Finite Group Databasebeta

Subgroups — #33002376

Order 18688.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988363	C₂	1	1	yes	maximal, characteristic
2	#22610103	C₂	1	1	yes	maximal, characteristic
3	#22608024	C₂	1	4	yes	maximal
4	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924003	C₂²	1	2	yes	—
6	C₂×#14556527	—	2	1	no	—
7	C₂×#14556527	C₂²	1	2	yes	—
8	#17988611	—	2	1	no	—
9	#17988363	C₄	1	4	yes	—
10	C₂×#108×C₇₃	—	2	1	no	—
11	C₂×#108×C₇₃	C₂²	1	2	yes	—
12	#423×C₇₃	—	2	1	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#11924012	—	2	2	no	—
15	C₂×#11924003	—	2	2	no	—
16	C₂×#11924003	—	2	2	no	—
17	C₂×#11924003	C₂×C₄	1	4	yes	—
18	#14556527	—	2	4	no	—
19	#14556527	—	4	2	no	—
20	C₄⋊#10596831	—	4	2	no	—
21	C₂×M₁₆×C₇₃	—	2	2	no	—
22	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
25	#108×C₇₃	—	2	4	no	—
26	#108×C₇₃	—	4	2	no	—
27	C₄⋊C₈×C₇₃	—	4	2	no	—
28	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₃	Q₈	1	2	yes	—
30	#11924012	—	4	2	no	—
31	#11924012	—	4	2	no	—
32	C₂×#10596831	—	4	2	no	—
33	#11924003	—	2	4	no	—
34	C₂²×C₄×C₇₃	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₇₃	—	4	2	no	—
37	M₁₆×C₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	#41	1	2	yes	—
45	C₂×C₈×C₇₃	#41	1	2	yes	—
46	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₇₃	D₈	1	2	yes	—
48	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	#107	1	4	yes	—
51	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₇₃	—	8	2	no	—
58	C₈×C₇₃	#106	1	4	yes	—
59	C₈×C₇₃	#117	1	2	yes	—
60	C₈×C₇₃	#125	1	2	yes	—
61	#10596831	—	8	2	no	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
64	C₂²×C₇₃	#487	1	1	yes	characteristic
65	C₂²×C₇₃	#487	1	1	yes	characteristic
66	C₄×C₇₃	—	2	4	no	—
67	C₄×C₇₃	#382	1	1	yes	characteristic
68	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
69	C₄×C₇₃	#486	1	1	yes	characteristic
70	C₄×C₇₃	#486	1	1	yes	characteristic
71	#12892	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	—	2	1	no	—
74	C₂×C₇₃	#1834	1	1	yes	characteristic
75	C₂×C₇₃	#2161	1	2	yes	—
76	C₂×#346	—	73	1	no	—
77	#1848	—	73	1	no	—
78	#1848	D₇₃	1	1	yes	characteristic
79	#1381	—	73	4	no	—
80	C₇₃	#12892	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₁₆	—	73	2	no	—
82	C₂×#108	—	2	1	no	—
83	C₂×#108	—	73	2	no	—
84	C₂×#108	—	146	1	no	—
85	C₂×#108	C₂×D₇₃	1	2	yes	—
86	#423	—	2	1	no	—
87	#423	—	146	1	no	—
88	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
89	#346	—	73	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	146	2	no	—
92	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
95	C₂×C₁₆	—	73	4	no	—
96	C₂×C₁₆	—	146	2	no	—
97	C₂×C₁₆	—	146	2	no	—
98	#108	—	2	4	no	—
99	#108	—	4	2	no	—
100	#108	—	292	2	no	—
101	#108	—	146	4	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	292	2	no	—
104	C₄×C₈	Dic₁₄₆	1	2	yes	—
105	C₄×C₈	C₄×D₇₃	1	2	yes	—
106	C₂²×C₄	#11924018	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	292	2	no	—
111	M₁₆	—	292	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	292	2	no	—
114	C₂×C₈	#11924007	1	2	yes	—
115	C₂×C₈	#11924015	1	2	yes	—
116	C₂×C₈	#11924016	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
119	C₂×C₈	#11924027	1	2	yes	—
120	C₂×C₈	#11924029	1	2	yes	—
121	C₄²	#11924007	1	2	yes	—
122	C₄²	#11924013	1	2	yes	—
123	C₁₆	—	146	4	no	—
124	C₂³	C₂×#11924018	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#14556524	1	4	yes	—
127	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
128	C₂×C₄	#14556578	1	1	yes	characteristic
129	C₂×C₄	#14556580	1	1	yes	characteristic
130	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
131	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
132	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	584	2	no	—
135	C₈	#14556522	1	4	yes	—
136	C₈	#14556549	1	2	yes	—
137	C₈	#14556582	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#14556524	1	1	yes	characteristic
140	C₂²	#17988937	1	1	yes	characteristic
141	C₂²	#17988939	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	#17988452	1	1	yes	characteristic
144	C₄	C₂×#14556522	1	1	yes	characteristic
145	C₄	#17988933	1	1	yes	characteristic
146	C₄	#17988935	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#22610058	1	1	yes	characteristic
150	C₂	#22611725	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×C₇₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#12892
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
80. C₇₃	#12892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988363	2
#22610103	2
#22608024	2
C₇₃×#1848	2
#12892	73