Subgroups — #32979678

Order 18688.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610095	C₂	1	1	yes	maximal, characteristic
2	#22608214	C₂	1	1	yes	maximal, characteristic
3	#22607791	C₂	1	4	yes	maximal
4	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic
5	#17988640	C₂²	1	2	yes	—
6	C₂×#14556525	—	2	1	no	—
7	C₂×#14556523	—	2	1	no	—
8	#17988453	C₂²	1	2	yes	—
9	#17988308	C₄	1	2	yes	complemented
10	C₈×#10596831	C₄	1	2	yes	complemented
11	C₂×#108×C₇₃	—	2	1	no	—
12	C₂×#108×C₇₃	—	2	1	no	—
13	#423×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₃	—	2	2	no	—
16	#14556525	—	4	2	no	—
17	#14556523	—	4	2	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₂²⋊#10596831	—	2	2	no	—
20	C₄×#10596831	C₂×C₄	1	4	yes	—
21	#14556498	—	2	4	no	—
22	C₂×M₁₆×C₇₃	—	2	2	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
26	#108×C₇₃	—	4	2	no	—
27	#108×C₇₃	—	4	2	no	—
28	C₄⋊C₈×C₇₃	—	2	4	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₃	—	4	2	no	—
34	C₂×#10596831	—	2	4	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	#41	1	2	yes	—
39	M₁₆×C₇₃	—	4	2	no	—
40	M₁₆×C₇₃	—	4	2	no	—
41	C₂×C₈×C₇₃	—	4	2	no	—
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	D₈	1	2	yes	—
47	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
50	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇₃	#41	1	2	yes	—
53	C₂³×C₇₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#117	1	2	yes	—
57	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#125	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₃	—	8	4	no	—
64	C₂×Dic₇₃	—	8	2	no	—
65	C₈×C₇₃	—	8	2	no	—
66	C₈×C₇₃	#106	1	2	yes	—
67	C₈×C₇₃	#106	1	2	yes	—
68	C₈×C₇₃	#107	1	2	yes	—
69	C₈×C₇₃	#107	1	2	yes	—
70	#10596831	—	4	4	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	#382	1	1	yes	characteristic
73	C₂²×C₇₃	#486	1	1	yes	characteristic
74	C₂²×C₇₃	#487	1	1	yes	characteristic
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
78	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₇₃	#486	1	1	yes	characteristic
80	C₄×C₇₃	#487	1	1	yes	characteristic
81	Dic₇₃	—	16	2	no	—
82	#8793	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	#1833	1	2	yes	—
86	C₂×C₇₃	#1834	1	1	yes	characteristic
87	#1848	D₇₃	1	1	yes	characteristic
88	#1846	—	73	1	no	—
89	#1450	—	73	1	no	—
90	#1286	—	73	4	no	—
91	C₇₃	#8793	1	1	yes	characteristic, complemented, Sylow
92	#431	—	73	2	no	—
93	C₂×#108	—	2	1	no	—
94	C₂×#108	—	2	1	no	—
95	C₂×#107	—	146	1	no	—
96	C₂×#106	—	146	1	no	—
97	#423	C₂×D₇₃	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	#382	—	73	2	no	—
100	#322	—	73	2	no	—
101	C₈²	—	73	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	146	2	no	—
106	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
107	#108	—	4	2	no	—
108	#108	—	4	2	no	—
109	#107	—	292	2	no	—
110	#106	—	292	2	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	146	2	no	—
113	C₂²⋊C₈	—	146	2	no	—
114	C₄×C₈	—	73	4	no	—
115	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
116	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
117	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
118	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
119	#95	—	146	4	no	—
120	C₂²×C₄	—	292	2	no	—
121	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂²×C₄	#11924029	1	2	yes	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	146	4	no	—
127	C₂×C₈	#11924007	1	1	yes	characteristic
128	C₂×C₈	#11924007	1	1	yes	characteristic
129	C₂×C₈	#11924007	1	1	yes	characteristic
130	C₂×C₈	#11924007	1	1	yes	characteristic
131	C₂×C₈	#11924013	1	2	yes	—
132	C₂×C₈	#11924015	1	2	yes	—
133	C₂×C₈	#11924016	1	2	yes	—
134	C₂×C₈	#11924018	1	2	yes	—
135	C₄⋊C₄	—	292	2	no	—
136	C₄⋊C₄	—	292	2	no	—
137	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
139	C₄²	#11924027	1	2	yes	—
140	C₂³	#14556580	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	584	4	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
145	C₂×C₄	#14556549	1	2	yes	—
146	C₂×C₄	#14556578	1	1	yes	characteristic
147	C₂×C₄	#14556582	1	2	yes	—
148	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
149	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
150	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
151	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	292	4	no	—
154	C₈	#14556522	1	2	yes	—
155	C₈	#14556522	1	2	yes	—
156	C₈	#14556524	1	2	yes	—
157	C₈	#14556524	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#17988452	1	1	yes	characteristic
160	C₂²	#17988935	1	1	yes	characteristic
161	C₂²	#17988939	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	1168	2	no	—
165	C₄	C₂×#14556522	1	1	yes	characteristic
166	C₄	C₂×#14556524	1	1	yes	characteristic
167	C₄	#17988933	1	1	yes	characteristic
168	C₄	#17988937	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#22610056	1	2	yes	—
172	C₂	#22610058	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988452
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	#382
prime radical	C₂³×C₇₃	#124
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#8793
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #17988308	C₄	semidirect
10. C₈×#10596831	C₄	semidirect
91. C₇₃	#8793	semidirect

Maximal subgroups

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

subgroup	index
#22610095	2
#22608214	2
#22607791	2
C₇₃×#1848	2
#8793	73