Finite Group Databasebeta

← #17949807

Subgroups — #17949807

Order 4352.

612 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 122 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#14544675C211yesmaximal, characteristic
2#14544447C211yesmaximal, characteristic
3#14543602C214yesmaximal, complemented
4C2×#11762708C211yesmaximal, characteristic
5C22×#63247—21no—
6C2×C22⋊#3624—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C22×C4⋊Dic17—21no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762708—21no—
15#11762708—21no—
16#11762708C412yes—
17#11762708C2212yes—
18C2×#10587502—21no—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D411yescharacteristic
24C23×Dic17D411yescharacteristic
25C2×#63247—24no—
26C2×#63247—42no—
27C2×#63247—42no—
28C22×#3624—22no—
29C22×#3624—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—24no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—22no—
36C2×C4⋊Dic17—44no—
37C2×C4⋊Dic17—42no—
38C2×C4⋊Dic17C2×C412yes—
39C2×C4×Dic17—42no—
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44C22⋊#3624—42no—
45#10587502—22no—
46#10587502—22no—
47C24×C17C22⋊C411yescharacteristic
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—24no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—42no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—41no—
63C22×Dic17—42no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17SD1612yes—
69C22×Dic17Q1612yes—
70C22×Dic17C2×D411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72#63247—48no—
73C2×#3624—24no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C4⋊C4×C17—24no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17M1614yescomplemented
82C4⋊Dic17—44no—
83C4⋊Dic17—42no—
84C4⋊Dic17—44no—
85C4⋊Dic17—42no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88C4×Dic17—41no—
89C4×Dic17—41no—
90#7607—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17#9912yes—
96C23×C17#10312yes—
97C23×C17#10412yes—
98C23×C17C2×C22⋊C411yescharacteristic
99C2×C4×C17—28no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—24no—
105C2×C4×C17—48no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17#10112yes—
109C2×C4×C17C2×M1612yes—
110C2×Dic17—24no—
111C2×Dic17—24no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—84no—
126C2×Dic17#10312yes—
127C2×Dic17C2×SD1611yescharacteristic
128C2×Dic17C2×Q1611yescharacteristic
129C2×Dic17AGL(1,ℤ/8)12yes—
130#3624—44no—
131#3624—82no—
132C2×#389—171no—
133#1817—171no—
134#1728—171no—
135#1271—174no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17#32814yes—
143C22×C17#40711yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10111yescharacteristic
146C22×C17C2×#10311yescharacteristic
147C22×C17#41811yescharacteristic
148C22×C17C2×#10411yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—22no—
151C4×C17—22no—
152C4×C17—44no—
153Dic17—21no—
154Dic17—22no—
155Dic17—21no—
156Dic17—22no—
157Dic17—41no—
158Dic17—82no—
159C22×M16—341no—
160C22×C4⋊C4—171no—
161C22×C4⋊C4—341no—
162C22×C4⋊C4#59511yescharacteristic
163C22×C42—171no—
164C2×C22⋊C8—341no—
165#389—172no—
166#389—172no—
167#389—341no—
168#389—341no—
169C2×#95—341no—
170#336—172no—
171#336—172no—
172#336—341no—
173#336—341no—
174C2×C17—24no—
175C2×C17#146311yescharacteristic
176C2×C17#148711yescharacteristic
177C2×C17#149011yescharacteristic
178C2×C17C2×#32811yescharacteristic
179C2×C17#151011yescharacteristic
180C2×C17#152811yescharacteristic
181C2×C17#154911yescharacteristic
182C23×C4—21no—
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—341no—
186C23×C4C2×#59511yescharacteristic
187C2×M16—344no—
188C2×M16—682no—
189C2×M16—682no—
190C22×C8—342no—
191C22×C8—342no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—172no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—682no—
199C2×C4⋊C4—684no—
200C2×C4⋊C4C2×#59512yes—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—682no—
206C22⋊C8—682no—
207#95—342no—
208#95—342no—
209C17#760711yescharacteristic, complemented, Sylow
210C24C22⋊#59511yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—171no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—681no—
230C22×C4—682no—
231C22×C4—682no—
232C22×C4—682no—
233C22×C4C22⋊#59511yescharacteristic
234C22×C4C22×#59511yescharacteristic
235M16—688no—
236C2×C8—344no—
237C2×C8—682no—
238C2×C8—682no—
239C2×C8—682no—
240C2×C8—682no—
241C2×C8—682no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—682no—
245C4⋊C4—682no—
246C4⋊C4—684no—
247C4⋊C4—684no—
248C4⋊C4#6324714yescomplemented
249C42—681no—
250C42—681no—
251C42—681no—
252C42—681no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#1058753912yes—
258C23#1058754812yes—
259C23#1058754912yes—
260C23C2×C22⋊#59511yescharacteristic
261C2×C4—28no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—171no—
271C2×C4—172no—
272C2×C4—171no—
273C2×C4—172no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—344no—
281C2×C4—344no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—684no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—1364no—
290C2×C4#1058754512yes—
291C2×C4C2×#6324712yes—
292C8—684no—
293C8—1362no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22#1176366214yes—
301C22#1176377911yescharacteristic
302C22C2×#1058753911yescharacteristic
303C22C2×#1058754511yescharacteristic
304C22C2×#1058754911yescharacteristic
305C22#1176380111yescharacteristic
306C22C2×#1058754811yescharacteristic
307C22#1176380811yescharacteristic
308C4—22no—
309C4—22no—
310C4—44no—
311C4—341no—
312C4—342no—
313C4—341no—
314C4—342no—
315C4—681no—
316C4—1362no—
317C2—24no—
318C2#1454403211yescharacteristic
319C2#1454411211yescharacteristic
320C2#1454412311yescharacteristic
321C2C2×#1176366211yescharacteristic
322C2#1454419111yescharacteristic
323C2#1454424611yescharacteristic
324C2#1454429911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#7607
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543602C2semidirect
81. C4⋊C4×C17M16semidirect
209. C17#7607semidirect
248. C4⋊C4#63247semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145444472
#145436022
C2×#117627082
#760717