Finite Group Databasebeta

← #27275851

Subgroups — #27275851

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×#16811346C211yesmaximal, characteristic
2#20273397C212yesmaximal, complemented
3#20273139C212yesmaximal
4#20273039C211yesmaximal, characteristic, complemented
5C53×#1801C211yesmaximal, characteristic, complemented
6#16811346—24no—
7#16811346—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133212—22no—
10C2×#13133205C412yes—
11C2×#13133193—21no—
12C2×C4×Dic106C412yes—
13C2×C4×D212C2211yescharacteristic
14C2×#13133175—21no—
15#16810676C2212yes—
16#16810371—24no—
17#16810350—24no—
18#16810349—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#101×C53—21no—
22C53×#343—21no—
23C22×Dic106C2×C412yes—
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26#13133217—22no—
27#13133217D414yes—
28#13133212—216no—
29#13133205—24no—
30#13133205—24no—
31#13133193—28no—
32C4×Dic106—24no—
33C4×Dic106—24no—
34C4×D212—22no—
35C4×D212D414yes—
36#13133175—28no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—22no—
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—21no—
44C2×C4×Dic53—22no—
45C4⋊#88322—22no—
46C22⋊#88322—22no—
47C4×#88322—22no—
48C22×Q8×C53D411yescharacteristic
49C2×M16×C53—22no—
50C4×Q8×C53—22no—
51C4×Q8×C53—22no—
52C2×C4⋊C4×C53D412yes—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C42×C53C2×C412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58#101×C53—42no—
59C23×D53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—22no—
62C2×Q8×C53—21no—
63C2×Q8×C53—21no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—22no—
71C2×D212—21no—
72C2×D212—42no—
73C2×D212C22⋊C412yes—
74C2×C4×D53—24no—
75C2×C4×D53—42no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×Dic106—22no—
79C2×Dic106—44no—
80C2×Dic106C22⋊C414yes—
81M16×C53—42no—
82M16×C53—42no—
83C2×C8×C53—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—24no—
86C42×C53—22no—
87C42×C53—22no—
88C42×C53C2×D412yes—
89C42×C53C2×D412yes—
90#11719921—28no—
91#11719921—28no—
92#11719921—24no—
93C4⋊Dic53—21no—
94C4⋊Dic53—24no—
95C4⋊Dic53—24no—
96C4⋊Dic53C22⋊C412yes—
97#11719919—28no—
98C4×Dic53—28no—
99C2×#88322—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—22no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—41no—
105C22×D53—41no—
106Q8×C53—42no—
107Q8×C53—42no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53C2×C22⋊C411yescharacteristic
116C2×C4×C53C2×C22⋊C411yescharacteristic
117C2×C4×C53#12012yes—
118C2×C4×C53#12012yes—
119C2×C4×C53#13512yes—
120C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C53#13711yescharacteristic
122C2×Dic53—24no—
123C2×Dic53—22no—
124C2×Dic53—44no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127D212—24no—
128D212—42no—
129C4×D53—44no—
130C4×D53—44no—
131Dic106—44no—
132Dic106—44no—
133C8×C53—82no—
134#88322—82no—
135#8394—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53#37911yescharacteristic
139C22×C53#41611yescharacteristic
140C22×C53#41711yescharacteristic
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—42no—
144C2×D53—82no—
145C2×D53—82no—
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53#41312yes—
151C4×C53#44912yes—
152C4×C53#45212yes—
153Dic53—44no—
154Dic53—42no—
155C2×#556—531no—
156#1801D5311yescharacteristic, complemented
157#1548—532no—
158#1486—532no—
159#1461—531no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53#178511yescharacteristic
163C2×C53#178811yescharacteristic
164C2×C53#179611yescharacteristic
165D53—82no—
166#556—1064no—
167#556—1064no—
168C2×C4⋊Q8—531no—
169C2×#126—1062no—
170C2×#124—532no—
171C2×C22⋊Q8—1061no—
172C2×C4×Q8—532no—
173C2×C4×Q8C2×D5311yescharacteristic
174C2×C4×D4—531no—
175C2×#117—1061no—
176#431—532no—
177#423C2×D5312yes—
178C2×#101—21no—
179#343—21no—
180#333—1064no—
181D4⋊C8—1064no—
182#324—1062no—
183C53#839411yescharacteristic, complemented, Sylow
184C22×Q8—532no—
185C22×Q8C22⋊D5311yescharacteristic
186C22×D4—531no—
187C23×C4—1061no—
188C2×M16—22no—
189C4⋊Q8—534no—
190C4⋊Q8—1062no—
191#126—10616no—
192#124—1064no—
193#124—1064no—
194C22⋊Q8—1068no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—1064no—
198C4×Q8—1064no—
199C4×D4—534no—
200C4×D4—1062no—
201#117—1068no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1062no—
206C2×C4⋊C4D21212yes—
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—1062no—
211C2×C42—1062no—
212C2×C42C4×D5312yes—
213C2×C42C22×D5311yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—1062no—
217#101—42no—
218C22⋊C8—1062no—
219C4×C8—1062no—
220C24—1061no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—534no—
226C2×Q8—1062no—
227C2×Q8—2124no—
228C2×D4—532no—
229C2×D4—1061no—
230C2×D4—2122no—
231C22×C4—1061no—
232C22×C4—1062no—
233C22×C4—1064no—
234C22×C4—2122no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4#1171992112yes—
238C22×C4#1171992112yes—
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—2122no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—532no—
249C4⋊C4—1068no—
250C4⋊C4—1061no—
251C4⋊C4—1064no—
252C4⋊C4—1064no—
253C22⋊C4—1068no—
254C22⋊C4—1068no—
255C22⋊C4—1064no—
256C42—22no—
257C42—22no—
258C42—1068no—
259C42D4×D5312yes—
260C42D4×D5312yes—
261C23—1062no—
262C23—2122no—
263C23—2122no—
264C23—2121no—
265C23—2121no—
266C23C2×#1171992111yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—2124no—
270Q8—2124no—
271D4—1064no—
272D4—2122no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—1062no—
281C2×C4—1064no—
282C2×C4—2124no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4C22⋊C4×D5311yescharacteristic
288C2×C4C22⋊C4×D5311yescharacteristic
289C2×C4#1313318512yes—
290C2×C4#1313318712yes—
291C2×C4#1313323712yes—
292C2×C4#1313324311yescharacteristic
293C2×C4#1313324811yescharacteristic
294C8—82no—
295C8—4242no—
296C22—21no—
297C22—21no—
298C22—4242no—
299C22—4242no—
300C22—2122no—
301C22—2122no—
302C22—2122no—
303C22#1681047811yescharacteristic
304C22#1681061711yescharacteristic
305C22#1681062211yescharacteristic
306C4—22no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—2124no—
311C4—2122no—
312C4#1681060612yes—
313C4#1681076612yes—
314C4#1681078312yes—
315C2—21no—
316C2—21no—
317C2—4242no—
318C2#2027464211yescharacteristic
319C2#2027465311yescharacteristic
320C2#2027469111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#379
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8394
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #20273397C2semidirect
4. #20273039C2semidirect
5. C53×#1801C2semidirect
156. #1801D53semidirect
183. C53#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#168113462
#202733972
#202731392
#202730392
C53×#18012
#839453