Subgroups — C₂×C₇₃×#1281 #32917321

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₇₃	C₂	1	2	yes	maximal
2	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1281	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₇₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
6	#431×C₇₃	—	2	2	no	—
7	#431×C₇₃	—	2	2	no	—
8	#431×C₇₃	C₄	1	4	yes	—
9	#431×C₇₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₇₃	C₄	1	4	yes	—
12	C₂×#97×C₇₃	—	2	2	no	—
13	C₂×#97×C₇₃	—	2	2	no	—
14	C₂×#97×C₇₃	C₄	1	2	yes	—
15	C₂×#97×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	—	2	2	no	—
17	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	—	2	2	no	—
20	C₂²×C₈×C₇₃	—	2	2	no	—
21	C₂²×C₈×C₇₃	—	2	4	no	—
22	C₂²×C₈×C₇₃	—	2	4	no	—
23	C₂²×C₈×C₇₃	C₈	1	8	yes	complemented
24	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	D₄	1	2	yes	—
28	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₃	—	2	4	no	—
35	C₄⋊C₈×C₇₃	—	2	4	no	—
36	C₂²⋊C₈×C₇₃	—	2	4	no	—
37	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
38	#97×C₇₃	—	2	4	no	—
39	#97×C₇₃	—	2	4	no	—
40	C₄×C₈×C₇₃	—	2	4	no	—
41	C₄×C₈×C₇₃	—	2	4	no	—
42	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	—	4	4	no	—
44	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	8	no	—
59	C₂×C₈×C₇₃	—	2	8	no	—
60	C₂×C₈×C₇₃	—	2	4	no	—
61	C₂×C₈×C₇₃	—	2	4	no	—
62	C₂×C₈×C₇₃	—	2	8	no	—
63	C₂×C₈×C₇₃	—	4	8	no	—
64	C₂×C₈×C₇₃	C₂×C₈	1	8	yes	complemented
65	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇₃	C₂×D₄	1	2	yes	—
73	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₂³×C₇₃	—	2	4	no	—
76	C₂³×C₇₃	#95	1	1	yes	characteristic
77	C₂³×C₇₃	#95	1	1	yes	characteristic
78	C₂³×C₇₃	C₄×C₈	1	2	yes	—
79	C₂³×C₇₃	#97	1	2	yes	—
80	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	2	4	no	—
83	C₂×C₄×C₇₃	—	4	2	no	—
84	C₂×C₄×C₇₃	—	4	2	no	—
85	C₂×C₄×C₇₃	—	4	2	no	—
86	C₂×C₄×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	#95	1	2	yes	—
88	C₂×C₄×C₇₃	#95	1	2	yes	—
89	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₇₃	#97	1	2	yes	—
93	C₂×C₄×C₇₃	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₇₃	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
96	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
103	C₈×C₇₃	—	2	8	no	—
104	C₈×C₇₃	—	4	8	no	—
105	C₂²×C₇₃	—	2	4	no	—
106	C₂²×C₇₃	—	2	4	no	—
107	C₂²×C₇₃	—	2	2	no	—
108	C₂²×C₇₃	—	2	2	no	—
109	C₂²×C₇₃	#336	1	2	yes	—
110	C₂²×C₇₃	#336	1	2	yes	—
111	C₂²×C₇₃	#344	1	4	yes	—
112	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
115	C₄×C₇₃	—	2	2	no	—
116	C₄×C₇₃	—	2	2	no	—
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	—	2	2	no	—
119	C₄×C₇₃	#336	1	2	yes	—
120	C₄×C₇₃	#336	1	2	yes	—
121	C₄×C₇₃	C₂×C₂²⋊C₈	1	2	yes	—
122	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
123	C₂×#1281	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₇₃	—	2	2	no	—
125	C₂×C₇₃	—	2	1	no	—
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	#1281	1	4	yes	direct product factor
128	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
129	C₂×C₇₃	C₂×#344	1	2	yes	—
130	C₂×#431	C₂×C₇₃	1	2	yes	—
131	C₂²×#97	C₂×C₇₃	1	1	yes	characteristic
132	#1281	C₂×C₇₃	1	4	yes	direct product factor
133	C₇₃	C₂×#1281	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³×C₈	C₄×C₇₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
136	#431	—	2	2	no	—
137	#431	—	2	2	no	—
138	#431	C₄×C₇₃	1	4	yes	—
139	#431	C₂²×C₇₃	1	4	yes	—
140	C₂×C₄⋊C₈	—	2	2	no	—
141	C₂×C₂²⋊C₈	C₄×C₇₃	1	4	yes	—
142	C₂×#97	—	2	2	no	—
143	C₂×#97	—	2	2	no	—
144	C₂×#97	C₄×C₇₃	1	2	yes	—
145	C₂×#97	C₂²×C₇₃	1	2	yes	—
146	C₂×C₄×C₈	—	2	2	no	—
147	C₂³×C₄	C₂×C₄×C₇₃	1	2	yes	—
148	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₈×C₇₃	1	8	yes	complemented
154	C₂²×C₈	C₂×C₄×C₇₃	1	4	yes	—
155	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
157	C₂×C₄²	D₄×C₇₃	1	2	yes	—
158	C₂×C₄²	D₄×C₇₃	1	2	yes	—
159	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
161	C₂×C₄²	Q₈×C₇₃	1	1	yes	characteristic
162	C₂×C₄²	Q₈×C₇₃	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	4	no	—
166	C₂²⋊C₈	—	2	4	no	—
167	C₂²⋊C₈	C₂×C₄×C₇₃	1	8	yes	—
168	#97	—	2	4	no	—
169	#97	—	2	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₂⁴	C₄²×C₇₃	1	1	yes	characteristic
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	C₄²×C₇₃	1	2	yes	—
175	C₂²×C₄	C₄²×C₇₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
181	C₂²×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊C₄×C₇₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₈×C₇₃	1	4	yes	—
185	C₂²×C₄	M₁₆×C₇₃	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	4	8	no	—
194	C₂×C₈	C₂×C₈×C₇₃	1	8	yes	complemented
195	C₄²	C₂²⋊C₄×C₇₃	1	2	yes	—
196	C₄²	C₂²⋊C₄×C₇₃	1	2	yes	—
197	C₄²	C₂²⋊C₄×C₇₃	1	2	yes	—
198	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
199	C₄²	C₄⋊C₄×C₇₃	1	1	yes	characteristic
200	C₄²	C₄⋊C₄×C₇₃	1	2	yes	—
201	C₄²	C₄⋊C₄×C₇₃	1	2	yes	—
202	C₄²	C₂×D₄×C₇₃	1	2	yes	—
203	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
204	C₄²	C₂×Q₈×C₇₃	1	1	yes	characteristic
205	C₂³	—	2	4	no	—
206	C₂³	#95×C₇₃	1	1	yes	characteristic
207	C₂³	#95×C₇₃	1	1	yes	characteristic
208	C₂³	C₄×C₈×C₇₃	1	2	yes	—
209	C₂³	#97×C₇₃	1	2	yes	—
210	C₂³	C₂×C₄²×C₇₃	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	#95×C₇₃	1	2	yes	—
218	C₂×C₄	#95×C₇₃	1	2	yes	—
219	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
220	C₂×C₄	#95×C₇₃	1	1	yes	characteristic
221	C₂×C₄	C₄×C₈×C₇₃	1	2	yes	—
222	C₂×C₄	#97×C₇₃	1	2	yes	—
223	C₂×C₄	C₂²⋊C₈×C₇₃	1	4	yes	—
224	C₂×C₄	C₂²⋊C₈×C₇₃	1	4	yes	—
225	C₂×C₄	C₄⋊C₈×C₇₃	1	4	yes	—
226	C₂×C₄	C₄⋊C₈×C₇₃	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₈×C₇₃	1	2	yes	—
232	C₂×C₄	C₂×M₁₆×C₇₃	1	2	yes	—
233	C₈	—	2	8	no	—
234	C₈	—	4	8	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#336×C₇₃	1	2	yes	—
240	C₂²	#336×C₇₃	1	2	yes	—
241	C₂²	#344×C₇₃	1	4	yes	—
242	C₂²	C₂×#95×C₇₃	1	1	yes	characteristic
243	C₂²	C₂×C₄×C₈×C₇₃	1	1	yes	characteristic
244	C₂²	C₂×#97×C₇₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	#336×C₇₃	1	2	yes	—
250	C₄	#336×C₇₃	1	2	yes	—
251	C₄	C₂×C₂²⋊C₈×C₇₃	1	2	yes	—
252	C₄	C₂×C₄⋊C₈×C₇₃	1	2	yes	—
253	C₂	—	2	1	no	—
254	C₂	—	2	1	no	—
255	C₂	—	2	2	no	—
256	C₂	C₇₃×#1281	1	4	yes	direct product factor
257	C₂	C₂×#336×C₇₃	1	1	yes	characteristic
258	C₂	C₂×#344×C₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇₃	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂³×C₇₃	#95
prime radical	C₂⁴×C₇₃	C₄²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₃×#1281	C₂	direct (N × H)
23. C₂²×C₈×C₇₃	C₈	semidirect
64. C₂×C₈×C₇₃	C₂×C₈	semidirect
123. C₂×#1281	C₇₃	direct (N × H)
127. C₂×C₇₃	#1281	direct (N × H)
132. #1281	C₂×C₇₃	direct (N × H)
133. C₇₃	C₂×#1281	direct (N × H)
153. C₂²×C₈	C₈×C₇₃	semidirect
194. C₂×C₈	C₂×C₈×C₇₃	semidirect
256. C₂	C₇₃×#1281	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431×C₇₃	2
C₂²×#97×C₇₃	2
C₇₃×#1281	2
C₂×#1281	73

Subgroups — C2×C73×#1281 #32917321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₃×#1281 #32917321