Finite Group Databasebeta

← #28299631

Subgroups — #28299631

Order 13568.

462 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290643C211yesmaximal, characteristic
2#20290501C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5C4×#13133216C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133272—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×C4×Dic106—21no—
14#16810527—21no—
15#16810522C2211yescharacteristic
16#16810520—21no—
17#16810519C2211yescharacteristic
18#16810492C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26#13133272—42no—
27C22×Dic106—21no—
28#13133216—22no—
29#13133216—22no—
30C4×Dic106—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52#95×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×Dic106—24no—
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×#88324—24no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53M1612yes—
76C42×C53—22no—
77C42×C53—42no—
78C4⋊Dic53—21no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53M1612yes—
82#11719919—24no—
83#11719919—42no—
84#11719919—42no—
85#11719919—42no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C4×Dic53—42no—
93C4×Dic53SD1612yes—
94C4×Dic53Q1612yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53#10112yes—
109C2×C4×C53#10312yes—
110C2×C4×C53C2×M1611yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—22no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53#10312yes—
127C2×Dic53#10412yes—
128C2×Dic53C2×SD1611yescharacteristic
129C2×Dic53C2×Q1611yescharacteristic
130C2×Dic53AGL(1,ℤ/8)12yes—
131Dic106—48no—
132#88324—44no—
133#88324—82no—
134#7622—531nomaximal, Sylow
135C22×C53#40711yescharacteristic
136C22×C53#41011yescharacteristic
137C22×C53C2×#10111yescharacteristic
138C22×C53C2×#10311yescharacteristic
139C22×C53#41811yescharacteristic
140C22×C53C2×#10411yescharacteristic
141C22×C53#42111yescharacteristic
142C4×C53—22no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—22no—
146C4×C53—42no—
147C4×C53—81no—
148Dic53—22no—
149Dic53—22no—
150Dic53—21no—
151Dic53—21no—
152Dic53—22no—
153Dic53—42no—
154Dic53—81no—
155C4×C4⋊Q8—531no—
156#1839—531no—
157#1767—531no—
158#1271—532no—
159#1271—532no—
160C2×C53#146711yescharacteristic
161C2×C53#148711yescharacteristic
162C2×C53#149011yescharacteristic
163C2×C53#150311yescharacteristic
164C2×C53#151411yescharacteristic
165C2×C53#153111yescharacteristic
166C2×C53#154911yescharacteristic
167C2×C4⋊Q8—531no—
168C2×C4×Q8—1061no—
169C2×C4⋊C8—1061no—
170C2×#97—1061no—
171#391—1061no—
172#389—531no—
173#389—531no—
174#389—1061no—
175#384—531no—
176C4×C4⋊C4—531no—
177C4×C4⋊C4—1061no—
178C4×C4⋊C4#615711yescharacteristic
179C43—531no—
180#336—532no—
181#336—532no—
182#336—1061no—
183#336—1061no—
184C53#762211yescharacteristic, complemented, Sylow
185C22×Q8—1061no—
186C22×C8—1062no—
187C22×C8—1062no—
188C4⋊Q8—1062no—
189C4⋊Q8—1062no—
190C4×Q8—2124no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4C2×#615711yescharacteristic
199C2×C4⋊C4C2×#615711yescharacteristic
200C2×C42—21no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—1061no—
208C2×C42C2×#615711yescharacteristic
209C4⋊C8—1064no—
210#97—2122no—
211#95—21no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215C2×Q8—1064no—
216C2×Q8—2122no—
217C2×Q8—2122no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—1064no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—532no—
238C4⋊C4—1064no—
239C4⋊C4—1061no—
240C4⋊C4—2121no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C4⋊C4#1171993612yes—
246C42—22no—
247C42—42no—
248C42—532no—
249C42—532no—
250C42—1062no—
251C42—1062no—
252C42—1061no—
253C42—1061no—
254C42—2121no—
255C42—2121no—
256C42—2122no—
257C23C2×C22⋊#615711yescharacteristic
258Q8—2128no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—532no—
272C2×C4—531no—
273C2×C4—531no—
274C2×C4—532no—
275C2×C4—532no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1062no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4#1313327812yes—
291C2×C4#1313328012yes—
292C2×C4D53⋊M1611yescharacteristic
293C2×C4C2×#1171993611yescharacteristic
294C8—4242no—
295C8—2124no—
296C22#1681159411yescharacteristic
297C22#1681160211yescharacteristic
298C22C2×#1313327811yescharacteristic
299C22C2×#1313328011yescharacteristic
300C22#1681161511yescharacteristic
301C22D53⋊#10411yescharacteristic
302C22#1681162011yescharacteristic
303C4—22no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—42no—
308C4—81no—
309C4—1062no—
310C4—1061no—
311C4—1061no—
312C4—1062no—
313C4—1062no—
314C4—4241no—
315C4—2122no—
316C2#2028997711yescharacteristic
317C2#2029004411yescharacteristic
318C2#2029005511yescharacteristic
319C2#2029009711yescharacteristic
320C2#2029013411yescharacteristic
321C2#2029018711yescharacteristic
322C2#2029023011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7622
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
184. C53#7622semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202905012
#202895342
#202895332
C4×#131332162
#762253