Finite Group Databasebeta

← #28371491

Subgroups — #28371491

Order 13952.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812745C211yesmaximal, characteristic
2#20295500C212yesmaximal, complemented
3#20295000C211yesmaximal, characteristic, complemented
4#20294999C212yesmaximal, complemented
5#99×D109C218yesmaximal, complemented
6#410×C109C211yesmaximal, characteristic, complemented
7#16812745—24no—
8#16812745—24no—
9C2×#13182030C412yescomplemented
10C2×#13182029C412yescomplemented
11C2×#13182029C2211yescharacteristic
12C22×D436C412yescomplemented
13C22×D436C2211yescharacteristic
14C2×D4×D109C412yescomplemented
15C2×D4×D109C2214yes—
16C4×D436—24no—
17#16812661—24no—
18#16812658C2212yescomplemented
19C22⋊C4×D109C2218yescomplemented
20#16812614C2214yescomplemented
21#16812613C2218yescomplemented
22C2×#41×C109C2211yescharacteristic
23#117×C109C2212yescomplemented
24#99×C109C2214yescomplemented
25C23×D109C2×C414yescomplemented
26C23×D109C2312yes—
27#13182030—28no—
28#13182029—22no—
29#13182029—22no—
30#13182029—24no—
31#13182029—24no—
32C2×Dic218C2×C411yescharacteristic, complemented
33C2×D436—24no—
34C2×D436—24no—
35C2×D436C2×C412yes—
36C2×D436C2×C412yes—
37C2×D436C2×C414yescomplemented
38C2×D436C2×C411yescharacteristic, complemented
39C2×D436D414yes—
40C2×D436D414yes—
41C2×C22⋊D109C2×C414yescomplemented
42C2×C22⋊D109C2312yes—
43D4×D109—216no—
44D4×D109—28no—
45C2×C4×D109—28no—
46C2×C4×D109C2×C412yescomplemented
47C2×C4×D109C2×C414yescomplemented
48C4⋊Dic109—22no—
49C22⋊Dic109C2314yes—
50#13182006—28no—
51C4×Dic109—22no—
52#41×C109—22no—
53#41×C109—22no—
54C2×Q8×C109C2×C411yescharacteristic, complemented
55C2×D4×C109C2×C411yescharacteristic, complemented
56C2×D4×C109C2312yes—
57C22×C4×C109C2×C412yescomplemented
58C22×C4×C109C2311yescharacteristic
59C4⋊C4×C109—22no—
60C22⋊C4×C109C2314yes—
61C42×C109—22no—
62C23×C109C22×C412yes—
63C23×C109C2411yescharacteristic
64C22×D109—28no—
65C22×D109—82no—
66C22×D109—84no—
67C22×D109C22×C412yes—
68C22×D109C22×C414yes—
69C22×D109C2×D418yes—
70Q8×C109—22no—
71D4×C109—24no—
72D4×C109—22no—
73C2×C4×C109—22no—
74C2×C4×C109—22no—
75C2×C4×C109—24no—
76C2×C4×C109C22×C412yes—
77C2×C4×C109C22×C411yescharacteristic
78C2×C4×C109C22×C411yescharacteristic
79C2×C4×C109C2×D412yes—
80C2×C4×C109C2×D412yes—
81C22⋊D109—24no—
82C22⋊D109—28no—
83C2×Dic109—24no—
84C2×Dic109C22×C412yes—
85D436—28no—
86D436—21no—
87D436—21no—
88D436C22⋊C418yes—
89D436C22⋊C414yes—
90D436C22⋊C414yes—
91C4×D109—28no—
92C4×D109—24no—
93C4×D109—88no—
94Dic218—21no—
95Dic218—21no—
96C22×C109—22no—
97C22×C109—41no—
98C22×C109—42no—
99C22×C109C23×C411yescharacteristic
100C22×C109C22×D412yes—
101C2×D109—22no—
102C2×D109—48no—
103C2×D109—44no—
104C2×D109—42no—
105C2×D109—88no—
106C2×D109C2×C22⋊C418yes—
107C4×C109—22no—
108C4×C109—44no—
109C4×C109C2×C22⋊C411yescharacteristic
110C4×C109C2×C22⋊C411yescharacteristic
111C4×C109C2×C22⋊C412yes—
112Dic109—22no—
113Dic109—44no—
114C2×C109—22no—
115C2×C109—21no—
116C2×C109—42no—
117C2×C109C22×C22⋊C411yescharacteristic
118D109—28no—
119D109—42no—
120#2883—1091nomaximal, Sylow
121C109#288311yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1091no—
123#518—1092no—
124#410—1091no—
125#410—1092no—
126#410D10911yescharacteristic, complemented
127C2×#99—1098no—
128H(5,𝔽2)—2184no—
129H(5,𝔽2)—2184no—
130C2×#41—1092no—
131C2×#41—1092no—
132C2×#41—1091no—
133C2×#41C2×D10911yescharacteristic
134C22×D4—1092no—
135C22×D4—1094no—
136C22×D4—1092no—
137C22×D4—1091no—
138C4×D4—2184no—
139C4×D4—2184no—
140#117—1092no—
141#117C2×D10912yescomplemented
142C2×C22⋊C4—1098no—
143#99—1094no—
144#99—1098no—
145#99C2×D10914yescomplemented
146C24—1092no—
147C24—1094no—
148#41—22no—
149#41—22no—
150#41—2184no—
151#41—2184no—
152#41—2188no—
153#41—2182no—
154#41—2182no—
155C2×Q8—1091no—
156C2×Q8C4×D10911yescharacteristic, complemented
157C2×D4—1094no—
158C2×D4—1092no—
159C2×D4—1092no—
160C2×D4—1092no—
161C2×D4—1094no—
162C2×D4—1091no—
163C2×D4—1094no—
164C2×D4—1094no—
165C2×D4—2184no—
166C2×D4—2184no—
167C2×D4—2188no—
168C2×D4—21816no—
169C2×D4C4×D10911yescharacteristic, complemented
170C2×D4C22×D10912yes—
171C22×C4—1094no—
172C22×C4—1092no—
173C22×C4—2188no—
174C22×C4C4×D10912yescomplemented
175C22×C4C22×D10911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—2182no—
178C22⋊C4—1094no—
179C22⋊C4—2188no—
180C22⋊C4C22×D10914yes—
181C42—22no—
182C42—2182no—
183C23—8724no—
184C23—8722no—
185C23—1094no—
186C23—1098no—
187C23—1092no—
188C23—2188no—
189C23C2×C4×D10912yes—
190C23C23×D10911yescharacteristic
191Q8—22no—
192Q8—2181no—
193Q8—2181no—
194D4—24no—
195D4—22no—
196D4—1094no—
197D4—1094no—
198D4—1098no—
199D4—2188no—
200D4—2188no—
201D4—2184no—
202D4—2181no—
203D4—2181no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—8728no—
208C2×C4—1092no—
209C2×C4—2188no—
210C2×C4—2184no—
211C2×C4—2184no—
212C2×C4C2×C4×D10912yes—
213C2×C4C2×C4×D10911yescharacteristic
214C2×C4C2×C4×D10911yescharacteristic
215C2×C4D4×D10912yes—
216C2×C4D4×D10912yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—8728no—
221C22—1098no—
222C22—4362no—
223C22—4368no—
224C22—4364no—
225C22—2182no—
226C22C22×C4×D10911yescharacteristic
227C22C2×D4×D10912yes—
228C4—22no—
229C4—44no—
230C4—4364no—
231C4—2182no—
232C4C22⋊C4×D10911yescharacteristic
233C4C22⋊C4×D10911yescharacteristic
234C4C22⋊C4×D10912yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—4362no—
239C2—2188no—
240C2C2×C22⋊C4×D10911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D109
commutatorC22×C109C23×C4
FrattiniC23C23×D109
Fitting#410×C109C2
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812745C2
cosocleC23×C109C24
hypercenter#410D109
hypocenterC109#2883
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20295500C2semidirect
3. #20295000C2semidirect
4. #20294999C2semidirect
5. #99×D109C2semidirect
6. #410×C109C2semidirect
9. C2×#13182030C4semidirect
10. C2×#13182029C4semidirect
12. C22×D436C4semidirect
14. C2×D4×D109C4semidirect
18. #16812658C22semidirect
19. C22⋊C4×D109C22semidirect
20. #16812614C22semidirect
21. #16812613C22semidirect
23. #117×C109C22semidirect
24. #99×C109C22semidirect
25. C23×D109C2×C4semidirect
32. C2×Dic218C2×C4semidirect
37. C2×D436C2×C4semidirect
38. C2×D436C2×C4semidirect
41. C2×C22⋊D109C2×C4semidirect
46. C2×C4×D109C2×C4semidirect
47. C2×C4×D109C2×C4semidirect
54. C2×Q8×C109C2×C4semidirect
55. C2×D4×C109C2×C4semidirect
57. C22×C4×C109C2×C4semidirect
121. C109#2883semidirect
126. #410D109semidirect
141. #117C2×D109semidirect
145. #99C2×D109semidirect
156. C2×Q8C4×D109semidirect
169. C2×D4C4×D109semidirect
174. C22×C4C4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127452
#202955002
#202950002
#202949992
#99×D1092
#410×C1092
#2883109