Finite Group Databasebeta

← #16923627

Subgroups — #16923627

Order 4352.

494 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 83 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#14535810C211yesmaximal, characteristic
2#14530318C212yesmaximal, complemented
3#14526610C211yesmaximal, characteristic, complemented
4#14526604C212yesmaximal, complemented
5C17×#2095C211yesmaximal, characteristic, complemented
6C2×#10587633—21no—
7C2×#10587479—22no—
8C2×#10587480—22no—
9#11762726—21no—
10#11762719—22no—
11#11762711C2212yes—
12#11762701—22no—
13#11762564C2212yes—
14C2×C17×#107—21no—
15C2×C17×#103—21no—
16C17×#391C2211yescharacteristic
17C17×#384C2211yescharacteristic
18C17×#336C2211yescharacteristic
19C22×Q8×C17D411yescharacteristic
20C22×D68—21no—
21C22×D68—21no—
22C22×D68D412yes—
23C22×C8×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C42×C17C2311yescharacteristic
28#10587633—42no—
29#10587633—42no—
30C2×#63227—22no—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×#63227—22no—
34C2×C4⋊Dic17D412yes—
35C2×C4×Dic17—21no—
36C22×#3622—22no—
37C17×#107—42no—
38C17×#103—42no—
39C17×#95—22no—
40#10587502—21no—
41#10587480—44no—
42#10587479—44no—
43C23×D17—22no—
44C2×Q8×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17SD1612yes—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17—41no—
51C22×C4×C17C2×D412yes—
52C22×C4×C17C2×D411yescharacteristic
53C22×Dic17—22no—
54C2×D68—22no—
55C2×D68—22no—
56C2×D68—42no—
57C2×D68—42no—
58C2×D68—42no—
59C2×D68—81no—
60C2×D68—81no—
61C2×D68—81no—
62C2×D68—82no—
63C2×C8×C17—22no—
64C2×C8×C17—41no—
65C2×C8×C17—41no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—42no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C42×C17—41no—
72#63227—44no—
73#63227—42no—
74#63227—42no—
75#63227—42no—
76#63227—42no—
77#63227—82no—
78C4⋊Dic17—42no—
79C4×Dic17—42no—
80C2×#3622—24no—
81C2×#3622—42no—
82C2×#3622—42no—
83#7830—171nomaximal, Sylow
84C23×C17#12011yescharacteristic
85C22×D17—42no—
86C22×D17—42no—
87C22×D17—42no—
88C22×D17—42no—
89C22×D17—42no—
90C22×D17—42no—
91C22×D17—42no—
92Q8×C17—42no—
93Q8×C17—81no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—82no—
104C2×C4×C17—81no—
105C2×C4×C17C2×SD1611yescharacteristic
106C2×C4×C17#13512yes—
107C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
108C2×C4×C17#13712yes—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112D68—44no—
113D68—82no—
114D68—84no—
115D68—84no—
116C8×C17—42no—
117#3622—44no—
118#2733—171no—
119#2095D1711yescharacteristic, complemented
120#2094—172no—
121#1272—171no—
122#1270—172no—
123C22×C17#45011yescharacteristic
124C22×C17#45212yes—
125C22×C17#45312yes—
126C22×C17#45311yescharacteristic
127C22×C17#45811yescharacteristic
128C2×D17—82no—
129C2×D17—82no—
130C2×D17—82no—
131C2×D17—82no—
132C2×D17—82no—
133C2×D17—82no—
134C2×D17—82no—
135C4×C17—21no—
136C4×C17—21no—
137C4×C17—22no—
138C4×C17—22no—
139C4×C17—42no—
140C4×C17—81no—
141C4×C17—161no—
142Dic17—82no—
143C2×#124—341no—
144C2×#107—21no—
145C2×#106—342no—
146C2×#103—21no—
147C2×#102—342no—
148#394—341no—
149#392—342no—
150#391C2×D1711yescharacteristic
151#390—172no—
152#388—342no—
153#384C2×D1711yescharacteristic
154#336—172no—
155#336C2×D1711yescharacteristic
156C2×C17#161311yescharacteristic
157C2×C17#161812yes—
158C2×C17#163112yes—
159C2×C17#163411yescharacteristic
160C2×C17#165511yescharacteristic
161D17—162no—
162C22×Q8C22⋊D1711yescharacteristic
163C22×D4—172no—
164C22×D4—341no—
165C22×D4—341no—
166C22×C8—21no—
167C22×C8—342no—
168#124—682no—
169#124—682no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—172no—
173C2×C4⋊C4C22⋊D1711yescharacteristic
174C2×C22⋊C4—342no—
175C2×C22⋊C4—342no—
176C2×C22⋊C4—342no—
177C2×C22⋊C4—342no—
178C2×C42—341no—
179C2×C42C22×D1711yescharacteristic
180#107—42no—
181#106—684no—
182#103—42no—
183#102—684no—
184#95—22no—
185#95—341no—
186C17#783011yescharacteristic, complemented, Sylow
187C24—342no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8#6323012yes—
192C2×D4—342no—
193C2×D4—342no—
194C2×D4—682no—
195C2×D4—682no—
196C2×D4—682no—
197C2×D4—1361no—
198C2×D4—1361no—
199C2×D4—1362no—
200C2×D4—1361no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—342no—
205C22×C4D4×D1712yes—
206C22×C4C2×C22⋊D1711yescharacteristic
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—344no—
211C2×C8—682no—
212C2×C8—682no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—682no—
219C22⋊C4—684no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C22⋊C4—682no—
223C22⋊C4—682no—
224C22⋊C4—1362no—
225C42—41no—
226C42—682no—
227C23—682no—
228C23—682no—
229C23—682no—
230C23—682no—
231C23—682no—
232C23—682no—
233C23—682no—
234C23#1058762311yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—684no—
238D4—1362no—
239D4—1364no—
240D4—1364no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—682no—
253C2×C4—682no—
254C2×C4—682no—
255C2×C4#1058760412yes—
256C2×C4#1058760512yes—
257C2×C4C2×#6323011yescharacteristic
258C2×C4#1058763511yescharacteristic
259C8—42no—
260C8—684no—
261C22—1362no—
262C22—1362no—
263C22—1362no—
264C22—1362no—
265C22—1362no—
266C22—1362no—
267C22—1362no—
268C22#1176295911yescharacteristic
269C22#1176297012yes—
270C22#1176297512yes—
271C22#1176297611yescharacteristic
272C22#1176299811yescharacteristic
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—81no—
279C4—161no—
280C4—1362no—
281C2—2722no—
282C2#1452780311yescharacteristic
283C2#1452783112yes—
284C2#1452790912yes—
285C2#1452792711yescharacteristic
286C2#1452807211yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2095C2
socleC23×C17#120
prime radical#14535810C2
hypercenter#2095D17
hypocenterC17#7830
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
2. #14530318C2semidirect
3. #14526610C2semidirect
4. #14526604C2semidirect
5. C17×#2095C2semidirect
119. #2095D17semidirect
186. C17#7830semidirect

Maximal subgroups

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

subgroupindex
#145358102
#145303182
#145266102
#145266042
C17×#20952
#783017