Finite Group Databasebeta

← #17954574

Subgroups — #17954574

Order 4352.

566 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 116 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#14545631C211yesmaximal, characteristic
2#14544088C212yesmaximal, complemented
3#14544024C211yesmaximal, characteristic
4#14543608C212yesmaximal, complemented
5#14531494C211yesmaximal, characteristic
6D4×#3624—24no—
7#11763825C2212yes—
8#11763825C2212yes—
9C2×C22⋊#3624—22no—
10#11763659—24no—
11#11763652—22no—
12C2×#10587622C411yescharacteristic
13C2×#10587573C2211yescharacteristic
14C2×D4×Dic17C2211yescharacteristic
15#11762715C411yescharacteristic
16#11762703—22no—
17#11762681—22no—
18#11762678C411yescharacteristic
19#11762655C2211yescharacteristic
20C17×#390C411yescharacteristic
21C22×D4×C17C2×C411yescharacteristic
22C23×Dic17D412yes—
23C22×Dic34C2×C411yescharacteristic
24C22×#3624—24no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—22no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17D412yes—
30#10587622—22no—
31#10587622C814yescomplemented
32D4×Dic17—22no—
33D4×Dic17—22no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—22no—
36C2×#63225—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2311yescharacteristic
40#10587573—22no—
41#10587573—22no—
42C22⋊#3624—24no—
43C22⋊#3624—22no—
44C22⋊#3624—22no—
45C4⋊#3624—22no—
46C4⋊#3624—22no—
47C4×#3624—22no—
48C4×#3624D414yes—
49#10587502—22no—
50#10587502—21no—
51#10587502—21no—
52C24×C17C22⋊C412yes—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17C2×C812yes—
56C2×D4×C17M1612yes—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—21no—
61C22×Dic17—21no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—82no—
65C22×Dic17C22⋊C412yes—
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×Dic34—22no—
69C2×Dic34—22no—
70C2×Dic34—21no—
71C2×Dic34C2×C812yescomplemented
72C2×#3624—24no—
73C2×#3624—42no—
74C2×#3624—84no—
75C4⋊C4×C17—22no—
76C22⋊C4×C17—82no—
77C42×C17—22no—
78C22⋊Dic17—28no—
79C22⋊Dic17—24no—
80C4⋊Dic17—22no—
81#63225—41no—
82#63225—41no—
83#63225—82no—
84C4×Dic17—21no—
85C4×Dic17—42no—
86C4×Dic17C2×C812yescomplemented
87C4×Dic17C2×D412yes—
88C4×Dic17#4112yes—
89#9099—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—42no—
93C23×C17—42no—
94C23×C17C22⋊C814yes—
95C23×C17C2×C22⋊C411yescharacteristic
96D4×C17—24no—
97D4×C17—24no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17C4×D412yes—
104C2×C4×C17C22×C811yescharacteristic
105C2×C4×C17C2×M1611yescharacteristic
106C2×C4×C17#13112yes—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—41no—
111C2×Dic17—41no—
112C2×Dic17—44no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—42no—
118C2×Dic17—82no—
119C2×Dic17#12011yescharacteristic
120C2×Dic17#12112yes—
121C2×Dic17#12311yescharacteristic
122Dic34—24no—
123Dic34—42no—
124#3624—44no—
125#3624—82no—
126#2972—171no—
127#2326—171no—
128#1479—172no—
129#1460—171no—
130#1274—172no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—24no—
134C22×C17—44no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17—82no—
138C22×C17#38611yescharacteristic
139C22×C17C2×C22⋊C811yescharacteristic
140C22×C17#40811yescharacteristic
141C4×C17—81no—
142C4×C17—81no—
143C4×C17C8×D412yes—
144C4×C17#43512yes—
145Dic17—22no—
146Dic17—21no—
147Dic17—21no—
148Dic17—42no—
149Dic17—42no—
150Dic17—42no—
151C2×#124—171no—
152C2×C4×Q8—171no—
153C2×C4×D4—171no—
154C8×D4—344no—
155#431—172no—
156#431—172no—
157C2×C22⋊C8—342no—
158#391—171no—
159#390#59511yescharacteristic
160#388—342no—
161#385—342no—
162#384—171no—
163#376—171no—
164Q8⋊C8—344no—
165#323—342no—
166C2×C17—21no—
167C2×C17—21no—
168C2×C17—44no—
169C2×C17#186911yescharacteristic
170C2×C17#190911yescharacteristic
171C2×C17#191111yescharacteristic
172C22×Q8—171no—
173C22×D4C2×#59511yescharacteristic
174C23×C4—172no—
175C22×C8—344no—
176#124—174no—
177#124—342no—
178C4×Q8—342no—
179C4×Q8—342no—
180C4×D4—342no—
181C4×D4—342no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4C2×#59511yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—172no—
188C2×C22⋊C4—342no—
189C2×C42—171no—
190C2×C42—171no—
191C2×C42—341no—
192C2×C42C2×#59511yescharacteristic
193C4⋊C8—342no—
194C4⋊C8—342no—
195C22⋊C8—342no—
196C22⋊C8—344no—
197C22⋊C8—342no—
198C4×C8—174no—
199C4×C8—342no—
200#95—341no—
201#95—341no—
202#95—342no—
203C17#909911yescharacteristic, complemented, Sylow
204C24C22⋊#59512yes—
205C2×Q8—172no—
206C2×Q8—341no—
207C2×Q8—342no—
208C2×Q8—342no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4#6324212yes—
212C2×D4#6324312yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—172no—
218C22×C4—342no—
219C22×C4—342no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—1362no—
223C22×C4C22×#59511yescharacteristic
224C2×C8—344no—
225C2×C8—682no—
226C2×C8—1364no—
227C4⋊C4—22no—
228C4⋊C4—342no—
229C4⋊C4—681no—
230C4⋊C4—681no—
231C4⋊C4—1362no—
232C22⋊C4—82no—
233C22⋊C4—344no—
234C22⋊C4—348no—
235C42—22no—
236C42—172no—
237C42—172no—
238C42—172no—
239C42—341no—
240C42—682no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#1058754314yes—
246C23C2×C22⋊#59511yescharacteristic
247Q8—344no—
248Q8—682no—
249D4—24no—
250D4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—172no—
257C2×C4—171no—
258C2×C4—171no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—681no—
263C2×C4—681no—
264C2×C4—684no—
265C2×C4—681no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—682no—
270C2×C4—1362no—
271C2×C4C2×#6324211yescharacteristic
272C2×C4C2×#6324311yescharacteristic
273C2×C4#1058767812yes—
274C2×C4D4×#59512yes—
275C8—684no—
276C8—1362no—
277C22—21no—
278C22—21no—
279C22—24no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1176375711yescharacteristic
285C22C2×#1058754311yescharacteristic
286C22#1176378211yescharacteristic
287C4—81no—
288C4—81no—
289C4—342no—
290C4—341no—
291C4—341no—
292C4—682no—
293C4—682no—
294C4—682no—
295C4#1176383612yes—
296C4#1176383812yes—
297C2—21no—
298C2—21no—
299C2—44no—
300C2#1454475911yescharacteristic
301C2#1454482511yescharacteristic
302C2#1454483011yescharacteristic
subgroupG/H
centerC22#11763757
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#9099
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #14544088C2semidirect
4. #14543608C2semidirect
31. #10587622C8semidirect
71. C2×Dic34C2×C8semidirect
86. C4×Dic17C2×C8semidirect
203. C17#9099semidirect

Maximal subgroups

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

subgroupindex
#145456312
#145440882
#145440242
#145436082
#145314942
#909917