Finite Group Databasebeta

← #27304513

Subgroups — #27304513

Order 13568.

304 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 75 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#20276410C211yesmaximal, characteristic
2#20276404C211yesmaximal, characteristic
3#20272888C212yesmaximal, complemented
4#20272884C212yesmaximal, complemented
5C53×#3223C211yesmaximal, characteristic
6#16810607—21no—
7C2×#13133136—21no—
8#16810422—21no—
9#16810422C2212yes—
10#16810422C2212yes—
11#16810420—21no—
12C53×#541C411yescharacteristic
13C53×#536C2211yescharacteristic
14C53×#497C411yescharacteristic
15C53×#494C2211yescharacteristic
16C53×#491C2211yescharacteristic
17C53×#488C411yescharacteristic
18C53×#464—21no—
19C53×#462—21no—
20C53×#452—21no—
21C53×#451—21no—
22C53×#433C411yescharacteristic
23C2×#11719917—41no—
24C2×#11719917—41no—
25#13133136—22no—
26#13133136—22no—
27#13133136—41no—
28#13133134—41no—
29C2×#41×C53D411yescharacteristic
30C22×Q8×C53D411yescharacteristic
31C2×Q16×C53—21no—
32C2×Q16×C53—21no—
33C2×Q16×C53—21no—
34C2×SD16×C53—21no—
35C4⋊Q8×C53C2×C411yescharacteristic
36C4⋊Q8×C53D411yescharacteristic
37C4⋊Q8×C53C2311yescharacteristic
38#126×C53—41no—
39#125×C53—41no—
40#125×C53C2×C411yescharacteristic
41#124×C53C2×C411yescharacteristic
42#124×C53D411yescharacteristic
43#123×C53—41no—
44C22⋊Q8×C53—21no—
45C22⋊Q8×C53—41no—
46C4×Q8×C53—21no—
47#117×C53C2×C411yescharacteristic
48C4⋊C8×C53—21no—
49#103×C53—21no—
50#103×C53—21no—
51#103×C53—21no—
52#102×C53—21no—
53C22⋊C8×C53—21no—
54#97×C53C2×C411yescharacteristic
55C4×C8×C53C2×C411yescharacteristic
56#41×C53—81no—
57C2×Q8×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—81no—
60C2×Q8×C53C22⋊C411yescharacteristic
61C2×Q8×C53C2×D411yescharacteristic
62C2×Q8×C53C2×D411yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C22⋊C411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68Q16×C53—81no—
69Q16×C53—81no—
70Q16×C53—81no—
71SD16×C53—81no—
72C2×C8×C53—21no—
73C2×C8×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C42×C53—41no—
87C42×C53C22⋊C411yescharacteristic
88C42×C53C22×C411yescharacteristic
89#11719917—42no—
90#11719917—42no—
91#11719917—81no—
92#11719917—81no—
93C2×#88322—81no—
94C2×#88322—81no—
95C23×C53—21no—
96C23×C53#9911yescharacteristic
97Q8×C53—41no—
98Q8×C53—41no—
99Q8×C53—41no—
100Q8×C53—41no—
101Q8×C53—41no—
102Q8×C53—81no—
103D4×C53—41no—
104D4×C53—81no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C8×C53—41no—
120C8×C53—41no—
121#88322—82no—
122#88322—82no—
123#13612—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—81no—
128C22×C53C2×#9911yescharacteristic
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—81no—
137C4×C53—81no—
138#3223D5311yescharacteristic
139#2135—531no—
140#2133—531no—
141#1413—532no—
142#1411—532no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53—81no—
146C2×C53#212711yescharacteristic
147#541Dic5311yescharacteristic
148#536C2×D5311yescharacteristic
149#497Dic5311yescharacteristic
150#494C2×D5311yescharacteristic
151#491C2×D5311yescharacteristic
152#488Dic5311yescharacteristic
153#464—21no—
154#462—21no—
155#452—21no—
156#451—21no—
157#433Dic5311yescharacteristic
158#413—1061no—
159C2×#101—1061no—
160#356—532no—
161#356—532no—
162#356—1061no—
163#355—1061no—
164C53#1361211yescharacteristic, complemented, Sylow
165C2×#41C22⋊D5311yescharacteristic
166C22×Q8C22⋊D5311yescharacteristic
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×SD16—21no—
171C2×M16—2121no—
172C2×M16—2121no—
173C4⋊Q8C2×Dic5311yescharacteristic
174C4⋊Q8C22⋊D5311yescharacteristic
175C4⋊Q8C22×D5311yescharacteristic
176#126—41no—
177#125—41no—
178#125C2×Dic5311yescharacteristic
179#124C2×Dic5311yescharacteristic
180#124C22⋊D5311yescharacteristic
181#123—41no—
182C22⋊Q8—21no—
183C22⋊Q8—41no—
184C4×Q8—21no—
185#117C2×Dic5311yescharacteristic
186C4⋊C8—21no—
187#103—21no—
188#103—21no—
189#103—21no—
190#102—21no—
191#101—1062no—
192#101—1062no—
193#101—2121no—
194#100—2121no—
195C22⋊C8—21no—
196#97C2×Dic5311yescharacteristic
197C4×C8C2×Dic5311yescharacteristic
198#41—81no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—81no—
202C2×Q8C22⋊Dic5311yescharacteristic
203C2×Q8C2×C22⋊D5311yescharacteristic
204C2×Q8C2×C22⋊D5311yescharacteristic
205C2×D4—21no—
206C2×D4C22⋊Dic5311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4C22⋊Dic5311yescharacteristic
210Q16—81no—
211Q16—81no—
212Q16—81no—
213SD16—81no—
214M16—4241no—
215M16—4241no—
216M16—2122no—
217M16—2122no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—4241no—
221C2×C8—4241no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—21no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C42—41no—
235C42C22⋊Dic5311yescharacteristic
236C42C22×Dic5311yescharacteristic
237C23—21no—
238C23#1313313211yescharacteristic
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—81no—
245D4—41no—
246D4—81no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4#1313313211yescharacteristic
260C2×C4C2×C22⋊Dic5311yescharacteristic
261C8—41no—
262C8—41no—
263C8—4242no—
264C8—4242no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22C2×#1313313211yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—81no—
278C4—81no—
279C2—21no—
280C2—41no—
281C2—81no—
282C2#2027638411yescharacteristic
subgroupG/H
centerC2#20276384
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC53×#3223C2
socleC2×C53#2127
prime radicalC2×#41×C53D4
cosocleC4⋊Q8×C53C23
hypercenter#3223D53
hypocenterC53#13612
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3223

Complements

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

NH ≅ G/Ntype
3. #20272888C2semidirect
4. #20272884C2semidirect
164. C53#13612semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202764042
#202728882
#202728842
C53×#32232
#1361253