Finite Group Databasebeta

← #30758930

Subgroups — #30758930

Order 17152.

438 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 97 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#22568572C211yesmaximal, characteristic, complemented
2#22561037C211yesmaximal, characteristic
3#22558770C212yesmaximal
4#22558068C212yesmaximal, complemented
5C67×#1276C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7C2×#14524035—21no—
8#16863420C2211yescharacteristic
9C2×#14523971—21no—
10C2×#14523971—21no—
11#16863294—24no—
12#16863266—21no—
13#16863262C2212yes—
14#16863131—24no—
15#431×C67C2212yes—
16C43×C67C2211yescharacteristic
17C2×#11761628—21no—
18C2×#11761612—21no—
19#14524043Q811yescharacteristic
20#14524040Q811yescharacteristic
21#14524035—42no—
22#14524035—42no—
23#14524031—21no—
24#14524017—22no—
25C4×D268—22no—
26C2×C4⋊Dic67—24no—
27C2×C4⋊Dic67—22no—
28C2×Dic67⋊C4—24no—
29C4⋊(C67⋊C8)—22no—
30#14523971—42no—
31#14523971—42no—
32C67⋊#95—22no—
33C67⋊#95D414yes—
34C2×C42×C67—22no—
35C2×C42×C67D411yescharacteristic
36C2×C42×C67D411yescharacteristic
37C2×C42×C67C2311yescharacteristic
38C4⋊C8×C67—22no—
39C22⋊C8×C67—22no—
40C4×C8×C67—22no—
41#11761628—41no—
42#11761628—41no—
43#11761628—41no—
44#11761628—41no—
45C2×Dic134—21no—
46C2×D268—21no—
47C2×D268—22no—
48C2×C22⋊D67—41no—
49C22×Dic67—24no—
50C2×C4×D67—41no—
51#11761612—41no—
52#11761612—41no—
53C2×C67⋊C8—41no—
54C4⋊Dic67—22no—
55C4⋊Dic67—48no—
56C4⋊Dic67—42no—
57C4⋊Dic67—42no—
58Dic67⋊C4—48no—
59(C2×D67)⋊C4—42no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67C2×D412yes—
63C22×C4×C67C2×D411yescharacteristic
64C2×C8×C67—42no—
65C42×C67—21no—
66C42×C67—21no—
67C42×C67—44no—
68C42×C67D812yes—
69C42×C67SD1612yes—
70C42×C67C2×Q811yescharacteristic
71C42×C67#4111yescharacteristic
72C42×C67#4112yes—
73C23×C67#12011yescharacteristic
74C22×D67—41no—
75C2×C4×C67—24no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67C22⋊Q812yes—
81C2×C4×C67C22⋊Q811yescharacteristic
82C2×C4×C67#12412yes—
83C2×C4×C67#12411yescharacteristic
84C2×C4×C67C2×D811yescharacteristic
85C2×C4×C67C2×SD1611yescharacteristic
86C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C67#13711yescharacteristic
88C22⋊D67—81no—
89C22⋊D67—81no—
90C2×Dic67—48no—
91C2×Dic67—41no—
92C2×Dic67—44no—
93D268—41no—
94D268—41no—
95D268—82no—
96C4×D67—81no—
97C4×D67—81no—
98Dic134—41no—
99Dic134—41no—
100C8×C67—82no—
101C67⋊C8—81no—
102C22×C67—22no—
103C22×C67#39311yescharacteristic
104C22×C67#44711yescharacteristic
105C22×C67#44811yescharacteristic
106C2×D67—81no—
107C2×D67—82no—
108C4×C67—24no—
109C4×C67—22no—
110C4×C67—22no—
111C4×C67—42no—
112C4×C67#47411yescharacteristic
113C4×C67#47611yescharacteristic
114C4×C67#48611yescharacteristic
115C4×C67#48911yescharacteristic
116Dic67—84no—
117Dic67—81no—
118#10364—671nomaximal, Sylow
119C2×C67—21no—
120C2×C67—21no—
121C2×C67#202311yescharacteristic
122C2×C67#203112yes—
123D67—161no—
124#2859—671no—
125#1962—671no—
126#1517—672no—
127#1286—672no—
128#1276D6711yescharacteristic, complemented
129C67#1036411yescharacteristic, complemented, Sylow
130#532—671no—
131C2×#125—1341no—
132#431C2×D6712yes—
133#423—671no—
134C2×#104—1341no—
135C2×#104—1341no—
136#389—1344no—
137#384—1341no—
138#382—672no—
139C43C2×D6711yescharacteristic
140#327—1344no—
141C2×#41—1341no—
142C2×M16—1341no—
143C4⋊Q8—671no—
144#127—671no—
145#125—2682no—
146#125—2682no—
147#124—1341no—
148#121—1342no—
149C4×D4—1342no—
150C2×C4⋊C4—1344no—
151C2×C4⋊C4—1344no—
152C2×C4⋊C4—1342no—
153C2×C42—22no—
154C2×C42D26811yescharacteristic
155C2×C42D26811yescharacteristic
156C2×C42C22×D6711yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1342no—
159#104—2682no—
160#104—2682no—
161C22⋊C8—22no—
162C4×C8—22no—
163#95—674no—
164#95—1342no—
165#41—2681no—
166#41—2681no—
167#41—2681no—
168#41—2681no—
169C2×Q8—1341no—
170C2×D4—2681no—
171C2×D4—1342no—
172C2×D4—1341no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—2681no—
176C22×C4—1344no—
177C22×C4D4×D6712yes—
178C22×C4C2×D26811yescharacteristic
179M16—2681no—
180M16—2681no—
181C2×C8—42no—
182C2×C8—2681no—
183C4⋊C4—2688no—
184C4⋊C4—2688no—
185C4⋊C4—2682no—
186C4⋊C4—2682no—
187C4⋊C4—1342no—
188C22⋊C4—2682no—
189C42—21no—
190C42—21no—
191C42—44no—
192C42D53612yes—
193C42#1176161712yes—
194C42Q8×D6711yescharacteristic
195C42#1176162711yescharacteristic
196C42#1176162812yes—
197C23—2681no—
198C23#1452401311yescharacteristic
199Q8—2681no—
200Q8—2681no—
201D4—2681no—
202D4—2681no—
203D4—5362no—
204D4—5361no—
205D4—5361no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—2688no—
212C2×C4—2684no—
213C2×C4—2681no—
214C2×C4—5361no—
215C2×C4—5361no—
216C2×C4#1452402112yes—
217C2×C4#1452402311yescharacteristic
218C2×C4#1452402912yes—
219C2×C4#1452403111yescharacteristic
220C2×C4C2×D53611yescharacteristic
221C2×C4C2×#1176161711yescharacteristic
222C2×C4#1452407111yescharacteristic
223C2×C4#1452407611yescharacteristic
224C8—82no—
225C8—5361no—
226C22—22no—
227C22—5361no—
228C22—5362no—
229C22#1686330911yescharacteristic
230C22#1686353211yescharacteristic
231C22#1686353711yescharacteristic
232C4—24no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—5364no—
237C4—5361no—
238C4#1686367711yescharacteristic
239C4#1686368711yescharacteristic
240C4#1686374311yescharacteristic
241C4#1686375611yescharacteristic
242C2—21no—
243C2—21no—
244C2—10721no—
245C2#2256134611yescharacteristic
246C2#2256138212yes—
subgroupG/H
centerC22#16863309
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1276C2
socleC22×C67#393
prime radical#16863946C22
hypercenter#1276D67
hypocenterC67#10364
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #22568572C2semidirect
4. #22558068C2semidirect
5. C67×#1276C2semidirect
128. #1276D67semidirect
129. C67#10364semidirect

Maximal subgroups

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

subgroupindex
#225685722
#225610372
#225587702
#225580682
C67×#12762
#1036467