Finite Group Databasebeta

← #30755219

Subgroups — #30755219

Order 17152.

408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 115 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#22570532C211yesmaximal, characteristic
2#22560373C211yesmaximal, characteristic, complemented
3#22558774C212yesmaximal, complemented
4#22558770C212yesmaximal
5C67×#1452C211yesmaximal, characteristic, complemented
6#16863953C2211yescharacteristic
7#16863946C2211yescharacteristic
8#16863751—22no—
9#16863747C412yescomplemented
10#16863743C412yescomplemented
11C2×#14523977—21no—
12C2×#14523975—21no—
13#16863393—21no—
14#16863393—21no—
15#16863262C2212yes—
16#16863134—24no—
17#16863131—24no—
18#431×C67C2212yes—
19C2×C4×C8×C67C2211yescharacteristic
20#322×C67—22no—
21C2×#11761628—21no—
22#14524043C2×C411yescharacteristic
23#14524043C2×C412yes—
24#14524040C2×C411yescharacteristic
25#14524035—21no—
26#14524031—21no—
27#14524022—22no—
28#14524017—22no—
29C4×Dic134—22no—
30C4×D268—22no—
31#14524002—21no—
32C2×C4⋊Dic67—22no—
33#14523977—42no—
34#14523975—42no—
35#14523968—22no—
36#14523968—22no—
37#14523965—22no—
38#14523965—22no—
39C67⋊#95—22no—
40C67⋊#95—22no—
41C22×C8×C67D411yescharacteristic
42C22×C8×C67D411yescharacteristic
43C2×C42×C67C2311yescharacteristic
44C4⋊C8×C67—22no—
45C22⋊C8×C67D414yes—
46C4×C8×C67—22no—
47C4×C8×C67C2×C412yes—
48C4×C8×C67D412yes—
49#11761628—41no—
50#11761628—41no—
51#11761628—41no—
52#11761628—41no—
53C2×Dic134—21no—
54C2×Dic134—22no—
55C2×D268—21no—
56C2×D268—22no—
57C2×C22⋊D67—41no—
58C22×Dic67—42no—
59C2×C4×D67—41no—
60C4⋊Dic67—22no—
61C4⋊Dic67—22no—
62C4⋊Dic67—22no—
63C4⋊Dic67—42no—
64C4⋊Dic67—42no—
65Dic67⋊C4—42no—
66C22⋊Dic67—41no—
67(C2×D67)⋊C4—42no—
68C4×Dic67—41no—
69C22×C4×C67C2×D412yes—
70C22×C4×C67C2×D411yescharacteristic
71C2×C8×C67—21no—
72C2×C8×C67—21no—
73C2×C8×C67—21no—
74C2×C8×C67—21no—
75C2×C8×C67—24no—
76C2×C8×C67C22⋊C412yes—
77C2×C8×C67C22⋊C412yes—
78C42×C67C22×C411yescharacteristic
79C42×C67C2×D411yescharacteristic
80C42×C67#4112yes—
81C23×C67#12711yescharacteristic
82C22×D67—41no—
83C2×C4×C67—42no—
84C2×C4×C67C2×C22⋊C411yescharacteristic
85C2×C4×C67C4×D412yes—
86C2×C4×C67#12112yes—
87C2×C4×C67#12411yescharacteristic
88C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C67#13711yescharacteristic
91C2×C4×C67#13711yescharacteristic
92C22⋊D67—81no—
93C22⋊D67—81no—
94C2×Dic67—42no—
95C2×Dic67—41no—
96C2×Dic67—84no—
97C2×Dic67—82no—
98D268—41no—
99D268—41no—
100D268—82no—
101C4×D67—81no—
102C4×D67—81no—
103Dic134—41no—
104Dic134—41no—
105Dic134—82no—
106C8×C67—22no—
107C8×C67—22no—
108C8×C67—44no—
109C22×C67—22no—
110C22×C67#39011yescharacteristic
111C22×C67#49611yescharacteristic
112C22×C67#49711yescharacteristic
113C2×D67—81no—
114C2×D67—82no—
115C4×C67—22no—
116C4×C67—22no—
117C4×C67#41711yescharacteristic
118C4×C67#41711yescharacteristic
119C4×C67#48911yescharacteristic
120C4×C67#48911yescharacteristic
121Dic67—81no—
122Dic67—82no—
123Dic67—162no—
124#9758—671nomaximal, Sylow
125C2×C67—21no—
126C2×C67—21no—
127C2×C67#195911yescharacteristic
128C2×C67#197612yes—
129D67—161no—
130#3131—671no—
131#1846—671no—
132#1518—672no—
133#1517—672no—
134#1452D6711yescharacteristic, complemented
135C67#975811yescharacteristic, complemented, Sylow
136#533—671no—
137#532—671no—
138#488—1342no—
139#487—672no—
140#486—672no—
141#431C2×D6712yes—
142C2×#107—1341no—
143C2×#106—1341no—
144#416—1341no—
145#416—1341no—
146C2×C4×C8C2×D6711yescharacteristic
147#382—672no—
148#328—1344no—
149#327—1344no—
150#322—22no—
151C2×#41—1341no—
152C22×C8D26811yescharacteristic
153C22×C8D26811yescharacteristic
154C4⋊Q8—671no—
155C4⋊Q8—672no—
156#127—671no—
157#125—1341no—
158#124—1341no—
159C22⋊Q8—1342no—
160#121—1342no—
161C4×Q8—1342no—
162C4×D4—1342no—
163#117—1341no—
164C2×C4⋊C4—1342no—
165C2×C42C22×D6711yescharacteristic
166#107—2682no—
167#106—2682no—
168C4⋊C8—22no—
169#103—1342no—
170#103—1342no—
171#102—1342no—
172#102—1342no—
173C22⋊C8D26814yes—
174C4×C8—22no—
175C4×C8C4×D6712yes—
176C4×C8C22⋊D6712yes—
177#95—1342no—
178#95—1342no—
179#41—2681no—
180#41—2681no—
181#41—2681no—
182#41—2681no—
183C2×Q8—1341no—
184C2×Q8—1342no—
185C2×D4—2681no—
186C2×D4—1341no—
187C2×D4—1342no—
188C22×C4—2681no—
189C22×C4—2682no—
190C22×C4C2×D26812yes—
191C22×C4C2×D26811yescharacteristic
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—24no—
197C2×C8(C2×D67)⋊C412yes—
198C2×C8(C2×D67)⋊C412yes—
199C4⋊C4—2682no—
200C4⋊C4—2682no—
201C4⋊C4—2682no—
202C4⋊C4—1342no—
203C4⋊C4—1342no—
204C4⋊C4—1342no—
205C22⋊C4—2681no—
206C22⋊C4—2682no—
207C42—2681no—
208C42C2×C4×D6711yescharacteristic
209C42C2×C22⋊D6711yescharacteristic
210C42#1176162812yes—
211C23—2681no—
212C23#1452404011yescharacteristic
213Q8—2681no—
214Q8—2681no—
215Q8—5362no—
216D4—2681no—
217D4—2681no—
218D4—5362no—
219D4—5361no—
220D4—5361no—
221C2×C4—42no—
222C2×C4—2682no—
223C2×C4—2681no—
224C2×C4—5361no—
225C2×C4—5361no—
226C2×C4—5364no—
227C2×C4—5362no—
228C2×C4C2×(C2×D67)⋊C411yescharacteristic
229C2×C4C4×D26812yes—
230C2×C4#1452401712yes—
231C2×C4#1452403111yescharacteristic
232C2×C4#1452407111yescharacteristic
233C2×C4#1452407111yescharacteristic
234C2×C4#1452407611yescharacteristic
235C2×C4#1452407611yescharacteristic
236C8—22no—
237C8—22no—
238C8—44no—
239C22—22no—
240C22—5361no—
241C22—5362no—
242C22#1686329811yescharacteristic
243C22#1686378511yescharacteristic
244C22#1686379011yescharacteristic
245C4—22no—
246C4—22no—
247C4—5361no—
248C4—5362no—
249C4—10722no—
250C4#1686339811yescharacteristic
251C4#1686339811yescharacteristic
252C4#1686375611yescharacteristic
253C4#1686375611yescharacteristic
254C2—21no—
255C2—21no—
256C2—10721no—
257C2#2256102511yescharacteristic
258C2#2256110712yes—
subgroupG/H
centerC22#16863298
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1452C2
socleC22×C67#390
prime radical#16863946C22
cosocleC2×C42×C67C23
hypercenter#1452D67
hypocenterC67#9758
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #22560373C2semidirect
3. #22558774C2semidirect
5. C67×#1452C2semidirect
9. #16863747C4semidirect
10. #16863743C4semidirect
134. #1452D67semidirect
135. C67#9758semidirect

Maximal subgroups

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

subgroupindex
#225705322
#225603732
#225587742
#225587702
C67×#14522
#975867