Finite Group Databasebeta

← #27287730

Subgroups — #27287730

Order 13568.

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

normal subgroups only

This group has 376 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×#16810909C211yesmaximal, characteristic
2#20274708C211yesmaximal, characteristic
3#20273399C211yesmaximal, characteristic
4#20273397C211yesmaximal, characteristic
5#20273109C211yesmaximal, characteristic
6#20272612C211yesmaximal, characteristic
7C53×#1570C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10#16810909—22no—
11#16810909—22no—
12#16810676C2211yescharacteristic
13C2×#13133149—21no—
14#16810647C2211yescharacteristic
15C2×#13133142—21no—
16C2×#13133135—21no—
17#16810391—21no—
18#16810371—22no—
19#16810370—22no—
20#16810353—22no—
21C53×#431C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C53×#334—22no—
25C22×Dic106—21no—
26C22×Dic106D411yescharacteristic
27C2×#11719917—21no—
28#13133217—21no—
29#13133217—21no—
30#13133217—21no—
31#13133217D412yes—
32C4×Dic106—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—21no—
40C2×C4×Dic53—21no—
41#13133149—24no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133142—24no—
46#13133135—42no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C2×M16×C53—21no—
50C22×C8×C53—21no—
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53D412yes—
56C4⋊C8×C53D412yes—
57C22⋊C8×C53—21no—
58C4×C8×C53—21no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53#4111yescharacteristic
61C22×C4×C53#4111yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—41no—
65C2×Dic106—21no—
66C2×Dic106—21no—
67C2×Dic106—21no—
68C2×Dic106—21no—
69C2×Dic106—21no—
70C2×Dic106—22no—
71C2×Dic106—41no—
72C2×Dic106—82no—
73C2×Dic106SD1612yes—
74M16×C53—41no—
75M16×C53—41no—
76C2×C8×C53—22no—
77C2×C8×C53—22no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C42×C53C2×D411yescharacteristic
82C42×C53C2×D411yescharacteristic
83C42×C53C2×D411yescharacteristic
84C42×C53#4111yescharacteristic
85C4⋊Dic53—22no—
86C4⋊Dic53—21no—
87C4⋊Dic53—22no—
88C4⋊Dic53—21no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95#11719917—41no—
96#11719917—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53#12311yescharacteristic
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53#12011yescharacteristic
103C2×C4×C53#12111yescharacteristic
104C2×C4×C53#12111yescharacteristic
105C2×C4×C53#12111yescharacteristic
106C2×C4×C53#12311yescharacteristic
107C2×C4×C53#12411yescharacteristic
108C2×C4×C53C2×SD1611yescharacteristic
109C2×C4×C53#13511yescharacteristic
110C2×C4×C53#13711yescharacteristic
111C2×C4×C53#13711yescharacteristic
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—82no—
118C2×Dic53—81no—
119Dic106—22no—
120Dic106—22no—
121Dic106—41no—
122Dic106—41no—
123Dic106—82no—
124Dic106—82no—
125C8×C53—42no—
126C8×C53—42no—
127C8×C53—81no—
128#88322—81no—
129#88322—81no—
130#10450—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#39411yescharacteristic
134C22×C53#48311yescharacteristic
135C22×C53#48511yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53#44811yescharacteristic
140C4×C53#45211yescharacteristic
141C4×C53#46511yescharacteristic
142C4×C53#47011yescharacteristic
143C4×C53#47112yes—
144Dic53—42no—
145Dic53—81no—
146Dic53—82no—
147C2×#475—531no—
148#1801—531no—
149#1570D5311yescharacteristic
150#1549—531no—
151#1548—531no—
152#1479—531no—
153#1301—531no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53#203511yescharacteristic
157C2×C53#204611yescharacteristic
158C2×C53#204911yescharacteristic
159C2×C4⋊Q8—531no—
160C2×C4×Q8—531no—
161#475—1062no—
162#475—1062no—
163#431—531no—
164#431C2×D5311yescharacteristic
165C2×#106—1061no—
166#423—531no—
167#423C2×D5311yescharacteristic
168C2×C4⋊C8C2×D5311yescharacteristic
169C2×#103—1061no—
170C2×#101—1061no—
171#343—1061no—
172#334—22no—
173#333—1062no—
174#332—1062no—
175Q8⋊C8—1062no—
176C53#1045011yescharacteristic, complemented, Sylow
177C22×Q8—531no—
178C22×Q8—1061no—
179C2×M16—21no—
180C2×M16—1061no—
181C22×C8—21no—
182C4⋊Q8—532no—
183C4⋊Q8—1061no—
184C4⋊Q8—1061no—
185C4⋊Q8—1061no—
186C4×Q8—1061no—
187C4×Q8—1061no—
188C4×Q8—1061no—
189C4×Q8—1061no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C42—1061no—
195C2×C42C22×D5311yescharacteristic
196#106—1064no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—1061no—
201C4⋊C8—1061no—
202C4⋊C8—1061no—
203C4⋊C8D21212yes—
204C4⋊C8C22⋊D5312yes—
205#103—1064no—
206#101—2122no—
207C22⋊C8—21no—
208C22⋊C8—1061no—
209C4×C8—21no—
210C4×C8—1061no—
211C2×Q8—532no—
212C2×Q8—1061no—
213C2×Q8—1061no—
214C2×Q8—1061no—
215C2×Q8—1062no—
216C2×Q8—1061no—
217C2×Q8—1061no—
218C2×Q8—4242no—
219C2×Q8—2121no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—2121no—
223C22×C4#1171994511yescharacteristic
224C22×C4D4×D5311yescharacteristic
225C22×C4#1171994911yescharacteristic
226M16—41no—
227M16—41no—
228M16—2121no—
229M16—2121no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—2121no—
236C2×C8—2121no—
237C4⋊C4—1062no—
238C4⋊C4—1061no—
239C4⋊C4—1061no—
240C4⋊C4—1062no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C42—2121no—
246C42—2121no—
247C42C2×D21211yescharacteristic
248C42D4×D5311yescharacteristic
249C42#1171994911yescharacteristic
250C42C2×C22⋊D5311yescharacteristic
251C23#1313320211yescharacteristic
252Q8—1062no—
253Q8—1062no—
254Q8—4242no—
255Q8—4242no—
256Q8—2121no—
257Q8—2121no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—1062no—
261C2×C4—4242no—
262C2×C4—4241no—
263C2×C4—2121no—
264C2×C4—2122no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313318511yescharacteristic
268C2×C4#1313319111yescharacteristic
269C2×C4#1313319211yescharacteristic
270C2×C4#1313319211yescharacteristic
271C2×C4#1313320211yescharacteristic
272C2×C4#1313320611yescharacteristic
273C2×C4SD16×D5311yescharacteristic
274C2×C4#1313323811yescharacteristic
275C2×C4#1313324611yescharacteristic
276C2×C4#1313324611yescharacteristic
277C8—42no—
278C8—42no—
279C8—81no—
280C8—4241no—
281C8—4241no—
282C22—21no—
283C22—21no—
284C22#1681054011yescharacteristic
285C22#1681095511yescharacteristic
286C22#1681096711yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—4242no—
291C4—4241no—
292C4—2122no—
293C4#1681076011yescharacteristic
294C4#1681078211yescharacteristic
295C4#1681085311yescharacteristic
296C4#1681088211yescharacteristic
297C4#1681088612yes—
298C2—21no—
299C2—21no—
300C2#2027591311yescharacteristic
301C2#2027597811yescharacteristic
302C2#2027599711yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1570C2
socleC22×C53#394
prime radicalC23×C53#123
hypercenter#1570D53
hypocenterC53#10450
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
176. C53#10450semidirect

Maximal subgroups

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

subgroupindex
C2×#168109092
#202747082
#202733992
#202733972
#202731092
#202726122
C53×#15702
#1045053