Finite Group Databasebeta

← #30758920

Subgroups — #30758920

Order 17152.

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

normal subgroups only

This group has 454 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#22568587C211yesmaximal, characteristic, complemented
2#22561037C211yesmaximal, characteristic
3#22558770C212yesmaximal
4#22558068C212yesmaximal, complemented
5C67×#1276C211yesmaximal, characteristic, complemented
6#16863946C2211yescharacteristic
7C2×#14524043—21no—
8#16863420C2211yescharacteristic
9C2×#14523971—21no—
10C2×#14523971—21no—
11#16863302—24no—
12#16863262—21no—
13#16863262C2212yes—
14#16863131—24no—
15#431×C67C2212yes—
16C43×C67C2211yescharacteristic
17C2×#11761628—21no—
18C22×Dic134—22no—
19C2×#11761612—21no—
20#14524043—42no—
21#14524043—42no—
22#14524043Q811yescharacteristic, complemented
23#14524040Q811yescharacteristic, complemented
24#14524031—21no—
25#14524017—22no—
26C4×D268—22no—
27C2×C4⋊Dic67—22no—
28C2×C4⋊Dic67—22no—
29C4⋊(C67⋊C8)—22no—
30#14523971—42no—
31#14523971—42no—
32C67⋊#95—24no—
33C67⋊#95—22no—
34C67⋊#95D414yes—
35C2×C42×C67—22no—
36C2×C42×C67D411yescharacteristic
37C2×C42×C67D411yescharacteristic
38C2×C42×C67C2311yescharacteristic
39C4⋊C8×C67—22no—
40C22⋊C8×C67—22no—
41C4×C8×C67—22no—
42#11761628—41no—
43#11761628—41no—
44#11761628—41no—
45#11761628—41no—
46C2×Dic134—21no—
47C2×Dic134—44no—
48C2×Dic134—48no—
49C2×D268—21no—
50C2×D268—22no—
51C2×C22⋊D67—41no—
52C22×Dic67—24no—
53C2×C4×D67—41no—
54#11761612—41no—
55#11761612—41no—
56C2×C67⋊C8—41no—
57C4⋊Dic67—22no—
58C4⋊Dic67—44no—
59C4⋊Dic67—42no—
60C4⋊Dic67—42no—
61(C2×D67)⋊C4—42no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67C2×D412yes—
65C22×C4×C67C2×D411yescharacteristic
66C2×C8×C67—42no—
67C42×C67—21no—
68C42×C67—21no—
69C42×C67—44no—
70C42×C67D812yes—
71C42×C67SD1612yes—
72C42×C67C2×Q811yescharacteristic
73C42×C67#4111yescharacteristic
74C42×C67#4112yes—
75C23×C67#12011yescharacteristic
76C22×D67—41no—
77C2×C4×C67—24no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67C22⋊Q812yes—
83C2×C4×C67C22⋊Q811yescharacteristic
84C2×C4×C67#12412yes—
85C2×C4×C67#12411yescharacteristic
86C2×C4×C67C2×D811yescharacteristic
87C2×C4×C67C2×SD1611yescharacteristic
88C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C67#13711yescharacteristic
90C22⋊D67—81no—
91C22⋊D67—81no—
92C2×Dic67—48no—
93C2×Dic67—41no—
94C2×Dic67—44no—
95D268—41no—
96D268—41no—
97D268—82no—
98C4×D67—81no—
99C4×D67—81no—
100Dic134—41no—
101Dic134—41no—
102Dic134—88no—
103C8×C67—82no—
104C67⋊C8—81no—
105C22×C67—22no—
106C22×C67#39311yescharacteristic
107C22×C67#44711yescharacteristic
108C22×C67#44811yescharacteristic
109C2×D67—81no—
110C2×D67—82no—
111C4×C67—24no—
112C4×C67—22no—
113C4×C67—22no—
114C4×C67—42no—
115C4×C67#47411yescharacteristic
116C4×C67#47611yescharacteristic
117C4×C67#48611yescharacteristic
118C4×C67#48911yescharacteristic
119Dic67—84no—
120Dic67—81no—
121#10362—671nomaximal, Sylow
122C2×C67—21no—
123C2×C67—21no—
124C2×C67#202311yescharacteristic
125C2×C67#203112yes—
126D67—161no—
127#2862—671no—
128#1962—671no—
129#1517—672no—
130#1286—672no—
131#1276D6711yescharacteristic, complemented
132C67#1036211yescharacteristic, complemented, Sylow
133#532—671no—
134C2×C4⋊Q8—1341no—
135#431C2×D6712yes—
136#423—671no—
137C2×#104—1341no—
138C2×#104—1341no—
139#391—1344no—
140#382—672no—
141#382—1341no—
142C43C2×D6711yescharacteristic
143#327—1344no—
144C2×#41—1341no—
145C22×Q8—1342no—
146C2×M16—1341no—
147C4⋊Q8—2682no—
148C4⋊Q8—2682no—
149C4⋊Q8—671no—
150#127—671no—
151#124—1341no—
152#121—1342no—
153C4×D4—1342no—
154C2×C4⋊C4—1342no—
155C2×C4⋊C4—1342no—
156C2×C42—22no—
157C2×C42D26811yescharacteristic
158C2×C42D26811yescharacteristic
159C2×C42C22×D6711yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1342no—
162#104—2682no—
163#104—2682no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—674no—
167#95—1344no—
168#95—1342no—
169#41—2681no—
170#41—2681no—
171#41—2681no—
172#41—2681no—
173C2×Q8—2684no—
174C2×Q8—2688no—
175C2×Q8—1341no—
176C2×D4—2681no—
177C2×D4—1342no—
178C2×D4—1341no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—2681no—
182C22×C4—1344no—
183C22×C4D4×D6712yes—
184C22×C4C2×D26811yescharacteristic
185M16—2681no—
186M16—2681no—
187C2×C8—42no—
188C2×C8—2681no—
189C4⋊C4—2684no—
190C4⋊C4—2682no—
191C4⋊C4—2682no—
192C4⋊C4—1342no—
193C22⋊C4—2682no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42D53612yes—
198C42#1176161712yes—
199C42Q8×D6711yescharacteristic
200C42#1176162711yescharacteristic
201C42#1176162812yes—
202C23—2681no—
203C23#1452401311yescharacteristic
204Q8—2681no—
205Q8—2681no—
206Q8—5368no—
207D4—2681no—
208D4—2681no—
209D4—5362no—
210D4—5361no—
211D4—5361no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—2688no—
218C2×C4—2684no—
219C2×C4—2681no—
220C2×C4—5361no—
221C2×C4—5361no—
222C2×C4#1452402112yes—
223C2×C4#1452402311yescharacteristic
224C2×C4#1452402912yes—
225C2×C4#1452403111yescharacteristic
226C2×C4C2×D53611yescharacteristic
227C2×C4C2×#1176161711yescharacteristic
228C2×C4#1452407111yescharacteristic
229C2×C4#1452407611yescharacteristic
230C8—82no—
231C8—5361no—
232C22—22no—
233C22—5361no—
234C22—5362no—
235C22#1686330911yescharacteristic
236C22#1686353211yescharacteristic
237C22#1686353711yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—5364no—
243C4—5361no—
244C4#1686367711yescharacteristic
245C4#1686368711yescharacteristic
246C4#1686374311yescharacteristic
247C4#1686375611yescharacteristic
248C2—21no—
249C2—21no—
250C2—10721no—
251C2#2256134611yescharacteristic
252C2#2256138212yes—
subgroupG/H
centerC22#16863309
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1276C2
socleC22×C67#393
prime radical#16863946C22
hypercenter#1276D67
hypocenterC67#10362
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. #22568587C2semidirect
4. #22558068C2semidirect
5. C67×#1276C2semidirect
22. #14524043Q8semidirect
23. #14524040Q8semidirect
131. #1276D67semidirect
132. C67#10362semidirect

Maximal subgroups

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

subgroupindex
#225685872
#225610372
#225587702
#225580682
C67×#12762
#1036267