Finite Group Databasebeta

← #17959471

Subgroups — #17959471

Order 4352.

382 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 66 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#14544681C211yesmaximal, characteristic
2#14544037C211yesmaximal, characteristic
3#14543779C214yesmaximal
4#14534606C211yesmaximal, characteristic
5#11763829—21no—
6#11763752—21no—
7#11763687—21no—
8#11763687C2212yes—
9#11763655—21no—
10#11763653C2212yes—
11C2×#10587558C2211yescharacteristic
12#11762747—21no—
13#11762744—21no—
14#11762728C412yescomplemented
15#11762720C2212yes—
16#11762705—21no—
17#11762675C411yescharacteristic, complemented
18#11762666—21no—
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17C2×C412yes—
21C22×C4×D17D411yescharacteristic
22C2×C22⋊#595—22no—
23C17×C4⋊Q8—41no—
24C17×C4⋊Q8—41no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—21no—
27C2×C42×C17C2×C411yescharacteristic
28C2×#63227—22no—
29C2×#63227D411yescharacteristic
30C2×#63227C2311yescharacteristic
31C2×C4⋊Dic17—21no—
32C2×#63225—22no—
33C2×#63225C2×C411yescharacteristic
34C2×#63225D411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36#10587558—41no—
37#10587558—41no—
38C22⋊#3624—22no—
39(C2×#595)⋊C4—41no—
40(C2×#595)⋊C4—41no—
41#10587544—41no—
42#10587542—41no—
43#10587502—21no—
44#10587502—22no—
45#10587502—22no—
46#10587502C2×C412yes—
47C23×D17C2×D411yescharacteristic
48C22×#595—42no—
49C2×Q8×C17—42no—
50C2×Q8×C17—82no—
51C22×C4×C17—22no—
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22×C411yescharacteristic
55C22×Dic17—22no—
56C22×Dic17C22⋊C411yescharacteristic
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C2×D411yescharacteristic
59C2×C4×D17—22no—
60C2×C4×D17—21no—
61C2×C4×D17—21no—
62C2×C4×D17—41no—
63C22⋊#595—42no—
64C22⋊#595—42no—
65C2×#3624—42no—
66C4⋊C4×C17—42no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C42×C17—21no—
70C42×C17—21no—
71#63227—41no—
72#63227—41no—
73#63227—82no—
74C4⋊Dic17—42no—
75#63225—41no—
76#63225—41no—
77#63225—82no—
78C4×Dic17—21no—
79C4×Dic17—21no—
80#11812—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82C22×D17—21no—
83C22×D17—21no—
84C22×D17—41no—
85C22×D17—41no—
86C22×D17#13712yes—
87C2×#595—82no—
88C2×#595—82no—
89C2×#595—82no—
90Q8×C17—84no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17#9911yescharacteristic
101C2×C4×C17#9911yescharacteristic
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—41no—
108C2×Dic17—41no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C4×D17—42no—
113C4×D17—42no—
114#3624—82no—
115#2628—171no—
116#1820—171no—
117#1464—171no—
118#1351—174no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17C2×#9911yescharacteristic
122C22×C17#41911yescharacteristic
123C22×C17#42111yescharacteristic
124C2×D17—22no—
125C2×D17—41no—
126C2×D17—41no—
127C2×D17—42no—
128C2×D17—81no—
129#595—162no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—41no—
133C4×C17—41no—
134C4×C17—82no—
135Dic17—41no—
136Dic17—41no—
137Dic17—41no—
138Dic17—82no—
139C2×C4⋊Q8#59511yescharacteristic, complemented
140C2×#117—171no—
141#432—341no—
142#400—341no—
143#399—341no—
144#395—172no—
145#393—172no—
146#389—341no—
147#382—171no—
148#381—341no—
149#380—341no—
150#342—172no—
151#342—341no—
152#324—341no—
153#323—172no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17#151111yescharacteristic
157C2×C17#212211yescharacteristic
158C2×C17#212911yescharacteristic
159D17—42no—
160C22×Q8C2×#59512yes—
161C23×C4—171no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164#117—681no—
165#117—681no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—341no—
171C2×C4⋊C4—342no—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—171no—
174C2×C22⋊C4—342no—
175C2×C22⋊C4—342no—
176C2×C42—171no—
177C2×C42C2×#59511yescharacteristic
178C4⋊C8—681no—
179C22⋊C8—342no—
180#97—681no—
181#95—172no—
182#95—342no—
183#95—342no—
184#95—341no—
185#95—681no—
186#95—681no—
187C17#1181211yescharacteristic, complemented, Sylow
188C24—171no—
189C2×Q8—42no—
190C2×Q8—82no—
191C22×C4—22no—
192C22×C4—171no—
193C22×C4—171no—
194C22×C4—171no—
195C22×C4—342no—
196C22×C4—341no—
197C22×C4—341no—
198C22×C4—342no—
199C22×C4—681no—
200C22×C4—682no—
201C22×C4C22⋊#59511yescharacteristic
202C22×C4C22⋊#59511yescharacteristic
203C22×C4C22×#59511yescharacteristic
204C2×C8—682no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—681no—
209C4⋊C4—681no—
210C4⋊C4—682no—
211C4⋊C4—1362no—
212C22⋊C4—681no—
213C22⋊C4—681no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C22⋊C4—1362no—
217C42—21no—
218C42—21no—
219C42—341no—
220C42—341no—
221C23—172no—
222C23—341no—
223C23—341no—
224C23—681no—
225C23—681no—
226C23C2×C22⋊#59511yescharacteristic
227Q8—84no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—341no—
238C2×C4—341no—
239C2×C4—341no—
240C2×C4—341no—
241C2×C4—341no—
242C2×C4—682no—
243C2×C4—682no—
244C2×C4—681no—
245C2×C4—681no—
246C2×C4—682no—
247C2×C4—682no—
248C2×C4—682no—
249C2×C4—1362no—
250C2×C4—1362no—
251C2×C4—1362no—
252C2×C4#1058755111yescharacteristic
253C2×C4#1058755111yescharacteristic
254C8—1362no—
255C22—21no—
256C22—21no—
257C22—342no—
258C22—682no—
259C22—681no—
260C22—681no—
261C22—1361no—
262C22C2×#1058755111yescharacteristic
263C22#1176380211yescharacteristic
264C22#1176380811yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—82no—
270C4—2722no—
271C4—681no—
272C4—681no—
273C4—681no—
274C4—1362no—
275C2—21no—
276C2—21no—
277C2—682no—
278C2#1454419411yescharacteristic
279C2#1454504311yescharacteristic
280C2#1454506611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×D17C2×D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11812
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #11762728C4semidirect
17. #11762675C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
139. C2×C4⋊Q8#595semidirect
187. C17#11812semidirect

Maximal subgroups

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

subgroupindex
#145446812
#145440372
#145437792
#145346062
#1181217