Subgroups — C₇₃×#33494 #32942898

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#2168	C₂	1	8	yes	maximal, complemented
3	C₇₃×#2163	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2113	C₂	1	4	yes	maximal, complemented
5	C₂×#131×C₇₃	C₂²	1	2	yes	—
6	#532×C₇₃	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₇₃	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₇₃	C₂²	1	4	yes	—
9	#436×C₇₃	C₄	1	4	yes	—
10	C₈×D₄×C₇₃	C₂²	1	4	yes	—
11	#423×C₇₃	C₄	1	1	yes	characteristic
12	#408×C₇₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₃	C₂²	1	4	yes	—
15	C₂²⋊C₁₆×C₇₃	C₂²	1	4	yes	—
16	C₂²⋊C₁₆×C₇₃	C₂²	1	8	yes	—
17	#346×C₇₃	C₂²	1	4	yes	—
18	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
19	#131×C₇₃	—	4	2	no	—
20	#131×C₇₃	—	4	2	no	—
21	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
26	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
27	#127×C₇₃	C₈	1	1	yes	characteristic
28	#124×C₇₃	C₈	1	2	yes	—
29	#121×C₇₃	C₈	1	4	yes	—
30	C₄×D₄×C₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₃	—	4	4	no	—
33	C₂×C₁₆×C₇₃	—	4	4	no	—
34	C₂×C₁₆×C₇₃	C₂³	1	8	yes	—
35	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
36	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	C₂³	1	2	yes	—
39	#41×C₇₃	—	4	2	no	—
40	#41×C₇₃	—	4	2	no	—
41	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
42	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
43	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
44	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	4	2	no	—
49	M₁₆×C₇₃	—	4	4	no	—
50	C₂×C₈×C₇₃	—	2	4	no	—
51	C₂×C₈×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	4	1	no	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
60	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
63	C₁₆×C₇₃	—	2	8	no	—
64	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₇₃	—	4	2	no	—
67	D₄×C₇₃	—	4	4	no	—
68	D₄×C₇₃	—	4	2	no	—
69	D₄×C₇₃	—	4	4	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	4	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	4	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₇₃	—	2	4	no	—
81	C₈×C₇₃	—	2	1	no	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	2	8	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	—	2	4	no	—
88	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	4	no	—
93	C₄×C₇₃	#518	1	1	yes	characteristic
94	C₄×C₇₃	#568	1	1	yes	characteristic
95	#33494	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₇₃	—	4	1	no	—
97	C₂×C₇₃	—	4	4	no	—
98	C₂×C₇₃	#2972	1	1	yes	characteristic
99	C₂×C₇₃	#3407	1	2	yes	—
100	#2931	C₂×C₇₃	1	1	yes	characteristic
101	#2168	C₂×C₇₃	1	8	yes	complemented
102	#2163	C₂×C₇₃	1	2	yes	complemented
103	#2113	C₂×C₇₃	1	4	yes	complemented
104	C₇₃	#33494	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#131	C₂²×C₇₃	1	2	yes	—
106	#532	C₄×C₇₃	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₇₃	1	4	yes	—
108	C₂²×C₁₆	C₂²×C₇₃	1	4	yes	—
109	#436	C₄×C₇₃	1	4	yes	—
110	C₈×D₄	C₂²×C₇₃	1	4	yes	—
111	#423	C₄×C₇₃	1	1	yes	characteristic
112	#408	C₄×C₇₃	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₇₃	1	4	yes	—
115	C₂²⋊C₁₆	C₂²×C₇₃	1	4	yes	—
116	C₂²⋊C₁₆	C₂²×C₇₃	1	8	yes	—
117	#346	C₂²×C₇₃	1	4	yes	—
118	C₂×#41	C₂×C₄×C₇₃	1	2	yes	—
119	#131	—	4	2	no	—
120	#131	—	4	2	no	—
121	C₂×M₁₆	C₂×C₄×C₇₃	1	2	yes	—
122	C₂×M₁₆	C₂×C₄×C₇₃	1	4	yes	—
123	C₂²×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
126	C₄⋊Q₈	C₈×C₇₃	1	1	yes	characteristic
127	#127	C₈×C₇₃	1	1	yes	characteristic
128	#124	C₈×C₇₃	1	2	yes	—
129	#121	C₈×C₇₃	1	4	yes	—
130	C₄×D₄	C₂×C₄×C₇₃	1	4	yes	—
131	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
132	M₃₂	—	4	4	no	—
133	C₂×C₁₆	—	4	4	no	—
134	C₂×C₁₆	C₂³×C₇₃	1	8	yes	—
135	C₄⋊C₈	C₂×C₄×C₇₃	1	4	yes	—
136	C₂²⋊C₈	C₂×C₄×C₇₃	1	8	yes	—
137	C₄×C₈	C₂×C₄×C₇₃	1	2	yes	—
138	C₄×C₈	C₂³×C₇₃	1	2	yes	—
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₂×C₈×C₇₃	1	2	yes	—
142	C₂×D₄	C₂×C₈×C₇₃	1	4	yes	—
143	C₂×D₄	C₂×C₈×C₇₃	1	4	yes	—
144	C₂×D₄	C₂×C₈×C₇₃	1	2	yes	—
145	C₂²×C₄	C₂×C₈×C₇₃	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
147	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂²×C₄×C₇₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×C₄×C₇₃	1	2	yes	—
156	C₂×C₈	C₂²×C₄×C₇₃	1	4	yes	—
157	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×C₈×C₇₃	1	4	yes	—
159	C₂²⋊C₄	C₂×C₈×C₇₃	1	8	yes	—
160	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
161	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
162	C₄²	C₂²×C₄×C₇₃	1	2	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	C₂²×C₈×C₇₃	1	4	yes	—
165	C₂³	C₂²×C₈×C₇₃	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂²×C₈×C₇₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₈×C₇₃	1	2	yes	—
177	C₂×C₄	C₂²×C₈×C₇₃	1	2	yes	—
178	C₂×C₄	C₂²×C₈×C₇₃	1	4	yes	—
179	C₂×C₄	C₂³×C₄×C₇₃	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	4	no	—
182	C₈	—	2	2	no	—
183	C₈	H(5,𝔽₂)×C₇₃	1	2	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	8	no	—
188	C₂²	C₂³×C₈×C₇₃	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	#518×C₇₃	1	1	yes	characteristic
194	C₄	#568×C₇₃	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	C₇₃×#2972	1	1	yes	characteristic
198	C₂	C₇₃×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₇₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₇₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
socle	C₂²×C₇₃	C₂³×C₈
prime radical	#532×C₇₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₃×#2168	C₂	semidirect
3. C₇₃×#2163	C₂	semidirect
4. C₇₃×#2113	C₂	semidirect
95. #33494	C₇₃	direct (N × H)
101. #2168	C₂×C₇₃	semidirect
102. #2163	C₂×C₇₃	semidirect
103. #2113	C₂×C₇₃	semidirect
104. C₇₃	#33494	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₃×#2931	2
C₇₃×#2168	2
C₇₃×#2163	2
C₇₃×#2113	2
#33494	73

Subgroups — C73×#33494 #32942898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33494 #32942898