1674	C₂×C₇₃	1	2	yes	—
131	#1670	C₂×C₇₃	1	2	yes	—
132	#1648	C₂×C₇₃	1	4	yes	—
133	C₈×Q₁₆	C₂×C₇₃	1	1	yes	characteristic
134	C₈×Q₁₆	C₂×C₇₃	1	1	yes	characteristic
135	C₇₃	#20010	1	1	yes	characteristic, direct product factor, Sylow
136	#557	C₂²×C₇₃	1	4	yes	—
137	#500	C₂²×C₇₃	1	2	yes	—
138	#500	C₂²×C₇₃	1	2	yes	—
139	#499	—	2	2	no	—
140	#494	—	2	2	no	—
141	#494	C₂²×C₇₃	1	2	yes	—
142	#479	—	2	4	no	—
143	#475	C₂²×C₇₃	1	4	yes	—
144	#462	—	2	4	no	—
145	#462	C₂²×C₇₃	1	4	yes	—
146	C₈×Q₈	C₂²×C₇₃	1	2	yes	—
147	C₈×Q₈	C₂²×C₇₃	1	2	yes	—
148	C₄×Q₁₆	—	2	2	no	—
149	C₄×Q₁₆	—	2	2	no	—
150	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
151	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
152	#335	C₂²×C₇₃	1	1	yes	characteristic
153	#335	C₂²×C₇₃	1	1	yes	characteristic
154	#332	C₂²×C₇₃	1	4	yes	—
155	Q₈⋊C₈	C₂²×C₇₃	1	2	yes	—
156	Q₈⋊C₈	C₂²×C₇₃	1	2	yes	—
157	C₈²	C₂²×C₇₃	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	4	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
162	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	4	no	—
164	C₄⋊Q₈	C₂³×C₇₃	1	4	yes	—
165	#125	—	2	4	no	—
166	#125	—	4	4	no	—
167	C₄×Q₈	—	2	4	no	—
168	C₄×Q₈	—	2	4	no	—
169	C₄×Q₈	C₂³×C₇₃	1	2	yes	—
170	C₄×Q₈	C₂³×C₇₃	1	2	yes	—
171	#107	—	2	2	no	—
172	#107	—	2	2	no	—
173	#107	—	4	2	no	—
174	#107	D₄×C₇₃	1	1	yes	characteristic
175	#107	D₄×C₇₃	1	1	yes	characteristic
176	#106	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂³×C₇₃	1	2	yes	—
180	C₄⋊C₈	C₂³×C₇₃	1	2	yes	—
181	#103	—	2	4	no	—
182	#103	—	2	4	no	—
183	#103	D₄×C₇₃	1	2	yes	—
184	#103	D₄×C₇₃	1	2	yes	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	C₂×D₄×C₇₃	1	2	yes	—
192	C₂×Q₈	C₂×D₄×C₇₃	1	2	yes	—
193	Q₁₆	—	2	2	no	—
194	Q₁₆	—	4	4	no	—
195	Q₁₆	—	8	2	no	—
196	Q₁₆	—	8	4	no	—
197	Q₁₆	Q₁₆×C₇₃	1	4	yes	complemented
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	C₂×D₄×C₇₃	1	2	yes	—
213	C₄⋊C₄	C₂×D₄×C₇₃	1	2	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂⁴×C₇₃	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	C₂×Q₁₆×C₇₃	1	4	yes	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
231	C₂×C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	—	4	2	no	—
237	C₈	C₂×Q₁₆×C₇₃	1	2	yes	—
238	C₈	#135×C₇₃	1	2	yes	—
239	C₂²	D₄²×C₇₃	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	8	4	no	—
246	C₄	C₂²×Q₁₆×C₇₃	1	1	yes	characteristic
247	C₄	C₂×#135×C₇₃	1	1	yes	characteristic
248	C₄	#576×C₇₃	1	1	yes	characteristic
249	C₄	#578×C₇₃	1	1	yes	characteristic
250	C₂	D₄×Q₁₆×C₇₃	1	1	yes	characteristic
251	C₂	C₇₃×#3287	1	1	yes	characteristic
252	C₂	C₇₃×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
70. Q₁₆×C₇₃	Q₁₆	semidirect
123. #20010	C₇₃	direct (N × H)
135. C₇₃	#20010	direct (N × H)
197. Q₁₆	Q₁₆×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3395	2
C₇₃×#3383	2
C₇₃×#1713	2
C₇₃×#1674	2
C₇₃×#1670	2
C₇₃×#1648	2
C₈×Q₁₆×C₇₃	2
#20010	73

Subgroups — C73×#20010 #32929414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#20010 #32929414