Finite Group Databasebeta

← #16935266

Subgroups — #16935266

Order 4352.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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#14538902C212yesmaximal, complemented
2#14529723C211yesmaximal, characteristic, complemented
3#14527395C212yesmaximal, complemented
4#14526654C211yesmaximal, characteristic, complemented
5C17×#3245C211yesmaximal, characteristic
6C2×C4×Dic34C2211yescharacteristic
7#11763120—22no—
8#11763101—22no—
9#11763006—22no—
10#11762947—22no—
11#11762909—24no—
12#11762866C2212yes—
13C2×#10587474C2211yescharacteristic
14#11762563—22no—
15#11762549—24no—
16C17×#532C2212yes—
17C17×#474—22no—
18C17×#466—22no—
19C17×#459—22no—
20C2×C17×C4⋊C8C2211yescharacteristic
21C2×#41×C17—22no—
22C22×Dic34D411yescharacteristic
23C2×D8×C17—22no—
24C22×C8×C17D412yes—
25C17×C4⋊Q8D412yes—
26C17×#127D412yes—
27C17×#124—22no—
28C17×#121—22no—
29C17×#121—22no—
30C4×D4×C17—22no—
31C4×D4×C17—22no—
32C2×C42×C17C2311yescharacteristic
33C2×#63229—24no—
34C2×C4⋊Dic17D411yescharacteristic
35C2×#63225—21no—
36C2×C4×Dic17—21no—
37C22×#3622—21no—
38C4×Dic34—22no—
39C4×Dic34—22no—
40C17×#107—22no—
41C17×C4⋊C8—22no—
42C17×#102—24no—
43#10587503—24no—
44C22⋊#3622—22no—
45#10587481—24no—
46#10587478—24no—
47C4⋊#3622—22no—
48#10587474—22no—
49#10587474D414yes—
50#41×C17—84no—
51C2×Q8×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—42no—
56C22×C4×C17—42no—
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17#4111yescharacteristic
60C22×Dic17—21no—
61C22×Dic17—21no—
62C2×Dic34—22no—
63C2×Dic34—42no—
64D8×C17—82no—
65C2×C8×C17—22no—
66C2×C8×C17—42no—
67C4⋊C4×C17—22no—
68C4⋊C4×C17—22no—
69C22⋊C4×C17—42no—
70C22⋊C4×C17—42no—
71C42×C17C2×D412yes—
72C42×C17C2×D412yes—
73#63229—48no—
74C4⋊Dic17—22no—
75#63225—42no—
76C4×Dic17—41no—
77C4×Dic17—41no—
78C2×#3622—22no—
79C2×#3622—24no—
80C2×#3622—41no—
81C2×#3622—41no—
82#10028—171nomaximal, Sylow
83C23×C17—42no—
84C23×C17#12111yescharacteristic
85Q8×C17—82no—
86D4×C17—44no—
87D4×C17—42no—
88D4×C17—42no—
89D4×C17—82no—
90D4×C17—84no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—84no—
97C2×C4×C17#12012yes—
98C2×C4×C17#12011yescharacteristic
99C2×C4×C17#12112yes—
100C2×C4×C17#12711yescharacteristic
101C2×Dic17—22no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—42no—
106Dic34—24no—
107Dic34—42no—
108C8×C17—42no—
109#3622—42no—
110#3622—44no—
111#3245D1711yescharacteristic
112#3112—172no—
113#1985—171no—
114#1530—172no—
115#1290—171no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—82no—
119C22×C17—82no—
120C22×C17—82no—
121C22×C17#39211yescharacteristic
122C22×C17#47212yes—
123C4×C17—21no—
124C4×C17—22no—
125C4×C17—22no—
126C4×C17—82no—
127C4×C17#45511yescharacteristic
128C4×C17#45611yescharacteristic
129Dic17—42no—
130Dic17—81no—
131#532C2×D1712yes—
132C2×C4×Q8—171no—
133#480—342no—
134#477—342no—
135#474—22no—
136#466—22no—
137#460—342no—
138#459—22no—
139#448—342no—
140#440—344no—
141#432—172no—
142C2×C4⋊C8C2×D1711yescharacteristic
143C2×#97—171no—
144#336—342no—
145#329—344no—
146C2×C17—22no—
147C2×C17—162no—
148C2×C17#200711yescharacteristic
149C2×C17#201011yescharacteristic
150C2×C17#201811yescharacteristic
151C2×#41—22no—
152C22×Q8—171no—
153C2×SD16—344no—
154C2×D8—22no—
155C22×C8—341no—
156C22×C8C22⋊D1712yes—
157C4⋊Q8C22⋊D1712yes—
158#127C22⋊D1712yes—
159#124—22no—
160#121—22no—
161#121—22no—
162C4×Q8—342no—
163C4×Q8—342no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—341no—
168C2×C42—341no—
169C2×C42C22×D1711yescharacteristic
170#107—22no—
171#107—344no—
172C4⋊C8—22no—
173C4⋊C8—342no—
174#103—344no—
175#102—24no—
176#102—344no—
177C22⋊C8—342no—
178#97—174no—
179#97—342no—
180C17#1002811yescharacteristic, complemented, Sylow
181#41—84no—
182C2×Q8—22no—
183C2×Q8—342no—
184C2×Q8—682no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C22×C4—42no—
190C22×C4—341no—
191C22×C4—341no—
192C22×C4D4×D1711yescharacteristic
193C22×C4#6325411yescharacteristic
194C22×C4C2×C22⋊D1711yescharacteristic
195SD16—688no—
196D8—82no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—342no—
200C2×C8—344no—
201C2×C8—681no—
202C2×C8—681no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—342no—
206C4⋊C4—682no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—681no—
210C42—681no—
211C42D4×D1712yes—
212C42C2×C22⋊D1712yes—
213C23—42no—
214C23#1058762411yescharacteristic
215Q8—82no—
216Q8—344no—
217Q8—682no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—82no—
222D4—84no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—342no—
230C2×C4—681no—
231C2×C4—681no—
232C2×C4—681no—
233C2×C4—682no—
234C2×C4#1058762312yes—
235C2×C4#1058762512yes—
236C2×C4#1058762611yescharacteristic
237C2×C4#1058763911yescharacteristic
238C8—42no—
239C8—682no—
240C8—684no—
241C22—21no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#1176271711yescharacteristic
247C22#1176307612yes—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—82no—
252C4—682no—
253C4—1361no—
254C4#1176298711yescharacteristic
255C4#1176299211yescharacteristic
256C2—22no—
257C2—162no—
258C2#1452983111yescharacteristic
259C2#1452984811yescharacteristic
260C2#1452989111yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
Fitting, prime radicalC17×#3245C2
socleC22×C17#392
hypercenter#3245D17
hypocenterC17#10028
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #14538902C2semidirect
2. #14529723C2semidirect
3. #14527395C2semidirect
4. #14526654C2semidirect
180. C17#10028semidirect

Maximal subgroups

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

subgroupindex
#145389022
#145297232
#145273952
#145266542
C17×#32452
#1002817