Finite Group Databasebeta

← #30755215

Subgroups — #30755215

Order 17152.

400 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 119 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#22560340C211yesmaximal, characteristic, complemented
2#22558773C212yesmaximal, complemented
3#22558769C212yesmaximal
4#22558492C211yesmaximal, characteristic, complemented
5C67×#3131C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863262C2212yes—
8#16863159—22no—
9#16863133—24no—
10#16863130—24no—
11#16863117—21no—
12C8×C67⋊C8—21no—
13#533×C67C2211yescharacteristic
14#532×C67C2211yescharacteristic
15#488×C67—21no—
16#488×C67—21no—
17#487×C67C412yescomplemented
18#486×C67C412yescomplemented
19#416×C67—22no—
20C2×C4×C8×C67C2211yescharacteristic
21C2×C4⋊Dic67—22no—
22C4⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)D414yes—
24C4×C67⋊C8—22no—
25C67⋊#95—22no—
26C67⋊#95—22no—
27C2×#41×C67—21no—
28C22×C8×C67D412yes—
29C4⋊Q8×C67C2×C412yes—
30C4⋊Q8×C67C2×C411yescharacteristic
31#127×C67C2×C411yescharacteristic
32#125×C67—21no—
33#124×C67—21no—
34C22⋊Q8×C67—22no—
35#121×C67—22no—
36C4×Q8×C67—22no—
37C4×D4×C67—22no—
38#117×C67—21no—
39C2×C42×C67C2311yescharacteristic
40#103×C67—22no—
41#103×C67—22no—
42#102×C67—22no—
43#102×C67—22no—
44C4×C8×C67C2×C412yes—
45C4×C8×C67D411yescharacteristic
46C4×C8×C67D411yescharacteristic
47C22×Dic67—42no—
48C2×C67⋊C8—24no—
49C4⋊Dic67—44no—
50#41×C67—44no—
51C2×Q8×C67—21no—
52C2×Q8×C67—22no—
53C2×D4×C67—21no—
54C2×D4×C67—22no—
55C2×D4×C67—41no—
56C22×C4×C67—41no—
57C22×C4×C67C2×D412yes—
58C22×C4×C67C2×D411yescharacteristic
59C2×C8×C67—24no—
60C2×C8×C67C22⋊C412yes—
61C2×C8×C67C22⋊C412yes—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—22no—
64C4⋊C4×C67—22no—
65C4⋊C4×C67—42no—
66C22⋊C4×C67—41no—
67C22⋊C4×C67—42no—
68C42×C67—41no—
69C42×C67C22×C411yescharacteristic
70C42×C67C2×D411yescharacteristic
71C42×C67#4112yes—
72C23×C67—41no—
73C23×C67#12711yescharacteristic
74Q8×C67—42no—
75Q8×C67—82no—
76D4×C67—42no—
77D4×C67—82no—
78D4×C67—82no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—41no—
82C2×C4×C67—82no—
83C2×C4×C67C2×C22⋊C411yescharacteristic
84C2×C4×C67C4×D412yes—
85C2×C4×C67#12112yes—
86C2×C4×C67#12411yescharacteristic
87C2×C4×C67#13514yes—
88C2×Dic67—84no—
89C2×Dic67—82no—
90C8×C67—22no—
91C8×C67—22no—
92C67⋊C8—44no—
93C22×C67—22no—
94C22×C67—81no—
95C22×C67—82no—
96C22×C67#39011yescharacteristic
97C22×C67#49512yes—
98C4×C67—22no—
99C4×C67—22no—
100C4×C67—81no—
101C4×C67—82no—
102C4×C67#41612yes—
103C4×C67#48812yes—
104Dic67—162no—
105#9757—671nomaximal, Sylow
106C2×C67—22no—
107C2×C67—161no—
108C2×C67#195811yescharacteristic
109C2×C67#197612yes—
110#3131D6711yescharacteristic
111#1835—671no—
112#1518—672no—
113#1517—672no—
114#1450—671no—
115C67#975711yescharacteristic, complemented, Sylow
116#533C2×D6711yescharacteristic
117#532C2×D6711yescharacteristic
118#488—21no—
119#488—21no—
120#487Dic6712yescomplemented
121#486Dic6712yescomplemented
122#431—672no—
123#416—22no—
124C2×C4×C8C2×D6711yescharacteristic
125#382—672no—
126#340—1342no—
127#328—1344no—
128#327—1344no—
129#322—1341no—
130C82—1341no—
131C2×#41—21no—
132C22×C8C22⋊D6712yes—
133C4⋊Q8C2×Dic6711yescharacteristic
134C4⋊Q8C2×Dic6712yes—
135#127C2×Dic6711yescharacteristic
136#125—21no—
137#124—21no—
138C22⋊Q8—22no—
139#121—22no—
140C4×Q8—22no—
141C4×D4—22no—
142#117—21no—
143C2×C4⋊C4—1342no—
144C2×C42C22×D6711yescharacteristic
145C4⋊C8—1342no—
146#103—22no—
147#103—22no—
148#102—22no—
149#102—22no—
150C22⋊C8—674no—
151C4×C8—1342no—
152C4×C8C2×Dic6712yes—
153C4×C8C22⋊D6711yescharacteristic
154C4×C8C22⋊D6711yescharacteristic
155#95—1342no—
156#95—1342no—
157#41—44no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C22×C4—41no—
164C22×C4—2682no—
165C22×C4D4×D6712yes—
166C22×C4C2×C22⋊D6711yescharacteristic
167C2×C8—24no—
168C2×C8—1344no—
169C2×C8C22⋊Dic6712yes—
170C2×C8C22⋊Dic6712yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—2684no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—41no—
179C42C22×Dic6711yescharacteristic
180C42C2×C22⋊D6711yescharacteristic
181C42#1176162712yes—
182C23—41no—
183C23#1452403911yescharacteristic
184Q8—42no—
185Q8—82no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—5364no—
194C2×C4—5362no—
195C2×C4C2×C22⋊Dic6711yescharacteristic
196C2×C4D4×Dic6712yes—
197C2×C4#1452401412yes—
198C2×C4#1452403011yescharacteristic
199C2×C4#1452406314yes—
200C8—22no—
201C8—22no—
202C8—2684no—
203C22—22no—
204C22—82no—
205C22—81no—
206C22#1686329911yescharacteristic
207C22#1686377812yes—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—81no—
212C4—10722no—
213C4#1686339212yes—
214C4#1686375012yes—
215C2—22no—
216C2—161no—
217C2#2256102111yescharacteristic
218C2#2256110812yes—
subgroupG/H
centerC22#16863299
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#3131C2
socleC22×C67#390
prime radical#532×C67C22
cosocleC2×C42×C67C23
hypercenter#3131D67
hypocenterC67#9757
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #22560340C2semidirect
2. #22558773C2semidirect
4. #22558492C2semidirect
17. #487×C67C4semidirect
18. #486×C67C4semidirect
115. C67#9757semidirect
120. #487Dic67semidirect
121. #486Dic67semidirect

Maximal subgroups

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

subgroupindex
#225603402
#225587732
#225587692
#225584922
C67×#31312
#975767