Finite Group Databasebeta

← #16945457

Subgroups — #16945457

Order 4352.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533957C211yesmaximal, characteristic
2#14528772C211yesmaximal, characteristic
3#14526806C214yesmaximal, complemented
4C17×#1459C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587505—21no—
7#11762721C412yescomplemented
8#11762721C2212yes—
9#11762712—21no—
10#11762674—21no—
11#11762669—21no—
12#11762669—21no—
13#11762669—21no—
14C4×C4⋊Dic17C411yescharacteristic, complemented
15#11762576C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34C2×C412yes—
24C22×C8×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—41no—
38C2×C4×Dic17—21no—
39#10587555—41no—
40#10587555—41no—
41C17×C22⋊C8—22no—
42C17×C22⋊C8—41no—
43#10587505—42no—
44#10587502—22no—
45#10587502—22no—
46#10587502—41no—
47#10587502C2×C412yes—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C2×D411yescharacteristic
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17SD1612yes—
53C2×D4×C17Q1612yes—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—42no—
68C2×Dic34—42no—
69C2×Dic34—82no—
70C2×C8×C17—42no—
71C2×C8×C17—82no—
72C4⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C42×C17—21no—
76C42×C17—21no—
77C22⋊Dic17—42no—
78C22⋊Dic17—42no—
79C22⋊Dic17—82no—
80C22⋊Dic17—82no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85C4⋊Dic17—82no—
86C4×Dic17—42no—
87#11900—171nomaximal, Sylow
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17#10312yes—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17C2×SD1611yescharacteristic
99C23×C17C2×Q1611yescharacteristic
100D4×C17—42no—
101D4×C17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—81no—
114C2×C4×C17—81no—
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#10312yes—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—82no—
125C2×Dic17—82no—
126C2×Dic17—82no—
127Dic34—84no—
128C8×C17—82no—
129#2563—171no—
130#1799—171no—
131#1459D1711yescharacteristic
132#1351—174no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—21no—
138C22×C17—42no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17—81no—
147C22×C17C2×#9911yescharacteristic
148C22×C17C2×#10311yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—21no—
151C4×C17—22no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17#35412yes—
156Dic17—82no—
157Dic17—82no—
158Dic17—162no—
159C2×C4⋊Q8—171no—
160C2×C4×D4C2×D1711yescharacteristic
161C2×#99—341no—
162C2×C22⋊C8—21no—
163#393—172no—
164#393—172no—
165#390—341no—
166#382—341no—
167#380—341no—
168#380—341no—
169#380—341no—
170C4×C4⋊C4—171no—
171#342—172no—
172#324—21no—
173#323C2×D1712yes—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—41no—
177C2×C17—42no—
178C2×C17—41no—
179C2×C17#152011yescharacteristic
180C2×C17#212211yescharacteristic
181C2×C17C2×#35411yescharacteristic
182C22×Q8—172no—
183C22×D4D6811yescharacteristic
184C23×C4D6811yescharacteristic
185C23×C4C22⋊D1711yescharacteristic
186C22×C8—41no—
187C4⋊Q8—681no—
188C4⋊Q8—681no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—681no—
195C2×C4⋊C4C22⋊D1711yescharacteristic
196C2×C22⋊C4—342no—
197C2×C22⋊C4—342no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C4×D1711yescharacteristic
200C2×C22⋊C4C22×D1711yescharacteristic
201C2×C42—341no—
202C2×C42C4×D1711yescharacteristic
203#99—682no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206#95—172no—
207#95—342no—
208#95—342no—
209#95—681no—
210C17#1190011yescharacteristic, complemented, Sylow
211C24#6322711yescharacteristic
212C24C2×D6811yescharacteristic
213C2×Q8—682no—
214C2×Q8—1362no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#6321912yes—
218C2×D4Dic6812yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—682no—
228C22×C4#6322711yescharacteristic
229C22×C4#6322711yescharacteristic
230C22×C4#6322711yescharacteristic
231C22×C4C2×C4×D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233C2×C8—42no—
234C2×C8—82no—
235C4⋊C4—41no—
236C4⋊C4—682no—
237C4⋊C4—682no—
238C4⋊C4—681no—
239C4⋊C4—681no—
240C4⋊C4—1362no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—1362no—
246C22⋊C4—1362no—
247C42—21no—
248C42—21no—
249C42—682no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1058748712yes—
259C23C2×#6321911yescharacteristic
260C23C2×Dic6811yescharacteristic
261C23C2×#6322711yescharacteristic
262Q8—1364no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—1362no—
285C2×C4—1362no—
286C2×C4—1362no—
287C2×C4#1058747711yescharacteristic
288C2×C4#1058747711yescharacteristic
289C2×C4#1058748712yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1058747711yescharacteristic
306C22C2×#1058748711yescharacteristic
307C22#1176282511yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—2722no—
314C4—1362no—
315C4—1362no—
316C4#1176260312yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#1452735711yescharacteristic
323C2#1453043511yescharacteristic
324C2C2×#1176260311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11900
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526806C2semidirect
5. C2×#10587555C4semidirect
7. #11762721C4semidirect
14. C4×C4⋊Dic17C4semidirect
210. C17#11900semidirect

Maximal subgroups

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

subgroupindex
#145339572
#145287722
#145268062
C17×#14592
#1190017