Finite Group Databasebeta

← #30770228

Subgroups — #30770228

Order 17152.

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

normal subgroups only

This group has 172 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#22562485C211yesmaximal, characteristic, complemented
2#22562386C211yesmaximal, characteristic
3#22562382C211yesmaximal, characteristic
4#22562045C211yesmaximal, characteristic
5#22558284C211yesmaximal, characteristic, complemented
6#22558271C211yesmaximal, characteristic, complemented
7C67×#1366C211yesmaximal, characteristic, complemented
8#16863814—21no—
9#16863798C2211yescharacteristic
10#16863514C2211yescharacteristic
11#16863223—21no—
12#16863221—21no—
13#16863219—21no—
14#16863218—21no—
15#16863217C2211yescharacteristic
16#16863207—21no—
17#16863205—21no—
18#16863181—21no—
19C4×C67⋊C16C2211yescharacteristic
20#499×C67C2211yescharacteristic
21#364×C67C2211yescharacteristic
22#364×C67C2211yescharacteristic
23#14524065D411yescharacteristic
24#14524049—21no—
25#14524035—21no—
26#14523983—21no—
27#14523982—41no—
28C2×C67⋊C16D411yescharacteristic
29C2×C67⋊C16D411yescharacteristic
30C16×D67—41no—
31#14523979D411yescharacteristic
32#14523977—21no—
33#14523975—21no—
34#14523971—21no—
35#125×C67—21no—
36M32×C67—21no—
37M32×C67—21no—
38#107×C67—21no—
39#106×C67—21no—
40C4×C8×C67C2311yescharacteristic
41#11761628—21no—
42Dic268Q1611yescharacteristic
43#11761617—21no—
44D536Q1611yescharacteristic
45#11761612—21no—
46#11761611—41no—
47C8×D67—41no—
48C4⋊Dic67—41no—
49Dic67⋊C4—81no—
50C67⋊C16—21no—
51C67⋊C16—41no—
52C67⋊C16SD1611yescharacteristic
53C67⋊C16SD1611yescharacteristic
54C2×C8×C67C2×D411yescharacteristic
55C2×C8×C67C2×D411yescharacteristic
56C4⋊C4×C67—41no—
57C4⋊C4×C67—81no—
58C42×C67#4111yescharacteristic
59C16×C67—41no—
60C16×C67—41no—
61C2×C4×C67—21no—
62C2×C4×C67—81no—
63C2×C4×C67#12111yescharacteristic
64C22⋊D67—41no—
65C2×Dic67—81no—
66D268—21no—
67C4×D67—41no—
68Dic134—21no—
69C8×C67C2×SD1611yescharacteristic
70C8×C67C2×Q1611yescharacteristic
71C8×C67#13511yescharacteristic
72C8×C67#13711yescharacteristic
73C67⋊C8—41no—
74C22×C67#47211yescharacteristic
75C2×D67—41no—
76C4×C67—41no—
77C4×C67—161no—
78C4×C67#46111yescharacteristic
79C4×C67#46711yescharacteristic
80Dic67—41no—
81Dic67—161no—
82#12464—671nomaximal, Sylow
83C2×C67—21no—
84C2×C67#167511yescharacteristic
85D67—81no—
86#2253—671no—
87#2231—671no—
88#2231—671no—
89#2170—671no—
90#1373—671no—
91#1366—671no—
92#1366D6711yescharacteristic, complemented
93C67#1246411yescharacteristic, complemented, Sylow
94#504—1341no—
95#499—671no—
96#499C2×D6711yescharacteristic
97#443—671no—
98#367—1341no—
99#366—1341no—
100#365—1341no—
101#365—1341no—
102#364—671no—
103#364C2×D6711yescharacteristic
104#364C2×D6711yescharacteristic
105#360—1341no—
106#360—1341no—
107#350—1341no—
108C4×C16—671no—
109#135—671no—
110#131—1341no—
111#125—21no—
112#125—1341no—
113M32—21no—
114M32—21no—
115M32—2681no—
116M32—1341no—
117C2×C16—2681no—
118C2×C16—671no—
119C2×C16—671no—
120#108—671no—
121#107—21no—
122#107—1341no—
123#106—21no—
124#106—1341no—
125#104—1341no—
126C4×C8C22×D6711yescharacteristic
127#41—1341no—
128Q16—671no—
129SD16—1341no—
130D8—671no—
131M16—2681no—
132M16—1341no—
133C2×C8—2681no—
134C2×C8D4×D6711yescharacteristic
135C2×C8D4×D6711yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—81no—
138C4⋊C4—2681no—
139C4⋊C4—5361no—
140C42#1176162811yescharacteristic
141C16—41no—
142C16—41no—
143C16—2681no—
144C16—671no—
145C16—671no—
146C16—1341no—
147Q8—1341no—
148D4—2681no—
149D4—1341no—
150C2×C4—21no—
151C2×C4—81no—
152C2×C4—2681no—
153C2×C4—5361no—
154C2×C4#1452401611yescharacteristic
155C8—2681no—
156C8SD16×D6711yescharacteristic
157C8Q16×D6711yescharacteristic
158C8#1452406411yescharacteristic
159C8#1452407511yescharacteristic
160C22—2681no—
161C22#1686366511yescharacteristic
162C4—41no—
163C4—2681no—
164C4—161no—
165C4—10721no—
166C4#1686360011yescharacteristic
167C4#1686363611yescharacteristic
168C2—21no—
169C2—5361no—
170C2#2255960711yescharacteristic
subgroupG/H
centerC2#22559607
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1366C2
socleC2×C67#1675
prime radical#14524065D4
hypercenter#1366D67
hypocenterC67#12464
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. #22562485C2semidirect
5. #22558284C2semidirect
6. #22558271C2semidirect
7. C67×#1366C2semidirect
92. #1366D67semidirect
93. C67#12464semidirect

Maximal subgroups

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

subgroupindex
#225624852
#225623862
#225623822
#225620452
#225582842
#225582712
C67×#13662
#1246467