Finite Group Databasebeta

← #17191210

Subgroups — #17191210

Order 4352.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541177C211yesmaximal, characteristic
2#14531011C211yesmaximal, characteristic, complemented
3#14530984C212yesmaximal, complemented
4#14530936C211yesmaximal, characteristic, complemented
5#14530873C212yesmaximal, complemented
6#14530831C211yesmaximal, characteristic, complemented
7#14530718C212yesmaximal, complemented
8C4×#10587500C211yesmaximal, characteristic, complemented
9D4×#63214C211yesmaximal, characteristic
10C2×#11762613C212yesmaximal, complemented
11D4×Q16×C17C211yesmaximal, characteristic
12#11763516—21no—
13C2×#10587500C2211yescharacteristic
14C2×#10587500C2212yes—
15C22×#63214C2212yes—
16#11763214C2211yescharacteristic
17#11763145—22no—
18#11763105—21no—
19#11763050C2212yes—
20C4×Dic68C2211yescharacteristic
21C2×#10587489C2212yes—
22#11762634C2212yes—
23#11762634C2211yescharacteristic
24C4⋊#63214C2211yescharacteristic
25#11762613C2212yes—
26#11762613C2212yes—
27#11762613C2211yescharacteristic
28#11762613C2212yes—
29#11762613C2212yes—
30#11762612C2211yescharacteristic
31C22⋊#63214C2212yes—
32C4×#63214C2211yescharacteristic
33C22×Q16×C17C2212yes—
34D4×Q8×C17—21no—
35C17×#494C2211yescharacteristic
36C17×#467C2212yes—
37C17×#462—21no—
38C17×#451—22no—
39C4×Q16×C17C2211yescharacteristic
40C8×D4×C17C2211yescharacteristic
41C22×Q8×C17—22no—
42C2×Q16×C17—24no—
43C2×Q16×C17—22no—
44C2×Q16×C17—21no—
45C2×Q16×C17—42no—
46C2×Q16×C17C2312yes—
47C2×Q16×C17C2311yescharacteristic
48C22×C8×C17C2312yes—
49C17×C4⋊Q8—21no—
50C17×C4⋊Q8—41no—
51C17×C22⋊Q8—22no—
52C17×C22⋊Q8—42no—
53C4×Q8×C17—21no—
54C4×D4×C17—41no—
55C4×D4×C17D411yescharacteristic
56D4×Dic17—41no—
57C2×C4⋊Dic17—22no—
58#10587610—22no—
59C2×Dic68C2311yescharacteristic
60#10587576—41no—
61#10587564—42no—
62#10587555—21no—
63C4×Dic34—21no—
64C17×#107C2311yescharacteristic
65C17×C4⋊C8D411yescharacteristic
66C17×#103—21no—
67C17×#103—22no—
68C17×C22⋊C8D412yes—
69C4×C8×C17C2311yescharacteristic
70#10587500—22no—
71#10587500—42no—
72#10587489—21no—
73#10587489—22no—
74#10587489C2312yes—
75#10587489C2311yescharacteristic
76#10587487—22no—
77#10587487—21no—
78C2×#63214—42no—
79C2×#63214C2311yescharacteristic
80C2×#63214C2312yes—
81C2×#63214C2311yescharacteristic
82C2×Q8×C17—21no—
83C2×Q8×C17—22no—
84C2×Q8×C17—44no—
85C2×Q8×C17—42no—
86C2×Q8×C17—82no—
87C2×D4×C17D811yescharacteristic
88C22×C4×C17—42no—
89C22×C4×C17C2×D412yes—
90C22×Dic17—42no—
91C2×Dic34—21no—
92C2×Dic34—41no—
93Q16×C17—22no—
94Q16×C17—21no—
95Q16×C17—22no—
96Q16×C17—44no—
97Q16×C17—81no—
98Q16×C17#4112yes—
99C2×C8×C17—22no—
100C2×C8×C17—22no—
101C2×C8×C17C2×D412yes—
102C2×C8×C17C2×D411yescharacteristic
103C2×C8×C17C2411yescharacteristic
104C4⋊C4×C17—21no—
105C4⋊C4×C17—42no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C4⋊C4×C17—41no—
109C4⋊C4×C17D811yescharacteristic
110C22⋊C4×C17—42no—
111C22⋊C4×C17D812yes—
112C42×C17—41no—
113C42×C17C2×D411yescharacteristic
114C22⋊Dic17—42no—
115C4⋊Dic17—21no—
116C4⋊Dic17—22no—
117C4⋊Dic17—41no—
118C4⋊Dic17—42no—
119#63225—41no—
120#63225—42no—
121C4×Dic17—41no—
122Dic68—21no—
123Dic68—21no—
124#63214—21no—
125#63214—21no—
126#63214—22no—
127#33866—171nomaximal, Sylow
128C23×C17C2×D812yes—
129Q8×C17—22no—
130Q8×C17—41no—
131Q8×C17—42no—
132Q8×C17—42no—
133Q8×C17—84no—
134D4×C17Q3214yescomplemented
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—42no—
140C2×C4×C17—84no—
141C2×C4×C17C2×D811yescharacteristic
142C2×C4×C17C2×D812yes—
143C2×C4×C17C2×D811yescharacteristic
144C2×C4×C17C22×D411yescharacteristic
145C2×Dic17—41no—
146C2×Dic17—42no—
147C2×Dic17—41no—
148C2×Dic17—162no—
149Dic34—41no—
150Dic34—41no—
151Dic34—161no—
152C8×C17—22no—
153C8×C17—21no—
154C8×C17C2×#4111yescharacteristic
155C8×C17H(5,𝔽2)11yescharacteristic
156#3338—171no—
157D4×Q16D1711yescharacteristic
158#2241—171no—
159#2236—172no—
160#2227—171no—
161#2216—172no—
162#2209—171no—
163#2188—172no—
164C4×Q32—171no—
165D4×C16—171no—
166C2×#358—172no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17C2×Q3214yes—
170C22×C17C22×D811yescharacteristic
171C4×C17—21no—
172C4×C17—22no—
173C4×C17—42no—
174C4×C17—81no—
175C4×C17—82no—
176C4×C17C2×Q3212yes—
177C4×C17#54611yescharacteristic
178C4×C17#57711yescharacteristic
179Dic17—82no—
180Dic17—81no—
181Dic17—81no—
182C22×Q16C2×D1712yes—
183#554—341no—
184D4×Q8—21no—
185C2×Q32—171no—
186C2×Q32—172no—
187C22×C16—172no—
188#500—171no—
189#494C2×D1711yescharacteristic
190#484—342no—
191#477—341no—
192#467—172no—
193#467C2×D1712yes—
194#462—21no—
195#451—22no—
196C4×Q16—171no—
197C4×Q16C2×D1711yescharacteristic
198C8×D4C2×D1711yescharacteristic
199C2×#107—172no—
200#366—171no—
201#366—172no—
202C4⋊C16—171no—
203#358—171no—
204#358—172no—
205#358—172no—
206#358—172no—
207#358—172no—
208#358—171no—
209C22⋊C16—172no—
210C4×C16—171no—
211C2×C17—24no—
212C2×C17#329411yescharacteristic
213C2×C17C22×Q3211yescharacteristic
214C2×C17#341611yescharacteristic
215C22×Q8—22no—
216C2×Q16—24no—
217C2×Q16—22no—
218C2×Q16—21no—
219C2×Q16—42no—
220C2×Q16—171no—
221C2×Q16C22×D1712yes—
222C2×Q16C22×D1711yescharacteristic
223C22×C8C22×D1712yes—
224C4⋊Q8—21no—
225C4⋊Q8—41no—
226C4⋊Q8—341no—
227#125—681no—
228C22⋊Q8—22no—
229C22⋊Q8—42no—
230C22⋊Q8—342no—
231C22⋊Q8—682no—
232C4×Q8—21no—
233C4×Q8—341no—
234C4×D4—41no—
235C4×D4—681no—
236C4×D4C22⋊D1711yescharacteristic
237C2×C4⋊C4—342no—
238Q32—342no—
239Q32—682no—
240C2×C16—171no—
241C2×C16—171no—
242C2×C16—172no—
243C2×C16—682no—
244#107—171no—
245#107—172no—
246#107—342no—
247#107—341no—
248#107C22×D1711yescharacteristic
249C4⋊C8C22⋊D1711yescharacteristic
250#103—21no—
251#103—22no—
252#103—341no—
253#103—342no—
254C22⋊C8C22⋊D1712yes—
255C4×C8C22×D1711yescharacteristic
256C17#3386611yescharacteristic, complemented, Sylow
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—82no—
262C2×Q8—341no—
263C2×Q8—681no—
264C2×D4#6322811yescharacteristic
265C22×C4—42no—
266C22×C4—682no—
267C22×C4C2×C22⋊D1712yes—
268Q16—22no—
269Q16—21no—
270Q16—22no—
271Q16—44no—
272Q16—81no—
273Q16—341no—
274Q16—341no—
275Q16#6325412yes—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×C22⋊D1711yescharacteristic
279C2×C8C2×C22⋊D1712yes—
280C2×C8C23×D1711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—341no—
287C4⋊C4—342no—
288C4⋊C4—682no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—682no—
292C4⋊C4#6322811yescharacteristic
293C22⋊C4—42no—
294C22⋊C4—682no—
295C22⋊C4#6322812yes—
296C42—41no—
297C42—681no—
298C42C2×C22⋊D1711yescharacteristic
299C16—341no—
300C16—341no—
301C16—342no—
302C23C2×#6322812yes—
303Q8—22no—
304Q8—41no—
305Q8—42no—
306Q8—42no—
307Q8—84no—
308Q8—2721no—
309Q8—681no—
310Q8—681no—
311D4#1058750014yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—2722no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—682no—
322C2×C4C2×#6322811yescharacteristic
323C2×C4C2×#6322812yes—
324C2×C4C2×#6322811yescharacteristic
325C2×C4C22×C22⋊D1711yescharacteristic
326C8—21no—
327C8—22no—
328C8C2×#6325411yescharacteristic
329C8#1058769111yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×#1058750014yes—
333C22C22×#6322811yescharacteristic
334C4—22no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—1361no—
340C4—1361no—
341C4—1362no—
342C4C2×#1058750012yes—
343C4#1176345311yescharacteristic
344C4#1176362011yescharacteristic
345C2—24no—
346C2#1454059711yescharacteristic
347C2C22×#1058750011yescharacteristic
348C2#1454192811yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingD4×Q16×C17C2
socleC22×C17C22×D8
prime radicalC2×D4×C17D8
hypercenterD4×Q16D17
hypocenterC17#33866
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #14531011C2semidirect
3. #14530984C2semidirect
4. #14530936C2semidirect
5. #14530873C2semidirect
6. #14530831C2semidirect
7. #14530718C2semidirect
8. C4×#10587500C2semidirect
10. C2×#11762613C2semidirect
134. D4×C17Q32semidirect
256. C17#33866semidirect
311. D4#10587500semidirect

Maximal subgroups

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

subgroupindex
#145411772
#145310112
#145309842
#145309362
#145308732
#145308312
#145307182
C4×#105875002
D4×#632142
C2×#117626132
D4×Q16×C172
#3386617