Subgroups — #33000992

Order 18688.

338 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
2	#22609617	C₂	1	1	yes	maximal, characteristic
3	#22608010	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556531	C₂²	1	2	yes	—
6	C₂×#14556502	C₄	1	1	yes	characteristic
7	C₂×#14556501	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
9	#17988433	C₄	1	1	yes	characteristic
10	#17988363	C₄	1	2	yes	complemented
11	C₄×#11924003	C₄	1	2	yes	complemented
12	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#346×C₇₃	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₇₃	C₄	1	2	yes	complemented
16	C₂²×#10596831	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
18	#14556531	—	2	4	no	—
19	C₂×#11924003	C₂×C₄	1	4	yes	—
20	#14556502	—	2	2	no	—
21	#14556501	—	2	4	no	—
22	C₈×Dic₇₃	C₈	1	8	yes	—
23	#14556498	—	2	2	no	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₇₃	—	2	4	no	—
28	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₇₃	—	2	2	no	—
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₂×C₈	1	4	yes	—
37	C₄×Dic₇₃	—	2	2	no	—
38	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
39	#11924003	—	2	4	no	—
40	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	#41	1	4	yes	—
48	C₄²×C₇₃	C₄²	1	1	yes	characteristic
49	C₄²×C₇₃	C₄²	1	1	yes	characteristic
50	C₄²×C₇₃	M₁₆	1	2	yes	—
51	C₁₆×C₇₃	—	2	4	no	—
52	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	4	2	no	—
54	C₂×C₄×C₇₃	#97	1	2	yes	—
55	C₂×C₄×C₇₃	#97	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#117	1	2	yes	—
58	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×Dic₇₃	—	2	4	no	—
61	C₂×Dic₇₃	—	4	4	no	—
62	C₈×C₇₃	C₄×C₈	1	2	yes	—
63	C₈×C₇₃	C₄×C₈	1	2	yes	—
64	C₈×C₇₃	#117	1	2	yes	—
65	C₈×C₇₃	#117	1	2	yes	—
66	#10596831	—	2	4	no	—
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
69	C₂²×C₇₃	#431	1	2	yes	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	2	2	no	—
72	C₄×C₇₃	#376	1	1	yes	characteristic
73	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₇₃	#431	1	2	yes	—
75	Dic₇₃	—	4	4	no	—
76	#12593	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	#1744	1	1	yes	characteristic
79	C₂×C₇₃	#2110	1	2	yes	—
80	C₄×M₃₂	—	73	1	no	—
81	C₄×M₃₂	D₇₃	1	1	yes	characteristic
82	#1726	—	73	1	no	—
83	#1375	—	73	4	no	—
84	C₇₃	#12593	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	73	2	no	—
86	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
87	C₂×#97	—	73	2	no	—
88	C₂×#97	—	73	1	no	—
89	C₂×C₄×C₈	—	73	2	no	—
90	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
91	#376	—	73	1	no	—
92	#346	—	73	2	no	—
93	#346	Dic₇₃	1	2	yes	complemented
94	C₄×C₁₆	—	73	2	no	—
95	C₄×C₁₆	Dic₇₃	1	2	yes	complemented
96	C₂²×C₈	—	73	2	no	—
97	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
99	C₂×C₄²	—	73	2	no	—
100	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	146	4	no	—
103	C₂×C₁₆	—	73	4	no	—
104	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
105	#97	—	146	4	no	—
106	#97	—	146	2	no	—
107	C₄×C₈	—	73	8	no	—
108	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
109	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
112	#95	—	146	2	no	—
113	C₂²×C₄	—	146	2	no	—
114	C₂²×C₄	#11924011	1	2	yes	—
115	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	—	146	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
121	C₂×C₈	C₈×D₇₃	1	4	yes	—
122	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
123	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
124	C₂×C₈	#11924028	1	4	yes	—
125	C₄²	—	73	4	no	—
126	C₄²	—	146	2	no	—
127	C₄²	C₄×Dic₇₃	1	1	yes	characteristic
128	C₄²	C₄×Dic₇₃	1	1	yes	characteristic
129	C₄²	#11924011	1	2	yes	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	146	4	no	—
132	C₂³	C₂×#11924011	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	292	4	no	—
135	C₂×C₄	—	146	4	no	—
136	C₂×C₄	#14556501	1	2	yes	—
137	C₂×C₄	#14556501	1	2	yes	—
138	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
139	C₂×C₄	#14556548	1	2	yes	—
140	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
141	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
142	C₈	—	146	4	no	—
143	C₈	C₈×Dic₇₃	1	2	yes	—
144	C₈	C₈×Dic₇₃	1	2	yes	—
145	C₈	#14556548	1	2	yes	—
146	C₈	#14556550	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#14556501	1	1	yes	characteristic
149	C₂²	#17988638	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	292	4	no	—
153	C₄	#17988433	1	1	yes	characteristic
154	C₄	C₂×C₈×Dic₇₃	1	1	yes	characteristic
155	C₄	#17988638	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#22609676	1	1	yes	characteristic
158	C₂	#22611535	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#12593
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #17988363	C₄	semidirect
11. C₄×#11924003	C₄	semidirect
14. #346×C₇₃	C₄	semidirect
15. C₄×C₁₆×C₇₃	C₄	semidirect
84. C₇₃	#12593	semidirect
93. #346	Dic₇₃	semidirect
95. C₄×C₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556531	2
#22609617	2
#22608010	2
C₄×M₃₂×C₇₃	2
#12593	73