Finite Group Databasebeta

← #27207012

Subgroups — #27207012

Order 13184.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#20268456C211yesmaximal, characteristic, complemented
2#20268446C211yesmaximal, characteristic
3#20268141C211yesmaximal, characteristic, complemented
4#20268120C212yesmaximal, complemented
5#20268077C211yesmaximal, characteristic, complemented
6#20268017C212yesmaximal, complemented
7#20267928C212yesmaximal, complemented
8C4×#13110093C211yesmaximal, characteristic, complemented
9#20267856C211yesmaximal, characteristic, complemented
10#103×D103C212yesmaximal, complemented
11#461×C103C211yesmaximal, characteristic, complemented
12C2×#13110105—21no—
13C2×Q8×D103C2212yes—
14C2×#13110093C2211yescharacteristic
15C2×C8×D103C2212yes—
16#15693021C2211yescharacteristic
17#15693020—21no—
18#15693003—22no—
19#15693000—21no—
20#15692999—22no—
21#15692997C2212yes—
22#15692992C2212yes—
23C4×Dic206C2211yescharacteristic
24C4×D412C2211yescharacteristic
25#15692983—21no—
26C4⋊C4×D103C2212yes—
27#15692953C2212yes—
28#15692952C2211yescharacteristic
29#15692946C2212yes—
30#15692945C2212yes—
31#15692945C2212yes—
32#15692944C2211yescharacteristic
33#15692942C2211yescharacteristic
34(C2×D103)⋊C8C2212yes—
35C4×C103⋊C8C2211yescharacteristic
36C2×SD16×C103C2212yes—
37C4⋊Q8×C103C2211yescharacteristic
38C4×D4×C103C2211yescharacteristic
39C4⋊C8×C103C2211yescharacteristic
40#103×C103C2212yes—
41#13110105—44no—
42C2×Dic206—21no—
43C2×Dic206—22no—
44C2×Dic206C2311yescharacteristic
45Q8×D103—24no—
46Q8×D103—42no—
47C2×D412D411yescharacteristic
48C2×C22⋊D103—22no—
49C2×C4×D103—22no—
50C2×C4×D103—22no—
51C2×C4×D103C2312yes—
52#13110093—22no—
53C2×C103⋊C8C2311yescharacteristic
54C2×C103⋊C8C2311yescharacteristic
55C8×D103—42no—
56C4⋊Dic103—21no—
57C4⋊Dic103D411yescharacteristic
58C4⋊Dic103C2311yescharacteristic
59C4⋊Dic103C2312yes—
60Dic103⋊C4—21no—
61Dic103⋊C4—22no—
62Dic103⋊C4—22no—
63C22⋊Dic103—22no—
64(C2×D103)⋊C4—22no—
65(C2×D103)⋊C4D412yes—
66C4×Dic103—21no—
67C2×Q8×C103C2312yes—
68C2×D4×C103C2311yescharacteristic
69C22×C4×C103—21no—
70SD16×C103—42no—
71C2×C8×C103C2312yes—
72C4⋊C4×C103—21no—
73C4⋊C4×C103—21no—
74C4⋊C4×C103C2311yescharacteristic
75C22⋊C4×C103—21no—
76C42×C103C2311yescharacteristic
77C23×C103—21no—
78C22×D103C2×D412yes—
79Q8×C103—22no—
80Q8×C103—22no—
81D4×C103—21no—
82D4×C103—21no—
83C2×C4×C103—21no—
84C2×C4×C103—22no—
85C2×C4×C103—42no—
86C2×C4×C103C2×D411yescharacteristic
87C2×C4×C103C2×D411yescharacteristic
88C2×C4×C103C2411yescharacteristic
89C22⋊D103—44no—
90C2×Dic103—21no—
91C2×Dic103—21no—
92C2×Dic103—22no—
93C2×Dic103C2×D412yes—
94D412SD1614yescomplemented
95C4×D103—22no—
96C4×D103—22no—
97C4×D103—44no—
98C4×D103—44no—
99Dic206—21no—
100Dic206—44no—
101Dic206—42no—
102Dic206#4112yescomplemented
103C8×C103—22no—
104C103⋊C8—21no—
105C103⋊C8—21no—
106C22×C103—41no—
107C22×C103—41no—
108C22×C103—41no—
109C22×C103C22×D411yescharacteristic
110C2×D103—22no—
111C2×D103—22no—
112C2×D103C2×SD1614yes—
113C4×C103—21no—
114C4×C103—42no—
115C4×C103—41no—
116C4×C103C2×SD1612yes—
117C4×C103C2×#4111yescharacteristic
118C4×C103H(5,𝔽2)11yescharacteristic
119Dic103—22no—
120Dic103—22no—
121Dic103—42no—
122Dic103—41no—
123C2×C103—81no—
124C2×C103#54611yescharacteristic
125C2×C103C22×SD1611yescharacteristic
126C2×C103#57811yescharacteristic
127D103—24no—
128#3298—1031nomaximal, Sylow
129C103#329811yescharacteristic, complemented, Sylow
130D4×Q8—1031no—
131#547—1031no—
132#487—1031no—
133#483—1032no—
134#475—1031no—
135#465—1032no—
136#461D10311yescharacteristic, complemented
137#448—1032no—
138C4×SD16—1031no—
139C8×D4—1031no—
140C2×#103—1032no—
141C2×#41—2061no—
142C22×Q8—1032no—
143C2×SD16—1031no—
144C2×SD16C2×D10312yes—
145C22×C8—1032no—
146C4⋊Q8—1031no—
147C4⋊Q8—2061no—
148C4⋊Q8C2×D10311yescharacteristic
149#123—2062no—
150C22⋊Q8—1032no—
151C22⋊Q8—2062no—
152C22⋊Q8—2061no—
153#121—1032no—
154C4×Q8—1031no—
155C4×D4—1031no—
156C4×D4—2061no—
157C4×D4C2×D10311yescharacteristic
158C2×C4⋊C4—1032no—
159#106—1032no—
160#106—1031no—
161C4⋊C8C2×D10311yescharacteristic
162#103—1032no—
163#103—1032no—
164#103—1032no—
165#103—1031no—
166#103C2×D10312yes—
167#102—1031no—
168C22⋊C8—1032no—
169C4×C8—1031no—
170#41—4124no—
171C2×Q8—1031no—
172C2×Q8—4122no—
173C2×Q8—2062no—
174C2×Q8—2064no—
175C2×Q8—2061no—
176C2×Q8C22×D10312yes—
177C2×D4—1031no—
178C2×D4—2062no—
179C2×D4C22×D10311yescharacteristic
180C22×C4—21no—
181C22×C4—1032no—
182C22×C4—2062no—
183C22×C4—2062no—
184SD16—42no—
185SD16—2062no—
186C2×C8—1031no—
187C2×C8—1031no—
188C2×C8—4122no—
189C2×C8C22×D10312yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—1031no—
193C4⋊C4—1031no—
194C4⋊C4—1032no—
195C4⋊C4—2062no—
196C4⋊C4—2062no—
197C4⋊C4—2061no—
198C4⋊C4—2061no—
199C4⋊C4C22×D10311yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—1032no—
202C22⋊C4—2062no—
203C22⋊C4—2062no—
204C42—2061no—
205C42C22×D10311yescharacteristic
206C23—21no—
207C23—1032no—
208Q8—22no—
209Q8—22no—
210Q8—1032no—
211Q8—4124no—
212Q8—4122no—
213Q8—2061no—
214D4—21no—
215D4—21no—
216D4—1034no—
217D4—4124no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—1032no—
222C2×C4—4124no—
223C2×C4—4124no—
224C2×C4—2061no—
225C2×C4—2062no—
226C2×C4—2061no—
227C2×C4—2062no—
228C2×C4—2062no—
229C2×C4D4×D10311yescharacteristic
230C2×C4D4×D10311yescharacteristic
231C2×C4C23×D10311yescharacteristic
232C8—22no—
233C8—2061no—
234C8—2061no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1034no—
239C22—2062no—
240C22—2062no—
241C22C2×D4×D10311yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—4122no—
246C4—4121no—
247C4—2062no—
248C4—2062no—
249C4SD16×D10312yes—
250C4#41×D10311yescharacteristic
251C4#1569306611yescharacteristic
252C2—81no—
253C2—2064no—
254C2#2026843811yescharacteristic
255C2C2×SD16×D10311yescharacteristic
256C2#2026860711yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#461×C103C2
socleC22×C103C22×D4
prime radical#20268446C2
hypercenter#461D103
hypocenterC103#3298
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20268456C2semidirect
3. #20268141C2semidirect
4. #20268120C2semidirect
5. #20268077C2semidirect
6. #20268017C2semidirect
7. #20267928C2semidirect
8. C4×#13110093C2semidirect
9. #20267856C2semidirect
10. #103×D103C2semidirect
11. #461×C103C2semidirect
94. D412SD16semidirect
102. Dic206#41semidirect
129. C103#3298semidirect
136. #461D103semidirect

Maximal subgroups

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

subgroupindex
#202684562
#202684462
#202681412
#202681202
#202680772
#202680172
#202679282
C4×#131100932
#202678562
#103×D1032
#461×C1032
#3298103