Finite Group Databasebeta

← #17193808

Subgroups — #17193808

Order 4352.

2610 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531733C216yesmaximal, complemented
2Q8×C4⋊Dic17C212yesmaximal, complemented
3#14531669C216yesmaximal, complemented
4#14531625C212yesmaximal
5#14531249C216yesmaximal, complemented
6#14531191C213yesmaximal, complemented
7C4×Q8×Dic17C212yesmaximal, complemented
8C4×#10587612C213yesmaximal, complemented
9C2×C4×#41×C17C211yesmaximal, characteristic
10C2×Q8×Dic17C2214yes—
11C2×#10587612C2216yes—
12C2×#10587612C2216yes—
13#11762712C22112yes—
14#11762708C22112yes—
15#11762702C22124yes—
16#11762679C2216yes—
17#11762675C2216yes—
18#11762669C2216yes—
19#11762669C2216yes—
20C4⋊C4×Dic17C22112yes—
21C4×C4⋊Dic17C2211yescharacteristic
22C4×C4⋊Dic17C2216yes—
23C4×C4⋊Dic17C2213yes—
24C4×C4⋊Dic17C2216yes—
25C4×C22⋊Dic17C2216yes—
26C4×C22⋊Dic17C2216yes—
27#11762655C2216yes—
28C42×Dic17C2216yes—
29C22×#41×C17C2211yescharacteristic
30C4×#41×C17C4116yescomplemented
31C2×C4×Q8×C17C2211yescharacteristic
32C2×C4×Q8×C17C2211yescharacteristic
33C2×C4×D4×C17C2213yes—
34C2×C4×D4×C17C2213yes—
35C2×C17×#117C2213yes—
36C22×C42×C17C2213yes—
37C2×#41×C17—28no—
38C2×#41×C17C2×C418yes—
39C22×Q8×C17C2311yescharacteristic
40C22×D4×C17C2313yes—
41C23×C4×C17C2313yes—
42C23×C4×C17C2313yes—
43C23×C4×C17C2313yes—
44C4×Q8×C17C2×C418yes—
45C4×Q8×C17C2×C418yes—
46C4×D4×C17C2×C4124yes—
47C4×D4×C17C2×C4124yes—
48C17×#117C2×C4124yes—
49C2×C4⋊C4×C17C2313yes—
50C2×C4⋊C4×C17C2313yes—
51C2×C22⋊C4×C17C2316yes—
52C2×C42×C17C2×C4124yes—
53C2×C42×C17C2311yescharacteristic
54C2×C42×C17C2313yes—
55C2×C42×C17C2313yes—
56C2×C42×C17C2313yes—
57C2×C22⋊Dic17C23112yes—
58C2×C22⋊Dic17C23112yes—
59Q8×Dic17—216no—
60#10587612—224no—
61#10587612—412no—
62C2×C4⋊Dic17C23112yes—
63C2×C4⋊Dic17C2316yes—
64C2×C4⋊Dic17C23112yes—
65C2×C4⋊Dic17C2312yes—
66C2×C4×Dic17C23112yes—
67C2×C4×Dic17C23112yes—
68C2×C4×Dic17C2316yes—
69C2×C4×Dic17C2312yes—
70#10587502C23112yes—
71#10587502C23112yes—
72#10587502C2312yes—
73#10587502C2316yes—
74C24×C17C2413yes—
75#41×C17—232no—
76C2×Q8×C17C22×C414yes—
77C2×Q8×C17#4118yes—
78C2×D4×C17—212no—
79C2×D4×C17C22×C4112yes—
80C22×C4×C17—212no—
81C22×C4×C17—212no—
82C22×C4×C17—26no—
83C22×C4×C17—26no—
84C22×C4×C17C22×C4112yes—
85C22×C4×C17C22×C4112yes—
86C22×C4×C17C22×C4112yes—
87C22×C4×C17C2411yescharacteristic
88C22×C4×C17C2413yes—
89C22×C4×C17C2413yes—
90C22×C4×C17C2411yescharacteristic
91C22×C4×C17C2413yes—
92C22×C4×C17C2411yescharacteristic
93C22×Dic17C24112yes—
94C22×Dic17C2414yes—
95C4⋊C4×C17C22×C4112yes—
96C4⋊C4×C17C22×C4112yes—
97C22⋊C4×C17C22×C4124yes—
98C42×C17C22×C414yes—
99C42×C17C22×C4112yes—
100C42×C17C22×C4112yes—
101C42×C17C22×C4112yes—
102C22⋊Dic17—224no—
103C22⋊Dic17—412no—
104C4⋊Dic17—212no—
105C4⋊Dic17—224no—
106C4⋊Dic17—412no—
107C4⋊Dic17#4118yescomplemented
108C4×Dic17—224no—
109C4×Dic17—212no—
110C4×Dic17—24no—
111C4×Dic17—224no—
112#34129—171nomaximal, Sylow
113C23×C17—26no—
114C23×C17—26no—
115C23×C17—23no—
116C23×C17—23no—
117C23×C17C23×C416yes—
118C23×C17C2511yescharacteristic
119Q8×C17#117116yescomplemented
120D4×C17—224no—
121C2×C4×C17—22no—
122C2×C4×C17—26no—
123C2×C4×C17—23no—
124C2×C4×C17—23no—
125C2×C4×C17—224no—
126C2×C4×C17—224no—
127C2×C4×C17—224no—
128C2×C4×C17C23×C412yes—
129C2×C4×C17C23×C416yes—
130C2×C4×C17C23×C416yes—
131C2×C4×C17C23×C416yes—
132C2×C4×C17C23×C412yes—
133C2×C4×C17C23×C412yes—
134C2×C4×C17C2×#41112yes—
135C2×C4×C17C2×#4114yes—
136C2×C4×C17H(5,𝔽2)12yes—
137C2×C4×C172−1+412yes—
138C2×Dic17—24no—
139C2×Dic17—24no—
140C2×Dic17—212no—
141C2×Dic17—212no—
142C2×Dic17—212no—
143C2×Dic17C2×#4118yes—
144C2×C4×#41D1711yescharacteristic
145#2354—176no—
146Q8×C4⋊C4—172no—
147#2346—176no—
148#2341—172no—
149#2293—176no—
150#2285—173no—
151C42×Q8—172no—
152C4×#117—173no—
153C22×C17—26no—
154C22×C17—26no—
155C22×C17—26no—
156C22×C17—46no—
157C22×C17—46no—
158C22×C17C24×C411yescharacteristic
159C22×C17C22×#4112yes—
160C22×C17C22×#4112yes—
161C22×C17C2×H(5,𝔽2)11yescharacteristic
162C22×C17C2×2−1+411yescharacteristic
163C4×C17—22no—
164C4×C17—26no—
165C4×C17—26no—
166C4×C17C2×#117112yes—
167C4×C17C4×#4114yes—
168C4×C17#52014yes—
169Dic17—28no—
170Dic17—412no—
171C22×#41C2×D1711yescharacteristic
172C4×#41Dic17116yescomplemented
173C2×C4×Q8—174no—
174C2×C4×Q8C2×D1711yescharacteristic
175C2×C4×Q8C2×D1711yescharacteristic
176C2×C4×D4C2×D1713yes—
177C2×C4×D4C2×D1713yes—
178C2×#117—176no—
179C2×#117—176no—
180C2×#117C2×D1713yes—
181C22×C42C2×D1713yes—
182#390—1712no—
183#389—1712no—
184#388—1724no—
185#384—176no—
186#382—176no—
187#380—176no—
188#380—176no—
189C4×C4⋊C4—1712no—
190C4×C4⋊C4—176no—
191C4×C4⋊C4—171no—
192C4×C4⋊C4—173no—
193C4×C4⋊C4—176no—
194C4×C22⋊C4—176no—
195C4×C22⋊C4—176no—
196#376—176no—
197C43—176no—
198C2×C17—46no—
199C2×C17C22×#11711yescharacteristic
200C2×C17C2×C4×#4111yescharacteristic
201C2×C17C2×#52011yescharacteristic
202C2×C17#348212yes—
203C2×C17#348312yes—
204C2×#41—28no—
205C2×#41C2×Dic1718yes—
206C22×Q8C22×D1711yescharacteristic
207C22×D4C22×D1713yes—
208C23×C4C22×D1713yes—
209C23×C4C22×D1713yes—
210C23×C4C22×D1713yes—
211C4×Q8—3416no—
212C4×Q8C2×Dic1718yes—
213C4×Q8C2×Dic1718yes—
214C4×D4C2×Dic17124yes—
215C4×D4C2×Dic17124yes—
216#117—3424no—
217#117—6812no—
218#117C2×Dic17124yes—
219C2×C4⋊C4—172no—
220C2×C4⋊C4—176no—
221C2×C4⋊C4—1712no—
222C2×C4⋊C4—1712no—
223C2×C4⋊C4C22×D1713yes—
224C2×C4⋊C4C22×D1713yes—
225C2×C22⋊C4—1712no—
226C2×C22⋊C4—1712no—
227C2×C22⋊C4C22×D1716yes—
228C2×C42—1712no—
229C2×C42—1712no—
230C2×C42—176no—
231C2×C42—172no—
232C2×C42C2×Dic17124yes—
233C2×C42C22×D1713yes—
234C2×C42C22×D1713yes—
235C2×C42C22×D1713yes—
236C2×C42C22×D1711yescharacteristic
237#95—172no—
238#95—176no—
239#95—1712no—
240#95—1712no—
241C17#3412911yescharacteristic, complemented, Sylow
242C24C23×D1713yes—
243#41—232no—
244C2×Q8#6325218yes—
245C2×Q8C22×Dic1714yes—
246C2×D4—212no—
247C2×D4C22×Dic17112yes—
248C22×C4—212no—
249C22×C4—212no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—174no—
253C22×C4—1712no—
254C22×C4C22×Dic17112yes—
255C22×C4C22×Dic17112yes—
256C22×C4C22×Dic17112yes—
257C22×C4C23×D1711yescharacteristic
258C22×C4C23×D1713yes—
259C22×C4C23×D1713yes—
260C22×C4C23×D1713yes—
261C22×C4C23×D1711yescharacteristic
262C22×C4C23×D1711yescharacteristic
263C4⋊C4—178no—
264C4⋊C4—3424no—
265C4⋊C4—3412no—
266C4⋊C4—6812no—
267C4⋊C4C22×Dic17112yes—
268C4⋊C4C22×Dic17112yes—
269C22⋊C4—3424no—
270C22⋊C4—6812no—
271C22⋊C4C22×Dic17124yes—
272C42—3424no—
273C42—3424no—
274C42—3412no—
275C42—344no—
276C42C22×Dic1714yes—
277C42C22×Dic17112yes—
278C42C22×Dic17112yes—
279C42C22×Dic17112yes—
280C23—26no—
281C23—26no—
282C23—23no—
283C23—23no—
284C23C23×Dic1716yes—
285C23C24×D1711yescharacteristic
286Q8#10587612116yescomplemented
287D4—224no—
288C2×C4—26no—
289C2×C4—22no—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—224no—
293C2×C4—224no—
294C2×C4—224no—
295C2×C4—178no—
296C2×C4—344no—
297C2×C4—344no—
298C2×C4—3412no—
299C2×C4—3412no—
300C2×C4—3412no—
301C2×C4C2×#63252112yes—
302C2×C4#41×D1714yes—
303C2×C4#1058769612yes—
304C2×C4#1058769712yes—
305C2×C4C23×Dic1712yes—
306C2×C4C23×Dic1716yes—
307C2×C4C23×Dic1716yes—
308C2×C4C23×Dic1716yes—
309C2×C4C23×Dic1712yes—
310C2×C4C23×Dic1712yes—
311C22—26no—
312C22—26no—
313C22—26no—
314C22—46no—
315C22—46no—
316C22C24×Dic1711yescharacteristic
317C22C2×#41×D1712yes—
318C22C22×#6325212yes—
319C22C2×#1058769611yescharacteristic
320C22C2×#1058769711yescharacteristic
321C4—22no—
322C4—26no—
323C4—26no—
324C4—348no—
325C4—6812no—
326C4C2×#10587612112yes—
327C4#41×Dic1714yes—
328C4#1176329214yes—
329C2—46no—
330C2C22×#1058761211yescharacteristic
331C2C2×#41×Dic1711yescharacteristic
332C2C2×#1176329211yescharacteristic
333C2#1454240612yes—
334C2#1454241012yes—
subgroupG/H
centerC22×C4C23×D17
commutatorC22×C17C24×C4
FrattiniC23C24×D17
FittingC2×C4×#41×C17C2
socle, cosocleC23×C17C25
prime radicalC22×#41×C17C22
hypercenterC2×C4×#41D17
hypocenterC17#34129
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #14531733C2semidirect
2. Q8×C4⋊Dic17C2semidirect
3. #14531669C2semidirect
5. #14531249C2semidirect
6. #14531191C2semidirect
7. C4×Q8×Dic17C2semidirect
8. C4×#10587612C2semidirect
30. C4×#41×C17C4semidirect
107. C4⋊Dic17#41semidirect
119. Q8×C17#117semidirect
172. C4×#41Dic17semidirect
241. C17#34129semidirect
286. Q8#10587612semidirect

Maximal subgroups

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

subgroupindex
#145317332
Q8×C4⋊Dic172
#145316692
#145316252
#145312492
#145311912
C4×Q8×Dic172
C4×#105876122
C2×C4×#41×C172
#3412917