Subgroups — C₇₃×#62652 #32972056

Order 18688.

11674 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₄×H(5,𝔽₂)×C₇₃	C₂	1	32	yes	maximal, complemented
3	C₇₃×#3428	C₂	1	12	yes	maximal, complemented
4	C₂×#518×C₇₃	C₂	1	18	yes	maximal, complemented
5	C₂×H(5,𝔽₂)×C₇₃	C₄	1	32	yes	complemented
6	C₂×H(5,𝔽₂)×C₇₃	C₂²	1	16	yes	—
7	C₂²×#41×C₇₃	C₂²	1	6	yes	—
8	C₂³×D₄×C₇₃	C₂²	1	9	yes	—
9	#520×C₇₃	C₂²	1	96	yes	complemented
10	#519×C₇₃	C₂²	1	8	yes	complemented
11	#518×C₇₃	C₂²	1	72	yes	complemented
12	#518×C₇₃	C₂²	1	144	yes	complemented
13	C₄×#41×C₇₃	C₂²	1	96	yes	complemented
14	C₂×C₄×D₄×C₇₃	C₂²	1	144	yes	complemented
15	C₂×#117×C₇₃	C₂²	1	36	yes	complemented
16	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	24	yes	complemented
17	C₂⁵×C₇₃	C₂³	1	6	yes	—
18	H(5,𝔽₂)×C₇₃	C₂×C₄	1	256	yes	complemented
19	C₂×#41×C₇₃	C₂×C₄	1	96	yes	complemented
20	C₂×#41×C₇₃	C₂³	1	48	yes	—
21	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
22	C₂²×D₄×C₇₃	C₂×C₄	1	144	yes	complemented
23	C₂²×D₄×C₇₃	C₂³	1	72	yes	—
24	C₂²×D₄×C₇₃	C₂³	1	18	yes	—
25	C₂³×C₄×C₇₃	C₂³	1	9	yes	—
26	C₂³×C₄×C₇₃	C₂³	1	48	yes	complemented
27	C₄×Q₈×C₇₃	C₂³	1	64	yes	complemented
28	C₄×D₄×C₇₃	C₂³	1	576	yes	complemented
29	#117×C₇₃	C₂³	1	48	yes	—
30	#117×C₇₃	C₂³	1	144	yes	complemented
31	C₂×C₄⋊C₄×C₇₃	C₂³	1	72	yes	complemented
32	C₂×C₂²⋊C₄×C₇₃	C₂³	1	72	yes	—
33	C₂×C₂²⋊C₄×C₇₃	C₂³	1	144	yes	complemented
34	C₂×C₄²×C₇₃	C₂³	1	72	yes	complemented
35	C₂⁴×C₇₃	—	8	12	no	—
36	C₂⁴×C₇₃	C₂²×C₄	1	48	yes	complemented
37	C₂⁴×C₇₃	C₂⁴	1	9	yes	—
38	C₂⁴×C₇₃	C₂⁴	1	24	yes	—
39	#41×C₇₃	C₂²×C₄	1	384	yes	complemented
40	C₂×Q₈×C₇₃	C₂²×C₄	1	16	yes	complemented
41	C₂×Q₈×C₇₃	C₂⁴	1	8	yes	—
42	C₂×D₄×C₇₃	C₂²×C₄	1	144	yes	complemented
43	C₂×D₄×C₇₃	C₂²×C₄	1	576	yes	complemented
44	C₂×D₄×C₇₃	C₂⁴	1	72	yes	—
45	C₂²×C₄×C₇₃	—	8	48	no	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	72	yes	complemented
47	C₂²×C₄×C₇₃	C₂⁴	1	144	yes	—
48	C₂²×C₄×C₇₃	C₂⁴	1	36	yes	—
49	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
50	C₄⋊C₄×C₇₃	C₂⁴	1	96	yes	—
51	C₂²⋊C₄×C₇₃	C₂⁴	1	16	yes	—
52	C₂²⋊C₄×C₇₃	C₂⁴	1	144	yes	—
53	C₄²×C₇₃	C₂⁴	1	96	yes	—
54	C₂³×C₇₃	—	4	48	no	—
55	C₂³×C₇₃	—	4	18	no	—
56	C₂³×C₇₃	—	8	48	no	—
57	C₂³×C₇₃	C₂³×C₄	1	36	yes	—
58	C₂³×C₇₃	C₂³×C₄	1	96	yes	—
59	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
60	C₂³×C₇₃	C₂⁵	1	18	yes	—
61	Q₈×C₇₃	C₂³×C₄	1	32	yes	—
62	D₄×C₇₃	C₂³×C₄	1	288	yes	—
63	C₂×C₄×C₇₃	—	4	144	no	—
64	C₂×C₄×C₇₃	C₂³×C₄	1	24	yes	—
65	C₂×C₄×C₇₃	C₂³×C₄	1	144	yes	—
66	C₂×C₄×C₇₃	C₂⁵	1	32	yes	—
67	C₂×C₄×C₇₃	C₂⁵	1	12	yes	—
68	C₂²×C₇₃	—	2	36	no	—
69	C₂²×C₇₃	—	2	2	no	—
70	C₂²×C₇₃	—	4	36	no	—
71	C₂²×C₇₃	—	4	96	no	—
72	C₂²×C₇₃	C₂⁴×C₄	1	2	yes	—
73	C₂²×C₇₃	C₂⁴×C₄	1	36	yes	—
74	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
75	C₄×C₇₃	—	2	32	no	—
76	C₄×C₇₃	C₂⁴×C₄	1	24	yes	—
77	#62652	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	—	2	36	no	—
80	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
81	C₂×C₇₃	H(7,𝔽₂)	1	2	yes	—
82	C₂²×H(5,𝔽₂)	C₂×C₇₃	1	1	yes	characteristic
83	C₄×H(5,𝔽₂)	C₂×C₇₃	1	32	yes	complemented
84	#3428	C₂×C₇₃	1	12	yes	complemented
85	C₂×#518	C₂×C₇₃	1	18	yes	complemented
86	C₇₃	#62652	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×H(5,𝔽₂)	C₄×C₇₃	1	32	yes	complemented
88	C₂×H(5,𝔽₂)	C₂²×C₇₃	1	16	yes	—
89	C₂²×#41	C₂²×C₇₃	1	6	yes	—
90	C₂³×D₄	C₂²×C₇₃	1	9	yes	—
91	#520	C₂²×C₇₃	1	96	yes	complemented
92	#519	C₂²×C₇₃	1	8	yes	complemented
93	#518	C₂²×C₇₃	1	72	yes	complemented
94	#518	C₂²×C₇₃	1	144	yes	complemented
95	C₄×#41	C₂²×C₇₃	1	96	yes	complemented
96	C₂×C₄×D₄	C₂²×C₇₃	1	144	yes	complemented
97	C₂×#117	C₂²×C₇₃	1	36	yes	complemented
98	C₂²×C₂²⋊C₄	C₂²×C₇₃	1	24	yes	complemented
99	C₂⁵	C₂³×C₇₃	1	6	yes	—
100	H(5,𝔽₂)	C₂×C₄×C₇₃	1	256	yes	complemented
101	C₂×#41	C₂×C₄×C₇₃	1	96	yes	complemented
102	C₂×#41	C₂³×C₇₃	1	48	yes	—
103	C₂²×Q₈	C₂³×C₇₃	1	2	yes	—
104	C₂²×D₄	C₂×C₄×C₇₃	1	144	yes	complemented
105	C₂²×D₄	C₂³×C₇₃	1	72	yes	—
106	C₂²×D₄	C₂³×C₇₃	1	18	yes	—
107	C₂³×C₄	C₂³×C₇₃	1	9	yes	—
108	C₂³×C₄	C₂³×C₇₃	1	48	yes	complemented
109	C₄×Q₈	C₂³×C₇₃	1	64	yes	complemented
110	C₄×D₄	C₂³×C₇₃	1	576	yes	complemented
111	#117	C₂³×C₇₃	1	48	yes	—
112	#117	C₂³×C₇₃	1	144	yes	complemented
113	C₂×C₄⋊C₄	C₂³×C₇₃	1	72	yes	complemented
114	C₂×C₂²⋊C₄	C₂³×C₇₃	1	72	yes	—
115	C₂×C₂²⋊C₄	C₂³×C₇₃	1	144	yes	complemented
116	C₂×C₄²	C₂³×C₇₃	1	72	yes	complemented
117	C₂⁴	—	8	12	no	—
118	C₂⁴	C₂²×C₄×C₇₃	1	48	yes	complemented
119	C₂⁴	C₂⁴×C₇₃	1	9	yes	—
120	C₂⁴	C₂⁴×C₇₃	1	24	yes	—
121	#41	C₂²×C₄×C₇₃	1	384	yes	complemented
122	C₂×Q₈	C₂²×C₄×C₇₃	1	16	yes	complemented
123	C₂×Q₈	C₂⁴×C₇₃	1	8	yes	—
124	C₂×D₄	C₂²×C₄×C₇₃	1	144	yes	complemented
125	C₂×D₄	C₂²×C₄×C₇₃	1	576	yes	complemented
126	C₂×D₄	C₂⁴×C₇₃	1	72	yes	—
127	C₂²×C₄	—	8	48	no	—
128	C₂²×C₄	C₂²×C₄×C₇₃	1	72	yes	complemented
129	C₂²×C₄	C₂⁴×C₇₃	1	144	yes	—
130	C₂²×C₄	C₂⁴×C₇₃	1	36	yes	—
131	C₂²×C₄	C₂⁴×C₇₃	1	6	yes	—
132	C₄⋊C₄	C₂⁴×C₇₃	1	96	yes	—
133	C₂²⋊C₄	C₂⁴×C₇₃	1	16	yes	—
134	C₂²⋊C₄	C₂⁴×C₇₃	1	144	yes	—
135	C₄²	C₂⁴×C₇₃	1	96	yes	—
136	C₂³	—	4	48	no	—
137	C₂³	—	4	18	no	—
138	C₂³	—	8	48	no	—
139	C₂³	C₂³×C₄×C₇₃	1	36	yes	—
140	C₂³	C₂³×C₄×C₇₃	1	96	yes	—
141	C₂³	C₂⁵×C₇₃	1	1	yes	characteristic
142	C₂³	C₂⁵×C₇₃	1	18	yes	—
143	Q₈	C₂³×C₄×C₇₃	1	32	yes	—
144	D₄	C₂³×C₄×C₇₃	1	288	yes	—
145	C₂×C₄	—	4	144	no	—
146	C₂×C₄	C₂³×C₄×C₇₃	1	24	yes	—
147	C₂×C₄	C₂³×C₄×C₇₃	1	144	yes	—
148	C₂×C₄	C₂⁵×C₇₃	1	32	yes	—
149	C₂×C₄	C₂⁵×C₇₃	1	12	yes	—
150	C₂²	—	2	36	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	36	no	—
153	C₂²	—	4	96	no	—
154	C₂²	C₂⁴×C₄×C₇₃	1	2	yes	—
155	C₂²	C₂⁴×C₄×C₇₃	1	36	yes	—
156	C₂²	C₂⁶×C₇₃	1	1	yes	characteristic
157	C₄	—	2	32	no	—
158	C₄	C₂⁴×C₄×C₇₃	1	24	yes	—
159	C₂	—	2	2	no	—
160	C₂	—	2	36	no	—
161	C₂	C₂⁵×C₄×C₇₃	1	1	yes	characteristic
162	C₂	C₇₃×H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂⁶
commutator	C₂	C₂⁵×C₄×C₇₃
Frattini, cosocle	C₂²	C₂⁶×C₇₃
prime radical	C₂²×H(5,𝔽₂)×C₇₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×H(5,𝔽₂)×C₇₃	C₂	semidirect
3. C₇₃×#3428	C₂	semidirect
4. C₂×#518×C₇₃	C₂	semidirect
5. C₂×H(5,𝔽₂)×C₇₃	C₄	semidirect
9. #520×C₇₃	C₂²	semidirect
10. #519×C₇₃	C₂²	semidirect
11. #518×C₇₃	C₂²	semidirect
12. #518×C₇₃	C₂²	semidirect
13. C₄×#41×C₇₃	C₂²	semidirect
14. C₂×C₄×D₄×C₇₃	C₂²	semidirect
15. C₂×#117×C₇₃	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₇₃	C₂²	semidirect
18. H(5,𝔽₂)×C₇₃	C₂×C₄	semidirect
19. C₂×#41×C₇₃	C₂×C₄	semidirect
22. C₂²×D₄×C₇₃	C₂×C₄	semidirect
26. C₂³×C₄×C₇₃	C₂³	semidirect
27. C₄×Q₈×C₇₃	C₂³	semidirect
28. C₄×D₄×C₇₃	C₂³	semidirect
30. #117×C₇₃	C₂³	semidirect
31. C₂×C₄⋊C₄×C₇₃	C₂³	semidirect
33. C₂×C₂²⋊C₄×C₇₃	C₂³	semidirect
34. C₂×C₄²×C₇₃	C₂³	semidirect
36. C₂⁴×C₇₃	C₂²×C₄	semidirect
39. #41×C₇₃	C₂²×C₄	semidirect
40. C₂×Q₈×C₇₃	C₂²×C₄	semidirect
42. C₂×D₄×C₇₃	C₂²×C₄	semidirect
43. C₂×D₄×C₇₃	C₂²×C₄	semidirect
46. C₂²×C₄×C₇₃	C₂²×C₄	semidirect
77. #62652	C₇₃	direct (N × H)
83. C₄×H(5,𝔽₂)	C₂×C₇₃	semidirect
84. #3428	C₂×C₇₃	semidirect
85. C₂×#518	C₂×C₇₃	semidirect
86. C₇₃	#62652	direct (N × H)
87. C₂×H(5,𝔽₂)	C₄×C₇₃	semidirect
91. #520	C₂²×C₇₃	semidirect
92. #519	C₂²×C₇₃	semidirect
93. #518	C₂²×C₇₃	semidirect
94. #518	C₂²×C₇₃	semidirect
95. C₄×#41	C₂²×C₇₃	semidirect
96. C₂×C₄×D₄	C₂²×C₇₃	semidirect
97. C₂×#117	C₂²×C₇₃	semidirect
98. C₂²×C₂²⋊C₄	C₂²×C₇₃	semidirect
100. H(5,𝔽₂)	C₂×C₄×C₇₃	semidirect
101. C₂×#41	C₂×C₄×C₇₃	semidirect
104. C₂²×D₄	C₂×C₄×C₇₃	semidirect
108. C₂³×C₄	C₂³×C₇₃	semidirect
109. C₄×Q₈	C₂³×C₇₃	semidirect
110. C₄×D₄	C₂³×C₇₃	semidirect
112. #117	C₂³×C₇₃	semidirect
113. C₂×C₄⋊C₄	C₂³×C₇₃	semidirect
115. C₂×C₂²⋊C₄	C₂³×C₇₃	semidirect
116. C₂×C₄²	C₂³×C₇₃	semidirect
118. C₂⁴	C₂²×C₄×C₇₃	semidirect
121. #41	C₂²×C₄×C₇₃	semidirect
122. C₂×Q₈	C₂²×C₄×C₇₃	semidirect
124. C₂×D₄	C₂²×C₄×C₇₃	semidirect
125. C₂×D₄	C₂²×C₄×C₇₃	semidirect
128. C₂²×C₄	C₂²×C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₇₃	2
C₄×H(5,𝔽₂)×C₇₃	2
C₇₃×#3428	2
C₂×#518×C₇₃	2
#62652	73

Subgroups — C73×#62652 #32972056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#62652 #32972056