Finite Group Databasebeta

← #16923626

Subgroups — #16923626

Order 4352.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#14535815C211yesmaximal, characteristic
2#14530348C211yesmaximal, characteristic
3#14530347C211yesmaximal, characteristic
4#14530321C211yesmaximal, characteristic
5#14526609C211yesmaximal, characteristic
6#14526608C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic
8C2×#10587574—21no—
9C2×#10587489—21no—
10C2×#10587487—21no—
11#11762742—21no—
12#11762730—21no—
13#11762729—21no—
14#11762715C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762695—21no—
17#11762692—21no—
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762572—21no—
21#11762571—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Dic34—21no—
28C22×Dic34D411yescharacteristic
29C22×C8×C17—21no—
30C22×C8×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17D411yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C22×#3622—21no—
47#10587574—42no—
48#10587574—42no—
49C17×#95—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587489—42no—
57#10587487—42no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—41no—
68C2×Dic34—21no—
69C2×Dic34—41no—
70C2×Dic34—41no—
71C2×Dic34—42no—
72C2×Dic34—81no—
73C2×Dic34—81no—
74C2×Dic34SD1612yes—
75C2×C8×C17—22no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C2×C8×C17—41no—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17Q1612yes—
86C42×C17—41no—
87C4⋊Dic17—21no—
88C4⋊Dic17—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95C4⋊Dic17Q1612yes—
96#63225—42no—
97#63225—41no—
98#63225—41no—
99#63225—81no—
100C4×Dic17—42no—
101C2×#3622—41no—
102C2×#3622—41no—
103C2×#3622—41no—
104#7829—171nomaximal, Sylow
105C23×C17#12011yescharacteristic
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17C2×SD1611yescharacteristic
119C2×C4×C17C2×Q1611yescharacteristic
120C2×C4×C17C2×Q1611yescharacteristic
121C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C17#13711yescharacteristic
123C2×C4×C17#13711yescharacteristic
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—81no—
137C2×Dic17—81no—
138C2×Dic17—81no—
139Dic34—42no—
140Dic34—81no—
141Dic34—84no—
142C8×C17—42no—
143C8×C17—81no—
144#3622—81no—
145#2734—171no—
146#2100—171no—
147#2100—171no—
148#2095—171no—
149#1272—171no—
150#1271—171no—
151#1271D1711yescharacteristic
152C22×C17#45011yescharacteristic
153C22×C17#45111yescharacteristic
154C22×C17#45111yescharacteristic
155C22×C17#45311yescharacteristic
156C22×C17#45411yescharacteristic
157C22×C17#45411yescharacteristic
158C22×C17#45811yescharacteristic
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—42no—
166C4×C17—81no—
167Dic17—42no—
168Dic17—42no—
169Dic17—81no—
170Dic17—81no—
171Dic17—161no—
172C2×C4⋊Q8—341no—
173C2×#107—341no—
174C2×#103—341no—
175#398—341no—
176#395—341no—
177#395—341no—
178#391—171no—
179#389—171no—
180#389C2×D1711yescharacteristic
181#387—341no—
182#387—341no—
183#384—171no—
184#340—341no—
185#340—341no—
186#340—341no—
187#340—341no—
188#336—171no—
189#336C2×D1711yescharacteristic
190#336C2×D1711yescharacteristic
191C2×C17#161311yescharacteristic
192C2×C17#161711yescharacteristic
193C2×C17#161711yescharacteristic
194C2×C17#163311yescharacteristic
195C2×C17#163711yescharacteristic
196C2×C17#163711yescharacteristic
197C2×C17#165711yescharacteristic
198C22×Q8—171no—
199C22×Q8—341no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—341no—
203C4⋊Q8—682no—
204C4⋊Q8—682no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4D6811yescharacteristic
217C2×C4⋊C4D6811yescharacteristic
218C2×C42—341no—
219C2×C42C22×D1711yescharacteristic
220#107—682no—
221#103—682no—
222#95—21no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229C17#782911yescharacteristic, complemented, Sylow
230C2×Q8—172no—
231C2×Q8—341no—
232C2×Q8—681no—
233C2×Q8—681no—
234C2×Q8—682no—
235C2×Q8—1361no—
236C2×Q8—1361no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—681no—
244C22×C4C2×D6811yescharacteristic
245C22×C4D4×D1711yescharacteristic
246C22×C4D4×D1711yescharacteristic
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—681no—
254C2×C8—681no—
255C2×C8—681no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—172no—
261C4⋊C4—341no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—682no—
267C4⋊C4—682no—
268C4⋊C4—682no—
269C4⋊C4—682no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—1361no—
273C4⋊C4Dic6812yes—
274C42—41no—
275C42—682no—
276C23#1058756811yescharacteristic
277Q8—682no—
278Q8—1361no—
279Q8—1364no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—1361no—
305C2×C4—1361no—
306C2×C4—1361no—
307C2×C4C2×Dic6811yescharacteristic
308C2×C4#1058759511yescharacteristic
309C2×C4#1058759711yescharacteristic
310C2×C4SD16×D1711yescharacteristic
311C2×C4#1058760111yescharacteristic
312C2×C4Q16×D1711yescharacteristic
313C8—42no—
314C8—81no—
315C8—1361no—
316C22#1176296111yescharacteristic
317C22#1176296611yescharacteristic
318C22#1176296711yescharacteristic
319C22#1176297811yescharacteristic
320C22#1176298311yescharacteristic
321C22#1176298411yescharacteristic
322C22#1176300011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—81no—
331C4—2721no—
332C4—682no—
333C4—682no—
334C4—1361no—
335C4—1361no—
336C2#1452780511yescharacteristic
337C2#1452782711yescharacteristic
338C2#1452782811yescharacteristic
339C2#1452792611yescharacteristic
340C2#1452795011yescharacteristic
341C2#1452795211yescharacteristic
342C2#1452808411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socle, prime radicalC23×C17#120
hypercenter#1271D17
hypocenterC17#7829
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
229. C17#7829semidirect

Maximal subgroups

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

subgroupindex
#145358152
#145303482
#145303472
#145303212
#145266092
#145266082
C17×#12712
#782917