Finite Group Databasebeta

← #27287709

Subgroups — #27287709

Order 13568.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16810834C211yesmaximal, characteristic
2#20274708C211yesmaximal, characteristic
3#20273388C211yesmaximal, characteristic
4#20273386C211yesmaximal, characteristic
5#20273109C211yesmaximal, characteristic
6#20272612C211yesmaximal, characteristic
7C53×#1570C211yesmaximal, characteristic
8C22×Dic212—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810834—22no—
12#16810834—22no—
13#16810676C2211yescharacteristic
14#16810647C2211yescharacteristic
15C2×#13133142—21no—
16C2×#13133135—21no—
17#16810391—21no—
18#16810370—22no—
19#16810368—22no—
20#16810353—22no—
21C53×#431C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C53×#334—22no—
25C22×Dic106—21no—
26C22×Dic106D411yescharacteristic
27C2×Dic212—24no—
28C2×Dic212—42no—
29C2×Dic212—42no—
30C2×#11719917—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217D412yes—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—21no—
43C2×C4×Dic53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47#13133142—24no—
48#13133135—42no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C2×M16×C53—21no—
52C22×C8×C53—21no—
53C2×C42×C53C2311yescharacteristic
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C4⋊C8×C53D412yes—
58C4⋊C8×C53D412yes—
59C22⋊C8×C53—21no—
60C4×C8×C53—21no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53#4111yescharacteristic
63C22×C4×C53#4111yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—41no—
67C2×Dic106—21no—
68C2×Dic106—21no—
69C2×Dic106—21no—
70C2×Dic106—21no—
71C2×Dic106—21no—
72C2×Dic106—22no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106SD1612yes—
79M16×C53—41no—
80M16×C53—41no—
81C2×C8×C53—22no—
82C2×C8×C53—22no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C42×C53C2×D411yescharacteristic
87C42×C53C2×D411yescharacteristic
88C42×C53C2×D411yescharacteristic
89C42×C53#4111yescharacteristic
90C4⋊Dic53—21no—
91C4⋊Dic53—22no—
92C4⋊Dic53—21no—
93C4⋊Dic53—22no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96#11719919—41no—
97#11719919—41no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100#11719917—41no—
101#11719917—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104Dic212—48no—
105C23×C53#12311yescharacteristic
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53#12011yescharacteristic
109C2×C4×C53#12111yescharacteristic
110C2×C4×C53#12111yescharacteristic
111C2×C4×C53#12111yescharacteristic
112C2×C4×C53#12311yescharacteristic
113C2×C4×C53#12411yescharacteristic
114C2×C4×C53C2×SD1611yescharacteristic
115C2×C4×C53#13511yescharacteristic
116C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
118C2×Dic53—22no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—42no—
122C2×Dic53—41no—
123C2×Dic53—81no—
124C2×Dic53—81no—
125C2×Dic53—81no—
126Dic106—22no—
127Dic106—22no—
128Dic106—41no—
129Dic106—41no—
130Dic106—42no—
131Dic106—42no—
132Dic106—81no—
133Dic106—81no—
134C8×C53—42no—
135C8×C53—42no—
136C8×C53—81no—
137#88322—81no—
138#88322—81no—
139#10447—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#39411yescharacteristic
143C22×C53#48111yescharacteristic
144C22×C53#48211yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53#44911yescharacteristic
149C4×C53#45011yescharacteristic
150C4×C53#46511yescharacteristic
151C4×C53#46911yescharacteristic
152C4×C53#47312yes—
153Dic53—42no—
154Dic53—81no—
155Dic53—82no—
156C2×#462—531no—
157#1801—531no—
158#1570D5311yescharacteristic
159#1546—531no—
160#1546—531no—
161#1479—531no—
162#1301—531no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#203411yescharacteristic
166C2×C53#204711yescharacteristic
167C2×C53#204911yescharacteristic
168C22×Q16—1061no—
169C2×C4⋊Q8—531no—
170C2×C4×Q8—531no—
171#462—1062no—
172#462—1062no—
173#431—531no—
174#431C2×D5311yescharacteristic
175#423—531no—
176#423C2×D5311yescharacteristic
177C2×C4⋊C8C2×D5311yescharacteristic
178C2×#103—1061no—
179C2×#101—1061no—
180#343—1061no—
181#334—22no—
182#332—1062no—
183#332—1062no—
184Q8⋊C8—1062no—
185C53#1044711yescharacteristic, complemented, Sylow
186C22×Q8—531no—
187C22×Q8—1061no—
188C2×Q16—1064no—
189C2×Q16—2122no—
190C2×Q16—2122no—
191C2×M16—21no—
192C2×M16—1061no—
193C22×C8—21no—
194C4⋊Q8—532no—
195C4⋊Q8—1061no—
196C4⋊Q8—1061no—
197C4⋊Q8—1061no—
198C4×Q8—1061no—
199C4×Q8—1061no—
200C4×Q8—1061no—
201C4×Q8—1061no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C42—1061no—
207C2×C42C22×D5311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—1061no—
212C4⋊C8—1061no—
213C4⋊C8—1061no—
214C4⋊C8D21212yes—
215C4⋊C8C22⋊D5312yes—
216#103—1064no—
217#101—2122no—
218C22⋊C8—21no—
219C22⋊C8—1061no—
220C4×C8—21no—
221C4×C8—1061no—
222C2×Q8—532no—
223C2×Q8—1061no—
224C2×Q8—1061no—
225C2×Q8—1062no—
226C2×Q8—1061no—
227C2×Q8—1061no—
228C2×Q8—1061no—
229C2×Q8—2121no—
230C2×Q8—2121no—
231C2×Q8—2121no—
232C2×Q8—2121no—
233C2×Q8—2121no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—2121no—
237C22×C4#1171994511yescharacteristic
238C22×C4D4×D5311yescharacteristic
239C22×C4#1171994911yescharacteristic
240Q16—2128no—
241M16—41no—
242M16—41no—
243M16—2121no—
244M16—2121no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—2121no—
251C2×C8—2121no—
252C4⋊C4—1061no—
253C4⋊C4—1061no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C42—2121no—
261C42—2121no—
262C42C2×D21211yescharacteristic
263C42D4×D5311yescharacteristic
264C42#1171994911yescharacteristic
265C42C2×C22⋊D5311yescharacteristic
266C23#1313320211yescharacteristic
267Q8—1062no—
268Q8—1062no—
269Q8—4241no—
270Q8—4241no—
271Q8—2122no—
272Q8—2122no—
273Q8—2121no—
274Q8—2121no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—1062no—
278C2×C4—4241no—
279C2×C4—4241no—
280C2×C4—4241no—
281C2×C4—2121no—
282C2×C4—2122no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4#1313318511yescharacteristic
286C2×C4#1313319111yescharacteristic
287C2×C4#1313319211yescharacteristic
288C2×C4#1313319211yescharacteristic
289C2×C4#1313320211yescharacteristic
290C2×C4#1313320611yescharacteristic
291C2×C4SD16×D5311yescharacteristic
292C2×C4#1313323811yescharacteristic
293C2×C4#1313324411yescharacteristic
294C2×C4#1313324411yescharacteristic
295C8—42no—
296C8—42no—
297C8—81no—
298C8—4241no—
299C8—4241no—
300C22—21no—
301C22—21no—
302C22#1681054011yescharacteristic
303C22#1681094311yescharacteristic
304C22#1681094911yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—4241no—
309C4—4242no—
310C4—2122no—
311C4#1681076711yescharacteristic
312C4#1681077211yescharacteristic
313C4#1681085311yescharacteristic
314C4#1681087711yescharacteristic
315C4#1681089812yes—
316C2—21no—
317C2—21no—
318C2#2027590611yescharacteristic
319C2#2027598311yescharacteristic
320C2#2027599711yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1570C2
socleC22×C53#394
prime radicalC23×C53#123
hypercenter#1570D53
hypocenterC53#10447
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
185. C53#10447semidirect

Maximal subgroups

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

subgroupindex
C2×#168108342
#202747082
#202733882
#202733862
#202731092
#202726122
C53×#15702
#1044753