Finite Group Databasebeta

← #17953239

Subgroups — #17953239

Order 4352.

516 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 140 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#14544717C212yesmaximal
2C2×#11763669C212yesmaximal
3#14543662C212yesmaximal
4#14531353C211yesmaximal, characteristic
5C2×#10587544—22no—
6C2×#10587544C2214yes—
7C2×C4⋊#3624—22no—
8#11763669—24no—
9C2×#10587630C2211yescharacteristic
10C2×Q8×Dic17C411yescharacteristic
11C2×#10587573C411yescharacteristic
12#11762715C411yescharacteristic
13#11762706—22no—
14#11762678C2211yescharacteristic
15#11762660—22no—
16C42×Dic17C2211yescharacteristic
17C17×#391C411yescharacteristic
18C22×Q8×C17C2×C411yescharacteristic
19C22×Dic34C2×C411yescharacteristic
20C22×#3624—24no—
21C2×C4⋊C4×C17C2×C411yescharacteristic
22C2×C42×C17C2×C411yescharacteristic
23Q8×Dic17—42no—
24#10587630—22no—
25#10587630D412yes—
26#10587630Q812yes—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17C2×C411yescharacteristic
29C2×#63225—21no—
30C2×#63225—22no—
31C2×#63225—22no—
32C2×C4×Dic17—21no—
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D412yes—
36C2×C4×Dic17D412yes—
37C2×C4×Dic17C2311yescharacteristic
38#10587573—42no—
39#10587544—24no—
40#10587544—24no—
41#10587544—44no—
42C4⋊#3624—44no—
43C17×#95—22no—
44#10587502—22no—
45C2×Q8×C17—21no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17C4⋊C412yes—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17—21no—
55C22×Dic17C22⋊C411yescharacteristic
56C22×Dic17C22⋊C411yescharacteristic
57C22×Dic17C2×D412yes—
58C2×Dic34—21no—
59C2×Dic34—41no—
60C2×Dic34—41no—
61C2×Dic34C4⋊C412yes—
62C2×#3624—44no—
63C2×#3624—44no—
64C2×#3624—44no—
65C4⋊C4×C17—41no—
66C42×C17—41no—
67C4⋊Dic17—42no—
68C4⋊Dic17—41no—
69#63225—24no—
70#63225—24no—
71#63225—42no—
72#63225—42no—
73#63225—42no—
74C4×Dic17—22no—
75C4×Dic17—22no—
76C4×Dic17—22no—
77C4×Dic17—42no—
78C4×Dic17—42no—
79C4×Dic17SD1614yes—
80C4×Dic17Q1614yes—
81C4×Dic17C2×D411yescharacteristic
82C4×Dic17C2×Q811yescharacteristic
83C4×Dic17#4112yes—
84#8587—171nomaximal, Sylow
85C23×C17C2×C22⋊C411yescharacteristic
86Q8×C17—42no—
87Q8×C17—42no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—42no—
97C2×C4×C17#10314yes—
98C2×C4×C17C2×C4⋊C411yescharacteristic
99C2×C4×C17#11711yescharacteristic
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—42no—
107C2×Dic17#10012yes—
108C2×Dic17#10314yes—
109C2×Dic17#12112yes—
110C2×Dic17C22⋊Q812yes—
111C2×Dic17C2×SD1612yes—
112C2×Dic17C2×Q1612yes—
113C2×Dic17AGL(1,ℤ/8)12yes—
114C2×Dic17#13712yes—
115Dic34—42no—
116Dic34—42no—
117#3624—84no—
118C4×C4⋊Q8—171no—
119#1842—172no—
120C2×#333—172no—
121#1296—172no—
122C22×C17#38011yescharacteristic
123C22×C17C2×#10011yescharacteristic
124C22×C17#41311yescharacteristic
125C22×C17C2×#10312yes—
126C22×C17#41712yes—
127C4×C17—21no—
128C4×C17—21no—
129C4×C17—24no—
130C4×C17—42no—
131C4×C17—81no—
132Dic17—42no—
133Dic17—41no—
134Dic17—81no—
135Dic17#33314yescomplemented
136Dic17#46012yes—
137Dic17#46212yes—
138Dic17#47512yes—
139Dic17#47712yes—
140C2×C4⋊Q8—171no—
141C2×C4×Q8—171no—
142C2×C4×Q8—171no—
143C2×C4⋊C8—174no—
144C2×C4⋊C8—342no—
145C2×C4⋊C8—342no—
146#391—171no—
147#391#59511yescharacteristic
148#389—342no—
149#384—171no—
150C4×C4⋊C4—342no—
151C43—171no—
152#333—344no—
153C2×C17C2×#33311yescharacteristic
154C2×C17#155012yes—
155C2×C17#156011yescharacteristic
156C2×C17#180011yescharacteristic
157C2×C17#180312yes—
158C22×Q8—171no—
159C22×Q8C2×#59511yescharacteristic
160C22×C8—344no—
161C4⋊Q8—172no—
162C4⋊Q8—172no—
163C4⋊Q8—342no—
164C4×Q8—682no—
165C4×Q8—682no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—342no—
168C2×C4⋊C4—341no—
169C2×C4⋊C4—341no—
170C2×C4⋊C4—342no—
171C2×C4⋊C4C2×#59511yescharacteristic
172C2×C42—172no—
173C2×C42—171no—
174C2×C42—172no—
175C2×C42—171no—
176C2×C42—341no—
177C2×C42—341no—
178C2×C42C2×#59511yescharacteristic
179C4⋊C8—344no—
180C4⋊C8—344no—
181C4⋊C8—684no—
182C4⋊C8—684no—
183#95—22no—
184#95—342no—
185C17#858711yescharacteristic, complemented, Sylow
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—172no—
190C2×Q8—341no—
191C2×Q8—681no—
192C2×Q8—681no—
193C2×Q8C4⋊#59512yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—171no—
197C22×C4—172no—
198C22×C4—171no—
199C22×C4—341no—
200C22×C4—341no—
201C22×C4C22⋊#59512yes—
202C22×C4C22×#59511yescharacteristic
203C2×C8—684no—
204C2×C8—684no—
205C2×C8—684no—
206C4⋊C4—41no—
207C4⋊C4—344no—
208C4⋊C4—344no—
209C4⋊C4—681no—
210C4⋊C4—682no—
211C4⋊C4—682no—
212C4⋊C4—682no—
213C4⋊C4—682no—
214C42—41no—
215C42—171no—
216C42—171no—
217C42—172no—
218C42—174no—
219C42—174no—
220C42—342no—
221C42—342no—
222C42—342no—
223C42—682no—
224C42—682no—
225C23C2×C22⋊#59511yescharacteristic
226Q8—42no—
227Q8—42no—
228Q8—682no—
229Q8—682no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—172no—
240C2×C4—172no—
241C2×C4—172no—
242C2×C4—172no—
243C2×C4—174no—
244C2×C4—172no—
245C2×C4—172no—
246C2×C4—172no—
247C2×C4—342no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—681no—
251C2×C4—681no—
252C2×C4—681no—
253C2×C4—682no—
254C2×C4#1058754914yes—
255C2×C4C2×C4⋊#59511yescharacteristic
256C2×C4#1058767711yescharacteristic
257C8—1364no—
258C22#1176375111yescharacteristic
259C22D17⋊#10011yescharacteristic
260C22#1176379611yescharacteristic
261C22C2×#1058754912yes—
262C22#1176380012yes—
263C4—21no—
264C4—21no—
265C4—24no—
266C4—42no—
267C4—81no—
268C4—172no—
269C4—172no—
270C4—172no—
271C4—172no—
272C4—174no—
273C4—682no—
274C4—681no—
275C4—1361no—
276C2D17⋊#33311yescharacteristic
277C2#1454430212yes—
278C2#1454432211yescharacteristic
279C2#1454464911yescharacteristic
280C2#1454465312yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#8587
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
135. Dic17#333semidirect
185. C17#8587semidirect

Maximal subgroups

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

subgroupindex
#145447172
C2×#117636692
#145436622
#145313532
#858717