Subgroups — C₇₃×#10618 #32920022

Order 18688.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#1984	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1803	C₂	1	2	yes	maximal
4	C₇₃×#1296	C₂	1	2	yes	maximal
5	C₇₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×#107×C₇₃	—	2	1	no	—
8	C₂×#107×C₇₃	—	2	1	no	—
9	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
11	C₂×#103×C₇₃	—	2	2	no	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	#384×C₇₃	C₂²	1	2	yes	—
14	#340×C₇₃	—	2	2	no	—
15	#336×C₇₃	—	2	1	no	—
16	#336×C₇₃	—	2	1	no	—
17	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₃	—	2	2	no	—
19	C₂²×C₈×C₇₃	—	2	1	no	—
20	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	—	2	2	no	—
23	C₄×Q₈×C₇₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
25	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	—	2	1	no	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	#107×C₇₃	—	4	2	no	—
32	#107×C₇₃	—	4	2	no	—
33	C₄⋊C₈×C₇₃	—	2	2	no	—
34	C₄⋊C₈×C₇₃	—	2	2	no	—
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	#103×C₇₃	—	4	4	no	—
37	C₄×C₈×C₇₃	—	2	2	no	—
38	C₂×Q₈×C₇₃	—	2	1	no	—
39	C₂×Q₈×C₇₃	—	2	2	no	—
40	C₂×Q₈×C₇₃	—	2	2	no	—
41	C₂×Q₈×C₇₃	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₇₃	—	2	1	no	—
43	C₂²×C₄×C₇₃	—	2	1	no	—
44	C₂²×C₄×C₇₃	—	4	2	no	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	—	4	1	no	—
54	C₂×C₈×C₇₃	—	4	2	no	—
55	C₂×C₈×C₇₃	—	4	2	no	—
56	C₂×C₈×C₇₃	—	4	1	no	—
57	C₂×C₈×C₇₃	—	4	1	no	—
58	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₄⋊C₄×C₇₃	—	4	2	no	—
62	C₄⋊C₄×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₄⋊C₄×C₇₃	—	8	2	no	—
67	C₄²×C₇₃	—	4	1	no	—
68	C₄²×C₇₃	—	4	1	no	—
69	C₄²×C₇₃	#41	1	2	yes	—
70	C₄²×C₇₃	#41	1	2	yes	—
71	C₂³×C₇₃	#121	1	1	yes	characteristic
72	Q₈×C₇₃	—	2	4	no	—
73	Q₈×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	—	2	1	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	4	1	no	—
78	C₂×C₄×C₇₃	—	4	1	no	—
79	C₂×C₄×C₇₃	—	4	1	no	—
80	C₂×C₄×C₇₃	—	4	1	no	—
81	C₂×C₄×C₇₃	—	4	1	no	—
82	C₂×C₄×C₇₃	—	8	2	no	—
83	C₂×C₄×C₇₃	—	8	2	no	—
84	C₂×C₄×C₇₃	—	8	2	no	—
85	C₂×C₄×C₇₃	#123	1	2	yes	—
86	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#126	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
93	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
96	C₈×C₇₃	—	2	2	no	—
97	C₈×C₇₃	—	2	2	no	—
98	C₈×C₇₃	—	8	2	no	—
99	C₈×C₇₃	—	8	1	no	—
100	C₂²×C₇₃	#397	1	1	yes	characteristic
101	C₂²×C₇₃	#460	1	1	yes	characteristic
102	C₂²×C₇₃	#462	1	1	yes	characteristic
103	C₂²×C₇₃	#467	1	1	yes	characteristic
104	C₂²×C₇₃	#469	1	1	yes	characteristic
105	C₂²×C₇₃	#472	1	1	yes	characteristic
106	C₂²×C₇₃	#473	1	1	yes	characteristic
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	4	2	no	—
109	C₄×C₇₃	—	8	1	no	—
110	C₄×C₇₃	—	16	2	no	—
111	C₄×C₇₃	#482	1	2	yes	—
112	C₄×C₇₃	#483	1	1	yes	characteristic
113	C₄×C₇₃	#484	1	1	yes	characteristic
114	C₄×C₇₃	#484	1	2	yes	—
115	C₄×C₇₃	#487	1	1	yes	characteristic
116	C₄×C₇₃	#489	1	1	yes	characteristic
117	#10618	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₇₃	#1670	1	1	yes	characteristic
119	C₂×C₇₃	#1676	1	1	yes	characteristic
120	C₂×C₇₃	#1688	1	1	yes	characteristic
121	C₂×C₇₃	#1694	1	1	yes	characteristic
122	C₂×C₇₃	#2072	1	1	yes	characteristic
123	C₂×C₇₃	#2076	1	1	yes	characteristic
124	C₂×C₇₃	#2078	1	1	yes	characteristic
125	#1984	C₂×C₇₃	1	1	yes	characteristic
126	#1845	C₂×C₇₃	1	1	yes	characteristic
127	#1803	C₂×C₇₃	1	2	yes	—
128	#1296	C₂×C₇₃	1	2	yes	—
129	#1277	C₂×C₇₃	1	1	yes	characteristic
130	C₇₃	#10618	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄×Q₈	C₂²×C₇₃	1	1	yes	characteristic
132	C₂×#107	—	2	1	no	—
133	C₂×#107	—	2	1	no	—
134	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
136	C₂×#103	—	2	2	no	—
137	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
138	#384	C₂²×C₇₃	1	2	yes	—
139	#340	—	2	2	no	—
140	#336	—	2	1	no	—
141	#336	—	2	1	no	—
142	C₂²×Q₈	D₄×C₇₃	1	1	yes	characteristic
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	D₄×C₇₃	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
156	#107	—	4	2	no	—
157	#107	—	4	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	#103	—	4	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	Q₁₆×C₇₃	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
172	C₂²×C₄	#41×C₇₃	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	SD₁₆×C₇₃	1	2	yes	—
184	C₂×C₈	Q₁₆×C₇₃	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	8	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#41×C₇₃	1	2	yes	—
195	C₄²	#41×C₇₃	1	2	yes	—
196	C₂³	#121×C₇₃	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	#123×C₇₃	1	2	yes	—
211	C₂×C₄	#123×C₇₃	1	1	yes	characteristic
212	C₂×C₄	#124×C₇₃	1	1	yes	characteristic
213	C₂×C₄	#126×C₇₃	1	2	yes	—
214	C₂×C₄	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
215	C₂×C₄	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
216	C₂×C₄	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
217	C₂×C₄	#135×C₇₃	1	1	yes	characteristic
218	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
219	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
220	C₂×C₄	#137×C₇₃	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	8	1	no	—
225	C₂²	#397×C₇₃	1	1	yes	characteristic
226	C₂²	#460×C₇₃	1	1	yes	characteristic
227	C₂²	#462×C₇₃	1	1	yes	characteristic
228	C₂²	#467×C₇₃	1	1	yes	characteristic
229	C₂²	#469×C₇₃	1	1	yes	characteristic
230	C₂²	#472×C₇₃	1	1	yes	characteristic
231	C₂²	#473×C₇₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	16	2	no	—
236	C₄	#482×C₇₃	1	2	yes	—
237	C₄	#483×C₇₃	1	1	yes	characteristic
238	C₄	#484×C₇₃	1	1	yes	characteristic
239	C₄	#484×C₇₃	1	2	yes	—
240	C₄	#487×C₇₃	1	1	yes	characteristic
241	C₄	#489×C₇₃	1	1	yes	characteristic
242	C₂	C₇₃×#1670	1	1	yes	characteristic
243	C₂	C₇₃×#1676	1	1	yes	characteristic
244	C₂	C₇₃×#1688	1	1	yes	characteristic
245	C₂	C₇₃×#1694	1	1	yes	characteristic
246	C₂	C₇₃×#2072	1	1	yes	characteristic
247	C₂	C₇₃×#2076	1	1	yes	characteristic
248	C₂	C₇₃×#2078	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
117. #10618	C₇₃	direct (N × H)
130. C₇₃	#10618	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₃×#1984	2
C₇₃×#1845	2
C₇₃×#1803	2
C₇₃×#1296	2
C₇₃×#1277	2
#10618	73

Subgroups — C73×#10618 #32920022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#10618 #32920022