Finite Group Databasebeta

← #30755214

Subgroups — #30755214

Order 17152.

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

normal subgroups only

This group has 400 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#22560340C211yesmaximal, characteristic, complemented
3#22558774C212yesmaximal, complemented
4#22558770C212yesmaximal
5C67×#1450C211yesmaximal, characteristic, complemented
6#16863953C2211yescharacteristic
7#16863946C2211yescharacteristic
8#16863751—21no—
9#16863751—21no—
10#16863747C412yescomplemented
11#16863743C412yescomplemented
12#16863393—22no—
13#16863262C2212yes—
14#16863159—22no—
15#16863134—24no—
16#16863131—24no—
17#431×C67C2212yes—
18C2×C4×C8×C67C2211yescharacteristic
19#322×C67—21no—
20C82×C67—21no—
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#14523968—22no—
34#14523968—22no—
35#14523965—22no—
36#14523965—22no—
37C67⋊#95—22no—
38C67⋊#95—22no—
39C22×C8×C67D412yes—
40C2×C42×C67C2311yescharacteristic
41C4⋊C8×C67—22no—
42C22⋊C8×C67D414yes—
43C4×C8×C67—22no—
44C4×C8×C67C2×C412yes—
45C4×C8×C67D411yescharacteristic
46C4×C8×C67D411yescharacteristic
47#11761628—44no—
48C2×Dic134—21no—
49C2×Dic134—22no—
50C2×D268—21no—
51C2×D268—22no—
52C2×C22⋊D67—41no—
53C22×Dic67—42no—
54C2×C4×D67—41no—
55C4⋊Dic67—22no—
56C4⋊Dic67—22no—
57C4⋊Dic67—22no—
58C4⋊Dic67—44no—
59Dic67⋊C4—42no—
60C22⋊Dic67—41no—
61(C2×D67)⋊C4—42no—
62C4×Dic67—41no—
63C22×C4×C67C2×D412yes—
64C22×C4×C67C2×D411yescharacteristic
65C2×C8×C67—24no—
66C2×C8×C67—24no—
67C2×C8×C67C22⋊C412yes—
68C2×C8×C67C22⋊C412yes—
69C42×C67C22×C411yescharacteristic
70C42×C67C2×D411yescharacteristic
71C42×C67#4112yes—
72C23×C67#12711yescharacteristic
73C22×D67—41no—
74C2×C4×C67—42no—
75C2×C4×C67C2×C22⋊C411yescharacteristic
76C2×C4×C67C4×D412yes—
77C2×C4×C67#12112yes—
78C2×C4×C67#12411yescharacteristic
79C2×C4×C67#13514yes—
80C22⋊D67—82no—
81C2×Dic67—42no—
82C2×Dic67—41no—
83C2×Dic67—84no—
84C2×Dic67—82no—
85D268—42no—
86D268—82no—
87C4×D67—82no—
88Dic134—42no—
89Dic134—82no—
90C8×C67—22no—
91C8×C67—22no—
92C8×C67—44no—
93C22×C67—22no—
94C22×C67#39011yescharacteristic
95C22×C67#49512yes—
96C2×D67—81no—
97C2×D67—82no—
98C4×C67—22no—
99C4×C67—22no—
100C4×C67#41612yes—
101C4×C67#48812yes—
102Dic67—81no—
103Dic67—82no—
104Dic67—162no—
105#9757—671nomaximal, Sylow
106C2×C67—22no—
107C2×C67#195811yescharacteristic
108C2×C67#197612yes—
109D67—161no—
110#3131—671no—
111#1835—671no—
112#1518—672no—
113#1517—672no—
114#1450D6711yescharacteristic, complemented
115C67#975711yescharacteristic, complemented, Sylow
116#533—671no—
117#532—671no—
118#488—1341no—
119#488—1341no—
120#487—672no—
121#486—672no—
122#431C2×D6712yes—
123#416—1342no—
124C2×C4×C8C2×D6711yescharacteristic
125#382—672no—
126#340—1342no—
127#328—1344no—
128#327—1344no—
129#322—21no—
130C82—21no—
131C2×#41—1341no—
132C22×C8D26812yes—
133C4⋊Q8—671no—
134C4⋊Q8—672no—
135#127—671no—
136#125—1341no—
137#124—1341no—
138C22⋊Q8—1342no—
139#121—1342no—
140C4×Q8—1342no—
141C4×D4—1342no—
142#117—1341no—
143C2×C4⋊C4—1342no—
144C2×C42C22×D6711yescharacteristic
145C4⋊C8—22no—
146#103—1342no—
147#103—1342no—
148#102—1342no—
149#102—1342no—
150C22⋊C8D26814yes—
151C4×C8—22no—
152C4×C8C4×D6712yes—
153C4×C8C22⋊D6711yescharacteristic
154C4×C8C22⋊D6711yescharacteristic
155#95—1342no—
156#95—1342no—
157#41—2684no—
158C2×Q8—1341no—
159C2×Q8—1342no—
160C2×D4—2681no—
161C2×D4—1341no—
162C2×D4—1342no—
163C22×C4—2681no—
164C22×C4—2682no—
165C22×C4C2×D26812yes—
166C22×C4C2×D26811yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8(C2×D67)⋊C412yes—
170C2×C8(C2×D67)⋊C412yes—
171C4⋊C4—2682no—
172C4⋊C4—2684no—
173C4⋊C4—1342no—
174C4⋊C4—1342no—
175C4⋊C4—1342no—
176C22⋊C4—2681no—
177C22⋊C4—2682no—
178C42—2681no—
179C42C2×C4×D6711yescharacteristic
180C42C2×C22⋊D6711yescharacteristic
181C42#1176162812yes—
182C23—2681no—
183C23#1452404011yescharacteristic
184Q8—2682no—
185Q8—5362no—
186D4—2682no—
187D4—5362no—
188D4—5362no—
189C2×C4—42no—
190C2×C4—2682no—
191C2×C4—2681no—
192C2×C4—5362no—
193C2×C4—5364no—
194C2×C4—5362no—
195C2×C4C2×(C2×D67)⋊C411yescharacteristic
196C2×C4C4×D26812yes—
197C2×C4#1452401712yes—
198C2×C4#1452403111yescharacteristic
199C2×C4#1452406514yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—5362no—
205C22—5361no—
206C22#1686329811yescharacteristic
207C22#1686378012yes—
208C4—22no—
209C4—22no—
210C4—5362no—
211C4—5361no—
212C4—10722no—
213C4#1686339312yes—
214C4#1686375112yes—
215C2—22no—
216C2—10721no—
217C2#2256102011yescharacteristic
218C2#2256110712yes—
subgroupG/H
centerC22#16863298
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1450C2
socleC22×C67#390
prime radical#16863946C22
cosocleC2×C42×C67C23
hypercenter#1450D67
hypocenterC67#9757
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #22560340C2semidirect
3. #22558774C2semidirect
5. C67×#1450C2semidirect
10. #16863747C4semidirect
11. #16863743C4semidirect
114. #1450D67semidirect
115. C67#9757semidirect

Maximal subgroups

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

subgroupindex
#225705322
#225603402
#225587742
#225587702
C67×#14502
#975767