Finite Group Databasebeta

← #16935047

Subgroups — #16935047

Order 4352.

510 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 89 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#14539605C211yesmaximal, characteristic
2C2×#11763139C211yesmaximal, characteristic
3#14528864C211yesmaximal, characteristic
4#14527148C211yesmaximal, characteristic
5#14527147C211yesmaximal, characteristic
6#14526672C211yesmaximal, characteristic
7C2×C17×#484C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763139—21no—
10#11763139—22no—
11#11763139—22no—
12#11763137—21no—
13#11763065—22no—
14#11762965—21no—
15#11762964—21no—
16C2×#10587479—21no—
17C2×#10587481—21no—
18#11762773C2211yescharacteristic
19#11762773C2211yescharacteristic
20C2×C22⋊#3622C2211yescharacteristic
21#11762578—22no—
22#11762535—22no—
23#11762534—22no—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#484—22no—
27C17×#484—22no—
28C2×C17×#107—21no—
29C2×C17×#103—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17—21no—
33C23×C4×C17C2311yescharacteristic
34C22×Dic34D411yescharacteristic
35C22×C8×C17—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17D412yes—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17—21no—
49C2×C22⋊Dic17—21no—
50C2×#63231—22no—
51C2×C4⋊Dic17D411yescharacteristic
52#10587610—21no—
53#10587610—21no—
54#10587610—21no—
55#10587610—21no—
56C2×#63225—21no—
57C2×#63223—21no—
58C2×#63223D412yes—
59C22×#3622—21no—
60C17×#107—24no—
61C17×#103—24no—
62C17×C22⋊C8—21no—
63C17×C22⋊C8D412yes—
64#10587506—22no—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69C22⋊#3622—21no—
70C22⋊#3622D412yes—
71#10587481—22no—
72#10587481—24no—
73#10587479—22no—
74#10587479—24no—
75C24×C17#4111yescharacteristic
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—42no—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17—21no—
82C22×C4×C17—42no—
83C22×C4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17—82no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×Dic17—21no—
93C22×Dic17—21no—
94C2×Dic34—21no—
95C2×Dic34—21no—
96C2×Dic34—42no—
97C2×C8×C17—22no—
98C2×C8×C17—42no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111#63231—82no—
112C4⋊Dic17—21no—
113C4⋊Dic17—21no—
114#63225—41no—
115#63225—41no—
116#63223—42no—
117#63223—42no—
118C2×#3622—22no—
119C2×#3622—22no—
120C2×#3622—42no—
121C2×#3622—41no—
122#9995—171nomaximal, Sylow
123C23×C17—22no—
124C23×C17—22no—
125C23×C17—41no—
126C23×C17#12111yescharacteristic
127C23×C17#12111yescharacteristic
128C23×C17#12111yescharacteristic
129Q8×C17—42no—
130Q8×C17—42no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—82no—
142C2×C4×C17—82no—
143C2×C4×C17—82no—
144C2×C4×C17—81no—
145C2×C4×C17#12011yescharacteristic
146C2×C4×C17#12011yescharacteristic
147C2×C4×C17#12011yescharacteristic
148C2×C4×C17#12711yescharacteristic
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—42no—
152C2×Dic17—81no—
153Dic34—42no—
154Dic34—42no—
155C8×C17—42no—
156#3622—42no—
157#3622—42no—
158#3622—81no—
159#3195—171no—
160C2×#484D1711yescharacteristic
161C2×#483—171no—
162#1821—171no—
163#1472—171no—
164#1472—171no—
165#1298—171no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—44no—
174C22×C17#39211yescharacteristic
175C22×C17#47112yes—
176C22×C17#47312yes—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—81no—
181C4×C17—82no—
182C4×C17—81no—
183C4×C17#45812yes—
184Dic17—81no—
185Dic17—81no—
186C2×C22⋊Q8—171no—
187C2×C22⋊Q8C2×D1711yescharacteristic
188C22×C4⋊C4C2×D1711yescharacteristic
189#484—22no—
190#484—22no—
191#483—342no—
192#483—341no—
193#483—342no—
194#483—341no—
195#470—342no—
196#451—341no—
197#451—341no—
198C2×#107—21no—
199C2×#106—341no—
200C2×#103—21no—
201C2×#103—341no—
202#407—171no—
203#407—171no—
204C2×C22⋊C8—171no—
205C2×C22⋊C8C2×D1711yescharacteristic
206#343—342no—
207#324—342no—
208#324—342no—
209C2×C17—22no—
210C2×C17—22no—
211C2×C17—41no—
212C2×C17#201911yescharacteristic
213C2×C17#202011yescharacteristic
214C2×C17#202211yescharacteristic
215C22×Q8—171no—
216C22×Q8C22⋊D1711yescharacteristic
217C23×C4—21no—
218C23×C4C22×D1711yescharacteristic
219C2×Q16—342no—
220C2×M16—172no—
221C2×M16—341no—
222C22×C8—21no—
223C22×C8—341no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—341no—
229C22⋊Q8—341no—
230C22⋊Q8—341no—
231C22⋊Q8—341no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—341no—
240C2×C4⋊C4C22⋊D1712yes—
241C2×C4⋊C4C22⋊D1711yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—341no—
244#107—24no—
245#106—344no—
246#106—342no—
247#103—24no—
248#103—344no—
249#103—342no—
250#103—342no—
251C22⋊C8—21no—
252C22⋊C8—172no—
253C22⋊C8—341no—
254C22⋊C8—341no—
255C22⋊C8—341no—
256C22⋊C8—341no—
257C22⋊C8—341no—
258C22⋊C8C22⋊D1712yes—
259C17#999511yescharacteristic, complemented, Sylow
260C24#6325411yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—341no—
265C2×Q8—341no—
266C2×Q8—682no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—82no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4D4×D1711yescharacteristic
277C22×C4D4×D1711yescharacteristic
278C22×C4D4×D1711yescharacteristic
279C22×C4C2×C22⋊D1711yescharacteristic
280C22×C4C2×C22⋊D1711yescharacteristic
281C22×C4C2×C22⋊D1711yescharacteristic
282Q16—1362no—
283M16—682no—
284M16—682no—
285C2×C8—22no—
286C2×C8—42no—
287C2×C8—342no—
288C2×C8—342no—
289C2×C8—682no—
290C2×C8—681no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—341no—
300C4⋊C4—341no—
301C4⋊C4—681no—
302C4⋊C4—681no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23#1058762411yescharacteristic
311C23#1058762511yescharacteristic
312C23#1058762511yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—682no—
316Q8—682no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—682no—
334C2×C4—1361no—
335C2×C4#1058762311yescharacteristic
336C2×C4#1058762311yescharacteristic
337C2×C4#1058762611yescharacteristic
338C2×C4#1058763911yescharacteristic
339C8—42no—
340C8—682no—
341C8—682no—
342C8—1361no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#1176271711yescharacteristic
352C22#1176307012yes—
353C22#1176308112yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—82no—
359C4—81no—
360C4—1361no—
361C4—1361no—
362C4#1176299812yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2#1452989811yescharacteristic
367C2#1452990311yescharacteristic
368C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#484C2
socleC22×C17#392
prime radicalC24×C17#41
hypercenterC2×#484D17
hypocenterC17#9995
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
259. C17#9995semidirect

Maximal subgroups

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

subgroupindex
#145396052
C2×#117631392
#145288642
#145271482
#145271472
#145266722
C2×C17×#4842
#999517