Finite Group Databasebeta

← #27276626

Subgroups — #27276626

Order 13568.

694 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16811251C211yesmaximal, characteristic
2#20273431C212yesmaximal, complemented
3#20273341C212yesmaximal, complemented
4#20273049C211yesmaximal, characteristic
5C53×#1464C211yesmaximal, characteristic
6#16811251—24no—
7#16811251—24no—
8C2×#13133217C411yescharacteristic
9C2×#13133217C2211yescharacteristic
10C2×#13133216—21no—
11C2×#13133209C411yescharacteristic
12C2×#13133209C2211yescharacteristic
13C2×#13133208—21no—
14C2×#13133194—21no—
15C2×#13133194—21no—
16C2×#13133176C411yescharacteristic
17C22×C4⋊Dic53C411yescharacteristic
18#16810680C2212yes—
19#16810368—24no—
20#16810364—24no—
21#16810349—21no—
22#16810347—21no—
23C2×#117×C53C2211yescharacteristic
24C53×#432C2212yes—
25C53×#324—21no—
26C53×#323—21no—
27C22×Dic106—21no—
28C22×Dic106—21no—
29C23×Dic53—21no—
30#13133217—22no—
31#13133217—22no—
32#13133217—22no—
33#13133217D414yes—
34#13133216—28no—
35#13133209—22no—
36#13133209—22no—
37#13133209—22no—
38#13133209D414yes—
39#13133208—28no—
40#13133194—28no—
41#13133194—28no—
42#13133176—22no—
43#13133176—22no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53C2×C411yescharacteristic
50C2×C4⋊Dic53C2×C411yescharacteristic
51C2×C4⋊Dic53C2×C411yescharacteristic
52C2×C4⋊Dic53C2×C411yescharacteristic
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×C22⋊Dic53—21no—
58C2×C4×Dic53—21no—
59C4⋊#88322—22no—
60C22⋊#88322—22no—
61#13133128—22no—
62C23×C4×C53C2×C411yescharacteristic
63#117×C53—22no—
64#117×C53—22no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C42×C53C2×C411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—22no—
72C22⋊C8×C53—22no—
73#97×C53—22no—
74C24×C53C22⋊C411yescharacteristic
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—24no—
90C22×Dic53—82no—
91C2×Dic106—24no—
92C2×Dic106—24no—
93C2×Dic106—82no—
94C2×Dic106—82no—
95C2×C8×C53—42no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C42×C53—21no—
101C42×C53—21no—
102C42×C53C2×D412yes—
103C42×C53C2×D412yes—
104C22⋊Dic53—24no—
105C4⋊Dic53—24no—
106C4⋊Dic53—24no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—24no—
112C4⋊Dic53C22⋊C412yes—
113C4⋊Dic53C22⋊C412yes—
114C4⋊Dic53C22⋊C412yes—
115C4⋊Dic53C22⋊C412yes—
116#11719919—24no—
117#11719919—24no—
118#11719919—24no—
119#11719919—24no—
120C4×Dic53—24no—
121C2×#88322—42no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#10112yes—
127C23×C53C2×C22⋊C411yescharacteristic
128C2×C4×C53—24no—
129C2×C4×C53—24no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53#10112yes—
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53#12012yes—
144C2×C4×C53#12012yes—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—84no—
154Dic106—84no—
155Dic106—84no—
156C8×C53—82no—
157#88322—82no—
158#8525—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53#37911yescharacteristic
167C22×C53C2×#10111yescharacteristic
168C22×C53C2×#10111yescharacteristic
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176Dic53—42no—
177Dic53—42no—
178Dic53—42no—
179Dic53—42no—
180C2×#544—531no—
181#1556—532no—
182#1534—532no—
183#1464—531no—
184#1464D5311yescharacteristic
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—42no—
188C2×C53#178411yescharacteristic
189C2×C53#179511yescharacteristic
190C2×C53#179611yescharacteristic
191#544—1064no—
192#544—1064no—
193C2×C4⋊Q8—531no—
194C2×C4⋊Q8—531no—
195C2×C4⋊Q8—1061no—
196C2×#125—531no—
197C2×#125—531no—
198C2×#125—1061no—
199C2×C22⋊Q8—1061no—
200C2×C22⋊Q8—1061no—
201C2×#117—531no—
202C2×#117C2×D5311yescharacteristic
203C22×C4⋊C4—531no—
204#432—532no—
205#432C2×D5312yes—
206#332—1064no—
207#330—1064no—
208#324—21no—
209#324—1061no—
210#323—21no—
211#323—1061no—
212C53#852511yescharacteristic, complemented, Sylow
213C22×Q8—1061no—
214C22×Q8—1061no—
215C23×C4—1061no—
216C23×C4C4×D5311yescharacteristic
217C4⋊Q8—534no—
218C4⋊Q8—1062no—
219C4⋊Q8—1062no—
220C4⋊Q8—1068no—
221C4⋊Q8—1062no—
222#125—534no—
223#125—1062no—
224#125—1062no—
225#125—1068no—
226#125—1062no—
227C22⋊Q8—1068no—
228C22⋊Q8—1068no—
229#117—22no—
230#117—22no—
231#117—1062no—
232#117—1062no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4—1062no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4D21211yescharacteristic
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C22⋊C4—1061no—
249C2×C22⋊C4D21211yescharacteristic
250C2×C22⋊C4C22⋊D5311yescharacteristic
251C2×C42—1061no—
252C2×C42C4×D5311yescharacteristic
253C2×C42C22×D5311yescharacteristic
254C4⋊C8—22no—
255C4⋊C8—1062no—
256C22⋊C8—22no—
257C22⋊C8—1062no—
258#97—22no—
259#97—1062no—
260C24#1171992111yescharacteristic
261C2×Q8—1064no—
262C2×Q8—1064no—
263C2×Q8—4242no—
264C2×Q8—4242no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1064no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—4242no—
275C22×C4#1171992111yescharacteristic
276C22×C4#1171992111yescharacteristic
277C22×C4#1171992111yescharacteristic
278C22×C4C2×C4×D5311yescharacteristic
279C22×C4C2×D21211yescharacteristic
280C22×C4C2×C22⋊D5311yescharacteristic
281C2×C8—42no—
282C2×C8—2122no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—532no—
286C4⋊C4—532no—
287C4⋊C4—532no—
288C4⋊C4—532no—
289C4⋊C4—1064no—
290C4⋊C4—1064no—
291C4⋊C4—1064no—
292C4⋊C4—1064no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1064no—
298C4⋊C4—1064no—
299C4⋊C4—1064no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—1064no—
303C42—21no—
304C42—21no—
305C42—1064no—
306C42D4×D5312yes—
307C42D4×D5312yes—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23#1313313512yes—
313C23C2×#1171992111yescharacteristic
314Q8—4244no—
315Q8—4244no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4—4244no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4#1313313512yes—
338C2×C4C22⋊C4×D5311yescharacteristic
339C2×C4C22⋊C4×D5311yescharacteristic
340C2×C4#1313318512yes—
341C2×C4#1313318712yes—
342C8—82no—
343C8—4242no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22#1681047811yescharacteristic
352C22C2×#1313313511yescharacteristic
353C22C2×#1313313511yescharacteristic
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—2122no—
362C4—2122no—
363C4—2122no—
364C4—2122no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2#2027463811yescharacteristic
369C2#2027468611yescharacteristic
370C2#2027468811yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1464C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1464D53
hypocenterC53#8525
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20273431C2semidirect
3. #20273341C2semidirect
212. C53#8525semidirect

Maximal subgroups

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

subgroupindex
C2×#168112512
#202734312
#202733412
#202730492
C53×#14642
#852553