Finite Group Databasebeta

← #27275861

Subgroups — #27275861

Order 13568.

738 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic106C211yesmaximal, characteristic
2#20273401C212yesmaximal
3#20273143C212yesmaximal
4#20273039C211yesmaximal, characteristic
5C53×#1801C211yesmaximal, characteristic
6Q8×Dic106—24no—
7Q8×Dic106—24no—
8C2×#13133217C412yes—
9C2×#13133217C2211yescharacteristic
10C2×#13133216—22no—
11C2×#13133216—21no—
12C2×Q8×Dic53—21no—
13C2×C4×Dic106C412yes—
14C2×C4×Dic106C2211yescharacteristic
15#16810676C2212yes—
16#16810368—24no—
17#16810353—24no—
18#16810349—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#101×C53—21no—
22C53×#343—21no—
23C22×Dic106—22no—
24C22×Dic106—21no—
25C22×Dic106C2×C411yescharacteristic
26#13133217—22no—
27#13133217—24no—
28#13133217—24no—
29#13133217D414yes—
30#13133216—28no—
31#13133216—216no—
32Q8×Dic53—28no—
33C4×Dic106—22no—
34C4×Dic106—24no—
35C4×Dic106—24no—
36C4×Dic106D414yes—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C412yes—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719919—22no—
42C2×#11719919—21no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
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—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53SD1612yes—
62C2×Q8×C53Q1612yes—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—22no—
70C22×Dic53—21no—
71C2×Dic106—21no—
72C2×Dic106—24no—
73C2×Dic106—28no—
74C2×Dic106—24no—
75C2×Dic106—82no—
76C2×Dic106—84no—
77C2×Dic106C22⋊C412yes—
78M16×C53—42no—
79M16×C53—42no—
80C2×C8×C53—42no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—22no—
83C42×C53—22no—
84C42×C53—22no—
85C42×C53C2×D412yes—
86C42×C53C2×D412yes—
87C4⋊Dic53—28no—
88C4⋊Dic53—21no—
89C4⋊Dic53—22no—
90C4⋊Dic53—24no—
91C4⋊Dic53C22⋊C414yes—
92C4⋊Dic53C22⋊C412yes—
93#11719919—28no—
94#11719919—24no—
95C4×Dic53—28no—
96C4×Dic53—24no—
97C2×#88322—42no—
98C23×C53C2×C22⋊C411yescharacteristic
99Q8×C53—24no—
100Q8×C53—24no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53#10314yes—
108C2×C4×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53#12012yes—
111C2×C4×C53#12012yes—
112C2×C4×C53C2×SD1611yescharacteristic
113C2×C4×C53C2×Q1611yescharacteristic
114C2×C4×C53#13712yes—
115C2×Dic53—24no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—24no—
119C2×Dic53—44no—
120C2×Dic53—42no—
121Dic106—24no—
122Dic106—22no—
123Dic106—84no—
124Dic106—88no—
125Dic106#10114yescomplemented
126C8×C53—82no—
127#88322—82no—
128#8396—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53#37911yescharacteristic
132C22×C53C2×#10311yescharacteristic
133C22×C53#41911yescharacteristic
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—24no—
137C4×C53—42no—
138C4×C53C2×#10112yes—
139C4×C53#44812yes—
140C4×C53#45112yes—
141Dic53—24no—
142Dic53—44no—
143Dic53—42no—
144C2×Q82—531no—
145#1801D5311yescharacteristic
146#1549—532no—
147#1487—532no—
148#1461—531no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53#178411yescharacteristic
152C2×C53#178711yescharacteristic
153C2×C53#179511yescharacteristic
154Q82—1064no—
155Q82—1064no—
156C2×C4⋊Q8—532no—
157C2×C4⋊Q8—531no—
158C2×C4⋊Q8—1062no—
159C2×C4⋊Q8—1061no—
160C2×C4×Q8—532no—
161C2×C4×Q8—531no—
162C2×C4×Q8—1061no—
163C2×C4×Q8C2×D5311yescharacteristic
164#431—532no—
165#423C2×D5312yes—
166C2×#101—21no—
167#343—21no—
168#332—1064no—
169Q8⋊C8—1064no—
170#324—1062no—
171C53#839611yescharacteristic, complemented, Sylow
172C22×Q8—531no—
173C22×Q8—1061no—
174C22×Q8—1062no—
175C22×Q8C22⋊D5311yescharacteristic
176C2×M16—22no—
177C4⋊Q8—534no—
178C4⋊Q8—1064no—
179C4⋊Q8—1064no—
180C4⋊Q8—10616no—
181C4⋊Q8—1068no—
182C4⋊Q8—1062no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—534no—
186C4×Q8—1064no—
187C4×Q8—1064no—
188C4×Q8—1068no—
189C4×Q8—1062no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4D21212yes—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C42—1061no—
199C2×C42—1062no—
200C2×C42C4×D5312yes—
201C2×C42C22×D5311yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—1062no—
205#101—42no—
206C22⋊C8—1062no—
207C4×C8—1062no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—532no—
211C2×Q8—1064no—
212C2×Q8—1068no—
213C2×Q8—1061no—
214C2×Q8—1064no—
215C2×Q8—4242no—
216C2×Q8—4244no—
217C2×Q8#1171992312yes—
218C2×Q8#1171992512yes—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1062no—
222C22×C4#1171992112yes—
223C22×C4#1171992112yes—
224C22×C4C2×C4×D5311yescharacteristic
225C22×C4C2×D21211yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—42no—
228M16—42no—
229C2×C8—42no—
230C2×C8—2122no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—532no—
234C4⋊C4—534no—
235C4⋊C4—1068no—
236C4⋊C4—1068no—
237C4⋊C4—1061no—
238C4⋊C4—1062no—
239C4⋊C4—1064no—
240C4⋊C4—1064no—
241C42—22no—
242C42—22no—
243C42—1068no—
244C42—1064no—
245C42D4×D5312yes—
246C42D4×D5312yes—
247C23C2×#1171992111yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—534no—
251Q8—1064no—
252Q8—1062no—
253Q8—4244no—
254Q8—4248no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—1064no—
262C2×C4—1062no—
263C2×C4—1064no—
264C2×C4—1062no—
265C2×C4—2124no—
266C2×C4—2122no—
267C2×C4#1313314014yes—
268C2×C4C22⋊C4×D5311yescharacteristic
269C2×C4C22⋊C4×D5311yescharacteristic
270C2×C4#1313318512yes—
271C2×C4#1313318712yes—
272C2×C4C2×#1171992311yescharacteristic
273C2×C4C2×#1171992511yescharacteristic
274C2×C4#1313324712yes—
275C8—82no—
276C8—4242no—
277C22—21no—
278C22—21no—
279C22#1681047811yescharacteristic
280C22C2×#1313314011yescharacteristic
281C22#1681063111yescharacteristic
282C4—21no—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—1064no—
287C4—2124no—
288C4—2122no—
289C4#101×D5312yes—
290C4#1681076112yes—
291C4#1681077812yes—
292C2—21no—
293C2—21no—
294C2#2027463711yescharacteristic
295C2#2027464911yescharacteristic
296C2#2027468611yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#379
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8396
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
125. Dic106#101semidirect
171. C53#8396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1062
#202734012
#202731432
#202730392
C53×#18012
#839653