Finite Group Databasebeta

← #17191211

Subgroups — #17191211

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
2D4×Dic68C211yesmaximal, characteristic
3#14531012C211yesmaximal, characteristic, complemented
4#14530987C212yesmaximal, complemented
5#14530937C211yesmaximal, characteristic, complemented
6#14530875C212yesmaximal, complemented
7#14530835C211yesmaximal, characteristic, complemented
8#14530720C212yesmaximal, complemented
9C4×Dic136C211yesmaximal, characteristic, complemented
10C2×#11762614C212yesmaximal, complemented
11D4×C16×C17C211yesmaximal, characteristic
12C22×Dic68C2212yes—
13#11763516—21no—
14D4×Dic34—21no—
15C2×Dic136C2211yescharacteristic
16C2×Dic136C2212yes—
17#11763214C2211yescharacteristic
18#11763189C2211yescharacteristic
19#11763145—22no—
20#11763105—21no—
21#11763050C2212yes—
22#11763050C2212yes—
23#11763020—21no—
24#11762967—22no—
25C4×Dic68C2211yescharacteristic
26C4×Dic68C2211yescharacteristic
27C2×#10587489C2212yes—
28#11762635C2212yes—
29#11762635C2211yescharacteristic
30#11762614C2212yes—
31#11762614C2211yescharacteristic
32#11762614C2212yes—
33#11762614C2211yescharacteristic
34#11762614C2212yes—
35#11762614C2212yes—
36C22×C16×C17C2212yes—
37C8×D4×C17C2211yescharacteristic
38C17×C4⋊C16C2211yescharacteristic
39C17×C22⋊C16C2212yes—
40C4×C16×C17C2211yescharacteristic
41C22×Dic34—22no—
42C22×C8×C17C2312yes—
43C4×D4×C17D411yescharacteristic
44D4×Dic17—41no—
45D4×Dic17—41no—
46C2×C4⋊Dic17—22no—
47#10587610—22no—
48#10587610—22no—
49C2×Dic68—24no—
50C2×Dic68—22no—
51C2×Dic68—21no—
52C2×Dic68—42no—
53C2×Dic68C2311yescharacteristic
54C2×Dic68C2312yes—
55C2×Dic68C2311yescharacteristic
56#10587576—41no—
57#10587574—41no—
58#10587564—42no—
59#10587564—42no—
60#10587555—21no—
61#10587555—21no—
62C4×Dic34—21no—
63C4×Dic34—21no—
64C2×C16×C17—42no—
65C2×C16×C17C2311yescharacteristic
66C2×C16×C17C2312yes—
67C2×C16×C17C2311yescharacteristic
68C17×C4⋊C8D411yescharacteristic
69C17×C22⋊C8D412yes—
70C4×C8×C17C2311yescharacteristic
71#10587489—21no—
72#10587489—22no—
73#10587489C2312yes—
74#10587489C2311yescharacteristic
75#10587489C2311yescharacteristic
76#10587487—21no—
77#10587487—22no—
78#10587487—22no—
79#10587487—21no—
80Dic136—22no—
81Dic136—42no—
82C2×D4×C17D811yescharacteristic
83C22×C4×C17C2×D412yes—
84C22×Dic17—42no—
85C22×Dic17—42no—
86C2×Dic34—21no—
87C2×Dic34—22no—
88C2×Dic34—21no—
89C2×Dic34—44no—
90C2×Dic34—42no—
91C2×Dic34—41no—
92C2×Dic34—82no—
93C2×C8×C17—22no—
94C2×C8×C17—22no—
95C2×C8×C17C2×D412yes—
96C2×C8×C17C2×D411yescharacteristic
97C2×C8×C17C2411yescharacteristic
98C4⋊C4×C17D811yescharacteristic
99C22⋊C4×C17D812yes—
100C42×C17C2×D411yescharacteristic
101C22⋊Dic17—42no—
102C22⋊Dic17—42no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17—22no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108C4⋊Dic17—41no—
109C4⋊Dic17—42no—
110#63225—41no—
111#63225—41no—
112#63225—42no—
113#63225—42no—
114C4×Dic17—41no—
115C4×Dic17—41no—
116Dic68—21no—
117Dic68—22no—
118Dic68—21no—
119Dic68—21no—
120Dic68—22no—
121Dic68—44no—
122Dic68—81no—
123Dic68#4112yes—
124C16×C17—21no—
125C16×C17—22no—
126C16×C17—21no—
127#33866—171nomaximal, Sylow
128C23×C17C2×D812yes—
129D4×C17Q3214yescomplemented
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17C2×D811yescharacteristic
133C2×C4×C17C2×D812yes—
134C2×C4×C17C2×D811yescharacteristic
135C2×C4×C17C22×D411yescharacteristic
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—84no—
143C2×Dic17—162no—
144Dic34—22no—
145Dic34—41no—
146Dic34—42no—
147Dic34—41no—
148Dic34—41no—
149Dic34—42no—
150Dic34—84no—
151Dic34—161no—
152C8×C17—22no—
153C8×C17—21no—
154C8×C17C2×#4111yescharacteristic
155C8×C17H(5,𝔽2)11yescharacteristic
156#3338—171no—
157D4×Q16—171no—
158#2241—171no—
159#2236—172no—
160#2227—171no—
161#2216—172no—
162#2209—171no—
163#2188—172no—
164C4×Q32—171no—
165D4×C16D1711yescharacteristic
166C2×#358—172no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17C2×Q3214yes—
170C22×C17C22×D811yescharacteristic
171C4×C17—21no—
172C4×C17—22no—
173C4×C17C2×Q3212yes—
174C4×C17#54611yescharacteristic
175C4×C17#57711yescharacteristic
176Dic17—42no—
177Dic17—82no—
178Dic17—81no—
179Dic17—81no—
180Dic17—82no—
181Dic17—81no—
182C22×Q16—172no—
183#554—341no—
184D4×Q8—341no—
185C2×Q32—171no—
186C2×Q32—172no—
187C22×C16C2×D1712yes—
188#500—171no—
189#494—171no—
190#484—342no—
191#477—341no—
192#467—172no—
193#467—172no—
194#462—341no—
195#451—342no—
196C4×Q16—171no—
197C4×Q16—171no—
198C8×D4C2×D1711yescharacteristic
199C2×#107—172no—
200#366—171no—
201#366—172no—
202C4⋊C16C2×D1711yescharacteristic
203#358—171no—
204#358—172no—
205#358—172no—
206#358—172no—
207#358—171no—
208#358—172no—
209C22⋊C16C2×D1712yes—
210C4×C16C2×D1711yescharacteristic
211C2×C17—24no—
212C2×C17#329411yescharacteristic
213C2×C17C22×Q3211yescharacteristic
214C2×C17#341611yescharacteristic
215C22×Q8—342no—
216C2×Q16—171no—
217C2×Q16—172no—
218C2×Q16—171no—
219C2×Q16—344no—
220C2×Q16—342no—
221C2×Q16—341no—
222C2×Q16—682no—
223C22×C8C22×D1712yes—
224C4⋊Q8—341no—
225C4⋊Q8—341no—
226C4⋊Q8—681no—
227#125—681no—
228C22⋊Q8—342no—
229C22⋊Q8—342no—
230C22⋊Q8—682no—
231C22⋊Q8—682no—
232C4×Q8—341no—
233C4×Q8—341no—
234C4×D4—681no—
235C4×D4—681no—
236C4×D4D6811yescharacteristic
237C2×C4⋊C4—342no—
238Q32—342no—
239Q32—682no—
240C2×C16—42no—
241C2×C16C22×D1711yescharacteristic
242C2×C16C22×D1711yescharacteristic
243C2×C16C22×D1712yes—
244#107—171no—
245#107—172no—
246#107—171no—
247#107—342no—
248#107—341no—
249C4⋊C8D6811yescharacteristic
250#103—341no—
251#103—341no—
252#103—342no—
253#103—342no—
254C22⋊C8D6812yes—
255C4×C8C22×D1711yescharacteristic
256C17#3386611yescharacteristic, complemented, Sylow
257C2×Q8—341no—
258C2×Q8—341no—
259C2×Q8—342no—
260C2×Q8—684no—
261C2×Q8—681no—
262C2×Q8—682no—
263C2×Q8—1362no—
264C2×D4D13611yescharacteristic
265C22×C4—682no—
266C22×C4—682no—
267C22×C4C2×D6812yes—
268Q16—172no—
269Q16—341no—
270Q16—341no—
271Q16—342no—
272Q16—341no—
273Q16—342no—
274Q16—684no—
275Q16—1361no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×D6811yescharacteristic
279C2×C8C2×D6812yes—
280C2×C8C23×D1711yescharacteristic
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—342no—
284C4⋊C4—682no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—682no—
290C4⋊C4—682no—
291C4⋊C4—681no—
292C4⋊C4D13611yescharacteristic
293C22⋊C4—682no—
294C22⋊C4—682no—
295C22⋊C4D13612yes—
296C42—681no—
297C42—681no—
298C42C2×D6811yescharacteristic
299C16—21no—
300C16—22no—
301C16—21no—
302C23C2×D13612yes—
303Q8—2721no—
304Q8—342no—
305Q8—681no—
306Q8—682no—
307Q8—681no—
308Q8—682no—
309Q8—681no—
310Q8—1364no—
311D4Dic13614yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—2722no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—1364no—
322C2×C4C2×D13611yescharacteristic
323C2×C4C2×D13612yes—
324C2×C4C2×D13611yescharacteristic
325C2×C4C22×D6811yescharacteristic
326C8—21no—
327C8—22no—
328C8#1058769111yescharacteristic
329C8#41×D1711yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×Dic13614yes—
333C22C22×D13611yescharacteristic
334C4—22no—
335C4—21no—
336C4—682no—
337C4—1362no—
338C4—1361no—
339C4—1361no—
340C4—1362no—
341C4—1361no—
342C4C2×Dic13612yes—
343C4#1176345411yescharacteristic
344C4#1176362111yescharacteristic
345C2—24no—
346C2#1454059811yescharacteristic
347C2C22×Dic13611yescharacteristic
348C2#1454193211yescharacteristic
subgroupG/H
centerC22C22×D136
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingD4×C16×C17C2
socleC22×C17C22×D8
prime radicalC2×D4×C17D8
hypercenterD4×C16D17
hypocenterC17#33866
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
3. #14531012C2semidirect
4. #14530987C2semidirect
5. #14530937C2semidirect
6. #14530875C2semidirect
7. #14530835C2semidirect
8. #14530720C2semidirect
9. C4×Dic136C2semidirect
10. C2×#11762614C2semidirect
129. D4×C17Q32semidirect
256. C17#33866semidirect
311. D4Dic136semidirect

Maximal subgroups

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

subgroupindex
#145411772
D4×Dic682
#145310122
#145309872
#145309372
#145308752
#145308352
#145307202
C4×Dic1362
C2×#117626142
D4×C16×C172
#3386617