Finite Group Databasebeta

← #17950121

Subgroups — #17950121

Order 4352.

494 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 90 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#14544676C211yesmaximal, characteristic
2#14544448C211yesmaximal, characteristic
3#14543604C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14537125C211yesmaximal, characteristic
6C22×#63243—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587632C411yescharacteristic
13C2×C42×D17C2211yescharacteristic
14#11762716C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762707C411yescharacteristic
17#11762692—21no—
18#11762691—21no—
19#11762683—21no—
20#11762671—21no—
21#11762666C411yescharacteristic
22C17×#391C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×#63243—24no—
30C2×#63243—42no—
31C2×#63243—42no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34#10587632—42no—
35C2×#63227—21no—
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45C42×D17—42no—
46#10587543—42no—
47C17×#95—21no—
48C17×#95—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C23×D17C22⋊C411yescharacteristic
55C2×Q8×C17—22no—
56C2×Q8×C17—41no—
57C2×Q8×C17—41no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—21no—
71C2×C4×D17—21no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84#63243—48no—
85C4⋊C4×C17—22no—
86C42×C17—41no—
87#63227—42no—
88#63227—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63225—42no—
94#63225—42no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97C4×Dic17—41no—
98#7695—171nomaximal, Sylow
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17#9912yes—
107Q8×C17—82no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17#10412yes—
122C2×C4×C17#13112yes—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17AGL(1,ℤ/8)11yescharacteristic
137C2×Dic17#13711yescharacteristic
138C2×Dic17#13712yes—
139C4×D17—24no—
140C4×D17—24no—
141C4×D17—44no—
142C4×D17—44no—
143#3624—44no—
144#3624—82no—
145#2855—171no—
146#1817—171no—
147#1728—171no—
148#1272—172no—
149#1271—172no—
150C22×C17#40811yescharacteristic
151C22×C17C2×#9911yescharacteristic
152C22×C17#41311yescharacteristic
153C22×C17#41711yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17C2×#10411yescharacteristic
156C22×C17#42111yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C4×C17—22no—
165C4×C17—22no—
166C4×C17—41no—
167C4×C17—81no—
168C4×C17—81no—
169Dic17—21no—
170Dic17—21no—
171Dic17—22no—
172Dic17—41no—
173Dic17—81no—
174Dic17—81no—
175C22×M16—341no—
176C2×C22⋊Q8—171no—
177C22×C42—171no—
178C2×C22⋊C8—341no—
179#391—171no—
180#391#59511yescharacteristic
181#389—171no—
182#389—171no—
183#387—341no—
184#387—341no—
185#385—341no—
186#381—341no—
187#380—171no—
188#336—172no—
189#336—172no—
190#336—341no—
191#336—341no—
192C2×C17#146811yescharacteristic
193C2×C17#148511yescharacteristic
194C2×C17#149511yescharacteristic
195C2×C17#150411yescharacteristic
196C2×C17#151111yescharacteristic
197C2×C17#152911yescharacteristic
198C2×C17#155911yescharacteristic
199D17—42no—
200C22×Q8C2×#59511yescharacteristic
201C23×C4—171no—
202C23×C4—171no—
203C23×C4—171no—
204C2×M16—344no—
205C2×M16—682no—
206C2×M16—682no—
207C22×C8—342no—
208C22×C8—342no—
209C22⋊Q8—682no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4C2×#59511yescharacteristic
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C42—171no—
219C2×C42—171no—
220C2×C42—171no—
221C2×C42—682no—
222C2×C42C2×#59511yescharacteristic
223C22⋊C8—682no—
224#95—21no—
225#95—21no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230#95—341no—
231C17#769511yescharacteristic, complemented, Sylow
232C24—171no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4—681no—
252C22×C4—681no—
253C22×C4C22⋊#59511yescharacteristic
254C22×C4C22⋊#59511yescharacteristic
255C22×C4C22×#59511yescharacteristic
256M16—688no—
257C2×C8—344no—
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C2×C8—682no—
262C2×C8—682no—
263C4⋊C4—22no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C4⋊C4—682no—
269C4⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—682no—
272C42—41no—
273C42—681no—
274C42—681no—
275C42—681no—
276C23—172no—
277C23—341no—
278C23—341no—
279C23—341no—
280C23—341no—
281C23—341no—
282C23—341no—
283C23C2×C22⋊#59511yescharacteristic
284Q8—82no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—171no—
299C2×C4—171no—
300C2×C4—172no—
301C2×C4—344no—
302C2×C4—344no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—684no—
311C2×C4—684no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4#1058755012yes—
319C2×C4#1058767812yes—
320C8—684no—
321C8—1362no—
322C22—342no—
323C22—342no—
324C22—342no—
325C22—681no—
326C22—681no—
327C22—681no—
328C22—681no—
329C22#1176378311yescharacteristic
330C22D17⋊#9911yescharacteristic
331C22#1176379511yescharacteristic
332C22#1176380011yescharacteristic
333C22#1176380211yescharacteristic
334C22C2×#1058755011yescharacteristic
335C22#1176380611yescharacteristic
336C4—22no—
337C4—22no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—341no—
342C4—341no—
343C4—342no—
344C4—681no—
345C4—1361no—
346C4—1361no—
347C2—682no—
348C2#1454405011yescharacteristic
349C2#1454410711yescharacteristic
350C2#1454414011yescharacteristic
351C2#1454417111yescharacteristic
352C2#1454419311yescharacteristic
353C2#1454425111yescharacteristic
354C2#1454431911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7695
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #14543604C2semidirect
4. #14543602C2semidirect
231. C17#7695semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145444482
#145436042
#145436022
#145371252
#769517