Finite Group Databasebeta

← #30770244

Subgroups — #30770244

Order 17152.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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#22562456C211yesmaximal, characteristic, complemented
2#22562382C211yesmaximal, characteristic
3#22562291C211yesmaximal, characteristic
4#22562050C211yesmaximal, characteristic
5#22558276C211yesmaximal, characteristic, complemented
6#22558247C211yesmaximal, characteristic, complemented
7C67×#1366C211yesmaximal, characteristic, complemented
8#16863839—21no—
9#16863814—21no—
10#16863776C2211yescharacteristic
11#16863514C2211yescharacteristic
12#16863218—21no—
13#16863217C2211yescharacteristic
14#16863213—21no—
15#16863205—21no—
16#16863199—21no—
17#16863199—21no—
18#16863181—21no—
19#16863172C2211yescharacteristic
20#499×C67C2211yescharacteristic
21#364×C67C2211yescharacteristic
22#364×C67C2211yescharacteristic
23#14524065D411yescharacteristic
24C2×Dic268—21no—
25C2×Dic268—21no—
26#14524049—21no—
27#14524043—21no—
28Dic536—41no—
29#14523988—41no—
30#14523983—21no—
31#14523982—41no—
32C2×C67⋊C16D411yescharacteristic
33C2×C67⋊C16D411yescharacteristic
34C16×D67—41no—
35#14523979D411yescharacteristic
36#14523971—21no—
37#125×C67—21no—
38M32×C67—21no—
39M32×C67—21no—
40#107×C67—21no—
41#106×C67—21no—
42C4×C8×C67C2311yescharacteristic
43#11761628—21no—
44C2×Dic134—41no—
45Dic268—41no—
46Dic268—41no—
47Dic268—41no—
48Dic268—41no—
49Dic268SD1611yescharacteristic
50#11761617—21no—
51D536SD1611yescharacteristic
52#11761612—21no—
53#11761611—41no—
54C8×D67—41no—
55C4⋊Dic67—81no—
56C67⋊C16—21no—
57C67⋊C16—21no—
58C67⋊C16—41no—
59C2×C8×C67C2×D411yescharacteristic
60C2×C8×C67C2×D411yescharacteristic
61C4⋊C4×C67—41no—
62C4⋊C4×C67—81no—
63C42×C67#4111yescharacteristic
64C16×C67—41no—
65C16×C67—41no—
66C2×C4×C67—21no—
67C2×C4×C67—81no—
68C2×C4×C67#12111yescharacteristic
69C22⋊D67—41no—
70C2×Dic67—81no—
71D268—21no—
72C4×D67—41no—
73Dic134—21no—
74Dic134—81no—
75Dic134—81no—
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#46311yescharacteristic
86C4×C67#46511yescharacteristic
87Dic67—41no—
88Dic67—161no—
89#12466—671nomaximal, Sylow
90C2×C67—21no—
91C2×C67#167711yescharacteristic
92D67—81no—
93#2245—671no—
94#2231—671no—
95#2214—671no—
96#2171—671no—
97#1369—671no—
98#1366D6711yescharacteristic, complemented
99#1358—671no—
100C67#1246611yescharacteristic, complemented, Sylow
101#510—1341no—
102#504—1341no—
103#499C2×D6711yescharacteristic
104#494—671no—
105#443—671no—
106#365—1341no—
107#364—671no—
108#364C2×D6711yescharacteristic
109#364C2×D6711yescharacteristic
110#362—1341no—
111#360—1341no—
112#358—1341no—
113#358—1341no—
114#350—1341no—
115#346—671no—
116#135—671no—
117C2×Q16—1341no—
118C2×Q16—1341no—
119#131—1341no—
120C4⋊Q8—1341no—
121#125—21no—
122Q32—2681no—
123SD32—2681no—
124M32—21no—
125M32—21no—
126M32—2681no—
127M32—1341no—
128C2×C16—2681no—
129C2×C16—671no—
130C2×C16—671no—
131#108—671no—
132#107—21no—
133#106—21no—
134#104—1341no—
135C4×C8C22×D6711yescharacteristic
136#41—1341no—
137C2×Q8—2681no—
138Q16—2681no—
139Q16—2681no—
140Q16—2681no—
141Q16—2681no—
142Q16—671no—
143SD16—1341no—
144D8—671no—
145M16—2681no—
146M16—1341no—
147C2×C8—2681no—
148C2×C8D4×D6711yescharacteristic
149C2×C8D4×D6711yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—5361no—
153C42#1176162811yescharacteristic
154C16—41no—
155C16—41no—
156C16—2681no—
157C16—1341no—
158C16—1341no—
159Q8—5361no—
160Q8—5361no—
161Q8—1341no—
162D4—2681no—
163D4—1341no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—2681no—
167C2×C4—5361no—
168C2×C4#1452401611yescharacteristic
169C8—2681no—
170C8SD16×D6711yescharacteristic
171C8#1452406611yescharacteristic
172C8#1452406611yescharacteristic
173C8#1452407011yescharacteristic
174C22—2681no—
175C22#1686366511yescharacteristic
176C4—41no—
177C4—2681no—
178C4—161no—
179C4—10721no—
180C4#1686361411yescharacteristic
181C4#1686362811yescharacteristic
182C2—21no—
183C2—5361no—
184C2#2255962111yescharacteristic
subgroupG/H
centerC2#22559621
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1366C2
socleC2×C67#1677
prime radical#14524065D4
hypercenter#1366D67
hypocenterC67#12466
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
1. #22562456C2semidirect
5. #22558276C2semidirect
6. #22558247C2semidirect
7. C67×#1366C2semidirect
98. #1366D67semidirect
100. C67#12466semidirect

Maximal subgroups

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

subgroupindex
#225624562
#225623822
#225622912
#225620502
#225582762
#225582472
C67×#13662
#1246667