Finite Group Databasebeta

← #16936936

Subgroups — #16936936

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763161C211yesmaximal, characteristic
2C2×#11763101C211yesmaximal, characteristic
3C2×#11763095C211yesmaximal, characteristic
4#14529719C211yesmaximal, characteristic
5C2×#11762556C211yesmaximal, characteristic
6#14526618C211yesmaximal, characteristic
7C2×C17×#333C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×C4×Dic34C2211yescharacteristic
10#11763161—22no—
11#11763161—22no—
12#11763101—22no—
13#11763101—22no—
14#11763095—22no—
15#11763095—22no—
16C2×#10587478—21no—
17C2×#10587488—21no—
18C2×C4⋊#3622C2211yescharacteristic
19C2×#10587487—21no—
20C2×#10587481—22no—
21C2×#10587481—21no—
22C2×C4×#3622C2211yescharacteristic
23#11762563—22no—
24#11762556—22no—
25C2×C17×C4⋊Q8C2211yescharacteristic
26C2×C17×C4⋊C8C2211yescharacteristic
27C2×C17×C4⋊C8C2211yescharacteristic
28C17×#333—22no—
29C22×Q8×C17—21no—
30C22×Dic34—21no—
31C22×Dic34D411yescharacteristic
32C22×C8×C17—21no—
33C22×C8×C17—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8D412yes—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C42×C17C2311yescharacteristic
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17D411yescharacteristic
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C22×#3622—21no—
47C22×#3622D412yes—
48#10587555—21no—
49#10587555—21no—
50#10587555—21no—
51#10587555D412yes—
52C4×Dic34—21no—
53C4×Dic34—21no—
54C4×Dic34—21no—
55C4×Dic34—21no—
56C17×C4⋊C8—21no—
57C17×C4⋊C8—21no—
58C17×C4⋊C8—21no—
59C17×C4⋊C8D412yes—
60#10587488—24no—
61#10587487—24no—
62#10587481—24no—
63#10587481—28no—
64#10587478—24no—
65C4⋊#3622—21no—
66C4⋊#3622D412yes—
67C4×#3622—21no—
68C4×#3622—21no—
69C2×Q8×C17—22no—
70C2×Q8×C17—41no—
71C2×Q8×C17—82no—
72C22×C4×C17—41no—
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17#4111yescharacteristic
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—41no—
79C2×Dic34—21no—
80C2×Dic34—21no—
81C2×Dic34—21no—
82C2×Dic34—21no—
83C2×Dic34—21no—
84C2×Dic34—22no—
85C2×Dic34—42no—
86C2×Dic34—42no—
87C2×Dic34—41no—
88C2×Dic34Q1612yes—
89C2×C8×C17—22no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C2×C8×C17—42no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C42×C17C2×D411yescharacteristic
99C42×C17C2×D411yescharacteristic
100C42×C17C2×D411yescharacteristic
101C42×C17C2×D411yescharacteristic
102C4⋊Dic17—21no—
103C4⋊Dic17—22no—
104C4⋊Dic17—21no—
105C4⋊Dic17—22no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108#63225—41no—
109#63225—41no—
110C4×Dic17—41no—
111C4×Dic17—41no—
112C2×#3622—22no—
113C2×#3622—22no—
114C2×#3622—22no—
115C2×#3622—22no—
116C2×#3622—41no—
117C2×#3622—41no—
118#10277—171nomaximal, Sylow
119C23×C17#12111yescharacteristic
120Q8×C17—82no—
121Q8×C17—82no—
122C2×C4×C17—22no—
123C2×C4×C17—42no—
124C2×C4×C17—82no—
125C2×C4×C17#12011yescharacteristic
126C2×C4×C17#12011yescharacteristic
127C2×C4×C17#12011yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12711yescharacteristic
131C2×C4×C17C2×Q1611yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17#13512yes—
134C2×C4×C17#13711yescharacteristic
135C2×C4×C17#13711yescharacteristic
136C2×C4×C17#13712yes—
137C2×Dic17—22no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—42no—
143C2×Dic17—82no—
144Dic34—22no—
145Dic34—22no—
146Dic34—41no—
147Dic34—41no—
148Dic34—44no—
149Dic34—81no—
150Dic34—81no—
151C8×C17—42no—
152C8×C17—81no—
153#3622—42no—
154#3622—42no—
155C2×#487—171no—
156C2×#477—171no—
157C2×#475—171no—
158#1984—171no—
159C2×#333D1711yescharacteristic
160C2×#332—171no—
161#1277—171no—
162C22×C17#39211yescharacteristic
163C22×C17#46412yes—
164C22×C17#46711yescharacteristic
165C22×C17#47011yescharacteristic
166C22×C17#47112yes—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—82no—
170C4×C17#45111yescharacteristic
171C4×C17#45111yescharacteristic
172C4×C17#45211yescharacteristic
173C4×C17#45211yescharacteristic
174C4×C17#45212yes—
175C4×C17#49511yescharacteristic
176C4×C17#49711yescharacteristic
177Dic17—42no—
178Dic17—82no—
179Dic17—81no—
180C2×C4⋊Q8—171no—
181C2×C4⋊Q8C2×D1711yescharacteristic
182C2×C4×Q8—171no—
183#487—342no—
184#487—342no—
185#477—342no—
186#477—342no—
187#475—342no—
188#475—342no—
189C2×#107—341no—
190C2×#106—341no—
191C2×C4⋊C8—171no—
192C2×C4⋊C8C2×D1711yescharacteristic
193C2×C4⋊C8C2×D1711yescharacteristic
194C2×#103—342no—
195C2×#103—341no—
196C2×#103—341no—
197C2×C4×C8—171no—
198#336—342no—
199#333—22no—
200#332—342no—
201C2×C17#168412yes—
202C2×C17#170412yes—
203C2×C17#200111yescharacteristic
204C2×C17#201511yescharacteristic
205C2×C17#201911yescharacteristic
206C22×Q8—21no—
207C22×Q8—171no—
208C22×Q8—341no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—172no—
212C22×C8—341no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—172no—
217C4⋊Q8—341no—
218C4⋊Q8—341no—
219C4⋊Q8—341no—
220C4⋊Q8D6812yes—
221C4×Q8—341no—
222C4×Q8—341no—
223C4×Q8—341no—
224C4×Q8—341no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C42—341no—
232C2×C42C22×D1711yescharacteristic
233#107—344no—
234#106—344no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—172no—
239C4⋊C8—341no—
240C4⋊C8C22⋊D1712yes—
241#103—344no—
242#103—348no—
243#103—344no—
244C4×C8—341no—
245C4×C8—341no—
246C17#1027711yescharacteristic, complemented, Sylow
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—172no—
251C2×Q8—341no—
252C2×Q8—341no—
253C2×Q8—342no—
254C2×Q8—341no—
255C2×Q8—341no—
256C2×Q8—341no—
257C2×Q8—681no—
258C2×Q8—682no—
259C2×Q8—682no—
260C22×C4—41no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—681no—
264C22×C4#6325211yescharacteristic
265C22×C4D4×D1711yescharacteristic
266C22×C4D4×D1711yescharacteristic
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—342no—
273C2×C8—342no—
274C2×C8—342no—
275C2×C8—342no—
276C2×C8—681no—
277C2×C8—681no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—341no—
283C4⋊C4—342no—
284C4⋊C4—342no—
285C4⋊C4—341no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C42—681no—
291C42—681no—
292C42C2×D6811yescharacteristic
293C42D4×D1711yescharacteristic
294C42D4×D1711yescharacteristic
295C42C2×C22⋊D1711yescharacteristic
296C23#1058757011yescharacteristic
297Q8—82no—
298Q8—82no—
299Q8—342no—
300Q8—342no—
301Q8—681no—
302Q8—681no—
303Q8—684no—
304Q8—1361no—
305Q8—1361no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—342no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—682no—
314C2×C4—682no—
315C2×C4—1362no—
316C2×C4#1058756811yescharacteristic
317C2×C4#1058756811yescharacteristic
318C2×C4#1058757011yescharacteristic
319C2×C4#1058760111yescharacteristic
320C2×C4#1058760211yescharacteristic
321C2×C4#1058760212yes—
322C2×C4Q16×D1711yescharacteristic
323C2×C4#1058760412yes—
324C2×C4#1058760411yescharacteristic
325C2×C4#1058761611yescharacteristic
326C2×C4#1058762311yescharacteristic
327C2×C4#1058762611yescharacteristic
328C8—42no—
329C8—81no—
330C8—682no—
331C8—682no—
332C22#1176271811yescharacteristic
333C22#1176303412yes—
334C22#1176304911yescharacteristic
335C22#1176306611yescharacteristic
336C22#1176307112yes—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—682no—
341C4—1362no—
342C4—1361no—
343C4#1176296311yescharacteristic
344C4#1176296611yescharacteristic
345C4#1176296811yescharacteristic
346C4#1176297411yescharacteristic
347C4#1176297412yes—
348C4#1176319011yescharacteristic
349C4#1176320211yescharacteristic
350C2#1452824512yes—
351C2#1452836912yes—
352C2#1452979311yescharacteristic
353C2#1452987911yescharacteristic
354C2#1452990011yescharacteristic
subgroupG/H
centerC23#10587570
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#333C2
socle, prime radicalC23×C17#121
hypercenterC2×#333D17
hypocenterC17#10277
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
246. C17#10277semidirect

Maximal subgroups

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

subgroupindex
C2×#117631612
C2×#117631012
C2×#117630952
#145297192
C2×#117625562
#145266182
C2×C17×#3332
#1027717