Finite Group Databasebeta

← #17950144

Subgroups — #17950144

Order 4352.

470 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 92 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#14544638C211yesmaximal, characteristic
2#14544510C211yesmaximal, characteristic
3#14543605C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14537124C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9C2×#10587542—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587610C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762714C411yescharacteristic
15#11762714C2211yescharacteristic
16#11762708C411yescharacteristic
17#11762708C2211yescharacteristic
18#11762694—21no—
19#11762694—21no—
20#11762684—21no—
21#11762670—21no—
22C17×#380C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36#10587610—42no—
37C2×#63225—21no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43C22⋊#3624—24no—
44C22⋊#3624—42no—
45#10587542—42no—
46C4⋊#3624—42no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C24×C17C22⋊C411yescharacteristic
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—41no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—22no—
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17—22no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×Dic34—22no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—42no—
87C22⋊Dic17—42no—
88C4⋊Dic17—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92#63225—42no—
93C4×Dic17—22no—
94C4×Dic17—22no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97#7701—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17#10412yes—
105C23×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—44no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17#9912yes—
118C2×C4×C17#13112yes—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—44no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17AGL(1,ℤ/8)11yescharacteristic
139C2×Dic17#13711yescharacteristic
140C2×Dic17#13712yes—
141Dic34—82no—
142#3624—44no—
143#3624—82no—
144#2855—171no—
145#1797—171no—
146#1749—171no—
147#1272—172no—
148#1271—172no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17#40811yescharacteristic
157C22×C17C2×#9911yescharacteristic
158C22×C17#41311yescharacteristic
159C22×C17#41711yescharacteristic
160C22×C17#41911yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C22×C17#42111yescharacteristic
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—81no—
166C4×C17—81no—
167Dic17—22no—
168Dic17—21no—
169Dic17—21no—
170Dic17—41no—
171Dic17—41no—
172Dic17—81no—
173Dic17—81no—
174C2×C22⋊Q8—171no—
175C22×C42—171no—
176C2×C4⋊C8—341no—
177C2×C22⋊C8—341no—
178C2×C22⋊C8—341no—
179C2×#97—341no—
180#391—171no—
181#391—171no—
182#389—171no—
183#389—171no—
184#387—341no—
185#387—341no—
186#385—341no—
187#381—341no—
188#380#59511yescharacteristic
189#336—172no—
190#336—172no—
191C2×C17—42no—
192C2×C17#146811yescharacteristic
193C2×C17#148311yescharacteristic
194C2×C17#149711yescharacteristic
195C2×C17#150811yescharacteristic
196C2×C17#151311yescharacteristic
197C2×C17#153211yescharacteristic
198C2×C17#155911yescharacteristic
199C22×Q8—171no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4C2×#59511yescharacteristic
203C22×C8—342no—
204C22×C8—342no—
205C22⋊Q8—682no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4C2×#59511yescharacteristic
211C2×C4⋊C4C2×#59511yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—341no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42—682no—
219C4⋊C8—682no—
220C22⋊C8—344no—
221C22⋊C8—682no—
222#97—682no—
223#95—21no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#770111yescharacteristic, complemented, Sylow
231C24C22⋊#59511yescharacteristic
232C2×Q8—342no—
233C2×Q8—681no—
234C2×Q8—681no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—681no—
251C22×C4—681no—
252C22×C4—681no—
253C22×C4C22⋊#59511yescharacteristic
254C22×C4C22×#59511yescharacteristic
255C2×C8—344no—
256C2×C8—682no—
257C2×C8—682no—
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—682no—
267C4⋊C4—682no—
268C22⋊C4—42no—
269C22⋊C4—682no—
270C42—342no—
271C42—342no—
272C42—681no—
273C42—681no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1058754812yes—
281C23C2×C22⋊#59511yescharacteristic
282Q8—1362no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—171no—
295C2×C4—171no—
296C2×C4—172no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—684no—
308C2×C4—682no—
309C2×C4—682no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4#1058753912yes—
317C2×C4#1058767812yes—
318C8—684no—
319C8—1362no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1176378211yescharacteristic
328C22C2×#1058753911yescharacteristic
329C22#1176379611yescharacteristic
330C22#1176380011yescharacteristic
331C22#1176380211yescharacteristic
332C22C2×#1058754811yescharacteristic
333C22#1176380611yescharacteristic
334C4—22no—
335C4—22no—
336C4—81no—
337C4—81no—
338C4—342no—
339C4—341no—
340C4—341no—
341C4—681no—
342C4—681no—
343C4—1361no—
344C4—1361no—
345C2—42no—
346C2#1454405111yescharacteristic
347C2#1454409911yescharacteristic
348C2#1454414611yescharacteristic
349C2#1454418311yescharacteristic
350C2#1454420111yescharacteristic
351C2#1454426011yescharacteristic
352C2#1454432011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#380C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#380#595
hypocenterC17#7701
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543605C2semidirect
4. #14543602C2semidirect
230. C17#7701semidirect

Maximal subgroups

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

subgroupindex
#145446382
#145445102
#145436052
#145436022
#145371242
#770117