Subgroups — C₇₃×#3250 #22607403

Order 9344.

360 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529×C₇₃	C₂	1	2	yes	maximal, complemented
2	#479×C₇₃	C₂	1	2	yes	maximal, complemented
3	#478×C₇₃	C₂	1	2	yes	maximal, complemented
4	#467×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#465×C₇₃	C₂	1	2	yes	maximal, complemented
7	#464×C₇₃	C₂	1	2	yes	maximal, complemented
8	#463×C₇₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₇₃	C₂²	1	2	yes	—
12	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#126×C₇₃	—	2	2	no	—
16	#125×C₇₃	C₂²	1	2	yes	—
17	#124×C₇₃	C₂²	1	2	yes	—
18	#123×C₇₃	—	2	2	no	—
19	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
20	#121×C₇₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
22	C₄×D₄×C₇₃	—	2	2	no	—
23	C₄×D₄×C₇₃	C₂²	1	2	yes	—
24	#117×C₇₃	—	2	2	no	—
25	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
26	#107×C₇₃	C₂²	1	1	yes	characteristic
27	#107×C₇₃	C₂²	1	1	yes	characteristic
28	#106×C₇₃	C₂²	1	2	yes	—
29	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
31	#103×C₇₃	C₂²	1	2	yes	—
32	#103×C₇₃	C₂²	1	2	yes	—
33	#102×C₇₃	C₂²	1	2	yes	—
34	#102×C₇₃	C₂²	1	2	yes	—
35	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
36	C₂×D₄×C₇₃	—	2	2	no	—
37	C₂×D₄×C₇₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	Q₁₆×C₇₃	—	4	1	no	—
43	SD₁₆×C₇₃	—	4	2	no	—
44	D₈×C₇₃	—	4	1	no	—
45	C₂×C₈×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂³	1	2	yes	—
49	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₇₃	—	2	2	no	—
51	C₄⋊C₄×C₇₃	—	2	2	no	—
52	C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₄⋊C₄×C₇₃	—	2	2	no	—
54	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₇₃	—	2	2	no	—
57	C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₂²⋊C₄×C₇₃	—	2	2	no	—
59	C₂²⋊C₄×C₇₃	—	2	2	no	—
60	C₂²⋊C₄×C₇₃	—	2	2	no	—
61	C₄²×C₇₃	—	2	2	no	—
62	C₄²×C₇₃	D₄	1	2	yes	—
63	C₄²×C₇₃	C₂³	1	2	yes	—
64	C₂³×C₇₃	—	2	2	no	—
65	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₇₃	—	2	2	no	—
67	Q₈×C₇₃	—	2	2	no	—
68	D₄×C₇₃	—	2	2	no	—
69	D₄×C₇₃	—	2	2	no	—
70	D₄×C₇₃	—	4	4	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	4	1	no	—
78	C₂×C₄×C₇₃	—	4	4	no	—
79	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
83	C₈×C₇₃	—	2	1	no	—
84	C₈×C₇₃	—	2	2	no	—
85	C₈×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	—	4	2	no	—
89	C₂²×C₇₃	—	4	2	no	—
90	C₂²×C₇₃	—	4	2	no	—
91	C₂²×C₇₃	#135	1	2	yes	—
92	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	—	2	2	no	—
96	C₄×C₇₃	—	4	2	no	—
97	C₄×C₇₃	—	4	2	no	—
98	C₄×C₇₃	—	4	2	no	—
99	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
100	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×C₇₃	—	2	2	no	—
102	C₂×C₇₃	—	8	2	no	—
103	C₂×C₇₃	#537	1	1	yes	characteristic
104	C₂×C₇₃	C₂×#135	1	1	yes	characteristic
105	C₂×C₇₃	#575	1	1	yes	characteristic
106	#3250	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₇₃	#3250	1	1	yes	characteristic, direct product factor, Sylow
108	#529	C₂×C₇₃	1	2	yes	complemented
109	#479	C₂×C₇₃	1	2	yes	complemented
110	#478	C₂×C₇₃	1	2	yes	complemented
111	#467	C₂×C₇₃	1	1	yes	characteristic, complemented
112	#466	C₂×C₇₃	1	1	yes	characteristic
113	#465	C₂×C₇₃	1	2	yes	complemented
114	#464	C₂×C₇₃	1	2	yes	complemented
115	#463	C₂×C₇₃	1	2	yes	complemented
116	C₂×C₄⋊C₈	C₂×C₇₃	1	1	yes	characteristic, complemented
117	C₂×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
118	C₂×SD₁₆	C₂²×C₇₃	1	2	yes	—
119	C₂×D₈	C₂²×C₇₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₇₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×C₇₃	1	1	yes	characteristic
122	#126	—	2	2	no	—
123	#125	C₂²×C₇₃	1	2	yes	—
124	#124	C₂²×C₇₃	1	2	yes	—
125	#123	—	2	2	no	—
126	C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	—
127	#121	C₂²×C₇₃	1	2	yes	—
128	C₄×Q₈	C₂²×C₇₃	1	2	yes	—
129	C₄×D₄	—	2	2	no	—
130	C₄×D₄	C₂²×C₇₃	1	2	yes	—
131	#117	—	2	2	no	—
132	C₂×C₄²	C₂²×C₇₃	1	1	yes	characteristic
133	#107	C₂²×C₇₃	1	1	yes	characteristic
134	#107	C₂²×C₇₃	1	1	yes	characteristic
135	#106	C₂²×C₇₃	1	2	yes	—
136	C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
137	C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
138	#103	C₂²×C₇₃	1	2	yes	—
139	#103	C₂²×C₇₃	1	2	yes	—
140	#102	C₂²×C₇₃	1	2	yes	—
141	#102	C₂²×C₇₃	1	2	yes	—
142	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂³×C₇₃	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
147	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
149	Q₁₆	—	4	1	no	—
150	SD₁₆	—	4	2	no	—
151	D₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂³×C₇₃	1	1	yes	characteristic
155	C₂×C₈	C₂³×C₇₃	1	2	yes	—
156	C₂×C₈	C₂³×C₇₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
162	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	D₄×C₇₃	1	2	yes	—
170	C₄²	C₂³×C₇₃	1	2	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	Q₈	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
187	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
189	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	2	no	—
192	C₈	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	#135×C₇₃	1	2	yes	—
199	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
207	C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	#537×C₇₃	1	1	yes	characteristic
211	C₂	C₂×#135×C₇₃	1	1	yes	characteristic
212	C₂	#575×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
prime radical	#466×C₇₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #529×C₇₃	C₂	semidirect
2. #479×C₇₃	C₂	semidirect
3. #478×C₇₃	C₂	semidirect
4. #467×C₇₃	C₂	semidirect
6. #465×C₇₃	C₂	semidirect
7. #464×C₇₃	C₂	semidirect
8. #463×C₇₃	C₂	semidirect
9. C₂×C₄⋊C₈×C₇₃	C₂	semidirect
106. #3250	C₇₃	direct (N × H)
107. C₇₃	#3250	direct (N × H)
108. #529	C₂×C₇₃	semidirect
109. #479	C₂×C₇₃	semidirect
110. #478	C₂×C₇₃	semidirect
111. #467	C₂×C₇₃	semidirect
113. #465	C₂×C₇₃	semidirect
114. #464	C₂×C₇₃	semidirect
115. #463	C₂×C₇₃	semidirect
116. C₂×C₄⋊C₈	C₂×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
#529×C₇₃	2
#479×C₇₃	2
#478×C₇₃	2
#467×C₇₃	2
#466×C₇₃	2
#465×C₇₃	2
#464×C₇₃	2
#463×C₇₃	2
C₂×C₄⋊C₈×C₇₃	2
#3250	73

Subgroups — C73×#3250 #22607403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3250 #22607403