Finite Group Databasebeta

← #30770238

Subgroups — #30770238

Order 17152.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22562382C211yesmaximal, characteristic
2#22562287C211yesmaximal, characteristic
3#22562052C211yesmaximal, characteristic
4#22558277C211yesmaximal, characteristic, complemented
5#22558270C211yesmaximal, characteristic, complemented
6#22558248C211yesmaximal, characteristic, complemented
7C67×#2245C211yesmaximal, characteristic, complemented
8#16863837—21no—
9#16863812—21no—
10#16863514C2211yescharacteristic
11#16863218—21no—
12#16863217C2211yescharacteristic
13#16863217C2211yescharacteristic
14#16863217C2211yescharacteristic
15#16863212—21no—
16#16863205—21no—
17#16863179—21no—
18#499×C67C2211yescharacteristic
19#494×C67C2211yescharacteristic
20#358×C67—21no—
21#358×C67—21no—
22#346×C67C2211yescharacteristic
23#14524065D411yescharacteristic
24#14524049—21no—
25#14523989—41no—
26#14523986—41no—
27#14523983—21no—
28#14523983—21no—
29#14523983—21no—
30#14523982—41no—
31C16×D67—41no—
32#14523979D411yescharacteristic
33#14523971—21no—
34C2×Q16×C67—21no—
35C2×Q16×C67—21no—
36C4⋊Q8×C67—21no—
37#125×C67—21no—
38C2×C16×C67D411yescharacteristic
39C2×C16×C67D411yescharacteristic
40#107×C67—21no—
41#106×C67—21no—
42C4×C8×C67C2311yescharacteristic
43#11761628—21no—
44Dic268SD1611yescharacteristic
45#11761617—21no—
46D536SD1611yescharacteristic
47#11761612—21no—
48#11761611—41no—
49C8×D67—41no—
50C67⋊C16—41no—
51C67⋊C16—41no—
52C67⋊C16—41no—
53C2×Q8×C67—41no—
54Q16×C67—41no—
55Q16×C67—41no—
56Q16×C67—41no—
57Q16×C67—41no—
58C2×C8×C67C2×D411yescharacteristic
59C2×C8×C67C2×D411yescharacteristic
60C4⋊C4×C67—41no—
61C4⋊C4×C67—81no—
62C4⋊C4×C67—81no—
63C42×C67#4111yescharacteristic
64C16×C67—21no—
65C16×C67—21no—
66Q8×C67—81no—
67Q8×C67—81no—
68C2×C4×C67—21no—
69C2×C4×C67—81no—
70C2×C4×C67—81no—
71C2×C4×C67#12111yescharacteristic
72C22⋊D67—41no—
73D268—21no—
74C4×D67—41no—
75Dic134—21no—
76C8×C67C2×SD1611yescharacteristic
77C8×C67#13511yescharacteristic
78C8×C67#13511yescharacteristic
79C8×C67AGL(1,ℤ/8)11yescharacteristic
80C67⋊C8—41no—
81C22×C67#47211yescharacteristic
82C2×D67—41no—
83C4×C67—41no—
84C4×C67—161no—
85C4×C67—161no—
86C4×C67#46311yescharacteristic
87C4×C67#46511yescharacteristic
88Dic67—41no—
89#12466—671nomaximal, Sylow
90C2×C67—21no—
91C2×C67#167711yescharacteristic
92D67—81no—
93#2245D6711yescharacteristic, complemented
94#2231—671no—
95#2214—671no—
96#2171—671no—
97#1369—671no—
98#1366—671no—
99#1358—671no—
100C67#1246611yescharacteristic, complemented, Sylow
101#510—1341no—
102#504—1341no—
103#499C2×D6711yescharacteristic
104#494C2×D6711yescharacteristic
105#443—671no—
106#365—1341no—
107#364—671no—
108#364—671no—
109#364—671no—
110#362—1341no—
111#360—1341no—
112#358—21no—
113#358—21no—
114#350—1341no—
115#346C2×D6711yescharacteristic
116#135—671no—
117C2×Q16—21no—
118C2×Q16—21no—
119#131—1341no—
120C4⋊Q8—21no—
121#125—21no—
122Q32—2681no—
123SD32—2681no—
124M32—2681no—
125M32—1341no—
126M32—1341no—
127M32—1341no—
128C2×C16—2681no—
129C2×C16C22⋊D6711yescharacteristic
130C2×C16C22⋊D6711yescharacteristic
131#108—671no—
132#107—21no—
133#106—21no—
134#104—1341no—
135C4×C8C22×D6711yescharacteristic
136#41—1341no—
137C2×Q8—41no—
138Q16—41no—
139Q16—41no—
140Q16—41no—
141Q16—41no—
142Q16—671no—
143SD16—1341no—
144D8—671no—
145M16—2681no—
146M16—1341no—
147C2×C8—2681no—
148C2×C8D4×D6711yescharacteristic
149C2×C8C2×C22⋊D6711yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C42#1176162711yescharacteristic
154C16—21no—
155C16—21no—
156C16—2681no—
157C16—2681no—
158C16—2681no—
159Q8—81no—
160Q8—81no—
161Q8—1341no—
162D4—2681no—
163D4—1341no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—2681no—
168C2×C4#1452401411yescharacteristic
169C8—2681no—
170C8SD16×D6711yescharacteristic
171C8#1452406311yescharacteristic
172C8#1452406411yescharacteristic
173C8#1452406811yescharacteristic
174C22—2681no—
175C22#1686366311yescharacteristic
176C4—41no—
177C4—2681no—
178C4—161no—
179C4—161no—
180C4#1686360811yescharacteristic
181C4#1686362211yescharacteristic
182C2—21no—
183C2—5361no—
184C2#2255961511yescharacteristic
subgroupG/H
centerC2#22559615
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2245C2
socleC2×C67#1677
prime radical#14524065D4
hypercenter#2245D67
hypocenterC67#12466
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2245

Complements

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

NH ≅ G/Ntype
4. #22558277C2semidirect
5. #22558270C2semidirect
6. #22558248C2semidirect
7. C67×#2245C2semidirect
93. #2245D67semidirect
100. C67#12466semidirect

Maximal subgroups

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

subgroupindex
#225623822
#225622872
#225620522
#225582772
#225582702
#225582482
C67×#22452
#1246667