Finite Group Databasebeta

← #27304514

Subgroups — #27304514

Order 13568.

304 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 67 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#20285798C211yesmaximal, characteristic
2#20276404C211yesmaximal, characteristic
3#20272887C212yesmaximal, complemented
4#20272883C212yesmaximal, complemented
5C53×#2135C211yesmaximal, characteristic
6#16811235C411yescharacteristic
7#16811018C411yescharacteristic
8#16811002C2211yescharacteristic
9#16810991C2211yescharacteristic
10#16810977C411yescharacteristic
11#16810842—21no—
12#16810830—21no—
13#16810785—21no—
14#16810779—21no—
15#16810684C411yescharacteristic
16C2×#13133136—21no—
17#16810422C2212yes—
18#16810420—21no—
19C53×#536C2211yescharacteristic
20C53×#413—21no—
21C53×#356—21no—
22C53×#356C2212yes—
23C2×#11719925—21no—
24C2×#11719925—21no—
25C2×#11719925—21no—
26C2×#11719923—21no—
27C2×#11719917—41no—
28#13133217C2×C411yescharacteristic
29#13133210—41no—
30#13133209C2×C411yescharacteristic
31#13133207—41no—
32#13133196—21no—
33C4×Dic106—21no—
34C4⋊#88322—21no—
35#13133141—21no—
36#13133141—21no—
37#13133141—21no—
38#13133138—21no—
39#13133136—22no—
40#13133136—41no—
41C22⋊#88322—21no—
42#13133128C2×C411yescharacteristic
43C4×#88322C2×C411yescharacteristic
44C2×#41×C53D411yescharacteristic
45C22×Q8×C53D411yescharacteristic
46C2×M16×C53—41no—
47C4⋊Q8×C53D411yescharacteristic
48C4⋊Q8×C53C2311yescharacteristic
49#124×C53C2×C411yescharacteristic
50#124×C53D411yescharacteristic
51#123×C53—41no—
52C22⋊Q8×C53—41no—
53#117×C53C2×C411yescharacteristic
54#101×C53—22no—
55#100×C53—41no—
56#41×C53—81no—
57C2×Q8×C53—21no—
58C2×Q8×C53—81no—
59C2×Q8×C53C22⋊C411yescharacteristic
60C2×Q8×C53C2×D411yescharacteristic
61C2×Q8×C53C2×D411yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C22⋊C411yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C22⋊C411yescharacteristic
67C2×Dic106—21no—
68M16×C53—42no—
69M16×C53—81no—
70C2×C8×C53—81no—
71C4⋊C4×C53—21no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—21no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C42×C53C22⋊C411yescharacteristic
78C42×C53C22×C411yescharacteristic
79C22⋊Dic53—41no—
80#11719925—81no—
81#11719925—81no—
82#11719925—81no—
83#11719923—81no—
84C4⋊Dic53—21no—
85C4⋊Dic53—41no—
86#11719919—41no—
87#11719919—41no—
88#11719919—41no—
89C4×Dic53—41no—
90#11719917—42no—
91#11719917—81no—
92C2×#88322—21no—
93C2×#88322—21no—
94C2×#88322—81no—
95C23×C53—21no—
96C23×C53#9911yescharacteristic
97Q8×C53—41no—
98Q8×C53—41no—
99Q8×C53—41no—
100Q8×C53—81no—
101D4×C53—41no—
102D4×C53—81no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53C2×C22⋊C411yescharacteristic
115C2×Dic53—41no—
116C2×Dic53—41no—
117Dic106—41no—
118Dic106—41no—
119C8×C53—82no—
120#88322—41no—
121#88322—41no—
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—
136Dic53—81no—
137Dic53—81no—
138#3223—531no—
139#2135D5311yescharacteristic
140#2133—531no—
141#1413—532no—
142#1411—532no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53—81no—
146C2×C53#212711yescharacteristic
147#541—531no—
148#536C2×D5311yescharacteristic
149#497—531no—
150#494—531no—
151#491—531no—
152#488—531no—
153#464—1061no—
154#462—1061no—
155#452—1061no—
156#451—1061no—
157#433—531no—
158#413—21no—
159C2×#101—1061no—
160#356—21no—
161#356—532no—
162#356C2×D5312yes—
163#355—1061no—
164C53#1361211yescharacteristic, complemented, Sylow
165C2×#41C22⋊D5311yescharacteristic
166C22×Q8D21211yescharacteristic
167C2×Q16—1061no—
168C2×Q16—1061no—
169C2×Q16—1061no—
170C2×SD16—1061no—
171C2×M16—41no—
172C2×M16—2121no—
173C4⋊Q8—531no—
174C4⋊Q8C22⋊D5311yescharacteristic
175C4⋊Q8C22×D5311yescharacteristic
176#126—2121no—
177#125—531no—
178#125—2121no—
179#124C4×D5311yescharacteristic
180#124D21211yescharacteristic
181#123—41no—
182C22⋊Q8—41no—
183C22⋊Q8—1061no—
184C4×Q8—1061no—
185#117C4×D5311yescharacteristic
186C4⋊C8—1061no—
187#103—1061no—
188#103—1061no—
189#103—1061no—
190#102—1061no—
191#101—22no—
192#101—1062no—
193#101—2121no—
194#100—41no—
195C22⋊C8—1061no—
196#97—531no—
197C4×C8—531no—
198#41—81no—
199C2×Q8—21no—
200C2×Q8—81no—
201C2×Q8—1061no—
202C2×Q8#1171992111yescharacteristic
203C2×Q8C2×D21211yescharacteristic
204C2×Q8C2×C22⋊D5311yescharacteristic
205C2×D4—21no—
206C2×D4#1171992111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4#1171992111yescharacteristic
210Q16—4241no—
211Q16—4241no—
212Q16—4241no—
213SD16—4241no—
214M16—42no—
215M16—81no—
216M16—4241no—
217M16—2122no—
218C2×C8—81no—
219C2×C8—1061no—
220C2×C8—1061no—
221C2×C8—4241no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—1061no—
226C4⋊C4—2121no—
227C4⋊C4—2121no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C22⋊C4—21no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—2121no—
234C42—2121no—
235C42#1171992111yescharacteristic
236C42C2×C4×D5311yescharacteristic
237C23—21no—
238C23#1313313111yescharacteristic
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—81no—
243Q8—2121no—
244Q8—2121no—
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—2121no—
258C2×C4—2121no—
259C2×C4#1313313111yescharacteristic
260C2×C4C2×#1171992111yescharacteristic
261C8—82no—
262C8—4242no—
263C8—2121no—
264C8—2121no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22C2×#1313313111yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—4241no—
278C4—4241no—
279C2—21no—
280C2—41no—
281C2—81no—
282C2#2027638211yescharacteristic
subgroupG/H
centerC2#20276382
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC53×#2135C2
socleC2×C53#2127
prime radicalC2×#41×C53D4
cosocleC4⋊Q8×C53C23
hypercenter#2135D53
hypocenterC53#13612
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2135

Complements

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

NH ≅ G/Ntype
3. #20272887C2semidirect
4. #20272883C2semidirect
164. C53#13612semidirect

Maximal subgroups

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

subgroupindex
#202857982
#202764042
#202728872
#202728832
C53×#21352
#1361253