Finite Group Databasebeta

← #27275823

Subgroups — #27275823

Order 13568.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16811343C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic
3#20273386C212yesmaximal, complemented
4#20273145C212yesmaximal, complemented
5C53×#1460C211yesmaximal, characteristic
6#16811343—24no—
7#16811343—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133210—22no—
10C2×#13133204—21no—
11C2×#13133196C412yes—
12C2×C4×Dic106C2211yescharacteristic
13C2×#13133182—21no—
14C2×#13133176C412yes—
15#16810647C2212yes—
16C2×#13133134—21no—
17#16810392—21no—
18#16810368—24no—
19#16810355—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#431C2212yes—
22C53×#323—22no—
23C22×Dic106—21no—
24C22×Dic106C2×C411yescharacteristic
25C2×#11719917—22no—
26#13133217—22no—
27#13133217D414yes—
28#13133210—216no—
29#13133204—28no—
30#13133196—24no—
31#13133196—24no—
32C4×Dic106—22no—
33C4×Dic106D414yes—
34#13133182—28no—
35#13133176—24no—
36#13133176—24no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C412yes—
40C2×#11719919—22no—
41C2×#11719919—21no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—21no—
46C4⋊#88322—22no—
47C4⋊#88322—22no—
48#13133134—42no—
49C22×D4×C53D411yescharacteristic
50C23×C4×C53C2×C412yes—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C22⋊C8×C53—22no—
58C4×C8×C53—22no—
59C24×C53C22⋊C412yes—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—44no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C2×Dic106—21no—
75C2×Dic106—24no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106C22⋊C412yes—
81C2×C8×C53—42no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—24no—
84C42×C53C2×D412yes—
85C42×C53C2×D412yes—
86C22⋊Dic53—28no—
87C22⋊Dic53—28no—
88C4⋊Dic53—21no—
89C4⋊Dic53—22no—
90C4⋊Dic53—24no—
91C4⋊Dic53C22⋊C412yes—
92C4⋊Dic53C22⋊C414yes—
93#11719919—28no—
94#11719919—24no—
95C4×Dic53—28no—
96C4×Dic53—24no—
97#11719917—42no—
98#11719917—42no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53C2×C22⋊C411yescharacteristic
106D4×C53—42no—
107D4×C53—42no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—41no—
111C2×C4×C53—44no—
112C2×C4×C53—44no—
113C2×C4×C53—42no—
114C2×C4×C53C2×C22⋊C411yescharacteristic
115C2×C4×C53C2×C22⋊C411yescharacteristic
116C2×C4×C53#12012yes—
117C2×C4×C53#12012yes—
118C2×C4×C53#13512yes—
119C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C2×Dic53—24no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128Dic106—24no—
129Dic106—42no—
130Dic106—44no—
131Dic106—44no—
132C8×C53—82no—
133#88322—82no—
134#8389—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—82no—
141C22×C53—82no—
142C22×C53#37911yescharacteristic
143C22×C53#41611yescharacteristic
144C22×C53#41711yescharacteristic
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—42no—
148C4×C53#41312yes—
149C4×C53#44912yes—
150C4×C53#45212yes—
151Dic53—42no—
152Dic53—42no—
153Dic53—44no—
154C2×#556—531no—
155#1800—531no—
156#1546—532no—
157#1487—532no—
158#1460D5311yescharacteristic
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—82no—
162C2×C53#178511yescharacteristic
163C2×C53#178811yescharacteristic
164C2×C53#179611yescharacteristic
165#556—1064no—
166#556—1064no—
167C2×C4⋊Q8—531no—
168C2×#126—1062no—
169C2×#124—1061no—
170C2×C22⋊Q8—532no—
171C2×C4×Q8—531no—
172C2×C4×Q8—1061no—
173C2×C4×D4C2×D5311yescharacteristic
174C2×#117—532no—
175#431C2×D5312yes—
176#423—532no—
177C2×#100—1061no—
178#343—1061no—
179#332—1064no—
180Q8⋊C8—1064no—
181#323—22no—
182C53#838911yescharacteristic, complemented, Sylow
183C22×Q8—531no—
184C22×Q8—1061no—
185C22×D4D21211yescharacteristic
186C23×C4C4×D5312yes—
187C2×M16—1062no—
188C4⋊Q8—534no—
189C4⋊Q8—1062no—
190#126—10616no—
191#124—1068no—
192C22⋊Q8—1064no—
193C22⋊Q8—1064no—
194C4×Q8—534no—
195C4×Q8—1068no—
196C4×Q8—1062no—
197C4×D4—22no—
198C4×D4—22no—
199#117—1064no—
200#117—1064no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1062no—
206C2×C4⋊C4D21211yescharacteristic
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C22⋊C4C22⋊D5312yes—
210C2×C42—1061no—
211C2×C42—1062no—
212C2×C42C22×D5311yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—1062no—
215C4⋊C8—1062no—
216#100—2122no—
217C22⋊C8—22no—
218C4×C8—22no—
219C24#1171992112yes—
220C2×Q8—532no—
221C2×Q8—1061no—
222C2×Q8—1064no—
223C2×Q8—2122no—
224C2×Q8—2122no—
225C2×Q8—2121no—
226C2×Q8—2121no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—1061no—
235C22×C4—1062no—
236C22×C4—1061no—
237C22×C4#1171992112yes—
238C22×C4C2×C4×D5311yescharacteristic
239C22×C4C2×D21211yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—2122no—
242M16—2122no—
243C2×C8—42no—
244C2×C8—2122no—
245C4⋊C4—22no—
246C4⋊C4—532no—
247C4⋊C4—534no—
248C4⋊C4—1068no—
249C4⋊C4—1061no—
250C4⋊C4—1062no—
251C4⋊C4—1064no—
252C4⋊C4—1064no—
253C22⋊C4—24no—
254C22⋊C4—1068no—
255C22⋊C4—1068no—
256C42—1068no—
257C42—1064no—
258C42D4×D5312yes—
259C42D4×D5312yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×#1171992111yescharacteristic
266Q8—1064no—
267Q8—2122no—
268Q8—2124no—
269Q8—2124no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—1062no—
279C2×C4—1064no—
280C2×C4—1062no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4C22⋊C4×D5311yescharacteristic
286C2×C4C22⋊C4×D5311yescharacteristic
287C2×C4#1313318512yes—
288C2×C4#1313318712yes—
289C2×C4#1313323912yes—
290C2×C4#1313324511yescharacteristic
291C2×C4#1313325011yescharacteristic
292C8—82no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1681047811yescharacteristic
302C22#1681062011yescharacteristic
303C22#1681062511yescharacteristic
304C4—22no—
305C4—21no—
306C4—42no—
307C4—2122no—
308C4—2122no—
309C4—2124no—
310C4#1681060412yes—
311C4#1681077012yes—
312C4#1681078712yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2027464011yescharacteristic
317C2#2027465611yescharacteristic
318C2#2027468811yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1460C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1460D53
hypocenterC53#8389
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20273386C2semidirect
4. #20273145C2semidirect
182. C53#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#168113432
#202747072
#202733862
#202731452
C53×#14602
#838953