Finite Group Databasebeta

← #17949675

Subgroups — #17949675

Order 4352.

684 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C4×#10587544C212yesmaximal, complemented
2#14543601C212yesmaximal, complemented
3#14543600C212yesmaximal, complemented
4C4×#10587626C211yesmaximal, characteristic
5C2×#10587544—22no—
6C2×C4×#3624—22no—
7#11763675C2214yes—
8C2×#10587626C411yescharacteristic
9C2×C4×D68C411yescharacteristic
10C2×C4×C22⋊D17—22no—
11#11762712C2211yescharacteristic
12#11762710C2211yescharacteristic
13#11762709—22no—
14#11762678C411yescharacteristic
15C42×Dic17C2211yescharacteristic
16C2×C4×D4×C17C411yescharacteristic
17C22×D4×C17C2×C411yescharacteristic
18C23×C4×C17—21no—
19C22×C22⋊D17—22no—
20C22×D68C2×C411yescharacteristic
21C22×C4×D17—21no—
22C22×#3624—24no—
23C4×D4×C17—42no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C22⋊C4×C17—21no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C22⋊Dic17—22no—
28#10587626—22no—
29#10587626C814yescomplemented
30C4×C22⋊D17—44no—
31C4×C22⋊D17—44no—
32C2×#63227—21no—
33C2×#63227—22no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D412yes—
38C2×C4×Dic17D412yes—
39C2×C4×Dic17C2311yescharacteristic
40C4×D68—42no—
41#10587544—28no—
42C4×#3624—44no—
43C24×C17—21no—
44C23×D17—21no—
45C2×D4×C17—21no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17C2×C812yes—
49C22×C4×C17—21no—
50C22×C4×C17—42no—
51C22×C4×C17—44no—
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22×C411yescharacteristic
55C2×C22⋊D17—28no—
56C2×C22⋊D17—44no—
57C2×C22⋊D17—44no—
58C22×Dic17—21no—
59C22×Dic17C22⋊C412yes—
60C22×Dic17C2×D412yes—
61C2×D68—21no—
62C2×D68—41no—
63C2×D68—41no—
64C2×D68C2×C812yes—
65C2×C4×D17—42no—
66C2×C4×D17—44no—
67C2×#3624—28no—
68C2×#3624—44no—
69C2×#3624—44no—
70C4⋊C4×C17—41no—
71C22⋊C4×C17—42no—
72C42×C17—22no—
73C22⋊Dic17—42no—
74C22⋊Dic17—42no—
75#63227—42no—
76#63227—42no—
77#63227—42no—
78C4⋊Dic17—41no—
79#63225—44no—
80C4×Dic17—24no—
81C4×Dic17—41no—
82C4×Dic17C2×C812yes—
83C4×Dic17M1612yes—
84C4×Dic17D814yes—
85C4×Dic17SD1614yes—
86#7569—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—22no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17—42no—
97C22×D17—42no—
98D4×C17—42no—
99D4×C17—42no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—84no—
107C2×C4×C17#10012yes—
108C2×C4×C17#10214yes—
109C2×C4×C17C22×C811yescharacteristic
110C2×C4×C17C2×M1611yescharacteristic
111C22⋊D17—416no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—42no—
116C2×Dic17C22⋊C814yes—
117C2×Dic17#10214yes—
118C2×Dic17#10414yes—
119C2×Dic17C2×D812yes—
120C2×Dic17C2×SD1612yes—
121D68—42no—
122D68—42no—
123C4×D17—84no—
124#3624—48no—
125C4×#127—171no—
126C4×C4⋊C8—172no—
127#1270—172no—
128#1270—172no—
129C22×C17—42no—
130C22×C17—44no—
131C22×C17—81no—
132C22×C17—82no—
133C22×C17—81no—
134C22×C17C2×C22⋊C811yescharacteristic
135C22×C17#41011yescharacteristic
136C22×C17C2×#10011yescharacteristic
137C22×C17C2×#10212yes—
138C22×C17C2×#10412yes—
139C2×D17—42no—
140C2×D17—44no—
141C2×D17—81no—
142C2×D17—82no—
143C2×D17—81no—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—41no—
147C4×C17—81no—
148C4×C17#33114yes—
149Dic17—22no—
150Dic17—22no—
151Dic17—81no—
152Dic17D4⋊C818yes—
153C2×#127—171no—
154C2×C4×D4—171no—
155C2×C4×D4—342no—
156C2×C4×D4#59511yescharacteristic
157C2×C4⋊C8—342no—
158C2×C4×C8—342no—
159#390—171no—
160#390—171no—
161#390—342no—
162#384—171no—
163C43—171no—
164#336—174no—
165C2×C17—82no—
166C2×C17#146011yescharacteristic
167C2×C17C2×D4⋊C812yes—
168C2×C17#150212yes—
169C2×C17#152511yescharacteristic
170C2×C17C2×#33111yescharacteristic
171D17—82no—
172C22×D4—171no—
173C22×D4—342no—
174C22×D4C2×#59511yescharacteristic
175C23×C4—21no—
176C23×C4—341no—
177C22×C8—344no—
178#127—174no—
179#127—342no—
180C4×D4—42no—
181C4×D4—682no—
182C4×D4—684no—
183C4×D4—684no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—342no—
186C2×C4⋊C4C2×#59511yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—342no—
189C2×C22⋊C4—342no—
190C2×C22⋊C4—341no—
191C2×C42—172no—
192C2×C42—171no—
193C2×C42—172no—
194C2×C42—171no—
195C2×C42C2×#59511yescharacteristic
196C4⋊C8—348no—
197C4×C8—684no—
198C17#756911yescharacteristic, complemented, Sylow
199C24—21no—
200C24—341no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—172no—
205C2×D4—348no—
206C2×D4—341no—
207C2×D4—681no—
208C2×D4—681no—
209C2×D4—684no—
210C2×D4—684no—
211C2×D4#6324212yes—
212C22×C4—21no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—172no—
216C22×C4—172no—
217C22×C4—341no—
218C22×C4—684no—
219C22×C4—682no—
220C22×C4C22⋊#59511yescharacteristic
221C22×C4C22⋊#59511yescharacteristic
222C22×C4C22×#59511yescharacteristic
223C2×C8—348no—
224C2×C8—684no—
225C2×C8—684no—
226C4⋊C4—41no—
227C4⋊C4—681no—
228C4⋊C4—684no—
229C22⋊C4—42no—
230C22⋊C4—682no—
231C22⋊C4—682no—
232C22⋊C4—682no—
233C22⋊C4—682no—
234C22⋊C4—682no—
235C42—22no—
236C42—172no—
237C42—172no—
238C42—174no—
239C42—174no—
240C42—344no—
241C42—681no—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—342no—
248C23—681no—
249C23—681no—
250C23—682no—
251C23—682no—
252C23C2×C22⋊#59511yescharacteristic
253D4—42no—
254D4—42no—
255D4—682no—
256D4—682no—
257D4—6816no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—84no—
265C2×C4—174no—
266C2×C4—174no—
267C2×C4—174no—
268C2×C4—172no—
269C2×C4—172no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—681no—
273C2×C4—682no—
274C2×C4—1364no—
275C2×C4#1058754714yes—
276C2×C4#1058755312yes—
277C2×C4C2×#6324211yescharacteristic
278C2×C4D17⋊M1611yescharacteristic
279C8—688no—
280C22—42no—
281C22—44no—
282C22—82no—
283C22—81no—
284C22—81no—
285C22—682no—
286C22—684no—
287C22—1362no—
288C22—1361no—
289C22—1361no—
290C22#1176377711yescharacteristic
291C22#1176378811yescharacteristic
292C22C2×#1058755311yescharacteristic
293C22C2×#1058754712yes—
294C22D17⋊#10412yes—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—81no—
299C4—178no—
300C4—342no—
301C4—342no—
302C4—1361no—
303C4#1176366714yes—
304C2—82no—
305C2—1362no—
306C2#1454402311yescharacteristic
307C2#1454407012yes—
308C2#1454416312yes—
309C2#1454423811yescharacteristic
310C2C2×#1176366711yescharacteristic
subgroupG/H
centerC22×C4C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587626C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4×D4#595
hypocenterC17#7569
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×#10587544C2semidirect
2. #14543601C2semidirect
3. #14543600C2semidirect
29. #10587626C8semidirect
198. C17#7569semidirect

Maximal subgroups

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

subgroupindex
C4×#105875442
#145436012
#145436002
C4×#105876262
#756917