Finite Group Databasebeta

← #16946735

Subgroups — #16946735

Order 4352.

424 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 59 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#14534157C211yesmaximal, characteristic, complemented
2#14527787C212yesmaximal
3#14526789C212yesmaximal, complemented
4#14526730C211yesmaximal, characteristic, complemented
5C17×#2039C211yesmaximal, characteristic
6C2×C4×Dic34—21no—
7#11763120—22no—
8#11762739—21no—
9#11762739C2212yes—
10#11762714—22no—
11#11762708—22no—
12#11762543—22no—
13#11762535C2212yes—
14C17×#537C2211yescharacteristic
15C2×C17×#104—21no—
16C17×#417—21no—
17C17×#384C2211yescharacteristic
18C17×#343C2211yescharacteristic
19C2×#41×C17—21no—
20C22×Dic34—21no—
21C2×M16×C17—21no—
22C17×C22⋊Q8—41no—
23C17×#121—22no—
24C17×#121—41no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17D412yes—
32C2×#63225—22no—
33C2×C4×Dic17—22no—
34C4×Dic34—42no—
35C4×Dic34—42no—
36C17×#104—42no—
37C17×#103—41no—
38C17×#102—41no—
39#10587503—42no—
40#10587502—22no—
41#10587502—21no—
42#10587502—22no—
43#10587502D412yes—
44C22⋊#3622—22no—
45#10587478—42no—
46#41×C17—41no—
47#41×C17—41no—
48#41×C17—41no—
49#41×C17—41no—
50C2×Q8×C17—21no—
51C2×D4×C17—21no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—41no—
57C22×C4×C17C2×D412yes—
58C22×C4×C17C2×D411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17—22no—
61C2×Dic34—24no—
62C2×Dic34—42no—
63C2×Dic34—42no—
64M16×C17—41no—
65M16×C17—41no—
66C2×C8×C17—41no—
67C4⋊C4×C17—22no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—44no—
71C22⋊C4×C17—42no—
72C42×C17—21no—
73C42×C17D812yes—
74C4⋊Dic17—42no—
75C4⋊Dic17—42no—
76C4⋊Dic17—81no—
77#63225—42no—
78#63225—42no—
79C4×Dic17—44no—
80C2×#3622—42no—
81#12153—171nomaximal, Sylow
82C23×C17—41no—
83C23×C17#12011yescharacteristic
84Q8×C17—41no—
85Q8×C17—41no—
86D4×C17—41no—
87D4×C17—41no—
88D4×C17—82no—
89D4×C17—81no—
90D4×C17—81no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—81no—
102C2×C4×C17—81no—
103C2×C4×C17C2×D811yescharacteristic
104C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
105C2×Dic17—24no—
106C2×Dic17—42no—
107C2×Dic17—42no—
108C2×Dic17—42no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111Dic34—48no—
112C8×C17—81no—
113#3622—82no—
114#2582—171no—
115#2039D1711yescharacteristic
116#1610—172no—
117#1346—172no—
118#1325—171no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—81no—
122C22×C17—81no—
123C22×C17—81no—
124C22×C17#44711yescharacteristic
125C22×C17#45411yescharacteristic
126C22×C17B(4,𝔽2)11yescharacteristic
127C4×C17—24no—
128C4×C17—21no—
129C4×C17—21no—
130C4×C17—42no—
131C4×C17—81no—
132C4×C17—81no—
133Dic17—44no—
134Dic17—82no—
135#537C2×D1711yescharacteristic
136C2×C4×Q8—341no—
137#480—342no—
138C2×#104—21no—
139#417—21no—
140#398—172no—
141#398—341no—
142#391—342no—
143#389—342no—
144#384C2×D1711yescharacteristic
145#343C2×D1711yescharacteristic
146#327—342no—
147#324—172no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17—161no—
151C2×C17#159811yescharacteristic
152C2×C17#219811yescharacteristic
153C2×C17#220511yescharacteristic
154C2×#41—21no—
155C22×Q8—341no—
156C2×M16—21no—
157C22⋊Q8—41no—
158#121—22no—
159#121—41no—
160C4×Q8—682no—
161C4×Q8—682no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4—341no—
165C2×C4⋊C4—342no—
166C2×C4⋊C4—341no—
167C2×C4⋊C4C22⋊D1711yescharacteristic
168C2×C4⋊C4C22×D1711yescharacteristic
169C2×C42—342no—
170C2×C42C22⋊D1711yescharacteristic
171#107—682no—
172#104—42no—
173#103—41no—
174#102—41no—
175#102—682no—
176C22⋊C8—342no—
177#95—172no—
178#95—342no—
179#95—341no—
180#95—342no—
181C17#1215311yescharacteristic, complemented, Sylow
182#41—41no—
183#41—41no—
184#41—41no—
185#41—41no—
186C2×Q8—21no—
187C2×Q8—344no—
188C2×Q8—682no—
189C2×Q8—682no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—41no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—342no—
197C22×C4—342no—
198C22×C4D4×D1712yes—
199C22×C4C2×C22⋊D1711yescharacteristic
200M16—41no—
201M16—41no—
202C2×C8—41no—
203C2×C8—682no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—44no—
208C4⋊C4—682no—
209C4⋊C4—682no—
210C4⋊C4—682no—
211C4⋊C4—682no—
212C4⋊C4—1361no—
213C22⋊C4—42no—
214C42—21no—
215C42—684no—
216C42#6322812yes—
217C23—41no—
218C23#1058762311yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—688no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—344no—
240C2×C4—682no—
241C2×C4—682no—
242C2×C4—682no—
243C2×C4—682no—
244C2×C4—682no—
245C2×C4C2×#6322811yescharacteristic
246C2×C4#1058763511yescharacteristic
247C8—81no—
248C8—1362no—
249C22—21no—
250C22—21no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#1176294211yescharacteristic
255C22#1176298111yescharacteristic
256C22#1176299511yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4—684no—
264C4—1362no—
265C2—21no—
266C2—21no—
267C2—161no—
268C2#1452771811yescharacteristic
269C2#1453077911yescharacteristic
270C2#1453081311yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2039C2
socleC22×C17B(4,𝔽2)
prime radicalC17×#537C22
hypercenter#2039D17
hypocenterC17#12153
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #14534157C2semidirect
3. #14526789C2semidirect
4. #14526730C2semidirect
181. C17#12153semidirect

Maximal subgroups

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

subgroupindex
#145341572
#145277872
#145267892
#145267302
C17×#20392
#1215317