Subgroups — C₇₃×#13187 #32922591

Order 18688.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#365×C₇₃	C₂	1	2	yes	maximal
2	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1386	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
7	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
8	#427×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×#107×C₇₃	—	2	1	no	—
10	C₂×#106×C₇₃	C₂²	1	2	yes	—
11	#405×C₇₃	—	2	1	no	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	#382×C₇₃	—	2	1	no	—
14	#365×C₇₃	—	2	4	no	—
15	#364×C₇₃	C₄	1	4	yes	complemented
16	#339×C₇₃	—	2	1	no	—
17	C₂×M₁₆×C₇₃	—	2	1	no	—
18	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
20	#117×C₇₃	—	2	1	no	—
21	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
22	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
23	M₃₂×C₇₃	—	2	2	no	—
24	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₇₃	—	2	2	no	—
26	#108×C₇₃	—	2	1	no	—
27	#108×C₇₃	D₄	1	2	yes	—
28	#107×C₇₃	—	4	2	no	—
29	#107×C₇₃	Q₈	1	2	yes	—
30	#106×C₇₃	—	2	1	no	—
31	#106×C₇₃	—	2	2	no	—
32	#106×C₇₃	—	2	2	no	—
33	#97×C₇₃	—	4	1	no	—
34	C₄×C₈×C₇₃	—	4	1	no	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	Q₈	1	2	yes	—
37	#95×C₇₃	—	4	2	no	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	—	4	2	no	—
40	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₇₃	—	2	2	no	—
42	M₁₆×C₇₃	—	4	1	no	—
43	C₂×C₈×C₇₃	—	4	1	no	—
44	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₄⋊C₄×C₇₃	—	4	2	no	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₂²⋊C₄×C₇₃	—	4	1	no	—
58	C₄²×C₇₃	—	2	1	no	—
59	C₄²×C₇₃	—	4	1	no	—
60	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₁₆×C₇₃	—	2	4	no	—
62	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	—	8	4	no	—
67	C₂×C₄×C₇₃	—	8	2	no	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₈×C₇₃	—	4	2	no	—
75	C₈×C₇₃	#106	1	2	yes	—
76	C₈×C₇₃	#106	1	2	yes	—
77	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₇₃	#125	1	1	yes	characteristic
80	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	#428	1	1	yes	characteristic
83	C₂²×C₇₃	#460	1	1	yes	characteristic
84	C₂²×C₇₃	#476	1	1	yes	characteristic
85	C₄×C₇₃	—	4	2	no	—
86	C₄×C₇₃	—	8	2	no	—
87	C₄×C₇₃	—	16	2	no	—
88	C₄×C₇₃	#389	1	1	yes	characteristic
89	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₇₃	#461	1	1	yes	characteristic
91	C₄×C₇₃	#475	1	1	yes	characteristic
92	#13187	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₇₃	—	2	1	no	—
94	C₂×C₇₃	—	2	1	no	—
95	C₂×C₇₃	#1948	1	1	yes	characteristic
96	C₂×C₇₃	#2231	1	2	yes	—
97	C₂×#365	C₂×C₇₃	1	2	yes	—
98	C₂×#364	C₂×C₇₃	1	1	yes	characteristic
99	#1846	C₂×C₇₃	1	1	yes	characteristic
100	#1772	C₂×C₇₃	1	1	yes	characteristic
101	#1386	C₂×C₇₃	1	2	yes	—
102	C₇₃	#13187	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×M₃₂	C₂²×C₇₃	1	2	yes	—
104	C₂×#108	C₂²×C₇₃	1	1	yes	characteristic
105	#427	C₂²×C₇₃	1	1	yes	characteristic
106	C₂×#107	—	2	1	no	—
107	C₂×#106	C₂²×C₇₃	1	2	yes	—
108	#405	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
110	#382	—	2	1	no	—
111	#365	—	2	4	no	—
112	#364	C₄×C₇₃	1	4	yes	complemented
113	#339	—	2	1	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
117	#117	—	2	1	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	C₂×C₄×C₇₃	1	4	yes	—
122	C₂×C₁₆	—	2	2	no	—
123	#108	—	2	1	no	—
124	#108	D₄×C₇₃	1	2	yes	—
125	#107	—	4	2	no	—
126	#107	Q₈×C₇₃	1	2	yes	—
127	#106	—	2	1	no	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	#97	—	4	1	no	—
131	C₄×C₈	—	4	1	no	—
132	C₄×C₈	C₂×C₄×C₇₃	1	2	yes	—
133	C₄×C₈	Q₈×C₇₃	1	2	yes	—
134	#95	—	4	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	C₄⋊C₄×C₇₃	1	2	yes	—
142	C₂×C₈	SD₁₆×C₇₃	1	1	yes	characteristic
143	C₂×C₈	SD₁₆×C₇₃	1	1	yes	characteristic
144	C₂×C₈	SD₁₆×C₇₃	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₇₃	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₇₃	1	1	yes	characteristic
148	C₂×C₈	C₂×Q₈×C₇₃	1	1	yes	characteristic
149	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
150	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	C₄⋊C₄×C₇₃	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₂³	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
166	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
168	C₂×C₄	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
169	C₂×C₄	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
170	C₂×C₄	#137×C₇₃	1	1	yes	characteristic
171	C₈	—	4	2	no	—
172	C₈	#106×C₇₃	1	2	yes	—
173	C₈	#106×C₇₃	1	2	yes	—
174	C₈	C₄×D₄×C₇₃	1	1	yes	characteristic
175	C₈	C₄×Q₈×C₇₃	1	1	yes	characteristic
176	C₈	#125×C₇₃	1	1	yes	characteristic
177	C₈	C₄⋊Q₈×C₇₃	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#428×C₇₃	1	1	yes	characteristic
180	C₂²	#460×C₇₃	1	1	yes	characteristic
181	C₂²	#476×C₇₃	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	2	no	—
184	C₄	—	16	2	no	—
185	C₄	#389×C₇₃	1	1	yes	characteristic
186	C₄	C₂×#106×C₇₃	1	1	yes	characteristic
187	C₄	#461×C₇₃	1	1	yes	characteristic
188	C₄	#475×C₇₃	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₇₃×#1948	1	1	yes	characteristic
192	C₂	C₇₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₃	#476
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. #364×C₇₃	C₄	semidirect
92. #13187	C₇₃	direct (N × H)
102. C₇₃	#13187	direct (N × H)
112. #364	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×#365×C₇₃	2
C₂×#364×C₇₃	2
C₇₃×#1846	2
C₇₃×#1772	2
C₇₃×#1386	2
#13187	73

Subgroups — C73×#13187 #32922591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13187 #32922591