Finite Group Databasebeta

← #27294464

Subgroups — #27294464

Order 13568.

440 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277974C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272741C212yesmaximal, complemented
4#20272720C212yesmaximal, complemented
5C4×#100×C53C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×C22⋊#88322—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810551—21no—
12#16810548C411yescharacteristic
13#16810537—21no—
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#100×C53—21no—
21C4×M16×C53—21no—
22C53×#343C2212yes—
23C22×#88322—41no—
24#13133176—41no—
25#13133176—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—22no—
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#100×C53—24no—
55#97×C53—41no—
56C4×C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77M16×C53—24no—
78M16×C53—42no—
79C2×C8×C53—42no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—42no—
92#11719919—42no—
93C4×Dic53—42no—
94C4×Dic53—41no—
95C4×Dic53—41no—
96C2×#88322—42no—
97C2×#88322—82no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109D4×C53—44no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—81no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#13512yes—
126C2×Dic53—22no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C8×C53—44no—
139#88322—82no—
140#11735—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—42no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53C2×#9911yescharacteristic
154C22×C53#41611yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53#35212yes—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166#2395—531no—
167C4×#100D5311yescharacteristic
168#1459—531no—
169#1353—532no—
170#1346—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—81no—
175C2×C53#150111yescharacteristic
176C2×C53C2×#35211yescharacteristic
177C2×C53#213211yescharacteristic
178C2×C4×D4C2×D5311yescharacteristic
179C2×#117—531no—
180C2×#100—21no—
181C2×C22⋊C8—1061no—
182C4×M16—21no—
183#400—531no—
184#399—531no—
185#398—531no—
186#397—531no—
187#397—1061no—
188#394—1061no—
189C4×C4⋊C4—1061no—
190C4×C22⋊C4—531no—
191C4×C22⋊C4—1061no—
192#343C2×D5312yes—
193#324—532no—
194#323—1061no—
195C53#1173511yescharacteristic, complemented, Sylow
196C22×D4C22⋊D5311yescharacteristic
197C23×C4C4×D5311yescharacteristic
198C23×C4D21211yescharacteristic
199C2×M16—22no—
200C22×C8—2121no—
201C4×D4—41no—
202C4×D4—41no—
203#117—2121no—
204#117—2121no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4C4×D5311yescharacteristic
215C2×C22⋊C4D21211yescharacteristic
216C2×C42—1061no—
217C2×C42—1061no—
218C2×C42C22⋊D5311yescharacteristic
219#100—24no—
220C22⋊C8—1062no—
221C22⋊C8—2121no—
222#97—41no—
223C4×C8—41no—
224#95—531no—
225#95—531no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C24#1171992111yescharacteristic
231C24#1171992111yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4C2×D21211yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—24no—
251M16—42no—
252C2×C8—42no—
253C2×C8—4242no—
254C2×C8—2122no—
255C4⋊C4—41no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C42—21no—
267C42—21no—
268C42—2122no—
269C42—2121no—
270C42—2121no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1313314512yes—
281C23C2×#1171992111yescharacteristic
282D4—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—1062no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4#1313313111yescharacteristic
309C2×C4#1313313111yescharacteristic
310C2×C4#1313323712yes—
311C8—44no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#1313313111yescharacteristic
326C22#1681061711yescharacteristic
327C22C2×#1313314511yescharacteristic
328C4—22no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4241no—
334C4—4241no—
335C4—4241no—
336C4—2122no—
337C4#1681041412yes—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2027320411yescharacteristic
343C2C2×#1681041411yescharacteristic
344C2#2027639811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#100×C53C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#100D53
hypocenterC53#11735
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #20272741C2semidirect
4. #20272720C2semidirect
195. C53#11735semidirect

Maximal subgroups

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

subgroupindex
#202779742
#202730332
#202727412
#202727202
C4×#100×C532
#1173553