Finite Group Databasebeta

← #16947922

Subgroups — #16947922

Order 4352.

306 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 65 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#14541216C211yesmaximal, characteristic, complemented
2#14531007C211yesmaximal, characteristic, complemented
3#14528136C211yesmaximal, characteristic
4#14526838C211yesmaximal, characteristic, complemented
5#14526748C211yesmaximal, characteristic, complemented
6#14526738C211yesmaximal, characteristic, complemented
7C17×#2207C211yesmaximal, characteristic, complemented
8#11763532—21no—
9#11763214C2211yescharacteristic
10#11763149—22no—
11#11763094—21no—
12#11763047—21no—
13#11763001—21no—
14C8×D68C2211yescharacteristic
15#11762846—21no—
16#11762610—22no—
17C4×#63214C2211yescharacteristic
18#11762562C2211yescharacteristic
19#11762553—21no—
20C2×C17×SD32—21no—
21C17×#493C2211yescharacteristic
22C4×Q16×C17C2211yescharacteristic
23C17×C4⋊C16C2211yescharacteristic
24C17×#357—21no—
25C2×Q16×C17D411yescharacteristic
26C2×D8×C17—21no—
27C2×D8×C17—21no—
28C17×#127—21no—
29C4×Q8×C17—21no—
30#10587632—22no—
31C2×#63228—41no—
32C2×C8×D17—21no—
33#10587584—42no—
34#10587578—22no—
35#10587561—41no—
36C4×D68D411yescharacteristic
37#10587555—21no—
38C4×Dic34—41no—
39C17×SD32—42no—
40C2×C16×C17—21no—
41C17×#107D411yescharacteristic
42C17×#103—21no—
43C4×C8×C17C2311yescharacteristic
44#10587506—22no—
45#10587503—41no—
46#10587491—21no—
47#10587489—21no—
48#10587489D412yes—
49C2×#63214—21no—
50#10587479—22no—
51C4⋊#3622—21no—
52C4⋊#3622D411yescharacteristic
53C2×Q8×C17—21no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D68D811yescharacteristic
58C2×C4×D17—21no—
59C2×Dic34—41no—
60Q16×C17—21no—
61Q16×C17—21no—
62D8×C17—42no—
63D8×C17—41no—
64D8×C17—41no—
65C2×C8×C17C2×D411yescharacteristic
66C2×C8×C17C2×D411yescharacteristic
67C4⋊C4×C17—21no—
68C4⋊C4×C17—41no—
69C42×C17—41no—
70C42×C17C2×D411yescharacteristic
71#63228—82no—
72#63227—21no—
73#63227—42no—
74#63227—42no—
75C4⋊Dic17—22no—
76C4⋊Dic17—41no—
77C4⋊Dic17D811yescharacteristic
78#63225—42no—
79C4×Dic17—42no—
80C2×#3622—21no—
81C2×#3622—41no—
82C8×D17—41no—
83C8×D17—41no—
84C16×C17—41no—
85#63214—41no—
86#12341—171nomaximal, Sylow
87C23×C17—81no—
88C22×D17—21no—
89Q8×C17—41no—
90Q8×C17—41no—
91D4×C17—81no—
92D4×C17—81no—
93D4×C17—81no—
94D4×C17—81no—
95D4×C17—161no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17#12011yescharacteristic
99C2×C4×C17C2×D811yescharacteristic
100C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
101C2×Dic17—21no—
102C2×Dic17—42no—
103D68—21no—
104D68—21no—
105C4×D17—41no—
106C4×D17—41no—
107Dic34—81no—
108Dic34—81no—
109C8×C17—21no—
110C8×C17#13511yescharacteristic
111C8×C17AGL(1,ℤ/8)11yescharacteristic
112#3622—41no—
113#3622—81no—
114#3342—171no—
115#2240—171no—
116#2207D1711yescharacteristic, complemented
117#1667—171no—
118#1364—171no—
119#1332—171no—
120D4⋊C16—171no—
121C22×C17—161no—
122C22×C17—161no—
123C22×C17—161no—
124C22×C17#44711yescharacteristic
125C2×D17—41no—
126C2×D17—41no—
127C2×D17—41no—
128C4×C17—21no—
129C4×C17—81no—
130C4×C17—81no—
131C4×C17#44911yescharacteristic
132C4×C17#45411yescharacteristic
133C4×C17#50811yescharacteristic
134C4×C17#50911yescharacteristic
135Dic17—41no—
136Dic17—82no—
137#556—341no—
138C2×SD32—21no—
139#500—171no—
140#493C2×D1711yescharacteristic
141#485—342no—
142#475—341no—
143#467—341no—
144#459—341no—
145C4×Q16C2×D1711yescharacteristic
146C8×D4—171no—
147#427—341no—
148C4⋊C16C2×D1711yescharacteristic
149#357—21no—
150#357—342no—
151C4×C16—171no—
152#335—171no—
153#331—341no—
154C2×C17—321no—
155C2×C17#162511yescharacteristic
156C2×C17#218611yescharacteristic
157C2×C17#219311yescharacteristic
158D17—81no—
159C2×Q16C22⋊D1711yescharacteristic
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—681no—
163C22×C8—341no—
164C4⋊Q8—341no—
165#127—21no—
166#126—682no—
167#124—681no—
168C22⋊Q8—342no—
169C4×Q8—21no—
170C4×Q8—681no—
171C4×D4—171no—
172#117—342no—
173SD32—42no—
174C2×C16—21no—
175C2×C16—341no—
176#107—172no—
177#107—341no—
178#107C22⋊D1711yescharacteristic
179#106—342no—
180C4⋊C8—171no—
181C4⋊C8—341no—
182#103—21no—
183#103—342no—
184#102—681no—
185C22⋊C8—341no—
186C4×C8C22×D1711yescharacteristic
187C17#1234111yescharacteristic, complemented, Sylow
188C2×Q8—21no—
189C2×Q8—681no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—171no—
194C22×C4—341no—
195Q16—21no—
196Q16—21no—
197D8—42no—
198D8—41no—
199D8—41no—
200D8—1362no—
201C2×C8—341no—
202C2×C8—681no—
203C2×C8—681no—
204C2×C8—681no—
205C2×C8D4×D1711yescharacteristic
206C2×C8C2×C22⋊D1711yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—171no—
210C4⋊C4—342no—
211C4⋊C4—682no—
212C4⋊C4—681no—
213C22⋊C4—341no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C42—41no—
217C42—682no—
218C42D4×D1711yescharacteristic
219C16—41no—
220C16—681no—
221C23—81no—
222C23—341no—
223Q8—41no—
224Q8—41no—
225Q8—1361no—
226Q8—1361no—
227D4—81no—
228D4—81no—
229D4—81no—
230D4—81no—
231D4—161no—
232D4—341no—
233D4—341no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—341no—
237C2×C4—681no—
238C2×C4—681no—
239C2×C4—682no—
240C2×C4D8×D1711yescharacteristic
241C2×C4#1058759711yescharacteristic
242C2×C4#1058762311yescharacteristic
243C8—21no—
244C8—681no—
245C8—1361no—
246C8#1058761811yescharacteristic
247C8#1058763611yescharacteristic
248C22—161no—
249C22—161no—
250C22—161no—
251C22—681no—
252C22—681no—
253C22—681no—
254C22#1176294111yescharacteristic
255C4—21no—
256C4—81no—
257C4—81no—
258C4—681no—
259C4—1362no—
260C4#1176295211yescharacteristic
261C4#1176298011yescharacteristic
262C4#1176324011yescharacteristic
263C4#1176324611yescharacteristic
264C2—321no—
265C2—1361no—
266C2#1452787211yescharacteristic
267C2#1453070611yescharacteristic
268C2#1453074511yescharacteristic
subgroupG/H
centerC22#11762941
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2207C2
socleC22×C17#447
prime radical#14528136C2
hypercenter#2207D17
hypocenterC17#12341
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #14541216C2semidirect
2. #14531007C2semidirect
4. #14526838C2semidirect
5. #14526748C2semidirect
6. #14526738C2semidirect
7. C17×#2207C2semidirect
116. #2207D17semidirect
187. C17#12341semidirect

Maximal subgroups

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

subgroupindex
#145412162
#145310072
#145281362
#145268382
#145267482
#145267382
C17×#22072
#1234117