Finite Group Databasebeta

← #17949701

Subgroups — #17949701

Order 4352.

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

normal subgroups only

This group has 444 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#14544700C211yesmaximal, characteristic
2C4×#10587544C211yesmaximal, characteristic
3#14543604C212yesmaximal
4#14543603C212yesmaximal
5#14531259C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×#10587544—21no—
8C2×C4×#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×Q8×Dic17C411yescharacteristic
13#11762716C2211yescharacteristic
14#11762707C411yescharacteristic
15#11762705C2211yescharacteristic
16#11762694—21no—
17#11762692—21no—
18C4⋊C4×Dic17C411yescharacteristic
19#11762660—21no—
20#11762654—21no—
21C42×Dic17C2211yescharacteristic
22C17×#391C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C42×C17C2×C411yescharacteristic
28Q8×Dic17—22no—
29Q8×Dic17C814yescomplemented
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×#63225—21no—
34C2×#63225—21no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43#10587544—24no—
44#10587544—42no—
45C4×#3624—42no—
46C17×#95—21no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C2×Q8×C17—21no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17C2×C812yes—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×#3624—24no—
69C2×#3624—42no—
70C2×#3624—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C4⋊C4×C17—41no—
75C42×C17—41no—
76C4⋊Dic17—21no—
77C4⋊Dic17—24no—
78C4⋊Dic17—41no—
79C4⋊Dic17C2×C812yes—
80#63225—42no—
81#63225—41no—
82#63225—41no—
83C4×Dic17—24no—
84C4×Dic17—22no—
85C4×Dic17—21no—
86C4×Dic17—21no—
87C4×Dic17—41no—
88C4×Dic17—41no—
89C4×Dic17—42no—
90C4×Dic17C2×C812yes—
91C4×Dic17M1612yes—
92C4×Dic17SD1612yes—
93C4×Dic17Q1612yes—
94#7577—171nomaximal, Sylow
95C23×C17C2×C22⋊C411yescharacteristic
96Q8×C17—42no—
97Q8×C17—42no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17#10312yes—
112C2×C4×C17C22×C811yescharacteristic
113C2×C4×C17C2×M1611yescharacteristic
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—22no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17C22⋊C812yes—
127C2×Dic17C22⋊C812yes—
128C2×Dic17#10312yes—
129C2×Dic17#10412yes—
130C2×Dic17C2×SD1611yescharacteristic
131C2×Dic17C2×Q1611yescharacteristic
132C2×Dic17AGL(1,ℤ/8)11yescharacteristic
133C2×Dic17#13711yescharacteristic
134#3624—44no—
135#3624—82no—
136#2295—171no—
137#1831—171no—
138C4×C4⋊C8—171no—
139#1272—172no—
140#1271—172no—
141C22×C17C2×C22⋊C811yescharacteristic
142C22×C17#41011yescharacteristic
143C22×C17#41311yescharacteristic
144C22×C17C2×#10311yescharacteristic
145C22×C17#41711yescharacteristic
146C22×C17C2×#10411yescharacteristic
147C22×C17#42111yescharacteristic
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—42no—
152C4×C17—41no—
153C4×C17—81no—
154Dic17—22no—
155Dic17—21no—
156Dic17—21no—
157Dic17—21no—
158Dic17—21no—
159Dic17—42no—
160Dic17—81no—
161Dic17Q8⋊C814yes—
162C2×C4×Q8—171no—
163C2×C4⋊C8—341no—
164C2×C4⋊C8—341no—
165C2×C4×C8—341no—
166#391—171no—
167#391#59511yescharacteristic
168#389—171no—
169#389—171no—
170#387—341no—
171#387—341no—
172C4×C4⋊C4—171no—
173C4×C4⋊C4—341no—
174#376—341no—
175C43—171no—
176#336—172no—
177#336—172no—
178#336—341no—
179C2×C17#145811yescharacteristic
180C2×C17C2×Q8⋊C811yescharacteristic
181C2×C17#147711yescharacteristic
182C2×C17#150911yescharacteristic
183C2×C17#151811yescharacteristic
184C2×C17#152911yescharacteristic
185C2×C17#155011yescharacteristic
186C22×Q8C2×#59511yescharacteristic
187C22×C8—342no—
188C22×C8—342no—
189C4×Q8—174no—
190C4×Q8—342no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4C2×#59511yescharacteristic
197C2×C42—171no—
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—341no—
204C2×C42—341no—
205C2×C42C2×#59511yescharacteristic
206C4⋊C8—344no—
207C4⋊C8—682no—
208C4×C8—682no—
209#95—21no—
210#95—21no—
211#95—341no—
212#95—341no—
213#95—341no—
214#95—341no—
215#95—341no—
216C17#757711yescharacteristic, complemented, Sylow
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8#6324212yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—171no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4C22⋊#59511yescharacteristic
230C22×C4C22⋊#59511yescharacteristic
231C22×C4C22×#59511yescharacteristic
232C2×C8—344no—
233C2×C8—682no—
234C2×C8—682no—
235C2×C8—682no—
236C2×C8—682no—
237C2×C8—682no—
238C4⋊C4—41no—
239C4⋊C4—172no—
240C4⋊C4—344no—
241C4⋊C4—341no—
242C4⋊C4—681no—
243C4⋊C4—682no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C42—41no—
247C42—172no—
248C42—172no—
249C42—172no—
250C42—172no—
251C42—341no—
252C42—342no—
253C42—341no—
254C42—344no—
255C42—681no—
256C42—681no—
257C42—682no—
258C23C2×C22⋊#59511yescharacteristic
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—172no—
275C2×C4—172no—
276C2×C4—172no—
277C2×C4—172no—
278C2×C4—171no—
279C2×C4—171no—
280C2×C4—171no—
281C2×C4—171no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—342no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4#1058754912yes—
295C2×C4C2×#6324211yescharacteristic
296C2×C4D17⋊M1611yescharacteristic
297C8—684no—
298C8—1362no—
299C22#1176377711yescharacteristic
300C22#1176378911yescharacteristic
301C22#1176379611yescharacteristic
302C22C2×#1058754911yescharacteristic
303C22#1176380011yescharacteristic
304C22D17⋊#10411yescharacteristic
305C22#1176380811yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—42no—
311C4—81no—
312C4—174no—
313C4—341no—
314C4—341no—
315C4—341no—
316C4—341no—
317C4—342no—
318C4—682no—
319C4—1361no—
320C2#1454401611yescharacteristic
321C2#1454407311yescharacteristic
322C2#1454408211yescharacteristic
323C2#1454418511yescharacteristic
324C2#1454421811yescharacteristic
325C2#1454425211yescharacteristic
326C2#1454430211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#391C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7577
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
29. Q8×Dic17C8semidirect
216. C17#7577semidirect

Maximal subgroups

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

subgroupindex
#145447002
C4×#105875442
#145436042
#145436032
#145312592
#757717