Subgroups — C₇₃×#14347 #32923751

Order 18688.

1782 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#380×C₇₃	C₂	1	2	yes	maximal
2	C₂²×#95×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1442	C₂	1	8	yes	maximal, complemented
4	C₇₃×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₇₃	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	4	yes	—
9	#380×C₇₃	—	2	8	no	—
10	C₂×#95×C₇₃	C₄	1	8	yes	complemented
11	C₂×#95×C₇₃	C₂²	1	4	yes	—
12	C₂×#95×C₇₃	C₂²	1	8	yes	—
13	C₂×#95×C₇₃	C₂²	1	8	yes	—
14	C₂×#95×C₇₃	C₂²	1	4	yes	—
15	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₇₃	—	2	8	no	—
17	C₂³×C₄×C₇₃	—	4	4	no	—
18	C₂³×C₄×C₇₃	—	4	4	no	—
19	C₂³×C₄×C₇₃	—	4	4	no	—
20	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₇₃	D₄	1	4	yes	—
23	C₂³×C₄×C₇₃	Q₈	1	4	yes	—
24	C₂³×C₄×C₇₃	C₂³	1	8	yes	—
25	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
28	C₂×C₄⋊C₄×C₇₃	—	4	8	no	—
29	C₂×C₂²⋊C₄×C₇₃	—	2	8	no	—
30	C₂×C₂²⋊C₄×C₇₃	—	2	16	no	—
31	#95×C₇₃	—	2	8	no	—
32	#95×C₇₃	—	4	8	no	—
33	#95×C₇₃	—	4	8	no	—
34	#95×C₇₃	C₂×C₄	1	16	yes	—
35	C₂⁴×C₇₃	—	2	4	no	—
36	C₂⁴×C₇₃	—	2	2	no	—
37	C₂⁴×C₇₃	—	2	2	no	—
38	C₂⁴×C₇₃	C₂²×C₄	1	2	yes	—
39	C₂⁴×C₇₃	C₂×D₄	1	2	yes	—
40	C₂⁴×C₇₃	C₂×Q₈	1	2	yes	—
41	C₂⁴×C₇₃	#41	1	8	yes	—
42	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	—	2	8	no	—
44	C₂²×C₄×C₇₃	—	2	16	no	—
45	C₂²×C₄×C₇₃	—	2	8	no	—
46	C₂²×C₄×C₇₃	—	2	8	no	—
47	C₂²×C₄×C₇₃	—	2	8	no	—
48	C₂²×C₄×C₇₃	—	2	8	no	—
49	C₂²×C₄×C₇₃	—	2	16	no	—
50	C₂²×C₄×C₇₃	—	2	8	no	—
51	C₂²×C₄×C₇₃	—	2	16	no	—
52	C₂²×C₄×C₇₃	—	2	8	no	—
53	C₂²×C₄×C₇₃	—	4	8	no	—
54	C₂²×C₄×C₇₃	—	4	8	no	—
55	C₂²×C₄×C₇₃	—	4	8	no	—
56	C₂²×C₄×C₇₃	—	4	8	no	—
57	C₂²×C₄×C₇₃	—	4	8	no	—
58	C₂²×C₄×C₇₃	—	4	32	no	—
59	C₂²×C₄×C₇₃	C₄⋊C₄	1	16	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
64	C₄⋊C₄×C₇₃	—	4	16	no	—
65	C₂²⋊C₄×C₇₃	—	4	16	no	—
66	C₂³×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	4	no	—
68	C₂³×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	—	2	4	no	—
70	C₂³×C₇₃	—	2	4	no	—
71	C₂³×C₇₃	—	2	16	no	—
72	C₂³×C₇₃	—	2	16	no	—
73	C₂³×C₇₃	—	2	4	no	—
74	C₂³×C₇₃	—	2	4	no	—
75	C₂³×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂³×C₇₃	#117	1	8	yes	—
77	C₂³×C₇₃	#124	1	8	yes	—
78	C₂³×C₇₃	#125	1	8	yes	—
79	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
80	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
81	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂³×C₇₃	C₂×#41	1	4	yes	—
83	C₂³×C₇₃	H(5,𝔽₂)	1	4	yes	—
84	C₂³×C₇₃	H(5,𝔽₂)	1	2	yes	—
85	C₂³×C₇₃	2₋¹⁺⁴	1	2	yes	—
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	8	no	—
91	C₂×C₄×C₇₃	—	2	8	no	—
92	C₂×C₄×C₇₃	—	4	16	no	—
93	C₂×C₄×C₇₃	—	4	32	no	—
94	C₂×C₄×C₇₃	—	4	16	no	—
95	C₂×C₄×C₇₃	—	4	8	no	—
96	C₂×C₄×C₇₃	—	4	8	no	—
97	C₂×C₄×C₇₃	—	4	8	no	—
98	C₂×C₄×C₇₃	—	4	16	no	—
99	C₂×C₄×C₇₃	—	4	16	no	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
101	C₂²×C₇₃	—	2	8	no	—
102	C₂²×C₇₃	—	2	8	no	—
103	C₂²×C₇₃	—	2	4	no	—
104	C₂²×C₇₃	—	2	4	no	—
105	C₂²×C₇₃	—	2	16	no	—
106	C₂²×C₇₃	—	2	16	no	—
107	C₂²×C₇₃	#382	1	8	yes	—
108	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₇₃	C₂×#117	1	2	yes	—
110	C₂²×C₇₃	#518	1	1	yes	characteristic
111	C₂²×C₇₃	#518	1	2	yes	—
112	C₂²×C₇₃	#519	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×#124	1	2	yes	—
114	C₂²×C₇₃	C₂×#125	1	2	yes	—
115	C₂²×C₇₃	#534	1	2	yes	—
116	C₂²×C₇₃	#537	1	2	yes	—
117	C₂²×C₇₃	#538	1	8	yes	—
118	C₂²×C₇₃	#539	1	8	yes	—
119	C₂²×C₇₃	#543	1	2	yes	—
120	C₂²×C₇₃	#544	1	2	yes	—
121	C₄×C₇₃	—	2	8	no	—
122	C₄×C₇₃	—	4	8	no	—
123	C₄×C₇₃	—	8	8	no	—
124	#14347	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₇₃	—	2	8	no	—
126	C₂×C₇₃	C₂×#382	1	1	yes	characteristic
127	C₂×C₇₃	#2312	1	2	yes	—
128	C₂×C₇₃	#2313	1	2	yes	—
129	C₂×C₇₃	#2315	1	1	yes	characteristic
130	C₂×C₇₃	#2316	1	1	yes	characteristic
131	C₂×C₇₃	#2639	1	4	yes	—
132	C₂×C₇₃	#2646	1	4	yes	—
133	C₂×#380	C₂×C₇₃	1	2	yes	—
134	C₂²×#95	C₂×C₇₃	1	1	yes	characteristic
135	#1442	C₂×C₇₃	1	8	yes	complemented
136	C₂²⋊#95	C₂×C₇₃	1	4	yes	—
137	C₇₃	#14347	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴×C₄	C₂²×C₇₃	1	2	yes	—
139	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
141	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	4	yes	—
142	#380	—	2	8	no	—
143	C₂×#95	C₄×C₇₃	1	8	yes	complemented
144	C₂×#95	C₂²×C₇₃	1	4	yes	—
145	C₂×#95	C₂²×C₇₃	1	8	yes	—
146	C₂×#95	C₂²×C₇₃	1	8	yes	—
147	C₂×#95	C₂²×C₇₃	1	4	yes	—
148	C₂⁵	C₂³×C₇₃	1	1	yes	characteristic
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	4	4	no	—
151	C₂³×C₄	—	4	4	no	—
152	C₂³×C₄	—	4	4	no	—
153	C₂³×C₄	C₂×C₄×C₇₃	1	8	yes	—
154	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
155	C₂³×C₄	D₄×C₇₃	1	4	yes	—
156	C₂³×C₄	Q₈×C₇₃	1	4	yes	—
157	C₂³×C₄	C₂³×C₇₃	1	8	yes	—
158	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
159	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	8	no	—
161	C₂×C₄⋊C₄	—	4	8	no	—
162	C₂×C₂²⋊C₄	—	2	8	no	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	#95	—	2	8	no	—
165	#95	—	4	8	no	—
166	#95	—	4	8	no	—
167	#95	C₂×C₄×C₇₃	1	16	yes	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	2	no	—
170	C₂⁴	—	2	2	no	—
171	C₂⁴	C₂²×C₄×C₇₃	1	2	yes	—
172	C₂⁴	C₂×D₄×C₇₃	1	2	yes	—
173	C₂⁴	C₂×Q₈×C₇₃	1	2	yes	—
174	C₂⁴	#41×C₇₃	1	8	yes	—
175	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	16	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	—	4	8	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	—	4	32	no	—
192	C₂²×C₄	C₄⋊C₄×C₇₃	1	16	yes	—
193	C₂²×C₄	C₂²×C₄×C₇₃	1	8	yes	—
194	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
195	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
196	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
197	C₄⋊C₄	—	4	16	no	—
198	C₂²⋊C₄	—	4	16	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	2	16	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	4	no	—
208	C₂³	C₂×C₄⋊C₄×C₇₃	1	4	yes	—
209	C₂³	#117×C₇₃	1	8	yes	—
210	C₂³	#124×C₇₃	1	8	yes	—
211	C₂³	#125×C₇₃	1	8	yes	—
212	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
213	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
214	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
215	C₂³	C₂×#41×C₇₃	1	4	yes	—
216	C₂³	H(5,𝔽₂)×C₇₃	1	4	yes	—
217	C₂³	H(5,𝔽₂)×C₇₃	1	2	yes	—
218	C₂³	2₋¹⁺⁴×C₇₃	1	2	yes	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	4	16	no	—
226	C₂×C₄	—	4	32	no	—
227	C₂×C₄	—	4	16	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	4	16	no	—
232	C₂×C₄	—	4	16	no	—
233	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	8	yes	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	16	no	—
239	C₂²	—	2	16	no	—
240	C₂²	#382×C₇₃	1	8	yes	—
241	C₂²	C₂²×C₄⋊C₄×C₇₃	1	1	yes	characteristic
242	C₂²	C₂×#117×C₇₃	1	2	yes	—
243	C₂²	#518×C₇₃	1	1	yes	characteristic
244	C₂²	#518×C₇₃	1	2	yes	—
245	C₂²	#519×C₇₃	1	1	yes	characteristic
246	C₂²	C₂×#124×C₇₃	1	2	yes	—
247	C₂²	C₂×#125×C₇₃	1	2	yes	—
248	C₂²	#534×C₇₃	1	2	yes	—
249	C₂²	#537×C₇₃	1	2	yes	—
250	C₂²	#538×C₇₃	1	8	yes	—
251	C₂²	#539×C₇₃	1	8	yes	—
252	C₂²	#543×C₇₃	1	2	yes	—
253	C₂²	#544×C₇₃	1	2	yes	—
254	C₄	—	2	8	no	—
255	C₄	—	4	8	no	—
256	C₄	—	8	8	no	—
257	C₂	—	2	8	no	—
258	C₂	C₂×#382×C₇₃	1	1	yes	characteristic
259	C₂	C₇₃×#2312	1	2	yes	—
260	C₂	C₇₃×#2313	1	2	yes	—
261	C₂	C₇₃×#2315	1	1	yes	characteristic
262	C₂	C₇₃×#2316	1	1	yes	characteristic
263	C₂	C₇₃×#2639	1	4	yes	—
264	C₂	C₇₃×#2646	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₇₃
Frattini, cosocle	C₂⁴	C₂⁴×C₇₃
prime radical	C₂⁵×C₇₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₃×#1442	C₂	semidirect
10. C₂×#95×C₇₃	C₄	semidirect
124. #14347	C₇₃	direct (N × H)
135. #1442	C₂×C₇₃	semidirect
137. C₇₃	#14347	direct (N × H)
143. C₂×#95	C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#380×C₇₃	2
C₂²×#95×C₇₃	2
C₇₃×#1442	2
C₇₃×C₂²⋊#95	2
#14347	73

Subgroups — C73×#14347 #32923751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#14347 #32923751