Finite Group Databasebeta

← #17954198

Subgroups — #17954198

Order 4352.

586 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 118 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#14543612C212yesmaximal, complemented
5#14531331C211yesmaximal, 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#11762697—22no—
17#11762674C411yescharacteristic
18C22⋊C4×Dic17—22no—
19C42×Dic17C2211yescharacteristic
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—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2311yescharacteristic
41#10587573—22no—
42#10587573—22no—
43C22⋊#3624—24no—
44C22⋊#3624—22no—
45C22⋊#3624—22no—
46C4⋊#3624—22no—
47C4⋊#3624D414yes—
48C4×#3624—22no—
49C4×#3624—22no—
50#10587502—22no—
51#10587502—22no—
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—44no—
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—44no—
77C42×C17—22no—
78C22⋊Dic17—28no—
79C22⋊Dic17—24no—
80C4⋊Dic17—22no—
81#63225—42no—
82C4×Dic17—21no—
83C4×Dic17—42no—
84C4×Dic17—42no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17C2×C812yescomplemented
88C4×Dic17C2×D412yes—
89C4×Dic17#4112yes—
90#8957—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—42no—
94C23×C17—42no—
95C23×C17C22⋊C814yes—
96C23×C17C2×C22⋊C411yescharacteristic
97D4×C17—24no—
98D4×C17—24no—
99C2×C4×C17—22no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17C4×D412yes—
105C2×C4×C17C22×C811yescharacteristic
106C2×C4×C17C2×M1611yescharacteristic
107C2×C4×C17#13112yes—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—41no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—44no—
118C2×Dic17—42no—
119C2×Dic17—82no—
120C2×Dic17#12011yescharacteristic
121C2×Dic17#12112yes—
122C2×Dic17#12311yescharacteristic
123Dic34—24no—
124Dic34—42no—
125#3624—44no—
126#3624—82no—
127#2972—171no—
128C4×#124—171no—
129#1479—172no—
130#1460—171no—
131#1276—172no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—24no—
135C22×C17—44no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—82no—
139C22×C17#38611yescharacteristic
140C22×C17C2×C22⋊C811yescharacteristic
141C22×C17#40811yescharacteristic
142C4×C17—42no—
143C4×C17—81no—
144C4×C17C8×D412yes—
145C4×C17#43512yes—
146Dic17—22no—
147Dic17—21no—
148Dic17—42no—
149Dic17—42no—
150Dic17—42no—
151Dic17#45512yes—
152C2×#124—171no—
153C2×C4×Q8—171no—
154C2×C4×D4—171no—
155C8×D4—344no—
156#431—172no—
157#431—172no—
158C2×C22⋊C8—342no—
159#391—171no—
160#390#59511yescharacteristic
161#388—342no—
162#382—171no—
163C4×C22⋊C4—342no—
164C43—171no—
165Q8⋊C8—344no—
166#323—342no—
167C2×C17—21no—
168C2×C17—21no—
169C2×C17—44no—
170C2×C17#186911yescharacteristic
171C2×C17#188411yescharacteristic
172C2×C17#188511yescharacteristic
173C22×Q8—171no—
174C22×D4C2×#59511yescharacteristic
175C23×C4—172no—
176C22×C8—344no—
177#124—174no—
178#124—342no—
179C4×Q8—342no—
180C4×Q8—342no—
181C4×D4—342no—
182C4×D4—342no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—341no—
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×C42—341no—
193C2×C42—341no—
194C2×C42C2×#59511yescharacteristic
195C4⋊C8—174no—
196C4⋊C8—342no—
197C22⋊C8—342no—
198C22⋊C8—344no—
199C22⋊C8—342no—
200C4×C8—342no—
201C4×C8—342no—
202#95—342no—
203#95—342no—
204C17#895711yescharacteristic, complemented, Sylow
205C24C22⋊#59512yes—
206C2×Q8—172no—
207C2×Q8—341no—
208C2×Q8—342no—
209C2×Q8—342no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4#6324212yes—
213C2×D4#6324312yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—172no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—684no—
224C22×C4C22×#59511yescharacteristic
225C2×C8—344no—
226C2×C8—682no—
227C2×C8—1364no—
228C4⋊C4—22no—
229C4⋊C4—342no—
230C4⋊C4—682no—
231C22⋊C4—44no—
232C22⋊C4—344no—
233C22⋊C4—348no—
234C42—22no—
235C42—172no—
236C42—172no—
237C42—172no—
238C42—341no—
239C42—682no—
240C42—681no—
241C42—681no—
242C42—682no—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23#1058754314yes—
248C23C2×C22⋊#59511yescharacteristic
249Q8—344no—
250Q8—682no—
251D4—24no—
252D4—24no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—171no—
259C2×C4—171no—
260C2×C4—172no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—684no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—681no—
271C2×C4—682no—
272C2×C4—1362no—
273C2×C4C2×#6324211yescharacteristic
274C2×C4C2×#6324311yescharacteristic
275C2×C4#1058767812yes—
276C2×C4D4×#59512yes—
277C8—684no—
278C8—1362no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1176375711yescharacteristic
287C22C2×#1058754311yescharacteristic
288C22#1176378211yescharacteristic
289C4—42no—
290C4—81no—
291C4—172no—
292C4—342no—
293C4—341no—
294C4—682no—
295C4—682no—
296C4—682no—
297C4#1176383612yes—
298C4#1176383812yes—
299C2—21no—
300C2—21no—
301C2—44no—
302C2#1454475911yescharacteristic
303C2#1454478711yescharacteristic
304C2#1454478911yescharacteristic
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#8957
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. #14543612C2semidirect
31. #10587622C8semidirect
71. C2×Dic34C2×C8semidirect
87. C4×Dic17C2×C8semidirect
204. C17#8957semidirect

Maximal subgroups

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

subgroupindex
#145456312
#145440882
#145440242
#145436122
#145313312
#895717