Finite Group Databasebeta

← #16931703

Subgroups — #16931703

Order 4352.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#14537864C211yesmaximal, characteristic
2#14533999C211yesmaximal, characteristic
3#14528777C211yesmaximal, characteristic
4#14527470C211yesmaximal, characteristic
5#14527468C211yesmaximal, characteristic
6#14526687C211yesmaximal, characteristic
7C17×#1303C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×C4×Dic34C2211yescharacteristic
10C8×Dic34—22no—
11#11762933—22no—
12#11762862C2211yescharacteristic
13#11762859—22no—
14#11762833C2211yescharacteristic
15C2×#10587495—21no—
16#11762749C414yes—
17#11762715—21no—
18#11762714—22no—
19#11762675—21no—
20#11762577—21no—
21#11762556—22no—
22#11762554—22no—
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C4×C4⋊C4×C17C2211yescharacteristic
26C22×Dic34—21no—
27C22×Dic34D411yescharacteristic
28C2×M16×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C42×C17—21no—
31C2×C42×C17C2311yescharacteristic
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17C2×C412yes—
34C2×C4⋊Dic17D411yescharacteristic
35C2×#63225C2×C412yes—
36C2×C4×Dic17D412yes—
37C2×#63223—21no—
38#10587555—21no—
39#10587555—21no—
40#10587555—21no—
41#10587555—21no—
42C4×Dic34—21no—
43C4×Dic34—21no—
44C4×Dic34—21no—
45C4×Dic34—21no—
46C17×C4⋊C8—21no—
47C17×C4⋊C8—21no—
48C17×C4⋊C8D412yes—
49C17×C22⋊C8—21no—
50C4×C8×C17—21no—
51C17×#95C2×C412yes—
52#10587502—24no—
53#10587502—24no—
54C22⋊#3622—21no—
55#10587495—42no—
56#10587485—24no—
57C8×Dic17—24no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622D412yes—
61C4×#3622—21no—
62C22×C4×C17—22no—
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—22no—
67C22×Dic17C22⋊C412yes—
68C22×Dic17C22⋊C412yes—
69C2×Dic34—21no—
70C2×Dic34—21no—
71C2×Dic34—21no—
72C2×Dic34—21no—
73C2×Dic34—44no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34SD1612yes—
77C2×Dic34Q1612yes—
78M16×C17—41no—
79M16×C17—41no—
80C2×C8×C17—22no—
81C2×C8×C17—41no—
82C4⋊C4×C17—42no—
83C42×C17—41no—
84C42×C17—41no—
85C42×C17C2×D411yescharacteristic
86C42×C17C2×D411yescharacteristic
87C42×C17#4111yescharacteristic
88C42×C17#4111yescharacteristic
89C4⋊Dic17—21no—
90C4⋊Dic17—21no—
91C4⋊Dic17—22no—
92C4⋊Dic17—22no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95#63225—22no—
96#63225—22no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99#63223—41no—
100#63223—41no—
101C2×#3622—22no—
102C2×#3622—41no—
103#9396—171nomaximal, Sylow
104C23×C17C2×C22⋊C411yescharacteristic
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17C4×D411yescharacteristic
111C2×C4×C17C4×D411yescharacteristic
112C2×C4×C17#12011yescharacteristic
113C2×C4×C17#12111yescharacteristic
114C2×C4×C17#12111yescharacteristic
115C2×C4×C17#12311yescharacteristic
116C2×C4×C17C2×SD1611yescharacteristic
117C2×C4×C17C2×Q1611yescharacteristic
118C2×C4×C17#13511yescharacteristic
119C2×C4×C17#13511yescharacteristic
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—42no—
124C2×Dic17—44no—
125C2×Dic17—42no—
126C2×Dic17#10314yes—
127Dic34—22no—
128Dic34—22no—
129Dic34—22no—
130Dic34—22no—
131Dic34—82no—
132Dic34—82no—
133C8×C17—42no—
134C8×C17—81no—
135#3622—42no—
136#3622—81no—
137#2982—171no—
138#2568—171no—
139#1801—171no—
140#1549—171no—
141#1549—171no—
142#1303—171no—
143#1303D1711yescharacteristic
144C22×C17—21no—
145C22×C17—21no—
146C22×C17#38611yescharacteristic
147C22×C17C2×#10311yescharacteristic
148C22×C17#41611yescharacteristic
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—82no—
153C4×C17#45412yes—
154C4×C17#46511yescharacteristic
155C4×C17#46511yescharacteristic
156C4×C17#46711yescharacteristic
157C4×C17#46711yescharacteristic
158Dic17—24no—
159Dic17—42no—
160Dic17—82no—
161C2×C4⋊Q8—171no—
162C2×C4×Q8—171no—
163C8×Q8—342no—
164C8×Q8—342no—
165#431—171no—
166#431—342no—
167#431C2×D1711yescharacteristic
168#423—171no—
169#423C2×D1711yescharacteristic
170C2×#101—341no—
171#400—174no—
172#391—341no—
173#391—342no—
174#382—341no—
175C4×C4⋊C4C2×D1711yescharacteristic
176#343—341no—
177#332—342no—
178#332—342no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17#188811yescharacteristic
182C2×C17#189211yescharacteristic
183C2×C17#191511yescharacteristic
184C22×Q8—171no—
185C22×Q8—341no—
186C2×M16—21no—
187C2×M16—341no—
188C4⋊Q8—341no—
189C4⋊Q8—341no—
190C4⋊Q8—341no—
191C4⋊Q8—341no—
192C4×Q8—341no—
193C4×Q8—341no—
194C4×Q8—341no—
195C4×Q8—341no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—172no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—172no—
200C2×C4⋊C4—341no—
201C2×C42—21no—
202C2×C42—172no—
203C2×C42C22×D1711yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—172no—
207C4⋊C8—344no—
208C4⋊C8—341no—
209C4⋊C8—341no—
210C4⋊C8D6812yes—
211#101—682no—
212C22⋊C8—21no—
213C22⋊C8—341no—
214C4×C8—21no—
215C4×C8—344no—
216C4×C8—341no—
217#95—344no—
218#95—344no—
219#95C4×D1712yes—
220C17#939611yescharacteristic, complemented, Sylow
221C2×Q8—172no—
222C2×Q8—172no—
223C2×Q8—341no—
224C2×Q8—341no—
225C2×Q8—341no—
226C2×Q8—341no—
227C2×Q8—684no—
228C2×Q8—681no—
229C2×Q8—681no—
230C22×C4—22no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—342no—
234C22×C4C2×C4×D1711yescharacteristic
235C22×C4D4×D1711yescharacteristic
236C22×C4D4×D1711yescharacteristic
237M16—41no—
238M16—41no—
239M16—681no—
240M16—681no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—342no—
244C2×C8—681no—
245C4⋊C4—42no—
246C4⋊C4—342no—
247C4⋊C4—341no—
248C4⋊C4—342no—
249C4⋊C4—341no—
250C4⋊C4—342no—
251C4⋊C4—342no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C42—41no—
255C42—41no—
256C42—342no—
257C42—342no—
258C42C2×D6811yescharacteristic
259C42#6325211yescharacteristic
260C42D4×D1711yescharacteristic
261C42#6325611yescharacteristic
262C23C22⋊C4×D1711yescharacteristic
263Q8—342no—
264Q8—342no—
265Q8—342no—
266Q8—342no—
267Q8—1362no—
268Q8—1362no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—174no—
275C2×C4—342no—
276C2×C4—342no—
277C2×C4—342no—
278C2×C4—682no—
279C2×C4—684no—
280C2×C4—682no—
281C2×C4C4×D6811yescharacteristic
282C2×C4#1058756811yescharacteristic
283C2×C4#1058757011yescharacteristic
284C2×C4#1058757911yescharacteristic
285C2×C4#1058758011yescharacteristic
286C2×C4#1058758111yescharacteristic
287C2×C4#1058759811yescharacteristic
288C2×C4SD16×D1711yescharacteristic
289C2×C4#1058760211yescharacteristic
290C2×C4Q16×D1711yescharacteristic
291C8—42no—
292C8—81no—
293C8—682no—
294C8—1361no—
295C22—21no—
296C22—21no—
297C22#1176268711yescharacteristic
298C22#103×D1711yescharacteristic
299C22#1176280211yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—344no—
305C4—682no—
306C4—1362no—
307C4#1176298312yes—
308C4#1176303711yescharacteristic
309C4#1176303911yescharacteristic
310C4#1176304911yescharacteristic
311C4#1176305111yescharacteristic
312C2—21no—
313C2—21no—
314C2#1452918611yescharacteristic
315C2#1452920611yescharacteristic
316C2#1452933511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1303C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1303D17
hypocenterC17#9396
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
220. C17#9396semidirect

Maximal subgroups

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

subgroupindex
#145378642
#145339992
#145287772
#145274702
#145274682
#145266872
C17×#13032
#939617