Finite Group Databasebeta

← #17953594

Subgroups — #17953594

Order 4352.

630 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 140 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#14544640C211yesmaximal, characteristic
2#14544640C211yesmaximal, characteristic
3#14543662C214yesmaximal, complemented
4C2×#11762678C211yesmaximal, characteristic
5C2×#10587544C2212yes—
6C2×#10587544C2212yes—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9#11763675—21no—
10#11763675—21no—
11C22×C4⋊Dic17C411yescharacteristic
12C22×#63225—21no—
13C22×#63225—21no—
14C22×C4×Dic17C2211yescharacteristic
15#11762678—21no—
16#11762678—21no—
17#11762678C412yes—
18#11762678C2212yes—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D411yescharacteristic
24C23×Dic17D411yescharacteristic
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—42no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—42no—
32C2×C4⋊Dic17C2×C412yes—
33C2×#63225—22no—
34C2×#63225—22no—
35C2×#63225—22no—
36C2×#63225—22no—
37C2×#63225—44no—
38C2×#63225—42no—
39C2×#63225—42no—
40C2×C4×Dic17—24no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45C22⋊#3624—42no—
46C22⋊#3624—42no—
47#10587544—24no—
48#10587544—24no—
49C24×C17C22⋊C411yescharacteristic
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—42no—
55C22×C4×C17—42no—
56C22×C4×C17—42no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17D812yes—
70C22×Dic17SD1612yes—
71C22×Dic17SD1612yes—
72C22×Dic17Q1612yes—
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85#63225—44no—
86#63225—44no—
87#63225—42no—
88#63225—42no—
89#63225—44no—
90C4×Dic17—22no—
91C4×Dic17—22no—
92C4×Dic17—22no—
93C4×Dic17#4114yes—
94#8724—171nomaximal, Sylow
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17#10112yes—
100C23×C17#10212yes—
101C23×C17#10312yes—
102C23×C17C2×C22⋊C411yescharacteristic
103C2×C4×C17—28no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—84no—
112C2×C4×C17#10012yes—
113C2×C4×C17#11712yes—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—24no—
119C2×Dic17—24no—
120C2×Dic17—24no—
121C2×Dic17—24no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—84no—
126C2×Dic17#10212yes—
127C2×Dic17#10312yes—
128C2×Dic17#12312yes—
129C2×Dic17#12312yes—
130C2×Dic17C2×D811yescharacteristic
131C2×Dic17C2×SD1611yescharacteristic
132C2×Dic17C2×SD1611yescharacteristic
133C2×Dic17C2×Q1611yescharacteristic
134C2×Dic17AGL(1,ℤ/8)12yes—
135C2×Dic17#13712yes—
136#3624—82no—
137#3624—82no—
138C2×#384—171no—
139#1797—171no—
140#1797—171no—
141#1296—174no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17#33214yes—
149C22×C17#38111yescharacteristic
150C22×C17C2×#10011yescharacteristic
151C22×C17C2×#10111yescharacteristic
152C22×C17C2×#10211yescharacteristic
153C22×C17C2×#10311yescharacteristic
154C22×C17#41811yescharacteristic
155C22×C17#41911yescharacteristic
156C4×C17—22no—
157C4×C17—22no—
158C4×C17—82no—
159Dic17—22no—
160Dic17—22no—
161Dic17—82no—
162Dic17#48012yes—
163Dic17#48112yes—
164Dic17#48312yes—
165Dic17#48412yes—
166C22×C4⋊C4—171no—
167C22×C4⋊C4—341no—
168C22×C4⋊C4—341no—
169C22×C4⋊C4#59511yescharacteristic
170C22×C42—171no—
171C2×C4⋊C8—172no—
172C2×C4⋊C8—172no—
173C2×C22⋊C8—341no—
174C2×C22⋊C8—341no—
175#384—172no—
176#384—172no—
177#384—341no—
178#384—341no—
179#336—341no—
180#336—341no—
181C2×C17—24no—
182C2×C17C2×#33211yescharacteristic
183C2×C17#154511yescharacteristic
184C2×C17#154811yescharacteristic
185C2×C17#155611yescharacteristic
186C2×C17#182111yescharacteristic
187C2×C17#182311yescharacteristic
188C2×C17#182411yescharacteristic
189C23×C4—21no—
190C23×C4—171no—
191C23×C4—171no—
192C23×C4—341no—
193C23×C4C2×#59511yescharacteristic
194C22×C8—342no—
195C22×C8—342no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—172no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—682no—
205C2×C4⋊C4—682no—
206C2×C4⋊C4—684no—
207C2×C4⋊C4—682no—
208C2×C4⋊C4C2×#59512yes—
209C2×C42—171no—
210C2×C42—171no—
211C2×C42—171no—
212C2×C42—171no—
213C2×C42—344no—
214C4⋊C8—344no—
215C4⋊C8—344no—
216C22⋊C8—682no—
217C22⋊C8—682no—
218C17#872411yescharacteristic, complemented, Sylow
219C24C22⋊#59511yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—172no—
234C22×C4—172no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—682no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4C22⋊#59511yescharacteristic
244C22×C4C22×#59511yescharacteristic
245C2×C8—682no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—682no—
250C2×C8—682no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—682no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C4⋊C4—684no—
257C4⋊C4—684no—
258C4⋊C4—684no—
259C4⋊C4—682no—
260C42—174no—
261C42—342no—
262C42—342no—
263C42—342no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1058754512yes—
269C23#1058754712yes—
270C23#1058754912yes—
271C23C2×C22⋊#59511yescharacteristic
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—84no—
281C2×C4—172no—
282C2×C4—172no—
283C2×C4—171no—
284C2×C4—171no—
285C2×C4—171no—
286C2×C4—171no—
287C2×C4—172no—
288C2×C4—172no—
289C2×C4—172no—
290C2×C4—172no—
291C2×C4—344no—
292C2×C4—344no—
293C2×C4—344no—
294C2×C4—344no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—1364no—
303C2×C4#1058755312yes—
304C2×C4#1058767712yes—
305C8—1362no—
306C8—1362no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22#1176366814yes—
314C22#1176375211yescharacteristic
315C22C2×#1058755311yescharacteristic
316C22C2×#1058754511yescharacteristic
317C22C2×#1058754711yescharacteristic
318C22C2×#1058754911yescharacteristic
319C22#1176380111yescharacteristic
320C22#1176380211yescharacteristic
321C4—22no—
322C4—22no—
323C4—82no—
324C4—172no—
325C4—172no—
326C4—172no—
327C4—172no—
328C4—342no—
329C4—342no—
330C4—1362no—
331C2—24no—
332C2C2×#1176366811yescharacteristic
333C2#1454429111yescharacteristic
334C2#1454429711yescharacteristic
335C2#1454431311yescharacteristic
336C2#1454468311yescharacteristic
337C2#1454468711yescharacteristic
338C2#1454468811yescharacteristic
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#8724
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. #14543662C2semidirect
218. C17#8724semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145436622
C2×#117626782
#872417