Finite Group Databasebeta

← #16923401

Subgroups — #16923401

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#14530256C211yesmaximal, characteristic
3#14530193C211yesmaximal, characteristic
4#14526609C211yesmaximal, characteristic
5#14526607C211yesmaximal, characteristic
6#14526606C211yesmaximal, characteristic
7C17×#2099C211yesmaximal, characteristic
8C2×#10587574—21no—
9C2×#10587478—21no—
10C2×#10587481—21no—
11#11762742—21no—
12#11762730—21no—
13#11762729—21no—
14#11762715C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762695—21no—
17#11762692—21no—
18#11762571—21no—
19#11762571—21no—
20#11762564C2211yescharacteristic
21#11762564C2211yescharacteristic
22C17×#389C2211yescharacteristic
23C17×#384C2211yescharacteristic
24C17×#340—21no—
25C17×#340—21no—
26C17×#336C2211yescharacteristic
27C22×Dic34—21no—
28C22×Dic34D411yescharacteristic
29C22×C8×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C42×C17C2311yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17—21no—
45C22×#3622—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#10587481—42no—
57#10587478—42no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C2×Dic34—21no—
69C2×Dic34—41no—
70C2×Dic34—41no—
71C2×Dic34—42no—
72C2×Dic34—81no—
73C2×Dic34—81no—
74C2×Dic34D812yes—
75C2×C8×C17—41no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17Q1612yes—
88C42×C17—41no—
89C4⋊Dic17—21no—
90C4⋊Dic17—42no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17SD1612yes—
94#63225—42no—
95#63225—41no—
96#63225—41no—
97#63225—81no—
98C4×Dic17—42no—
99C2×#3622—22no—
100C2×#3622—41no—
101C2×#3622—41no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622—41no—
105#7794—171nomaximal, Sylow
106C23×C17#12011yescharacteristic
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—22no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—81no—
120C2×C4×C17—81no—
121C2×C4×C17—81no—
122C2×C4×C17C2×D811yescharacteristic
123C2×C4×C17C2×SD1611yescharacteristic
124C2×C4×C17C2×Q1611yescharacteristic
125C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C17#13711yescharacteristic
127C2×C4×C17#13711yescharacteristic
128C2×Dic17—22no—
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—41no—
140Dic34—42no—
141Dic34—81no—
142Dic34—84no—
143C8×C17—81no—
144#3622—42no—
145#3622—81no—
146#2734—171no—
147#2099D1711yescharacteristic
148#2083—171no—
149#2072—171no—
150#1272—171no—
151#1271—171no—
152#1271—171no—
153C22×C17#44711yescharacteristic
154C22×C17#44811yescharacteristic
155C22×C17#45111yescharacteristic
156C22×C17#45411yescharacteristic
157C22×C17#45511yescharacteristic
158C22×C17#45611yescharacteristic
159C22×C17#45811yescharacteristic
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—42no—
167C4×C17—81no—
168C4×C17—161no—
169Dic17—42no—
170Dic17—42no—
171Dic17—81no—
172Dic17—81no—
173C2×C4⋊Q8—341no—
174C2×#107—341no—
175C2×#103—341no—
176#398—341no—
177#395—341no—
178#395—341no—
179#391—171no—
180#389—171no—
181#389C2×D1711yescharacteristic
182#387—341no—
183#387—341no—
184#384C2×D1711yescharacteristic
185#340—21no—
186#340—21no—
187#340—341no—
188#340—341no—
189#336—171no—
190#336—171no—
191#336C2×D1711yescharacteristic
192C2×C17#160711yescharacteristic
193C2×C17#161111yescharacteristic
194C2×C17#161711yescharacteristic
195C2×C17#162711yescharacteristic
196C2×C17#164511yescharacteristic
197C2×C17#164911yescharacteristic
198C2×C17#166411yescharacteristic
199C22×Q8—171no—
200C22×Q8—341no—
201C22×C8—21no—
202C22×C8—341no—
203C22×C8—341no—
204C4⋊Q8—682no—
205C4⋊Q8—682no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4C22⋊D1711yescharacteristic
218C2×C4⋊C4C22⋊D1711yescharacteristic
219C2×C42—341no—
220C2×C42C22×D1711yescharacteristic
221#107—682no—
222#103—682no—
223#95—21no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#779411yescharacteristic, complemented, Sylow
231C2×Q8—172no—
232C2×Q8—341no—
233C2×Q8—681no—
234C2×Q8—681no—
235C2×Q8—682no—
236C2×Q8—1361no—
237C2×Q8—1361no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4D4×D1711yescharacteristic
246C22×C4D4×D1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—342no—
252C2×C8—681no—
253C2×C8—681no—
254C2×C8—681no—
255C2×C8—681no—
256C2×C8—681no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—172no—
267C4⋊C4—341no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4—682no—
271C4⋊C4—682no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—1361no—
275C4⋊C4#6323112yes—
276C42—41no—
277C42—682no—
278C23#1058762311yescharacteristic
279Q8—682no—
280Q8—1361no—
281Q8—1364no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—342no—
298C2×C4—342no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4D8×D1711yescharacteristic
310C2×C4#1058759711yescharacteristic
311C2×C4SD16×D1711yescharacteristic
312C2×C4#1058760111yescharacteristic
313C2×C4C2×#6323111yescharacteristic
314C2×C4#1058763711yescharacteristic
315C8—81no—
316C8—682no—
317C8—1361no—
318C22#1176294111yescharacteristic
319C22#1176294611yescharacteristic
320C22#1176296411yescharacteristic
321C22#1176298111yescharacteristic
322C22#1176298511yescharacteristic
323C22#1176299011yescharacteristic
324C22#1176299811yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—81no—
333C4—161no—
334C4—682no—
335C4—682no—
336C4—1361no—
337C4—1361no—
338C2#1452776811yescharacteristic
339C2#1452779011yescharacteristic
340C2#1452782511yescharacteristic
341C2#1452788411yescharacteristic
342C2#1452800411yescharacteristic
343C2#1452803211yescharacteristic
344C2#1452811511yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2099C2
socle, prime radicalC23×C17#120
hypercenter#2099D17
hypocenterC17#7794
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
230. C17#7794semidirect

Maximal subgroups

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

subgroupindex
#145358152
#145302562
#145301932
#145266092
#145266072
#145266062
C17×#20992
#779417