Finite Group Databasebeta

Subgroups — #33002435

Order 18688.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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×#17988416	C₂	1	4	yes	maximal
2	C₂×#17988363	C₂	1	1	yes	maximal, characteristic
3	#22610103	C₂	1	1	yes	maximal, characteristic
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	#17988416	—	2	8	no	—
10	#17988363	C₄	1	4	yes	—
11	C₂×#108×C₇₃	—	2	1	no	—
12	C₂×#108×C₇₃	C₂²	1	2	yes	—
13	#423×C₇₃	—	2	1	no	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924012	—	2	2	no	—
16	C₂×#11924003	C₂×C₄	1	4	yes	—
17	C₂×#11924003	D₄	1	4	yes	—
18	C₂×#11924003	Q₈	1	4	yes	—
19	#14556527	—	2	4	no	—
20	#14556527	—	4	2	no	—
21	C₄⋊#10596831	—	4	2	no	—
22	C₂×M₁₆×C₇₃	—	2	2	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
26	#108×C₇₃	—	2	4	no	—
27	#108×C₇₃	—	4	2	no	—
28	C₄⋊C₈×C₇₃	—	4	2	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₃	Q₈	1	2	yes	—
31	#11924012	—	4	2	no	—
32	#11924012	—	4	2	no	—
33	C₂×#10596831	—	4	2	no	—
34	#11924003	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₇₃	Q₁₆	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₇₃	—	4	2	no	—
38	M₁₆×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	—	4	2	no	—
40	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₃	D₈	1	2	yes	—
42	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₇₃	D₈	1	2	yes	—
49	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	#107	1	4	yes	—
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₇₃	—	8	2	no	—
59	C₈×C₇₃	#107	1	4	yes	—
60	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
62	#10596831	—	8	2	no	—
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₇₃	#494	1	1	yes	characteristic
66	C₂²×C₇₃	#500	1	1	yes	characteristic
67	C₄×C₇₃	—	2	4	no	—
68	C₄×C₇₃	#384	1	1	yes	characteristic
69	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
70	C₄×C₇₃	#493	1	1	yes	characteristic
71	C₄×C₇₃	#500	1	1	yes	characteristic
72	#12910	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	1	no	—
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	#1845	1	1	yes	characteristic
76	C₂×C₇₃	#2161	1	2	yes	—
77	C₂×#368	—	73	4	no	—
78	C₂×#346	—	73	1	no	—
79	#1848	—	73	1	no	—
80	#1848	D₇₃	1	1	yes	characteristic
81	C₇₃	#12910	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₁₆	—	73	2	no	—
83	C₂×#108	—	2	1	no	—
84	C₂×#108	—	73	2	no	—
85	C₂×#108	—	146	1	no	—
86	C₂×#108	C₂×D₇₃	1	2	yes	—
87	#423	—	2	1	no	—
88	#423	—	146	1	no	—
89	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
90	#368	—	146	8	no	—
91	#346	—	73	4	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	146	2	no	—
94	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
97	C₂×C₁₆	—	73	4	no	—
98	C₂×C₁₆	—	73	4	no	—
99	C₂×C₁₆	—	73	4	no	—
100	#108	—	2	4	no	—
101	#108	—	4	2	no	—
102	#108	—	292	2	no	—
103	#108	—	146	4	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	292	2	no	—
106	C₄×C₈	Dic₁₄₆	1	2	yes	—
107	C₄×C₈	C₄×D₇₃	1	2	yes	—
108	C₂²×C₄	#11924018	1	2	yes	—
109	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	292	2	no	—
113	M₁₆	—	292	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	292	2	no	—
116	C₂×C₈	#11924007	1	2	yes	—
117	C₂×C₈	#11924013	1	2	yes	—
118	C₂×C₈	#11924018	1	2	yes	—
119	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
120	C₂×C₈	D₄×D₇₃	1	2	yes	—
121	C₂×C₈	Q₈×D₇₃	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
123	C₄²	#11924007	1	2	yes	—
124	C₄²	#11924013	1	2	yes	—
125	C₁₆	—	73	8	no	—
126	C₂³	C₂×#11924018	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	#14556524	1	4	yes	—
129	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
130	C₂×C₄	#14556587	1	1	yes	characteristic
131	C₂×C₄	#14556589	1	1	yes	characteristic
132	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
133	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
134	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
135	C₈	—	8	2	no	—
136	C₈	—	584	2	no	—
137	C₈	#14556524	1	4	yes	—
138	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
139	C₈	#14556590	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	C₂×#14556524	1	1	yes	characteristic
142	C₂²	#17988966	1	1	yes	characteristic
143	C₂²	#17988990	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	#17988456	1	1	yes	characteristic
146	C₄	C₂×#14556524	1	1	yes	characteristic
147	C₄	#17988963	1	1	yes	characteristic
148	C₄	#17988990	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#22610090	1	1	yes	characteristic
152	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	#12910
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
81. C₇₃	#12910	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988416	2
C₂×#17988363	2
#22610103	2
C₇₃×#1848	2
#12910	73