Finite Group Databasebeta

← #27276237

Subgroups — #27276237

Order 13568.

746 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 119 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#20288518C211yesmaximal, characteristic
2#20273413C212yesmaximal, complemented
3#20273329C212yesmaximal, complemented
4#20273056C211yesmaximal, characteristic
5C53×#1955C211yesmaximal, characteristic
6#16811332—24no—
7#16811330—22no—
8D4×Dic106—22no—
9#16811254C412yes—
10#16811249—22no—
11#16811180C2212yes—
12C2×#13133196—22no—
13#16811098C412yes—
14#16810680C2212yes—
15#16810370—22no—
16#16810368—22no—
17#16810364—24no—
18#16810347—22no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C4⋊C8×C53C2211yescharacteristic
21C2×C22⋊C8×C53—22no—
22C2×#97×C53C2211yescharacteristic
23C22×Dic106—22no—
24#13133217D412yes—
25#13133217D412yes—
26#13133216—24no—
27#13133209D414yes—
28#13133208—24no—
29#13133196—24no—
30#13133196—24no—
31#13133196—24no—
32#13133196—22no—
33#13133196—22no—
34#13133194—28no—
35#13133194—28no—
36C4×Dic106—22no—
37C4×Dic106—22no—
38D4×Dic53—24no—
39D4×Dic53—24no—
40#13133176C2×C414yes—
41C2×C4⋊Dic53—22no—
42C2×#11719919—24no—
43C2×C22⋊Dic53—24no—
44C4⋊#88322—22no—
45C22⋊#88322—22no—
46#13133128—22no—
47C22×D4×C53C2×C411yescharacteristic
48C23×C4×C53D412yes—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C22⋊C8×C53—44no—
58#97×C53—22no—
59C24×C53C22⋊C412yes—
60C2×D4×C53—24no—
61C2×D4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—24no—
71C22×Dic53—24no—
72C2×Dic106—22no—
73C2×Dic106—24no—
74C2×Dic106—22no—
75C2×Dic106—82no—
76C2×Dic106—82no—
77C2×C8×C53—22no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—42no—
81C2×C8×C53—41no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—24no—
84C42×C53C2×D412yes—
85C42×C53C2×D412yes—
86C22⋊Dic53—28no—
87C22⋊Dic53—24no—
88C4⋊Dic53—24no—
89C4⋊Dic53—22no—
90C4⋊Dic53—22no—
91C4⋊Dic53C22⋊C414yes—
92C4⋊Dic53C22⋊C414yes—
93#11719919—28no—
94#11719919—24no—
95#11719919—24no—
96C4×Dic53—24no—
97C2×#88322—42no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53#10114yes—
103C23×C53C2×C22⋊C411yescharacteristic
104D4×C53—28no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—82no—
114C2×C4×C53C2×C22⋊C412yes—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53#12011yescharacteristic
117C2×C4×C53#12012yes—
118C2×Dic53—24no—
119C2×Dic53—24no—
120C2×Dic53—84no—
121C2×Dic53—84no—
122Dic106—42no—
123Dic106—42no—
124Dic106—82no—
125Dic106—84no—
126C8×C53—42no—
127C8×C53—81no—
128#88322—82no—
129#8457—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—24no—
133C22×C53—44no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—82no—
137C22×C53#37911yescharacteristic
138C22×C53C2×#10112yes—
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—42no—
143C4×C53#45311yescharacteristic
144C4×C53#45511yescharacteristic
145Dic53—44no—
146Dic53—44no—
147#3501—531no—
148#1955D5311yescharacteristic
149#1552—532no—
150#1531—532no—
151#1466—531no—
152C2×C53—22no—
153C2×C53—44no—
154C2×C53#178411yescharacteristic
155C2×C53#179211yescharacteristic
156C2×C53#179411yescharacteristic
157#554—1064no—
158#554—1062no—
159D4×Q8—1062no—
160#544—532no—
161#543—1062no—
162#533—532no—
163C2×C22⋊Q8—1062no—
164#518—532no—
165C2×C4×D4C2×D5311yescharacteristic
166#432—532no—
167C2×C4⋊C8C2×D5311yescharacteristic
168C2×C22⋊C8—22no—
169C2×#97C2×D5311yescharacteristic
170#332—1062no—
171#332—1062no—
172#330—1064no—
173#323—1062no—
174C53#845711yescharacteristic, complemented, Sylow
175C22×Q8—1062no—
176C22×D4C4×D5311yescharacteristic
177C23×C4C22⋊D5312yes—
178C22×C8—21no—
179C22×C8—21no—
180C4⋊Q8—532no—
181C4⋊Q8—532no—
182C4⋊Q8—1064no—
183#125—534no—
184#125—1064no—
185C22⋊Q8—1068no—
186C22⋊Q8—1068no—
187C22⋊Q8—1064no—
188C22⋊Q8—1064no—
189C22⋊Q8—1062no—
190C22⋊Q8—1062no—
191C22⋊Q8—1064no—
192C4×Q8—1062no—
193C4×Q8—1062no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—1064no—
197C4×D4—1064no—
198#117—534no—
199C2×C4⋊C4—1064no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4C4×D5311yescharacteristic
202C2×C22⋊C4—1064no—
203C2×C22⋊C4D21212yes—
204C2×C42C22×D5311yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—1062no—
207C22⋊C8—44no—
208C22⋊C8—1062no—
209#97—22no—
210#97—1062no—
211C24#1171992112yes—
212C2×Q8—1064no—
213C2×Q8—1062no—
214C2×Q8—1062no—
215C2×Q8—4242no—
216C2×Q8—4242no—
217C2×D4—24no—
218C2×D4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1064no—
224C22×C4—1064no—
225C22×C4#1171992112yes—
226C22×C4C2×C4×D5311yescharacteristic
227C22×C4C2×D21211yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—2122no—
235C4⋊C4—22no—
236C4⋊C4—534no—
237C4⋊C4—534no—
238C4⋊C4—1064no—
239C4⋊C4—1068no—
240C4⋊C4—1064no—
241C4⋊C4—1062no—
242C4⋊C4—1064no—
243C4⋊C4—1062no—
244C22⋊C4—24no—
245C22⋊C4—1068no—
246C22⋊C4—1064no—
247C42—1064no—
248C42D4×D5312yes—
249C42D4×D5312yes—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23#1313313514yes—
255C23C2×#1171992111yescharacteristic
256Q8—4242no—
257Q8—4244no—
258Q8—2122no—
259Q8—2122no—
260D4—28no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—1064no—
271C2×C4—1064no—
272C2×C4—4244no—
273C2×C4—4244no—
274C2×C4C22⋊C4×D5312yes—
275C2×C4#1313318511yescharacteristic
276C2×C4#1313318511yescharacteristic
277C2×C4#1313318712yes—
278C8—42no—
279C8—81no—
280C8—4242no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#1681047811yescharacteristic
289C22C2×#1313313512yes—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—2124no—
295C4—2124no—
296C4#1681079011yescharacteristic
297C4#1681080011yescharacteristic
298C2—22no—
299C2—44no—
300C2#2027463811yescharacteristic
301C2#2027466811yescharacteristic
302C2#2027467811yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1955C2
socleC22×C53#379
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1955D53
hypocenterC53#8457
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #20273413C2semidirect
3. #20273329C2semidirect
174. C53#8457semidirect

Maximal subgroups

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

subgroupindex
#202885182
#202734132
#202733292
#202730562
C53×#19552
#845753