Finite Group Databasebeta

← #17949966

Subgroups — #17949966

Order 4352.

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

normal subgroups only

This group has 456 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#14544701C211yesmaximal, characteristic
2#14544701C211yesmaximal, characteristic
3#14543603C212yesmaximal
4#14543602C212yesmaximal
5C4×#10587574C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×C4⋊#3624—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587574C411yescharacteristic
13C2×C4×Dic34—21no—
14#11762713—21no—
15#11762708C2211yescharacteristic
16#11762706—21no—
17#11762705C2211yescharacteristic
18#11762678C411yescharacteristic
19C4×#63225—21no—
20C4×C4⋊Dic17C411yescharacteristic
21C42×Dic17C2211yescharacteristic
22C4×C4⋊C4×C17C411yescharacteristic
23C22×Dic34—21no—
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C42×C17—21no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×#63225—21no—
34C2×#63225—21no—
35C2×#63225—21no—
36C2×#63225—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587574—22no—
45#10587574—22no—
46C4×Dic34—42no—
47C4×Dic34—42no—
48C4⋊#3624—42no—
49C4⋊#3624—42no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—21no—
60C22×Dic17—21no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C2×Dic34—24no—
66C2×Dic34—42no—
67C2×Dic34—42no—
68C2×#3624—42no—
69C2×#3624—42no—
70C2×#3624—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17M1612yes—
77C42×C17—22no—
78C42×C17—42no—
79C4⋊Dic17—21no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82C4⋊Dic17—41no—
83C4⋊Dic17M1612yes—
84#63225—24no—
85#63225—42no—
86#63225—42no—
87#63225—41no—
88#63225—41no—
89C4×Dic17—21no—
90C4×Dic17—21no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94C4×Dic17—42no—
95C4×Dic17D812yes—
96C4×Dic17SD1612yes—
97C4×Dic17SD1612yes—
98C4×Dic17Q1612yes—
99#7652—171nomaximal, Sylow
100C23×C17C2×C22⋊C411yescharacteristic
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17#10112yes—
113C2×C4×C17#10212yes—
114C2×C4×C17#10312yes—
115C2×C4×C17C2×M1611yescharacteristic
116C2×C4×C17C2×M1611yescharacteristic
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—22no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17#10212yes—
132C2×Dic17#10312yes—
133C2×Dic17C2×D811yescharacteristic
134C2×Dic17C2×SD1611yescharacteristic
135C2×Dic17C2×SD1611yescharacteristic
136C2×Dic17C2×Q1611yescharacteristic
137Dic34—48no—
138#3624—82no—
139#3624—82no—
140C4×C4⋊Q8—171no—
141#1831—171no—
142#1831—171no—
143#1271—172no—
144#1271—172no—
145C22×C17#40711yescharacteristic
146C22×C17#41011yescharacteristic
147C22×C17C2×#10111yescharacteristic
148C22×C17C2×#10211yescharacteristic
149C22×C17C2×#10311yescharacteristic
150C22×C17#42111yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—41no—
155C4×C17—42no—
156C4×C17—81no—
157C4×C17#33214yes—
158Dic17—22no—
159Dic17—22no—
160Dic17—21no—
161Dic17—21no—
162Dic17—21no—
163Dic17—21no—
164Dic17—42no—
165Dic17—81no—
166C2×C4⋊Q8—171no—
167C2×C4×Q8—341no—
168C2×C4⋊C8—341no—
169C2×C4⋊C8—341no—
170#391—341no—
171#389—171no—
172#389—171no—
173#389—341no—
174#384—171no—
175C4×C4⋊C4—171no—
176C4×C4⋊C4—341no—
177C4×C4⋊C4#59511yescharacteristic
178C43—171no—
179#336—172no—
180#336—172no—
181#336—341no—
182#336—341no—
183C2×C17#146711yescharacteristic
184C2×C17#148611yescharacteristic
185C2×C17#148711yescharacteristic
186C2×C17#151711yescharacteristic
187C2×C17#151811yescharacteristic
188C2×C17#153711yescharacteristic
189C2×C17C2×#33211yescharacteristic
190C22×Q8—341no—
191C22×C8—342no—
192C22×C8—342no—
193C4⋊Q8—342no—
194C4⋊Q8—342no—
195C4×Q8—682no—
196C4×Q8—682no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4C2×#59511yescharacteristic
205C2×C4⋊C4C2×#59511yescharacteristic
206C2×C42—21no—
207C2×C42—171no—
208C2×C42—171no—
209C2×C42—171no—
210C2×C42—171no—
211C2×C42—171no—
212C2×C42—171no—
213C2×C42—341no—
214C2×C42C2×#59511yescharacteristic
215C4⋊C8—682no—
216C4⋊C8—682no—
217#95—21no—
218#95—341no—
219#95—341no—
220#95—341no—
221C17#765211yescharacteristic, complemented, Sylow
222C2×Q8—344no—
223C2×Q8—682no—
224C2×Q8—682no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4C22⋊#59511yescharacteristic
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22×#59511yescharacteristic
236C2×C8—682no—
237C2×C8—682no—
238C2×C8—682no—
239C2×C8—682no—
240C2×C8—682no—
241C2×C8—682no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—172no—
245C4⋊C4—344no—
246C4⋊C4—341no—
247C4⋊C4—681no—
248C4⋊C4—681no—
249C4⋊C4—681no—
250C4⋊C4—682no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—682no—
254C4⋊C4#6324312yes—
255C42—22no—
256C42—42no—
257C42—172no—
258C42—172no—
259C42—172no—
260C42—172no—
261C42—341no—
262C42—341no—
263C42—681no—
264C42—681no—
265C42—681no—
266C42—682no—
267C23C2×C22⋊#59511yescharacteristic
268Q8—688no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—171no—
281C2×C4—171no—
282C2×C4—172no—
283C2×C4—171no—
284C2×C4—172no—
285C2×C4—171no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—342no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4#1058754512yes—
301C2×C4#1058754712yes—
302C2×C4#1058754912yes—
303C2×C4C2×#6324311yescharacteristic
304C2×C4D17⋊M1611yescharacteristic
305C8—1362no—
306C8—1362no—
307C22#1176378011yescharacteristic
308C22#1176378811yescharacteristic
309C22C2×#1058754511yescharacteristic
310C22C2×#1058754711yescharacteristic
311C22C2×#1058754911yescharacteristic
312C22#1176380611yescharacteristic
313C22#1176380611yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—42no—
318C4—81no—
319C4—341no—
320C4—341no—
321C4—341no—
322C4—341no—
323C4—342no—
324C4—342no—
325C4—682no—
326C4—1361no—
327C4#1176366814yes—
328C2#1454404611yescharacteristic
329C2#1454411011yescharacteristic
330C2#1454411311yescharacteristic
331C2#1454421411yescharacteristic
332C2#1454421711yescharacteristic
333C2#1454427511yescharacteristic
334C2C2×#1176366811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#7652
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
221. C17#7652semidirect

Maximal subgroups

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

subgroupindex
#145447012
#145436032
#145436022
C4×#105875742
#765217