Finite Group Databasebeta

← #30758931

Subgroups — #30758931

Order 17152.

400 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 101 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#22561186C211yesmaximal, characteristic
2#22558772C212yesmaximal
3#22558067C212yesmaximal
4#22558042C211yesmaximal, characteristic
5C67×#2859C211yesmaximal, characteristic
6#16863953C2211yescharacteristic
7#16863449C2212yes—
8#16863420C2211yescharacteristic
9#16863156—21no—
10#16863156—21no—
11#16863132—24no—
12C2×#125×C67—21no—
13#389×C67—24no—
14#384×C67—21no—
15#382×C67C2212yes—
16C43×C67C2211yescharacteristic
17C2×#11761612—21no—
18#14524043Q812yes—
19#14524035—21no—
20#14524022—22no—
21C4×Dic134—22no—
22#14524002—21no—
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—22no—
27#125×C67—42no—
28#125×C67—42no—
29C2×C4⋊C4×C67—24no—
30C2×C4⋊C4×C67—24no—
31C2×C4⋊C4×C67—22no—
32C2×C42×C67—22no—
33C2×C42×C67D411yescharacteristic
34C2×C42×C67D411yescharacteristic
35C2×C42×C67C2311yescharacteristic
36#95×C67—22no—
37#95×C67D414yes—
38C2×Dic134—22no—
39#11761612—41no—
40#11761612—41no—
41C2×C67⋊C8—42no—
42C2×C67⋊C8—41no—
43C4⋊Dic67—22no—
44C4⋊Dic67—22no—
45Dic67⋊C4—42no—
46C22⋊Dic67—41no—
47C4×Dic67—41no—
48C22×C4×C67—21no—
49C22×C4×C67—21no—
50C22×C4×C67—24no—
51C22×C4×C67C2×D412yes—
52C22×C4×C67C2×D411yescharacteristic
53C4⋊C4×C67—48no—
54C4⋊C4×C67—48no—
55C4⋊C4×C67—42no—
56C4⋊C4×C67—42no—
57C42×C67—21no—
58C42×C67—21no—
59C42×C67—44no—
60C42×C67SD1612yes—
61C42×C67Q1612yes—
62C42×C67C2×Q811yescharacteristic
63C42×C67#4111yescharacteristic
64C42×C67#4112yes—
65C23×C67#12011yescharacteristic
66C2×C4×C67—24no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—48no—
72C2×C4×C67—44no—
73C2×C4×C67C22⋊Q812yes—
74C2×C4×C67C22⋊Q811yescharacteristic
75C2×C4×C67#12412yes—
76C2×C4×C67#12411yescharacteristic
77C2×C4×C67C2×SD1611yescharacteristic
78C2×C4×C67C2×Q1611yescharacteristic
79C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C67#13711yescharacteristic
81C2×Dic67—42no—
82Dic134—82no—
83C67⋊C8—82no—
84C67⋊C8—81no—
85C22×C67—22no—
86C22×C67#39311yescharacteristic
87C22×C67#45011yescharacteristic
88C22×C67#45111yescharacteristic
89C4×C67—24no—
90C4×C67—22no—
91C4×C67—22no—
92C4×C67—42no—
93C4×C67—84no—
94C4×C67#47511yescharacteristic
95C4×C67#47711yescharacteristic
96C4×C67#48711yescharacteristic
97C4×C67#48911yescharacteristic
98Dic67—82no—
99#10365—671nomaximal, Sylow
100C2×C67—21no—
101C2×C67—21no—
102C2×C67#202411yescharacteristic
103C2×C67#203112yes—
104#2859D6711yescharacteristic
105#1994—671no—
106#1518—672no—
107#1286—672no—
108#1276—671no—
109C67#1036511yescharacteristic, complemented, Sylow
110#533—671no—
111C2×#125—21no—
112#431—672no—
113#423—671no—
114#389—24no—
115#384—21no—
116#382C2×D6712yes—
117C43C2×D6711yescharacteristic
118#339—1341no—
119#339—1341no—
120#328—1344no—
121C2×M16—1341no—
122C4⋊Q8—672no—
123#125—42no—
124#125—42no—
125#125—1341no—
126C22⋊Q8—1342no—
127C4×Q8—1342no—
128#117—1341no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C42—22no—
133C2×C42C22⋊D6711yescharacteristic
134C2×C42C22⋊D6711yescharacteristic
135C2×C42C22×D6711yescharacteristic
136C4⋊C8—1342no—
137C4⋊C8—1342no—
138C22⋊C8—1342no—
139C4×C8—1342no—
140#95—22no—
141#95C22⋊D6714yes—
142C2×Q8—1342no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4C2×C22⋊D6712yes—
147C22×C4C2×C22⋊D6711yescharacteristic
148M16—2681no—
149M16—2681no—
150C2×C8—2682no—
151C2×C8—2681no—
152C4⋊C4—48no—
153C4⋊C4—48no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—2682no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C22⋊C4—2681no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2681no—
164C42#1176161612yes—
165C42#1176161812yes—
166C42Q8×D6711yescharacteristic
167C42#1176162911yescharacteristic
168C42#1176162912yes—
169C23#1452401211yescharacteristic
170Q8—5362no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—48no—
177C2×C4—44no—
178C2×C4—2682no—
179C2×C4#1452401912yes—
180C2×C4#1452401911yescharacteristic
181C2×C4#1452403212yes—
182C2×C4#1452403211yescharacteristic
183C2×C4C2×#1176161611yescharacteristic
184C2×C4C2×#1176161811yescharacteristic
185C2×C4#1452406911yescharacteristic
186C2×C4#1452407411yescharacteristic
187C8—5362no—
188C8—5361no—
189C22—22no—
190C22#1686330711yescharacteristic
191C22#1686354711yescharacteristic
192C22#1686355211yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—84no—
198C4—5362no—
199C4#1686367811yescharacteristic
200C4#1686368811yescharacteristic
201C4#1686374811yescharacteristic
202C4#1686375811yescharacteristic
203C2—21no—
204C2—21no—
205C2#2256134711yescharacteristic
206C2#2256137812yes—
subgroupG/H
centerC22#16863307
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2859C2
socleC22×C67#393
prime radicalC23×C67#120
hypercenter#2859D67
hypocenterC67#10365
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2859

Complements

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

NH ≅ G/Ntype
109. C67#10365semidirect

Maximal subgroups

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

subgroupindex
#225611862
#225587722
#225580672
#225580422
C67×#28592
#1036567