Finite Group Databasebeta

← #27288475

Subgroups — #27288475

Order 13568.

574 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 99 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#20275552C212yesmaximal, complemented
2#20275144C212yesmaximal, complemented
3#20274892C211yesmaximal, characteristic, complemented
4#20272583C211yesmaximal, characteristic, complemented
5C53×#3186C211yesmaximal, characteristic
6C23×#88322C2211yescharacteristic
7C2×#13133146—21no—
8C2×#13133138—22no—
9C2×#13133132—22no—
10C2×#13133132—22no—
11#16810483C2212yes—
12C4×C22⋊Dic53C2211yescharacteristic
13#16810385—22no—
14#16810377—21no—
15C2×#121×C53C2212yes—
16C53×#480—22no—
17C53×#466—21no—
18C53×#466—21no—
19C53×#447—22no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C22×#88322—22no—
22C22×#88322—21no—
23C22×#88322—42no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53D411yescharacteristic
27C2×C22⋊Dic53D411yescharacteristic
28C2×C4×Dic53—21no—
29#13133146—42no—
30#13133138—44no—
31#13133132—44no—
32#13133132—44no—
33#13133124—21no—
34#13133124—22no—
35C22×D4×C53—22no—
36C22×D4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C2×D8×C53—21no—
39C2×D8×C53—21no—
40C22×C8×C53D411yescharacteristic
41C22×C8×C53D411yescharacteristic
42#121×C53—22no—
43#121×C53—22no—
44#121×C53—22no—
45#121×C53—22no—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—22no—
48#107×C53—21no—
49#107×C53—21no—
50#102×C53—22no—
51#102×C53—22no—
52C22⋊C8×C53—22no—
53C24×C53—22no—
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C22×C4×C53—22no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×C4×C53#4112yes—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75D8×C53—81no—
76D8×C53—81no—
77C2×C8×C53—21no—
78C2×C8×C53—21no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊Dic53—21no—
86C22⋊Dic53—22no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53D812yescomplemented
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C2×#88322—24no—
95C2×#88322—22no—
96C2×#88322—42no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—82no—
105C23×C53—82no—
106C23×C53#12111yescharacteristic
107C23×C53#12112yes—
108C23×C53C2×D811yescharacteristic
109C23×C53#13511yescharacteristic
110C23×C53AGL(1,ℤ/8)12yes—
111D4×C53—44no—
112D4×C53—42no—
113D4×C53—42no—
114D4×C53—82no—
115D4×C53—82no—
116D4×C53—82no—
117D4×C53—82no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53#12312yes—
129C2×C4×C53#12411yescharacteristic
130C2×Dic53—22no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C8×C53—41no—
140C8×C53—41no—
141#88322—42no—
142#10564—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—42no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53—162no—
154C22×C53—162no—
155C22×C53#39411yescharacteristic
156C22×C53#45912yes—
157C22×C53#46312yes—
158C22×C53#46611yescharacteristic
159C22×C53#46911yescharacteristic
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—82no—
163C4×C53B(4,𝔽2)11yescharacteristic
164C4×C53#45811yescharacteristic
165Dic53—42no—
166Dic53—81no—
167Dic53—162no—
168#3186D5311yescharacteristic
169#1964—532no—
170#1893—532no—
171#1850—531no—
172#1289—531no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—22no—
177C2×C53—162no—
178C2×C53#204411yescharacteristic
179C2×C53#205311yescharacteristic
180C2×C53#205411yescharacteristic
181C23×C8—531no—
182C2×#121C2×D5312yes—
183#480—22no—
184#466—21no—
185#466—21no—
186#447—22no—
187C2×C4⋊C8—1061no—
188C2×#102—1062no—
189C2×#99—1062no—
190C2×#99—1062no—
191C2×C22⋊C8C2×D5311yescharacteristic
192#380—532no—
193C4×C22⋊C4—531no—
194#340—1062no—
195#336—1061no—
196C53#1056411yescharacteristic, complemented, Sylow
197C22×D4—22no—
198C22×D4C22⋊D5312yes—
199C23×C4C22×D5311yescharacteristic
200C2×D8—21no—
201C2×D8—21no—
202C22×C8—1061no—
203C22×C8—1062no—
204C22×C8—2122no—
205C22×C8C22⋊D5311yescharacteristic
206C22×C8C22⋊D5311yescharacteristic
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4C22⋊D5312yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—1062no—
217C2×C42—1061no—
218#107—21no—
219#107—21no—
220C4⋊C8—2122no—
221#102—22no—
222#102—22no—
223#102—2124no—
224#99—2124no—
225#99—2124no—
226C22⋊C8—22no—
227#95—1061no—
228#95—1062no—
229C24—22no—
230C24D4×D5311yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—2122no—
247C22×C4#1171994711yescharacteristic
248C22×C4#1171994712yes—
249C22×C4C2×C22⋊D5312yes—
250C22×C4C2×C22⋊D5311yescharacteristic
251D8—81no—
252D8—81no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—1064no—
258C2×C8—1062no—
259C2×C8—2121no—
260C2×C8—2122no—
261C2×C8—2121no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—532no—
269C22⋊C4—1061no—
270C22⋊C4—1062no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C42—2121no—
274C42—2121no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—82no—
281C23—82no—
282C23#1313318811yescharacteristic
283C23#1313318912yes—
284C23D8×D5311yescharacteristic
285C23#1313323611yescharacteristic
286C23#1313324112yes—
287D4—44no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—1062no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4#1313318611yescharacteristic
313C2×C4#1313319812yes—
314C2×C4#1313320411yescharacteristic
315C8—41no—
316C8—41no—
317C8—2122no—
318C22—22no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22#1681053711yescharacteristic
331C22#1681081612yes—
332C22#1681083612yes—
333C22#1681085611yescharacteristic
334C22#1681087311yescharacteristic
335C4—22no—
336C4—21no—
337C4—82no—
338C4—8482no—
339C4—4241no—
340C4—2122no—
341C4#1681081011yescharacteristic
342C4#1681081311yescharacteristic
343C2—22no—
344C2—21no—
345C2—21no—
346C2—22no—
347C2—162no—
348C2#2027596311yescharacteristic
349C2#2027601811yescharacteristic
350C2#2027602311yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3186C2
socleC22×C53#394
hypercenter#3186D53
hypocenterC53#10564
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #20275552C2semidirect
2. #20275144C2semidirect
3. #20274892C2semidirect
4. #20272583C2semidirect
89. C22⋊Dic53D8semidirect
196. C53#10564semidirect

Maximal subgroups

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

subgroupindex
#202755522
#202751442
#202748922
#202725832
C53×#31862
#1056453