Finite Group Databasebeta

← #16932093

Subgroups — #16932093

Order 4352.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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#14533101C211yesmaximal, characteristic
2#14528875C212yesmaximal, complemented
3#14528772C211yesmaximal, characteristic
4#14526652C212yesmaximal, complemented
5C17×#2972C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C2×#10587505—21no—
8#11762748C412yescomplemented
9#11762745C412yescomplemented
10#11762707—21no—
11#11762689—21no—
12#11762681—21no—
13#11762669—21no—
14#11762669C2212yes—
15#11762669C2211yescharacteristic
16#11762576—21no—
17#11762572—24no—
18C2×C4×D4×C17C2211yescharacteristic
19C8×D4×C17—24no—
20C17×#431—21no—
21C2×C17×C22⋊C8—21no—
22C2×C17×C22⋊C8C2212yes—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C22×C8×C17—22no—
28C22×C8×C17D414yes—
29C4×D4×C17—22no—
30C4×D4×C17—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—22no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—22no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17C2×C412yes—
44C2×#63225—22no—
45C17×C4⋊C8—22no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48C17×C22⋊C8—22no—
49C4×C8×C17—22no—
50#10587505—42no—
51#10587502—22no—
52#10587502—22no—
53#10587502—22no—
54#10587502C2×C412yes—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17SD1612yes—
60C2×D4×C17Q1612yes—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D412yes—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17#4112yes—
72C22×Dic17—22no—
73C22×Dic17—22no—
74C22×Dic17—42no—
75C22×Dic17—42no—
76C22×Dic17—82no—
77C2×C8×C17—24no—
78C2×C8×C17—44no—
79C2×C8×C17—82no—
80C4⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C42×C17—21no—
84C42×C17—21no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—82no—
89C4⋊Dic17—24no—
90C4⋊Dic17—42no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—44no—
94#63225—82no—
95#9469—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17#10312yes—
103C23×C17C2×C22⋊C411yescharacteristic
104C23×C17#12112yes—
105C23×C17C2×SD1611yescharacteristic
106C23×C17C2×Q1611yescharacteristic
107C23×C17#13512yes—
108D4×C17—24no—
109D4×C17—24no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—81no—
122C2×C4×C17#10312yes—
123C2×C4×C17C4×D412yes—
124C2×C4×C17#12011yescharacteristic
125C2×C4×C17#12311yescharacteristic
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—84no—
133C2×Dic17—82no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136C8×C17—44no—
137#2972D1711yescharacteristic
138#2487—171no—
139#1824—172no—
140#1799—171no—
141#1289—172no—
142C22×C17—21no—
143C22×C17—22no—
144C22×C17—21no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17#38611yescharacteristic
155C22×C17C2×#10311yescharacteristic
156C22×C17#41611yescharacteristic
157C22×C17#46512yes—
158C22×C17#46712yes—
159C4×C17—21no—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17#45812yes—
165Dic17—82no—
166Dic17—82no—
167Dic17—162no—
168C2×C4×D4C2×D1711yescharacteristic
169C22×C4⋊C4—171no—
170C8×D4—24no—
171#431—21no—
172C2×#99—341no—
173C2×C22⋊C8—21no—
174C2×C22⋊C8C2×D1712yes—
175#400—172no—
176#399—172no—
177#389—341no—
178#386—341no—
179#385—341no—
180#380—171no—
181#380—172no—
182#380—341no—
183#342—341no—
184#340—344no—
185C2×C17—21no—
186C2×C17—22no—
187C2×C17—21no—
188C2×C17—22no—
189C2×C17—42no—
190C2×C17#189011yescharacteristic
191C2×C17#189211yescharacteristic
192C2×C17#191011yescharacteristic
193C22×D4D6811yescharacteristic
194C23×C4—341no—
195C23×C4C22⋊D1711yescharacteristic
196C23×C4C22×D1711yescharacteristic
197C22×C8—22no—
198C22×C8C22⋊D1714yes—
199C4×D4—22no—
200C4×D4—22no—
201C2×C4⋊C4—172no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—342no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—342no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4C4×D1711yescharacteristic
209C2×C22⋊C4—341no—
210C2×C22⋊C4—342no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4C4×D1711yescharacteristic
213C2×C22⋊C4D6811yescharacteristic
214C2×C42C22⋊D1711yescharacteristic
215C4⋊C8—22no—
216#99—682no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C4×C8—22no—
221#95—172no—
222#95—342no—
223#95—342no—
224#95—342no—
225C17#946911yescharacteristic, complemented, Sylow
226C24#6322711yescharacteristic
227C24C2×D6811yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#6321912yes—
231C2×D4Dic6812yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—682no—
239C22×C4—682no—
240C22×C4—1362no—
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4C2×C4×D1711yescharacteristic
245C22×C4#6325212yes—
246C22×C4C2×C22⋊D1712yes—
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—344no—
253C4⋊C4—682no—
254C4⋊C4—684no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—1362no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—682no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C22⋊C4—1362no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1058748712yes—
273C23C2×#6321911yescharacteristic
274C23#1058759012yes—
275C23C2×Dic6811yescharacteristic
276C23C2×#6322711yescharacteristic
277C23#1058761612yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—1362no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4—1364no—
302C2×C4#1058748712yes—
303C2×C4C4×C22⋊D1712yes—
304C2×C4#1058761511yescharacteristic
305C2×C4#1058763911yescharacteristic
306C8—44no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1176268811yescharacteristic
320C22C2×#1058748711yescharacteristic
321C22#1176280611yescharacteristic
322C22#1176303812yes—
323C22#1176305012yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—2722no—
330C4—1362no—
331C4—1362no—
332C4#1176299912yes—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#1452919711yescharacteristic
339C2#1452920711yescharacteristic
340C2#1452931111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2972C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#2972D17
hypocenterC17#9469
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #14528875C2semidirect
4. #14526652C2semidirect
8. #11762748C4semidirect
9. #11762745C4semidirect
225. C17#9469semidirect

Maximal subgroups

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

subgroupindex
#145331012
#145288752
#145287722
#145266522
C17×#29722
#946917