Subgroups — C₇₃×#14154 #32923558

Order 18688.

3062 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	C₂²×#127×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#390×C₇₃	C₂	1	8	yes	maximal, complemented
3	C₂×C₄×C₄⋊C₄×C₇₃	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1438	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×C₇₃	C₂²	1	4	yes	—
6	C₂³×D₄×C₇₃	C₂²	1	2	yes	—
7	C₂×#127×C₇₃	—	2	8	no	—
8	C₂×#127×C₇₃	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	16	yes	—
11	C₂²×C₄²×C₇₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	#390×C₇₃	—	2	32	no	—
14	C₄×C₄⋊C₄×C₇₃	—	2	8	no	—
15	C₂×#95×C₇₃	C₂²	1	4	yes	—
16	C₂⁵×C₇₃	C₂³	1	4	yes	—
17	C₂²×D₄×C₇₃	—	2	16	no	—
18	C₂²×D₄×C₇₃	—	2	8	no	—
19	C₂²×D₄×C₇₃	—	2	8	no	—
20	C₂²×D₄×C₇₃	—	2	8	no	—
21	C₂²×D₄×C₇₃	—	2	16	no	—
22	C₂²×D₄×C₇₃	—	4	8	no	—
23	C₂²×D₄×C₇₃	C₂×C₄	1	16	yes	—
24	C₂²×D₄×C₇₃	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₇₃	C₂³	1	8	yes	—
26	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
28	#127×C₇₃	—	2	32	no	—
29	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
31	C₂×C₄⋊C₄×C₇₃	D₄	1	16	yes	complemented
32	C₂×C₂²⋊C₄×C₇₃	—	2	64	no	—
33	C₂×C₂²⋊C₄×C₇₃	—	4	16	no	—
34	C₂×C₄²×C₇₃	—	2	8	no	—
35	C₂×C₄²×C₇₃	—	2	16	no	—
36	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇₃	D₄	1	8	yes	—
38	#95×C₇₃	—	2	8	no	—
39	C₂⁴×C₇₃	—	2	16	no	—
40	C₂⁴×C₇₃	—	2	8	no	—
41	C₂⁴×C₇₃	—	2	16	no	—
42	C₂⁴×C₇₃	—	2	16	no	—
43	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
44	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₄×C₇₃	—	2	16	no	—
46	C₂×D₄×C₇₃	—	2	32	no	—
47	C₂×D₄×C₇₃	—	2	64	no	—
48	C₂×D₄×C₇₃	—	4	16	no	—
49	C₂×D₄×C₇₃	—	4	32	no	—
50	C₂×D₄×C₇₃	—	4	32	no	—
51	C₂×D₄×C₇₃	—	4	16	no	—
52	C₂×D₄×C₇₃	—	4	16	no	—
53	C₂²×C₄×C₇₃	—	2	4	no	—
54	C₂²×C₄×C₇₃	—	2	8	no	—
55	C₂²×C₄×C₇₃	—	2	16	no	—
56	C₂²×C₄×C₇₃	—	2	8	no	—
57	C₂²×C₄×C₇₃	—	2	16	no	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₃	C₂×D₄	1	16	yes	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₇₃	#41	1	8	yes	—
66	C₄⋊C₄×C₇₃	—	2	32	no	—
67	C₂²⋊C₄×C₇₃	—	4	64	no	—
68	C₄²×C₇₃	—	4	16	no	—
69	C₄²×C₇₃	C₂²⋊C₄	1	16	yes	complemented
70	C₂³×C₇₃	—	2	8	no	—
71	C₂³×C₇₃	—	2	32	no	—
72	C₂³×C₇₃	—	2	16	no	—
73	C₂³×C₇₃	—	4	16	no	—
74	C₂³×C₇₃	—	4	8	no	—
75	C₂³×C₇₃	—	4	8	no	—
76	C₂³×C₇₃	—	4	16	no	—
77	C₂³×C₇₃	—	4	16	no	—
78	C₂³×C₇₃	—	4	32	no	—
79	C₂³×C₇₃	—	4	16	no	—
80	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
81	C₂³×C₇₃	C₂²×D₄	1	2	yes	—
82	C₂³×C₇₃	C₂²×D₄	1	4	yes	—
83	C₂³×C₇₃	C₂×#41	1	4	yes	—
84	C₂³×C₇₃	H(5,𝔽₂)	1	4	yes	—
85	D₄×C₇₃	—	4	32	no	—
86	D₄×C₇₃	—	4	64	no	—
87	C₂×C₄×C₇₃	—	2	4	no	—
88	C₂×C₄×C₇₃	—	2	8	no	—
89	C₂×C₄×C₇₃	—	2	16	no	—
90	C₂×C₄×C₇₃	—	2	32	no	—
91	C₂×C₄×C₇₃	—	4	8	no	—
92	C₂×C₄×C₇₃	—	4	8	no	—
93	C₂×C₄×C₇₃	—	4	16	no	—
94	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
95	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	8	yes	—
96	C₂×C₄×C₇₃	C₄×D₄	1	16	yes	—
97	C₂×C₄×C₇₃	#121	1	16	yes	—
98	C₂×C₄×C₇₃	#124	1	8	yes	—
99	C₂×C₄×C₇₃	#127	1	8	yes	—
100	C₂²×C₇₃	—	4	8	no	—
101	C₂²×C₇₃	—	4	32	no	—
102	C₂²×C₇₃	—	4	16	no	—
103	C₂²×C₇₃	—	8	16	no	—
104	C₂²×C₇₃	—	8	16	no	—
105	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₇₃	C₂×C₄×D₄	1	4	yes	—
107	C₂²×C₇₃	#518	1	2	yes	—
108	C₂²×C₇₃	C₂×#121	1	4	yes	—
109	C₂²×C₇₃	C₂×#124	1	2	yes	—
110	C₂²×C₇₃	C₂×#127	1	2	yes	—
111	C₂²×C₇₃	#535	1	4	yes	—
112	C₂²×C₇₃	D₄²	1	4	yes	—
113	C₂²×C₇₃	#550	1	8	yes	—
114	C₂²×C₇₃	#559	1	4	yes	—
115	C₄×C₇₃	—	2	4	no	—
116	C₄×C₇₃	—	4	16	no	—
117	C₄×C₇₃	#390	1	16	yes	complemented
118	#14154	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₇₃	—	8	8	no	—
120	C₂×C₇₃	C₂×#390	1	2	yes	—
121	C₂×C₇₃	#2309	1	1	yes	characteristic
122	C₂×C₇₃	#2366	1	4	yes	—
123	C₂×C₇₃	#2541	1	4	yes	—
124	C₂×C₇₃	#2548	1	4	yes	—
125	C₂²×#127	C₂×C₇₃	1	1	yes	characteristic
126	C₂×#390	C₂×C₇₃	1	8	yes	complemented
127	C₂×C₄×C₄⋊C₄	C₂×C₇₃	1	2	yes	complemented
128	#1438	C₂×C₇₃	1	4	yes	complemented
129	C₇₃	#14154	1	1	yes	characteristic, direct product factor, Sylow
130	C₂³×D₄	C₂²×C₇₃	1	4	yes	—
131	C₂³×D₄	C₂²×C₇₃	1	2	yes	—
132	C₂×#127	—	2	8	no	—
133	C₂×#127	C₄×C₇₃	1	8	yes	complemented
134	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
135	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	16	yes	—
136	C₂²×C₄²	C₂²×C₇₃	1	4	yes	—
137	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
138	#390	—	2	32	no	—
139	C₄×C₄⋊C₄	—	2	8	no	—
140	C₂×#95	C₂²×C₇₃	1	4	yes	—
141	C₂⁵	C₂³×C₇₃	1	4	yes	—
142	C₂²×D₄	—	2	8	no	—
143	C₂²×D₄	—	2	8	no	—
144	C₂²×D₄	—	2	8	no	—
145	C₂²×D₄	—	2	16	no	—
146	C₂²×D₄	—	2	16	no	—
147	C₂²×D₄	—	4	8	no	—
148	C₂²×D₄	C₂×C₄×C₇₃	1	16	yes	—
149	C₂²×D₄	C₂×C₄×C₇₃	1	8	yes	—
150	C₂³×C₄	C₂³×C₇₃	1	8	yes	—
151	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
153	#127	—	2	32	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	D₄×C₇₃	1	16	yes	complemented
157	C₂×C₂²⋊C₄	—	2	64	no	—
158	C₂×C₂²⋊C₄	—	4	16	no	—
159	C₂×C₄²	—	2	8	no	—
160	C₂×C₄²	—	2	16	no	—
161	C₂×C₄²	C₂×C₄×C₇₃	1	4	yes	—
162	C₂×C₄²	D₄×C₇₃	1	8	yes	—
163	#95	—	2	8	no	—
164	C₂⁴	—	2	16	no	—
165	C₂⁴	—	2	8	no	—
166	C₂⁴	—	2	16	no	—
167	C₂⁴	—	2	16	no	—
168	C₂⁴	C₂²×C₄×C₇₃	1	8	yes	—
169	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
170	C₂×D₄	—	2	16	no	—
171	C₂×D₄	—	2	32	no	—
172	C₂×D₄	—	2	64	no	—
173	C₂×D₄	—	4	16	no	—
174	C₂×D₄	—	4	32	no	—
175	C₂×D₄	—	4	32	no	—
176	C₂×D₄	—	4	16	no	—
177	C₂×D₄	—	4	16	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
185	C₂²×C₄	C₂×D₄×C₇₃	1	16	yes	—
186	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
187	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
188	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
189	C₂²×C₄	C₂×D₄×C₇₃	1	8	yes	—
190	C₂²×C₄	#41×C₇₃	1	8	yes	—
191	C₄⋊C₄	—	2	32	no	—
192	C₂²⋊C₄	—	4	64	no	—
193	C₄²	—	4	16	no	—
194	C₄²	C₂²⋊C₄×C₇₃	1	16	yes	complemented
195	C₂³	—	2	8	no	—
196	C₂³	—	2	32	no	—
197	C₂³	—	2	16	no	—
198	C₂³	—	4	16	no	—
199	C₂³	—	4	8	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	16	no	—
202	C₂³	—	4	16	no	—
203	C₂³	—	4	32	no	—
204	C₂³	—	4	16	no	—
205	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
206	C₂³	C₂²×D₄×C₇₃	1	2	yes	—
207	C₂³	C₂²×D₄×C₇₃	1	4	yes	—
208	C₂³	C₂×#41×C₇₃	1	4	yes	—
209	C₂³	H(5,𝔽₂)×C₇₃	1	4	yes	—
210	D₄	—	4	32	no	—
211	D₄	—	4	64	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	32	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	—	4	16	no	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	4	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	8	yes	—
221	C₂×C₄	C₄×D₄×C₇₃	1	16	yes	—
222	C₂×C₄	#121×C₇₃	1	16	yes	—
223	C₂×C₄	#124×C₇₃	1	8	yes	—
224	C₂×C₄	#127×C₇₃	1	8	yes	—
225	C₂²	—	4	16	no	—
226	C₂²	—	4	8	no	—
227	C₂²	—	4	32	no	—
228	C₂²	—	8	16	no	—
229	C₂²	—	8	16	no	—
230	C₂²	C₂²×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
231	C₂²	C₂×C₄×D₄×C₇₃	1	4	yes	—
232	C₂²	#518×C₇₃	1	2	yes	—
233	C₂²	C₂×#121×C₇₃	1	4	yes	—
234	C₂²	C₂×#124×C₇₃	1	2	yes	—
235	C₂²	C₂×#127×C₇₃	1	2	yes	—
236	C₂²	#535×C₇₃	1	4	yes	—
237	C₂²	D₄²×C₇₃	1	4	yes	—
238	C₂²	#550×C₇₃	1	8	yes	—
239	C₂²	#559×C₇₃	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	16	no	—
242	C₄	#390×C₇₃	1	16	yes	complemented
243	C₂	—	8	8	no	—
244	C₂	C₂×#390×C₇₃	1	2	yes	—
245	C₂	C₇₃×#2309	1	1	yes	characteristic
246	C₂	C₇₃×#2366	1	4	yes	—
247	C₂	C₇₃×#2541	1	4	yes	—
248	C₂	C₇₃×#2548	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₇₃
Frattini, cosocle	C₂⁴	C₂⁴×C₇₃
prime radical	C₂²×#127×C₇₃	C₂
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
2. C₂×#390×C₇₃	C₂	semidirect
3. C₂×C₄×C₄⋊C₄×C₇₃	C₂	semidirect
4. C₇₃×#1438	C₂	semidirect
8. C₂×#127×C₇₃	C₄	semidirect
31. C₂×C₄⋊C₄×C₇₃	D₄	semidirect
69. C₄²×C₇₃	C₂²⋊C₄	semidirect
117. C₄×C₇₃	#390	semidirect
118. #14154	C₇₃	direct (N × H)
126. C₂×#390	C₂×C₇₃	semidirect
127. C₂×C₄×C₄⋊C₄	C₂×C₇₃	semidirect
128. #1438	C₂×C₇₃	semidirect
129. C₇₃	#14154	direct (N × H)
133. C₂×#127	C₄×C₇₃	semidirect
156. C₂×C₄⋊C₄	D₄×C₇₃	semidirect
194. C₄²	C₂²⋊C₄×C₇₃	semidirect
242. C₄	#390×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₇₃	2
C₂×#390×C₇₃	2
C₂×C₄×C₄⋊C₄×C₇₃	2
C₇₃×#1438	2
#14154	73

Subgroups — C73×#14154 #32923558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#14154 #32923558