Finite Group Databasebeta

← #27275836

Subgroups — #27275836

Order 13568.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic106C211yesmaximal, characteristic
2#20273397C212yesmaximal, complemented
3#20273143C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic
5C53×#1800C211yesmaximal, characteristic
6C23×Dic106C412yes—
7D4×Dic106—24no—
8D4×Dic106—24no—
9C2×#13133217C2211yescharacteristic
10C2×#13133216—21no—
11C2×#13133196C412yes—
12C2×#13133194—22no—
13C2×C4×Dic106C2211yescharacteristic
14C2×D4×Dic53—21no—
15#16810676C2212yes—
16#16810371—24no—
17#16810353—24no—
18#16810347—22no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#100×C53—21no—
22C53×#343—21no—
23C22×Dic106—22no—
24C22×Dic106—24no—
25C22×Dic106—24no—
26C22×Dic106—48no—
27C22×Dic106C2×C411yescharacteristic
28C22×Dic106C2×C412yes—
29C23×Dic53—22no—
30#13133217—22no—
31#13133217D414yes—
32#13133216—28no—
33#13133196—24no—
34#13133196—24no—
35#13133194—216no—
36C4×Dic106—22no—
37C4×Dic106D414yes—
38D4×Dic53—28no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919—22no—
43C2×#11719919—21no—
44C2×C22⋊Dic53—22no—
45C2×C4×Dic53—21no—
46C4⋊#88322—22no—
47C22⋊#88322—22no—
48C4×#88322—22no—
49C22×D4×C53D411yescharacteristic
50C23×C4×C53C2×C412yes—
51C2×M16×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—22no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53D412yes—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—22no—
58C4⋊C8×C53—22no—
59#100×C53—42no—
60C24×C53C22⋊C412yes—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53SD1612yes—
64C2×D4×C53Q1612yes—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—44no—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—28no—
74C22×Dic53—21no—
75C22×Dic53—22no—
76C22×Dic53—48no—
77C2×Dic106—21no—
78C2×Dic106—24no—
79C2×Dic106—28no—
80C2×Dic106—22no—
81C2×Dic106—416no—
82C2×Dic106—48no—
83C2×Dic106—48no—
84C2×Dic106—48no—
85C2×Dic106—44no—
86C2×Dic106C22⋊C412yes—
87C2×Dic106C22⋊C414yes—
88M16×C53—42no—
89M16×C53—42no—
90C2×C8×C53—42no—
91C4⋊C4×C53—22no—
92C22⋊C4×C53—24no—
93C42×C53C2×D412yes—
94C42×C53C2×D412yes—
95C22⋊Dic53—28no—
96C4⋊Dic53—21no—
97C4⋊Dic53—24no—
98C4⋊Dic53—24no—
99C4⋊Dic53C22⋊C412yes—
100#11719919—28no—
101#11719919—24no—
102C4×Dic53—24no—
103C2×#88322—42no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53#10314yes—
109C23×C53C2×C22⋊C411yescharacteristic
110D4×C53—24no—
111D4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—42no—
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53C2×C22⋊C411yescharacteristic
120C2×C4×C53#12012yes—
121C2×C4×C53#12012yes—
122C2×C4×C53C2×SD1611yescharacteristic
123C2×C4×C53C2×Q1611yescharacteristic
124C2×C4×C53#13712yes—
125C2×Dic53—24no—
126C2×Dic53—22no—
127C2×Dic53—24no—
128C2×Dic53—22no—
129C2×Dic53—416no—
130C2×Dic53—44no—
131C2×Dic53—42no—
132Dic106—24no—
133Dic106—22no—
134Dic106—416no—
135Dic106—44no—
136Dic106—44no—
137Dic106#10014yescomplemented
138C8×C53—82no—
139#88322—82no—
140#8391—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—24no—
144C22×C53—44no—
145C22×C53—44no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53#37911yescharacteristic
149C22×C53C2×#10311yescharacteristic
150C22×C53#41911yescharacteristic
151C4×C53—22no—
152C4×C53—21no—
153C4×C53—42no—
154C4×C53C2×#10012yes—
155C4×C53#44812yes—
156C4×C53#45112yes—
157Dic53—24no—
158Dic53—42no—
159Dic53—44no—
160C2×D4×Q8—531no—
161#1800D5311yescharacteristic
162#1548—532no—
163#1487—532no—
164#1460—531no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—44no—
168C2×C53#178311yescharacteristic
169C2×C53#178711yescharacteristic
170C2×C53#179611yescharacteristic
171C23×Q8—532no—
172D4×Q8—1064no—
173D4×Q8—1064no—
174C2×C4⋊Q8—531no—
175C2×C4⋊Q8—1061no—
176C2×C22⋊Q8—532no—
177C2×C22⋊Q8—1062no—
178C2×C4×Q8—531no—
179C2×C4×D4—1061no—
180C2×C4×D4C2×D5311yescharacteristic
181#431—532no—
182#423C2×D5312yes—
183C2×#100—21no—
184#343—21no—
185#333—1064no—
186Q8⋊C8—1064no—
187#323—1062no—
188C53#839111yescharacteristic, complemented, Sylow
189C22×Q8—531no—
190C22×Q8—532no—
191C22×Q8—1064no—
192C22×Q8—1064no—
193C22×Q8—1062no—
194C22×Q8—2128no—
195C22×D4C22⋊D5311yescharacteristic
196C23×C4—1062no—
197C23×C4C4×D5312yes—
198C2×M16—22no—
199C4⋊Q8—534no—
200C4⋊Q8—1068no—
201C4⋊Q8—1062no—
202C22⋊Q8—10616no—
203C22⋊Q8—1064no—
204C22⋊Q8—1064no—
205C4×Q8—534no—
206C4×Q8—1062no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—1068no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C22⋊C4—1062no—
217C2×C22⋊C4D21212yes—
218C2×C42—1061no—
219C2×C42C22×D5311yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—1062no—
223#100—42no—
224C22⋊C8—1062no—
225C4×C8—1062no—
226C24#1171992112yes—
227C2×Q8—532no—
228C2×Q8—534no—
229C2×Q8—1064no—
230C2×Q8—1068no—
231C2×Q8—1061no—
232C2×Q8—1062no—
233C2×Q8—2124no—
234C2×Q8—2128no—
235C2×Q8—2128no—
236C2×Q8—21216no—
237C2×Q8—2128no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4#1171992312yes—
241C2×D4#1171992512yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—1061no—
246C22×C4—1068no—
247C22×C4—1061no—
248C22×C4—1062no—
249C22×C4—2128no—
250C22×C4#1171992112yes—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4C2×D21211yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254M16—42no—
255M16—42no—
256C2×C8—42no—
257C2×C8—2122no—
258C4⋊C4—22no—
259C4⋊C4—532no—
260C4⋊C4—1068no—
261C4⋊C4—1061no—
262C4⋊C4—1064no—
263C4⋊C4—1064no—
264C4⋊C4—1064no—
265C22⋊C4—24no—
266C22⋊C4—1068no—
267C42—1064no—
268C42D4×D5312yes—
269C42D4×D5312yes—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23#1313314014yes—
275C23C2×#1171992111yescharacteristic
276Q8—534no—
277Q8—1064no—
278Q8—1062no—
279Q8—2124no—
280Q8—21216no—
281Q8—2124no—
282D4—24no—
283D4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—1064no—
291C2×C4—1062no—
292C2×C4—1062no—
293C2×C4—1064no—
294C2×C4—21216no—
295C2×C4—2122no—
296C2×C4—2124no—
297C2×C4C22⋊C4×D5311yescharacteristic
298C2×C4C22⋊C4×D5311yescharacteristic
299C2×C4#1313318512yes—
300C2×C4#1313318712yes—
301C2×C4C2×#1171992311yescharacteristic
302C2×C4C2×#1171992511yescharacteristic
303C2×C4#1313324712yes—
304C8—82no—
305C8—4242no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22#1681047811yescharacteristic
314C22C2×#1313314011yescharacteristic
315C22#1681063111yescharacteristic
316C4—22no—
317C4—21no—
318C4—42no—
319C4—1064no—
320C4—2122no—
321C4—2124no—
322C4#100×D5312yes—
323C4#1681076112yes—
324C4#1681077812yes—
325C2—21no—
326C2—21no—
327C2—44no—
328C2#2027463411yescharacteristic
329C2#2027464911yescharacteristic
330C2#2027469111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#8391
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20273397C2semidirect
3. #20273143C2semidirect
137. Dic106#100semidirect
188. C53#8391semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1062
#202733972
#202731432
#202730362
C53×#18002
#839153