Subgroups — C₇₃×#3223 #22607376

Order 9344.

364 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#536×C₇₃	C₂	1	1	yes	maximal, characteristic
3	#497×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#491×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#488×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#464×C₇₃	C₂	1	2	yes	maximal, complemented
8	#462×C₇₃	C₂	1	2	yes	maximal, complemented
9	#452×C₇₃	C₂	1	2	yes	maximal
10	#451×C₇₃	C₂	1	2	yes	maximal, complemented
11	#433×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₇₃	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	#126×C₇₃	—	2	2	no	—
22	#125×C₇₃	—	2	2	no	—
23	#125×C₇₃	C₂²	1	1	yes	characteristic
24	#124×C₇₃	C₂²	1	1	yes	characteristic
25	#124×C₇₃	C₂²	1	1	yes	characteristic
26	#123×C₇₃	—	2	2	no	—
27	C₂²⋊Q₈×C₇₃	—	2	2	no	—
28	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
29	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
30	#117×C₇₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
32	#103×C₇₃	C₂²	1	2	yes	—
33	#103×C₇₃	C₂²	1	2	yes	—
34	#103×C₇₃	C₂²	1	2	yes	—
35	#102×C₇₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
37	#97×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	#41×C₇₃	—	4	2	no	—
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	—	4	2	no	—
42	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
46	C₂×D₄×C₇₃	—	2	1	no	—
47	C₂×D₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	—	2	1	no	—
50	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	Q₁₆×C₇₃	—	4	2	no	—
52	Q₁₆×C₇₃	—	4	2	no	—
53	Q₁₆×C₇₃	—	4	2	no	—
54	SD₁₆×C₇₃	—	4	2	no	—
55	C₂×C₈×C₇₃	C₂³	1	2	yes	—
56	C₂×C₈×C₇₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₇₃	—	2	2	no	—
58	C₄⋊C₄×C₇₃	—	2	2	no	—
59	C₄⋊C₄×C₇₃	—	2	2	no	—
60	C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₄⋊C₄×C₇₃	—	2	2	no	—
63	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
64	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₇₃	—	2	2	no	—
66	C₂²⋊C₄×C₇₃	—	2	2	no	—
67	C₂²⋊C₄×C₇₃	—	2	2	no	—
68	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
69	C₄²×C₇₃	—	2	2	no	—
70	C₄²×C₇₃	C₂³	1	1	yes	characteristic
71	C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	C₂³×C₇₃	—	2	1	no	—
73	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₇₃	—	2	2	no	—
75	Q₈×C₇₃	—	2	2	no	—
76	Q₈×C₇₃	—	2	2	no	—
77	Q₈×C₇₃	—	2	2	no	—
78	Q₈×C₇₃	—	2	2	no	—
79	Q₈×C₇₃	—	4	2	no	—
80	D₄×C₇₃	—	2	2	no	—
81	D₄×C₇₃	—	4	2	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	4	2	no	—
89	C₂×C₄×C₇₃	—	4	2	no	—
90	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₇₃	—	2	2	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	2	no	—
98	C₂²×C₇₃	—	4	1	no	—
99	C₂²×C₇₃	—	4	2	no	—
100	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	2	no	—
103	C₄×C₇₃	—	2	2	no	—
104	C₄×C₇₃	—	4	1	no	—
105	C₄×C₇₃	—	4	1	no	—
106	C₄×C₇₃	—	4	1	no	—
107	C₄×C₇₃	—	4	2	no	—
108	C₄×C₇₃	—	4	2	no	—
109	C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
110	C₂×C₇₃	—	4	1	no	—
111	C₂×C₇₃	—	8	1	no	—
112	C₂×C₇₃	#535	1	1	yes	characteristic
113	C₂×C₇₃	#578	1	2	yes	—
114	#3223	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₇₃	#3223	1	1	yes	characteristic, direct product factor, Sylow
116	#541	C₂×C₇₃	1	1	yes	characteristic, complemented
117	#536	C₂×C₇₃	1	1	yes	characteristic
118	#497	C₂×C₇₃	1	1	yes	characteristic, complemented
119	#494	C₂×C₇₃	1	1	yes	characteristic, complemented
120	#491	C₂×C₇₃	1	1	yes	characteristic, complemented
121	#488	C₂×C₇₃	1	1	yes	characteristic, complemented
122	#464	C₂×C₇₃	1	2	yes	complemented
123	#462	C₂×C₇₃	1	2	yes	complemented
124	#452	C₂×C₇₃	1	2	yes	—
125	#451	C₂×C₇₃	1	2	yes	complemented
126	#433	C₂×C₇₃	1	1	yes	characteristic, complemented
127	C₂×#41	C₂²×C₇₃	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₇₃	1	1	yes	characteristic
129	C₂×Q₁₆	C₂²×C₇₃	1	2	yes	—
130	C₂×Q₁₆	C₂²×C₇₃	1	2	yes	—
131	C₂×Q₁₆	C₂²×C₇₃	1	2	yes	—
132	C₂×SD₁₆	C₂²×C₇₃	1	2	yes	—
133	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
134	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
135	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
136	#126	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	C₂²×C₇₃	1	1	yes	characteristic
139	#124	C₂²×C₇₃	1	1	yes	characteristic
140	#124	C₂²×C₇₃	1	1	yes	characteristic
141	#123	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	—
144	C₄×Q₈	C₂²×C₇₃	1	2	yes	—
145	#117	C₂²×C₇₃	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
147	#103	C₂²×C₇₃	1	2	yes	—
148	#103	C₂²×C₇₃	1	2	yes	—
149	#103	C₂²×C₇₃	1	2	yes	—
150	#102	C₂²×C₇₃	1	2	yes	—
151	C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
152	#97	C₂²×C₇₃	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
154	#41	—	4	2	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	C₂³×C₇₃	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₇₃	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₇₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	C₂³×C₇₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	Q₁₆	—	4	2	no	—
169	SD₁₆	—	4	2	no	—
170	C₂×C₈	C₂³×C₇₃	1	2	yes	—
171	C₂×C₈	C₂³×C₇₃	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₇₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	D₄×C₇₃	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂³×C₇₃	1	1	yes	characteristic
186	C₄²	C₂³×C₇₃	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	8	1	no	—
227	C₂	#535×C₇₃	1	1	yes	characteristic
228	C₂	#578×C₇₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
prime radical	C₂×#41×C₇₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ #117×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #541×C₇₃	C₂	semidirect
3. #497×C₇₃	C₂	semidirect
4. #494×C₇₃	C₂	semidirect
5. #491×C₇₃	C₂	semidirect
6. #488×C₇₃	C₂	semidirect
7. #464×C₇₃	C₂	semidirect
8. #462×C₇₃	C₂	semidirect
10. #451×C₇₃	C₂	semidirect
11. #433×C₇₃	C₂	semidirect
114. #3223	C₇₃	direct (N × H)
115. C₇₃	#3223	direct (N × H)
116. #541	C₂×C₇₃	semidirect
118. #497	C₂×C₇₃	semidirect
119. #494	C₂×C₇₃	semidirect
120. #491	C₂×C₇₃	semidirect
121. #488	C₂×C₇₃	semidirect
122. #464	C₂×C₇₃	semidirect
123. #462	C₂×C₇₃	semidirect
125. #451	C₂×C₇₃	semidirect
126. #433	C₂×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
#541×C₇₃	2
#536×C₇₃	2
#497×C₇₃	2
#494×C₇₃	2
#491×C₇₃	2
#488×C₇₃	2
#464×C₇₃	2
#462×C₇₃	2
#452×C₇₃	2
#451×C₇₃	2
#433×C₇₃	2
#3223	73

Subgroups — C73×#3223 #22607376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3223 #22607376