Finite Group Databasebeta

← #30772579

Subgroups — #30772579

Order 17152.

396 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#16863503C211yesmaximal, characteristic
2C2×#16863219C211yesmaximal, characteristic
3C2×#16863212C211yesmaximal, characteristic
4#22560963C211yesmaximal, characteristic
5#22558314C211yesmaximal, characteristic
6#22558288C211yesmaximal, characteristic
7C67×#2141C211yesmaximal, characteristic
8C2×Q8×Dic67—21no—
9C2×#14523983C2211yescharacteristic
10#16863503—22no—
11#16863503—22no—
12C2×#14523979C2211yescharacteristic
13C2×#14523975C2211yescharacteristic
14#16863423—21no—
15C2×#14523967—21no—
16C2×#14523953C2211yescharacteristic
17#16863219—22no—
18#16863212—22no—
19#16863155—21no—
20C22×Q16×C67C2211yescharacteristic
21C2×M32×C67C2211yescharacteristic
22#427×C67C2211yescharacteristic
23#358×C67C414yescomplemented
24C2×#11761612—21no—
25C22×C67⋊C8D411yescharacteristic
26Q8×Dic67—24no—
27Q8×Dic67—42no—
28C2×C4⋊Dic67—21no—
29C2×C4⋊Dic67—41no—
30C2×C4×Dic67—41no—
31C2×C4×Dic67D411yescharacteristic
32#14523983—21no—
33#14523983—21no—
34C2×C67⋊C16—21no—
35#14523979—21no—
36#14523979—21no—
37#14523975—21no—
38#14523975D412yes—
39C4⋊(C67⋊C8)—42no—
40#14523967—24no—
41#14523953—21no—
42#14523953—21no—
43C22×Q8×C67—21no—
44C2×Q16×C67—21no—
45C2×Q16×C67—21no—
46C2×Q16×C67—21no—
47C2×Q16×C67C2×C412yes—
48C2×Q16×C67D412yes—
49C2×Q16×C67Q812yes—
50C22×C8×C67C2311yescharacteristic
51#117×C67—21no—
52M32×C67—21no—
53M32×C67—21no—
54C2×C16×C67C2×C412yes—
55#107×C67C2×C412yes—
56#106×C67—21no—
57C22×Dic67—21no—
58C22×Dic67—41no—
59#11761612—41no—
60#11761612—41no—
61C2×C67⋊C8—21no—
62C2×C67⋊C8—21no—
63C2×C67⋊C8—21no—
64C2×C67⋊C8—41no—
65C4⋊Dic67—22no—
66C4⋊Dic67—44no—
67C4⋊Dic67—41no—
68C4×Dic67—21no—
69C4×Dic67—44no—
70C4×Dic67SD1612yes—
71C67⋊C16—41no—
72C2×Q8×C67—22no—
73C2×Q8×C67—22no—
74C2×Q8×C67—22no—
75C2×Q8×C67—41no—
76C2×Q8×C67—41no—
77C2×Q8×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67C2×D411yescharacteristic
80Q16×C67—22no—
81Q16×C67—22no—
82Q16×C67—22no—
83Q16×C67C4⋊C414yes—
84C2×C8×C67C22×C411yescharacteristic
85C2×C8×C67C2×D411yescharacteristic
86C2×C8×C67C2×D411yescharacteristic
87C2×C8×C67C2×Q811yescharacteristic
88C2×C8×C67#4111yescharacteristic
89C2×C8×C67#4111yescharacteristic
90C4⋊C4×C67—22no—
91C22⋊C4×C67—41no—
92C42×C67—41no—
93C16×C67—22no—
94C23×C67AGL(1,ℤ/8)11yescharacteristic
95Q8×C67—24no—
96Q8×C67—42no—
97Q8×C67—42no—
98Q8×C67—42no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67C4×D411yescharacteristic
104C2×C4×C67#12011yescharacteristic
105C2×C4×C67C22⋊Q811yescharacteristic
106C2×C4×C67C2×SD1611yescharacteristic
107C2×C4×C67#13511yescharacteristic
108C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
109C2×Dic67—22no—
110C2×Dic67—42no—
111C2×Dic67—42no—
112C2×Dic67—82no—
113C8×C67C2×C4⋊C411yescharacteristic
114C8×C67C4×D411yescharacteristic
115C8×C67C22⋊Q811yescharacteristic
116C8×C67#12311yescharacteristic
117C67⋊C8—41no—
118C67⋊C8—81no—
119C22×C67—22no—
120C22×C67#44211yescharacteristic
121C22×C67#45011yescharacteristic
122C22×C67#47811yescharacteristic
123C4×C67—44no—
124C4×C67—82no—
125C4×C67#38511yescharacteristic
126C4×C67C4×SD1611yescharacteristic
127C4×C67#44911yescharacteristic
128C4×C67#47611yescharacteristic
129Dic67—42no—
130Dic67—82no—
131#12957—671nomaximal, Sylow
132C2×C67—21no—
133C2×C67—21no—
134C2×C67#186411yescharacteristic
135C2×C67#218012yes—
136C2×#441—671no—
137C2×#365—671no—
138C2×#362—671no—
139#2141D6711yescharacteristic
140#1946—671no—
141#1386—671no—
142#1375—671no—
143C67#1295711yescharacteristic, complemented, Sylow
144C22×Q16C2×D6711yescharacteristic
145C2×C4×Q8—1341no—
146C2×M32—671no—
147C2×M32C2×D6711yescharacteristic
148#441—1342no—
149#441—1342no—
150C2×#108—671no—
151#427C2×D6711yescharacteristic
152C2×#106—671no—
153#424—1341no—
154C2×#103—1341no—
155C2×#97—671no—
156#365—1342no—
157#362—1342no—
158#358Dic6714yescomplemented
159#339—1341no—
160C22×Q8—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16Dic13412yes—
165C2×Q16D26812yes—
166C2×Q16C2×Dic6712yes—
167C2×M16—1341no—
168C22×C8—671no—
169C22×C8C22×D6711yescharacteristic
170C4×Q8—2682no—
171C4×Q8—1344no—
172#117—21no—
173C2×C4⋊C4—2681no—
174C2×C4⋊C4—1341no—
175C2×C42—2681no—
176C2×C42—671no—
177M32—21no—
178M32—21no—
179M32—1341no—
180M32—1341no—
181C2×C16—1341no—
182C2×C16C2×Dic6712yes—
183#108—1341no—
184#108—1341no—
185#107C2×Dic6712yes—
186#106—21no—
187#106—672no—
188#106—1341no—
189C4⋊C8—2682no—
190#103—1344no—
191#97—1341no—
192#97—1341no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C22×C4—41no—
200C22×C4—2681no—
201C22×C4—1341no—
202C22×C4D4×D6711yescharacteristic
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16C4⋊Dic6714yes—
207M16—2681no—
208M16—2681no—
209C2×C8—2681no—
210C2×C8—1341no—
211C2×C8—1341no—
212C2×C8—1341no—
213C2×C8C22×Dic6711yescharacteristic
214C2×C8D4×D6711yescharacteristic
215C2×C8C2×D26811yescharacteristic
216C2×C8C2×Dic13411yescharacteristic
217C2×C8#1176162711yescharacteristic
218C2×C8#1176162711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—2684no—
221C4⋊C4—2681no—
222C4⋊C4—1342no—
223C22⋊C4—41no—
224C42—41no—
225C42—2684no—
226C42—672no—
227C42—1341no—
228C16—22no—
229C16—2681no—
230C23#1452406711yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—2682no—
240C2×C4—2682no—
241C2×C4—5362no—
242C2×C4—1342no—
243C2×C4D4×Dic6711yescharacteristic
244C2×C4#1452401311yescharacteristic
245C2×C4#1452402011yescharacteristic
246C2×C4SD16×D6711yescharacteristic
247C2×C4#1452406411yescharacteristic
248C2×C4#1452406711yescharacteristic
249C8—2681no—
250C8—5361no—
251C8C2×C4⋊Dic6711yescharacteristic
252C8D4×Dic6711yescharacteristic
253C8#1452402011yescharacteristic
254C8#1452402711yescharacteristic
255C22—22no—
256C22#1686350811yescharacteristic
257C22#1686354811yescharacteristic
258C22#1686369811yescharacteristic
259C4—44no—
260C4—82no—
261C4—2682no—
262C4—5362no—
263C4#1686327111yescharacteristic
264C4SD16×Dic6711yescharacteristic
265C4#1686354211yescharacteristic
266C4#1686368411yescharacteristic
267C2—21no—
268C2—21no—
269C2#2256047211yescharacteristic
270C2#2256210112yes—
subgroupG/H
centerC22#16863508
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
FittingC67×#2141C2
socleC22×C67#442
prime radicalC23×C67AGL(1,ℤ/8)
cosocleC22×C8×C67C23
hypercenter#2141D67
hypocenterC67#12957
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
23. #358×C67C4semidirect
143. C67#12957semidirect
158. #358Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168635032
C2×#168632192
C2×#168632122
#225609632
#225583142
#225582882
C67×#21412
#1295767