Finite Group Databasebeta

← #28372511

Subgroups — #28372511

Order 13952.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16812727C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic, complemented
3C2×#16812695C211yesmaximal, characteristic, complemented
4#20295494C211yesmaximal, characteristic, complemented
5#20295225C214yesmaximal, complemented
6#20295220C214yesmaximal, complemented
7#20295049C211yesmaximal, characteristic, complemented
8#20295030C211yesmaximal, characteristic, complemented
9#418×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11C22×D436C2211yescharacteristic
12C22×C22⋊D109—22no—
13#16812727—24no—
14#16812725—24no—
15C2×#13182018C2212yescomplemented
16C2×#13182017C2212yescomplemented
17C2×D872C2212yescomplemented
18C2×#13182014C2212yescomplemented
19C2×#13182013C2211yescharacteristic
20#16812695—21no—
21#16812695—21no—
22#16812695C2214yes—
23#16812664C2212yescomplemented
24#16812661C2212yescomplemented
25#16812659—22no—
26#16812656—21no—
27C2×C4×Dic109C2211yescharacteristic
28C42×D109—21no—
29#16812628C2214yescomplemented
30#16812619C2214yescomplemented
31C22×D4×C109C2211yescharacteristic
32C2×M16×C109C2211yescharacteristic
33#117×C109C2211yescharacteristic
34#102×C109C2214yescomplemented
35C23×D109—21no—
36#13182029—22no—
37#13182029D414yescomplemented
38C2×Dic218C2311yescharacteristic
39C2×D436—22no—
40C2×D436—22no—
41C2×D436—21no—
42C2×D436C2312yes—
43C2×D436C2311yescharacteristic
44C2×C22⋊D109—21no—
45C2×C22⋊D109—24no—
46C2×C22⋊D109—24no—
47C2×C22⋊D109—48no—
48C22×Dic109D412yes—
49C2×C4×D109—21no—
50C2×C4×D109—22no—
51#13182018—24no—
52#13182017—24no—
53D872—24no—
54#13182014—24no—
55#13182013—22no—
56C2×#11724784C2312yes—
57#13182008—22no—
58#13182006—22no—
59C4×Dic109—22no—
60C4×Dic109D411yescharacteristic
61C4×Dic109D411yescharacteristic
62C4×Dic109C2312yes—
63C24×C109—21no—
64C2×D4×C109—22no—
65C2×D4×C109—21no—
66C2×D4×C109—22no—
67C2×D4×C109C2312yes—
68C22×C4×C109C2311yescharacteristic
69M16×C109—22no—
70C2×C8×C109C2312yes—
71C4⋊C4×C109C2312yes—
72C22⋊C4×C109—21no—
73C42×C109—21no—
74C23×C109—22no—
75C23×C109—44no—
76C23×C109—42no—
77C23×C109C2×D411yescharacteristic
78C22×D109—21no—
79C22×D109—22no—
80C22×D109—44no—
81C22×D109—42no—
82D4×C109—22no—
83D4×C109—22no—
84D4×C109—42no—
85C2×C4×C109—22no—
86C2×C4×C109C2×D411yescharacteristic
87C2×C4×C109C2×D412yes—
88C2×C4×C109#4112yes—
89C2×C4×C109C2411yescharacteristic
90C22⋊D109—24no—
91C22⋊D109—48no—
92C22⋊D109—48no—
93C2×Dic109—21no—
94C2×Dic109—24no—
95C2×Dic109C2×D412yes—
96C2×Dic109C2×D412yes—
97D436—22no—
98D436—21no—
99D436—22no—
100D436—42no—
101D436C2×D412yescomplemented
102C4×D109—24no—
103C4×D109—44no—
104Dic218—21no—
105Dic218C2×D412yescomplemented
106C8×C109—22no—
107#11724784—22no—
108C22×C109—22no—
109C22×C109—42no—
110C22×C109—44no—
111C22×C109—84no—
112C22×C109#12112yes—
113C22×C109C22×D411yescharacteristic
114C2×D109—22no—
115C2×D109—42no—
116C2×D109—42no—
117C2×D109—44no—
118C2×D109—84no—
119C4×C109—42no—
120C4×C109#12112yes—
121C4×C109C22×D411yescharacteristic
122C4×C109C2×#4111yescharacteristic
123Dic109—22no—
124Dic109—22no—
125Dic109AGL(1,ℤ/8)14yescomplemented
126C2×C109—22no—
127C2×C109—82no—
128C2×C109C2×#12111yescharacteristic
129C2×C109C2×AGL(1,ℤ/8)12yes—
130D109—42no—
131D109—82no—
132#3038—1091nomaximal, Sylow
133C109#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1091no—
135C2×AGL(1,ℤ/8)—1091no—
136C2×#127—1091no—
137C4×#41—1091no—
138#460—1094no—
139#459—1094no—
140#423—1091no—
141#418—1091no—
142#418D10911yescharacteristic, complemented
143C2×#41—1091no—
144C22×D4—1091no—
145C22×D4—2182no—
146C22×D4C2×D10911yescharacteristic
147AGL(1,ℤ/8)—2184no—
148AGL(1,ℤ/8)—2184no—
149C2×SD16—1092no—
150C2×SD16—1092no—
151C2×D8—1092no—
152C2×D8—1092no—
153C2×M16—1091no—
154C2×M16C2×D10911yescharacteristic
155#127—1094no—
156#127—2181no—
157#127—2181no—
158C4×Q8—1092no—
159C4×D4—1092no—
160C4×D4—2182no—
161#117—2181no—
162#117C2×D10911yescharacteristic
163C2×C42—1091no—
164C2×C42—2181no—
165C4⋊C8—1094no—
166#102—1094no—
167#102C2×D10914yescomplemented
168C24—21no—
169C24—2181no—
170#41—1094no—
171#41—2182no—
172C2×Q8—1091no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—1091no—
177C2×D4—1092no—
178C2×D4—4368no—
179C2×D4—2184no—
180C2×D4—2184no—
181C2×D4—2181no—
182C2×D4—2181no—
183C2×D4—2182no—
184C2×D4—2182no—
185C2×D4C22×D10912yes—
186C22×C4—1092no—
187C22×C4—2182no—
188C22×C4—2181no—
189C22×C4C22×D10911yescharacteristic
190SD16—2184no—
191SD16—2184no—
192D8—2184no—
193D8—2184no—
194M16—22no—
195M16—2182no—
196C2×C8—1092no—
197C2×C8C22×D10912yes—
198C4⋊C4—2182no—
199C4⋊C4C22×D10912yes—
200C22⋊C4—21no—
201C22⋊C4—2182no—
202C42—21no—
203C42—1091no—
204C42—1091no—
205C42—1092no—
206C42—2182no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—4364no—
211C23—4362no—
212C23—2182no—
213C23—2181no—
214C23D4×D10911yescharacteristic
215Q8—1092no—
216Q8—2181no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—1092no—
221D4—4368no—
222D4—4368no—
223D4—4362no—
224D4—2184no—
225D4—2182no—
226D4—2182no—
227D4—2181no—
228C2×C4—22no—
229C2×C4—1092no—
230C2×C4—1092no—
231C2×C4—4364no—
232C2×C4—2184no—
233C2×C4—2181no—
234C2×C4—2184no—
235C2×C4D4×D10911yescharacteristic
236C2×C4D4×D10912yes—
237C2×C4#1318202912yes—
238C2×C4C23×D10911yescharacteristic
239C8—22no—
240C8—2182no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—8724no—
246C22—4362no—
247C22—4364no—
248C22—4362no—
249C22—2182no—
250C22#1681267212yes—
251C22C2×D4×D10911yescharacteristic
252C4—42no—
253C4—1094no—
254C4—2182no—
255C4—2182no—
256C4#1681267212yes—
257C4C2×D4×D10911yescharacteristic
258C4C2×#1318202911yescharacteristic
259C2—22no—
260C2—82no—
261C2—8722no—
262C2—4362no—
263C2C2×#1681267211yescharacteristic
264C2AGL(1,ℤ/8)×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#418×C109C2
socleC22×C109C22×D4
hypercenter#418D109
hypocenterC109#3038
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812727C2semidirect
2. C2×#16812725C2semidirect
3. C2×#16812695C2semidirect
4. #20295494C2semidirect
5. #20295225C2semidirect
6. #20295220C2semidirect
7. #20295049C2semidirect
8. #20295030C2semidirect
9. #418×C109C2semidirect
15. C2×#13182018C22semidirect
16. C2×#13182017C22semidirect
17. C2×D872C22semidirect
18. C2×#13182014C22semidirect
23. #16812664C22semidirect
24. #16812661C22semidirect
29. #16812628C22semidirect
30. #16812619C22semidirect
34. #102×C109C22semidirect
37. #13182029D4semidirect
101. D436C2×D4semidirect
105. Dic218C2×D4semidirect
125. Dic109AGL(1,ℤ/8)semidirect
133. C109#3038semidirect
142. #418D109semidirect
167. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×#168127252
C2×#168126952
#202954942
#202952252
#202952202
#202950492
#202950302
#418×C1092
#3038109