Finite Group Databasebeta

← #28299601

Subgroups — #28299601

Order 13568.

504 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290632C211yesmaximal, characteristic, complemented
2#20290501C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20277177C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C4⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×D4×Dic53C411yescharacteristic
13C2×#13133173—21no—
14#16810526C2211yescharacteristic
15#16810521C411yescharacteristic
16#16810519C2211yescharacteristic
17#16810517—21no—
18#16810511—21no—
19C4⋊C4×Dic53C411yescharacteristic
20C22⋊C4×Dic53—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26C4⋊#88324—42no—
27#13133272—42no—
28C23×Dic53—21no—
29D4×Dic53—22no—
30D4×Dic53—22no—
31#13133173—42no—
32#13133173—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C42×C53C2×C411yescharacteristic
55C24×C53—21no—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—24no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C2×#88324—24no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C42×C53—41no—
83C22⋊Dic53—24no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C4⋊Dic53—21no—
89C4⋊Dic53—41no—
90C4⋊Dic53M1612yes—
91#11719919—42no—
92#11719919—41no—
93#11719919—41no—
94C4×Dic53—22no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53—42no—
102C4×Dic53D812yes—
103C4×Dic53SD1612yes—
104C23×C53—22no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53C2×C22⋊C411yescharacteristic
112D4×C53—42no—
113D4×C53—42no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53#10212yes—
124C2×C4×C53#10412yes—
125C2×C4×C53C2×M1611yescharacteristic
126C2×C4×C53C2×M1611yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—22no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—48no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53#10212yes—
144C2×Dic53C2×D811yescharacteristic
145C2×Dic53C2×SD1611yescharacteristic
146C2×Dic53AGL(1,ℤ/8)11yescharacteristic
147C2×Dic53#13711yescharacteristic
148#88324—44no—
149#88324—82no—
150#7614—531nomaximal, Sylow
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53#40711yescharacteristic
159C22×C53#41011yescharacteristic
160C22×C53#41311yescharacteristic
161C22×C53C2×#10211yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53C2×#10411yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—81no—
170Dic53—22no—
171Dic53—21no—
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—42no—
176Dic53—41no—
177Dic53—81no—
178#2293—531no—
179#1831—531no—
180#1767—531no—
181#1271—532no—
182#1270—532no—
183C2×C53—82no—
184C2×C53#146411yescharacteristic
185C2×C53#148611yescharacteristic
186C2×C53#148911yescharacteristic
187C2×C53#150211yescharacteristic
188C2×C53#151211yescharacteristic
189C2×C53#153211yescharacteristic
190C2×C53#154711yescharacteristic
191C2×C4×D4—531no—
192C2×#117—1061no—
193C2×C4⋊C8—1061no—
194C2×C4⋊C8—1061no—
195C2×#97—1061no—
196#390—531no—
197#390#615711yescharacteristic
198#389—531no—
199#389—531no—
200#388—1061no—
201#388—1061no—
202C4×C4⋊C4—531no—
203C4×C22⋊C4—1061no—
204C43—531no—
205#336—532no—
206#336—532no—
207#336—1061no—
208C53#761411yescharacteristic, complemented, Sylow
209C22×D4C2×#615711yescharacteristic
210C23×C4—1061no—
211C22×C8—1062no—
212C22×C8—1062no—
213C4×D4—1062no—
214C4×D4—1062no—
215#117—2122no—
216#117—2122no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4C2×#615711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42C2×#615711yescharacteristic
235C4⋊C8—1064no—
236C4⋊C8—2122no—
237#97—2122no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24—21no—
242C2×D4—21no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4#1171993612yes—
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1064no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4C22⋊#615711yescharacteristic
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22×#615711yescharacteristic
259C2×C8—1064no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C4⋊C4—41no—
266C4⋊C4—532no—
267C4⋊C4—1061no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2122no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—1064no—
275C22⋊C4—2121no—
276C22⋊C4—2121no—
277C22⋊C4—2121no—
278C22⋊C4—2121no—
279C42—41no—
280C42—532no—
281C42—532no—
282C42—1061no—
283C42—1061no—
284C42—1062no—
285C42—1061no—
286C42—1061no—
287C42—2121no—
288C42—2121no—
289C42—2122no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊#615711yescharacteristic
298D4—42no—
299D4—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—532no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1062no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—2128no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4#1313327912yes—
331C2×C4#1313328112yes—
332C2×C4D53⋊M1611yescharacteristic
333C2×C4C2×#1171993611yescharacteristic
334C8—4242no—
335C8—2124no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#1681159411yescharacteristic
344C22#1681160311yescharacteristic
345C22#1681161011yescharacteristic
346C22C2×#1313327911yescharacteristic
347C22#1681161411yescharacteristic
348C22C2×#1313328111yescharacteristic
349C22#1681162011yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—81no—
355C4—1061no—
356C4—1061no—
357C4—1061no—
358C4—1061no—
359C4—1062no—
360C4—4241no—
361C4—2121no—
362C4—2122no—
363C2—82no—
364C2#2028996711yescharacteristic
365C2#2029004111yescharacteristic
366C2#2029005111yescharacteristic
367C2#2029009511yescharacteristic
368C2#2029012711yescharacteristic
369C2#2029019111yescharacteristic
370C2#2029022711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#7614
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290632C2semidirect
2. #20290501C2semidirect
3. #20289534C2semidirect
208. C53#7614semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202905012
#202895342
#202895312
#202771772
#761453