Finite Group Databasebeta

← #30755285

Subgroups — #30755285

Order 17152.

398 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 127 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#22560347C211yesmaximal, characteristic
2#22558894C211yesmaximal, characteristic
3#22558890C211yesmaximal, characteristic
4#22558881C211yesmaximal, characteristic
5#22558881C211yesmaximal, characteristic
6#22558492C211yesmaximal, characteristic
7C2×#487×C67C211yesmaximal, characteristic
8#16863449C2211yescharacteristic
9#16863449C2211yescharacteristic
10#16863420C2211yescharacteristic
11#16863420C2211yescharacteristic
12#16863161—22no—
13#16863145—22no—
14#16863142—22no—
15#16863142—22no—
16#16863142—22no—
17#16863117—21no—
18C8×C67⋊C8—21no—
19C2×C4⋊Q8×C67C2211yescharacteristic
20C2×C4⋊Q8×C67C2211yescharacteristic
21#487×C67—21no—
22#487×C67—21no—
23#487×C67C414yes—
24C2×#103×C67—22no—
25C2×C4×C8×C67C2211yescharacteristic
26C2×#11761612—21no—
27C2×#11761612—21no—
28C4⋊(C67⋊C8)—21no—
29C4⋊(C67⋊C8)—21no—
30C4⋊(C67⋊C8)—21no—
31C4⋊(C67⋊C8)—21no—
32C4⋊(C67⋊C8)D412yes—
33C4⋊(C67⋊C8)D412yes—
34C22⋊(C67⋊C8)—21no—
35C22⋊(C67⋊C8)—21no—
36C4×C67⋊C8—21no—
37C4×C67⋊C8—21no—
38C22×Q8×C67—21no—
39C22×Q8×C67—21no—
40C22×C8×C67D412yes—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—21no—
44C4⋊Q8×C67—21no—
45C4⋊Q8×C67—21no—
46C4⋊Q8×C67—21no—
47C4⋊Q8×C67C2×C412yes—
48C4⋊Q8×C67C2×C412yes—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67—21no—
53C2×C42×C67C2311yescharacteristic
54#103×C67—24no—
55#103×C67—24no—
56C4×C8×C67C2×C412yes—
57C4×C8×C67D411yescharacteristic
58C4×C8×C67D411yescharacteristic
59#11761612—42no—
60#11761612—42no—
61C2×C67⋊C8—22no—
62C2×C67⋊C8—22no—
63C2×C67⋊C8—41no—
64C2×C67⋊C8—41no—
65C2×Q8×C67—22no—
66C2×Q8×C67—22no—
67C2×Q8×C67—41no—
68C2×Q8×C67—41no—
69C2×Q8×C67—42no—
70C2×Q8×C67—42no—
71C2×Q8×C67—82no—
72C22×C4×C67—41no—
73C22×C4×C67—41no—
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67#4111yescharacteristic
76C22×C4×C67#4111yescharacteristic
77C2×C8×C67—22no—
78C2×C8×C67—22no—
79C2×C8×C67C22⋊C412yes—
80C2×C8×C67C22⋊C412yes—
81C4⋊C4×C67—22no—
82C4⋊C4×C67—22no—
83C4⋊C4×C67—22no—
84C4⋊C4×C67—22no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C42×C67C22×C411yescharacteristic
90C42×C67C2×D411yescharacteristic
91C42×C67C2×D411yescharacteristic
92C42×C67C2×D411yescharacteristic
93C23×C67#12411yescharacteristic
94Q8×C67—44no—
95Q8×C67—82no—
96Q8×C67—82no—
97Q8×C67—81no—
98Q8×C67—81no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—82no—
104C2×C4×C67—82no—
105C2×C4×C67C2×C22⋊C411yescharacteristic
106C2×C4×C67C4×D411yescharacteristic
107C2×C4×C67C4×D411yescharacteristic
108C2×C4×C67#12111yescharacteristic
109C2×C4×C67#12111yescharacteristic
110C2×C4×C67#12711yescharacteristic
111C2×C4×C67#13512yes—
112C2×C4×C67#13512yes—
113C8×C67—22no—
114C8×C67—22no—
115C67⋊C8—42no—
116C67⋊C8—42no—
117C67⋊C8—81no—
118C67⋊C8—81no—
119C22×C67—21no—
120C22×C67—21no—
121C22×C67#39011yescharacteristic
122C22×C67#48812yes—
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—82no—
126C4×C67—82no—
127C4×C67#41612yes—
128C4×C67#47112yes—
129C4×C67#47312yes—
130C4×C67#49511yescharacteristic
131C4×C67#49511yescharacteristic
132#9770—671nomaximal, Sylow
133C2×C67—22no—
134C2×C67#195811yescharacteristic
135C2×C67#197711yescharacteristic
136C2×C67#197911yescharacteristic
137C2×#487D6711yescharacteristic
138#1837—671no—
139#1549—671no—
140#1548—671no—
141#1546—671no—
142#1546—671no—
143#1450—671no—
144C67#977011yescharacteristic, complemented, Sylow
145C2×C4⋊Q8C2×D6711yescharacteristic
146C2×C4⋊Q8C2×D6711yescharacteristic
147#487—21no—
148#487—21no—
149#487Dic6714yes—
150#431—671no—
151#431—671no—
152#423—671no—
153#423—671no—
154C2×#103—22no—
155C2×C4×C8C2×D6711yescharacteristic
156#341—1342no—
157#333—1342no—
158#332—1342no—
159#332—1342no—
160#332—1342no—
161#322—1341no—
162C82—1341no—
163C22×Q8—21no—
164C22×Q8—21no—
165C2×M16—1341no—
166C2×M16—1341no—
167C22×C8C22⋊D6712yes—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C2×Dic6712yes—
175C4⋊Q8C2×Dic6712yes—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C42C22×D6711yescharacteristic
181C4⋊C8—672no—
182C4⋊C8—672no—
183C4⋊C8—1341no—
184C4⋊C8—1341no—
185C4⋊C8—1341no—
186C4⋊C8—1341no—
187#103—24no—
188#103—24no—
189C22⋊C8—1341no—
190C22⋊C8—1341no—
191C4×C8—1341no—
192C4×C8—1341no—
193C4×C8C2×Dic6712yes—
194C4×C8C22⋊D6711yescharacteristic
195C4×C8C22⋊D6711yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—82no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4C2×C22⋊D6711yescharacteristic
206C22×C4#1176162711yescharacteristic
207C22×C4#1176162711yescharacteristic
208M16—2682no—
209M16—2682no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—2681no—
213C2×C8—2681no—
214C2×C8—1342no—
215C2×C8—1342no—
216C2×C8C22⋊Dic6712yes—
217C2×C8C22⋊Dic6712yes—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C42C22×Dic6711yescharacteristic
227C42D4×D6711yescharacteristic
228C42D4×D6711yescharacteristic
229C42C2×C22⋊D6711yescharacteristic
230C23#1452403011yescharacteristic
231Q8—44no—
232Q8—82no—
233Q8—81no—
234Q8—81no—
235Q8—82no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4C2×C22⋊Dic6711yescharacteristic
243C2×C4D4×Dic6711yescharacteristic
244C2×C4D4×Dic6711yescharacteristic
245C2×C4#1452401411yescharacteristic
246C2×C4#1452401411yescharacteristic
247C2×C4#1452403911yescharacteristic
248C2×C4#1452406312yes—
249C2×C4#1452406312yes—
250C8—22no—
251C8—22no—
252C8—2682no—
253C8—2682no—
254C8—5361no—
255C8—5361no—
256C22—21no—
257C22—21no—
258C22#1686329911yescharacteristic
259C22#1686375012yes—
260C4—21no—
261C4—21no—
262C4—82no—
263C4—82no—
264C4#1686339212yes—
265C4#1686365612yes—
266C4#1686366812yes—
267C4#1686377811yescharacteristic
268C4#1686377811yescharacteristic
269C2—22no—
270C2#2256102111yescharacteristic
271C2#2256111411yescharacteristic
272C2#2256112511yescharacteristic
subgroupG/H
centerC22#16863299
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC2×#487×C67C2
socleC22×C67#390
prime radicalC23×C67#124
cosocleC2×C42×C67C23
hypercenterC2×#487D67
hypocenterC67#9770
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
144. C67#9770semidirect

Maximal subgroups

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

subgroupindex
#225603472
#225588942
#225588902
#225588812
#225584922
C2×#487×C672
#977067