Finite Group Databasebeta

← #28373415

Subgroups — #28373415

Order 13952.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20295582C211yesmaximal, characteristic, complemented
2#124×D109C211yesmaximal, characteristic
3#20295387C212yesmaximal, complemented
4#20295384C212yesmaximal, complemented
5#20295383C211yesmaximal, characteristic, complemented
6#20295381C211yesmaximal, characteristic, complemented
7#20295380C211yesmaximal, characteristic, complemented
8#20295033C212yesmaximal, complemented
9#20295029C212yesmaximal
10C4×#13182012C211yesmaximal, characteristic, complemented
11#488×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2211yescharacteristic
13C2×D4×D109C2211yescharacteristic
14C2×#13182012C2212yes—
15#16812698C2212yescomplemented
16#16812691C2211yescharacteristic, complemented
17#16812688C2211yescharacteristic
18#16812687C2211yescharacteristic
19#16812686C2211yescharacteristic
20#16812685—22no—
21#16812677—22no—
22#16812664—22no—
23#16812657C2212yes—
24C22⋊C4×D109—22no—
25C42×D109C2211yescharacteristic
26#16812624C2212yescomplemented
27#16812623C2212yescomplemented
28#16812622C2212yescomplemented
29#16812621C2212yes—
30#16812620C2212yes—
31#16812619C2212yes—
32#16812609C2212yescomplemented
33C4×#11724784C2211yescharacteristic, complemented
34#125×C109C2211yescharacteristic, complemented
35#124×C109C2211yescharacteristic
36#103×C109C2212yescomplemented
37#102×C109C2212yes—
38C4×C8×C109C2211yescharacteristic, complemented
39C23×D109—21no—
40C2×Dic218—22no—
41C2×Dic218—21no—
42C2×Dic218C2311yescharacteristic
43Q8×D109—24no—
44C2×D436C2311yescharacteristic
45C2×C22⋊D109—21no—
46D4×D109—24no—
47C2×C4×D109—21no—
48C2×C4×D109D412yes—
49C2×C4×D109C2311yescharacteristic
50#13182012—24no—
51C2×#11724784C2312yes—
52C4⋊Dic109C2312yes—
53#13182008—22no—
54#13182008—22no—
55C22⋊Dic109—22no—
56#13182006—22no—
57#13182006—22no—
58#13182006—22no—
59C4×Dic109—22no—
60C4×Dic109D412yes—
61C4×Dic109C2311yescharacteristic
62C2×Q8×C109C2311yescharacteristic
63C2×D4×C109C2311yescharacteristic
64C2×C8×C109C2312yes—
65C4⋊C4×C109—22no—
66C4⋊C4×C109C2312yes—
67C22⋊C4×C109—22no—
68C42×C109C2311yescharacteristic
69C23×C109—21no—
70C22×D109—21no—
71C22×D109—44no—
72C22×D109—42no—
73C22×D109C2×D411yescharacteristic
74Q8×C109—22no—
75D4×C109—22no—
76C2×C4×C109—21no—
77C2×C4×C109—22no—
78C2×C4×C109C2×D412yes—
79C2×C4×C109C2411yescharacteristic
80C22⋊D109—42no—
81C2×Dic109—22no—
82C2×Dic109—21no—
83C2×Dic109C2×D412yes—
84C2×Dic109C2×D411yescharacteristic
85D436—22no—
86C4×D109—42no—
87C4×D109—42no—
88C4×D109#4114yes—
89Dic218—22no—
90Dic218—44no—
91Dic218—42no—
92C8×C109—22no—
93#11724784—22no—
94C22×C109—41no—
95C22×C109—42no—
96C22×C109C22×D411yescharacteristic
97C2×D109—22no—
98C2×D109—41no—
99C2×D109—42no—
100C2×D109—84no—
101C2×D109#12412yes—
102C4×C109—22no—
103C4×C109—41no—
104C4×C109—42no—
105C4×C109C2×#4112yes—
106Dic109—22no—
107Dic109—41no—
108Dic109—42no—
109Dic109#12412yes—
110C2×C109—81no—
111C2×C109C2×#12411yescharacteristic
112C2×C109#57512yes—
113D109—22no—
114D109—81no—
115#3140—1091nomaximal, Sylow
116C109#314011yescharacteristic, complemented, Sylow
117#533—1091no—
118C2×#124—1091no—
119#489—1092no—
120#489—1092no—
121#488—1091no—
122#488—1091no—
123#488—1091no—
124#488D10911yescharacteristic, complemented
125#419—1092no—
126#418—1092no—
127C4×M16—1091no—
128C22×Q8—1091no—
129C22×D4—1091no—
130C2×M16—1092no—
131C4⋊Q8—1092no—
132#125—1091no—
133#125C2×D10911yescharacteristic, complemented
134#124—1091no—
135#124—1091no—
136#124—1091no—
137#124—2182no—
138#124C2×D10911yescharacteristic
139C22⋊Q8—2182no—
140C4×Q8—2182no—
141#117—1092no—
142C2×C22⋊C4—2182no—
143C2×C42—1091no—
144#103—1092no—
145#103—1092no—
146#103—1092no—
147#103C2×D10912yescomplemented
148#102—1092no—
149#102—1092no—
150#102—1092no—
151#102C2×D10912yes—
152#97—1092no—
153C4×C8—1091no—
154C4×C8C2×D10911yescharacteristic, complemented
155C24—2181no—
156C2×Q8—1091no—
157C2×Q8—2182no—
158C2×Q8—2181no—
159C2×Q8—2184no—
160C2×Q8C22×D10911yescharacteristic
161C2×D4—1091no—
162C2×D4—2184no—
163C2×D4—2181no—
164C2×D4C22×D10911yescharacteristic
165C22×C4—1091no—
166C22×C4—1092no—
167C22×C4—2181no—
168M16—2184no—
169C2×C8—1092no—
170C2×C8C22×D10912yes—
171C4⋊C4—22no—
172C4⋊C4—1092no—
173C4⋊C4—2182no—
174C4⋊C4—2182no—
175C4⋊C4C22×D10912yes—
176C22⋊C4—22no—
177C22⋊C4—2182no—
178C22⋊C4—2182no—
179C22⋊C4—2182no—
180C22⋊C4—2182no—
181C42—1092no—
182C42—1091no—
183C42—2182no—
184C42C22×D10911yescharacteristic
185C23—21no—
186C23—1091no—
187C23—4362no—
188C23—4364no—
189C23—2181no—
190Q8—22no—
191Q8—4362no—
192Q8—4364no—
193Q8—2182no—
194D4—22no—
195D4—4362no—
196D4—2182no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—1091no—
200C2×C4—1094no—
201C2×C4—1092no—
202C2×C4—4362no—
203C2×C4—4362no—
204C2×C4—2182no—
205C2×C4—2181no—
206C2×C4D4×D10912yes—
207C2×C4C23×D10911yescharacteristic
208C8—22no—
209C8—2182no—
210C22—41no—
211C22—42no—
212C22—8724no—
213C22—1092no—
214C22—4361no—
215C22—4362no—
216C22—2182no—
217C22C2×D4×D10911yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—1092no—
222C4—4361no—
223C4—4362no—
224C4—2182no—
225C4#41×D10912yes—
226C2—81no—
227C2—8721no—
228C2—2182no—
229C2#124×D10911yescharacteristic
230C2#2029582312yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#488×C109C2
socleC22×C109C22×D4
prime radicalC2×D4×D109C22
hypercenter#488D109
hypocenterC109#3140
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20295582C2semidirect
3. #20295387C2semidirect
4. #20295384C2semidirect
5. #20295383C2semidirect
6. #20295381C2semidirect
7. #20295380C2semidirect
8. #20295033C2semidirect
10. C4×#13182012C2semidirect
11. #488×C109C2semidirect
15. #16812698C22semidirect
16. #16812691C22semidirect
26. #16812624C22semidirect
27. #16812623C22semidirect
28. #16812622C22semidirect
32. #16812609C22semidirect
33. C4×#11724784C22semidirect
34. #125×C109C22semidirect
36. #103×C109C22semidirect
38. C4×C8×C109C22semidirect
116. C109#3140semidirect
124. #488D109semidirect
133. #125C2×D109semidirect
147. #103C2×D109semidirect
154. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955822
#124×D1092
#202953872
#202953842
#202953832
#202953812
#202953802
#202950332
#202950292
C4×#131820122
#488×C1092
#3140109