Finite Group Databasebeta

← #16923218

Subgroups — #16923218

Order 4352.

392 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 81 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#14536739C211yesmaximal, characteristic
2#14530118C211yesmaximal, characteristic
3#14530031C211yesmaximal, characteristic
4#14529925C211yesmaximal, characteristic
5#14526608C211yesmaximal, characteristic
6#14526606C211yesmaximal, characteristic
7C17×#1272C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×#10587488—21no—
10C2×#10587487—21no—
11C2×#10587487—21no—
12C2×#10587506—21no—
13#11762751—21no—
14#11762749—21no—
15#11762728—21no—
16#11762707C2211yescharacteristic
17#11762707C2211yescharacteristic
18#11762706—21no—
19#11762692—21no—
20#11762690—21no—
21#11762572—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#391C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Q8×C17D411yescharacteristic
28C22×Dic34—21no—
29C22×C8×C17—21no—
30C22×C8×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17D411yescharacteristic
36C2×C4⋊Dic17D411yescharacteristic
37C2×C4⋊Dic17D411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44C22×#3622—21no—
45#10587555—42no—
46C17×#95—21no—
47C17×#95—21no—
48#10587506—42no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587488—42no—
57#10587487—42no—
58#10587487—42no—
59C2×Q8×C17—21no—
60C2×Q8×C17—21no—
61C2×Q8×C17—41no—
62C2×Q8×C17—41no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—41no—
73C2×Dic34—22no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×Dic34—41no—
79C2×C8×C17—22no—
80C2×C8×C17—41no—
81C2×C8×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17—41no—
85C4⋊C4×C17—41no—
86C42×C17—41no—
87C4⋊Dic17—21no—
88C4⋊Dic17—22no—
89C4⋊Dic17—22no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96C4⋊Dic17SD1612yes—
97#63225—41no—
98#63225—41no—
99#63225—41no—
100#63225—41no—
101#63225—81no—
102#63225—81no—
103C4×Dic17—81no—
104C2×#3622—41no—
105C2×#3622—41no—
106C2×#3622—41no—
107#7767—171nomaximal, Sylow
108C23×C17#12011yescharacteristic
109Q8×C17—42no—
110Q8×C17—81no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17C2×SD1611yescharacteristic
124C2×C4×C17#13511yescharacteristic
125C2×C4×C17#13511yescharacteristic
126C2×C4×C17#13711yescharacteristic
127C2×C4×C17#13711yescharacteristic
128C2×C4×C17#13711yescharacteristic
129C2×Dic17—22no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—42no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—81no—
143C2×Dic17—81no—
144Dic34—42no—
145Dic34—42no—
146Dic34—81no—
147Dic34—81no—
148C8×C17—42no—
149C8×C17—81no—
150#3622—81no—
151#2805—171no—
152#2058—171no—
153#2043—171no—
154#2024—171no—
155#1272D1711yescharacteristic
156#1271—171no—
157#1271—171no—
158C22×C17#44811yescharacteristic
159C22×C17#45211yescharacteristic
160C22×C17#45211yescharacteristic
161C22×C17#45411yescharacteristic
162C22×C17#45511yescharacteristic
163C22×C17#45611yescharacteristic
164C22×C17B(4,𝔽2)11yescharacteristic
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—81no—
172C4×C17—81no—
173Dic17—42no—
174Dic17—82no—
175Dic17—81no—
176Dic17—81no—
177Dic17—81no—
178C2×C4⋊Q8—171no—
179C2×#106—341no—
180C2×#103—341no—
181C2×#103—341no—
182C2×#103—341no—
183#400—341no—
184#400—341no—
185#395—341no—
186#391C2×D1711yescharacteristic
187#389—171no—
188#389—171no—
189#389—341no—
190#387—341no—
191#387—341no—
192#340—341no—
193#340—341no—
194#336—171no—
195#336C2×D1711yescharacteristic
196#336C2×D1711yescharacteristic
197C2×C17#159911yescharacteristic
198C2×C17#160311yescharacteristic
199C2×C17#160611yescharacteristic
200C2×C17#163811yescharacteristic
201C2×C17#164311yescharacteristic
202C2×C17#165311yescharacteristic
203C2×C17#166411yescharacteristic
204C22×Q8—341no—
205C22×Q8D6811yescharacteristic
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—341no—
209C4⋊Q8—682no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4D6811yescharacteristic
221C2×C42—341no—
222C2×C42C22×D1711yescharacteristic
223#106—682no—
224#103—682no—
225#103—682no—
226#103—682no—
227#95—21no—
228#95—21no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#776711yescharacteristic, complemented, Sylow
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—342no—
242C2×Q8—681no—
243C2×Q8—681no—
244C2×Q8—681no—
245C2×Q8—681no—
246C2×Q8—681no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—681no—
254C22×C4C2×D6811yescharacteristic
255C22×C4D4×D1711yescharacteristic
256C22×C4D4×D1711yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—681no—
264C2×C8—681no—
265C2×C8—681no—
266C4⋊C4—41no—
267C4⋊C4—172no—
268C4⋊C4—342no—
269C4⋊C4—342no—
270C4⋊C4—341no—
271C4⋊C4—341no—
272C4⋊C4—341no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—1361no—
282C4⋊C4—1361no—
283C42—41no—
284C42—1361no—
285C23#1058756811yescharacteristic
286Q8—42no—
287Q8—81no—
288Q8—682no—
289Q8—682no—
290Q8—1361no—
291Q8—1361no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—342no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—682no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—1361no—
318C2×C4—1361no—
319C2×C4#1058759011yescharacteristic
320C2×C4#1058759511yescharacteristic
321C2×C4SD16×D1711yescharacteristic
322C2×C4#1058760211yescharacteristic
323C2×C4#1058760411yescharacteristic
324C2×C4#1058760411yescharacteristic
325C8—42no—
326C8—81no—
327C8—1361no—
328C22#1176294911yescharacteristic
329C22#1176297311yescharacteristic
330C22#1176297411yescharacteristic
331C22#1176298311yescharacteristic
332C22#1176298911yescharacteristic
333C22#1176299311yescharacteristic
334C22#1176299711yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—682no—
344C4—1361no—
345C4—1361no—
346C4—1361no—
347C4—1362no—
348C2#1452772511yescharacteristic
349C2#1452775011yescharacteristic
350C2#1452776611yescharacteristic
351C2#1452795911yescharacteristic
352C2#1452799211yescharacteristic
353C2#1452806411yescharacteristic
354C2#1452812111yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socle, prime radicalC23×C17#120
hypercenter#1272D17
hypocenterC17#7767
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
236. C17#7767semidirect

Maximal subgroups

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

subgroupindex
#145367392
#145301182
#145300312
#145299252
#145266082
#145266062
C17×#12722
#776717