Finite Group Databasebeta

← #28372308

Subgroups — #28372308

Order 13952.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#13182016C211yesmaximal, characteristic, complemented
3#20295600C211yesmaximal, characteristic, complemented
4#20295187C214yesmaximal, complemented
5#20295175C214yesmaximal, complemented
6#20295032C211yesmaximal, characteristic, complemented
7#20295022C211yesmaximal, characteristic, complemented
8#20294994C211yesmaximal, characteristic, complemented
9C2×#137×C109C211yesmaximal, characteristic, complemented
10#16812745—24no—
11#16812745—24no—
12C2×#13182030C2212yes—
13C2×#13182029—21no—
14C2×#13182029C2211yescharacteristic
15C22×D436—21no—
16C22×D436C2211yescharacteristic
17C2×D4×D109—22no—
18C2×D4×D109C2212yes—
19C2×#13182016—24no—
20C2×#13182016C2214yescomplemented
21C22×#11724784C2211yescharacteristic, complemented
22#16812697—21no—
23#16812688—21no—
24#16812682—22no—
25#16812675C2214yescomplemented
26#16812658C2211yescharacteristic, complemented
27#16812623C2212yescomplemented
28#16812621C2214yescomplemented
29#16812620C2212yescomplemented
30#16812611C2214yescomplemented
31C2×#41×C109C2211yescharacteristic
32C22×Q8×C109C2211yescharacteristic, complemented
33#137×C109—24no—
34C2×Q16×C109C2212yescomplemented
35C2×SD16×C109C2212yescomplemented
36C2×M16×C109C2211yescharacteristic, complemented
37C23×D109—21no—
38C23×D109—22no—
39#13182030—24no—
40#13182030—24no—
41#13182029—28no—
42#13182029—22no—
43#13182029—22no—
44C2×Dic218D411yescharacteristic
45C2×D436—28no—
46C2×D436—22no—
47C2×D436—21no—
48C2×D436—22no—
49C2×D436D411yescharacteristic
50C2×D436D414yes—
51C2×D436C2312yes—
52C2×C22⋊D109—22no—
53C2×C22⋊D109D412yes—
54D4×D109—216no—
55D4×D109—24no—
56D4×D109—24no—
57C2×C4×D109—22no—
58C2×C4×D109C2312yes—
59#13182016—28no—
60C2×#11724784—22no—
61C2×#11724784C2312yes—
62C4⋊Dic109C2312yes—
63#13182008—24no—
64C22⋊Dic109—21no—
65#13182006—24no—
66C4×Dic109—21no—
67#41×C109—22no—
68#41×C109D414yes—
69C2×Q8×C109—21no—
70C2×Q8×C109—22no—
71C2×Q8×C109—22no—
72C2×Q8×C109C2312yes—
73C2×Q8×C109C2311yescharacteristic
74C2×D4×C109—21no—
75C2×D4×C109C2311yescharacteristic
76C22×C4×C109—21no—
77C22×C4×C109—21no—
78C22×C4×C109C2311yescharacteristic
79Q16×C109—24no—
80SD16×C109—24no—
81M16×C109—22no—
82C2×C8×C109C2312yes—
83C23×C109—21no—
84C23×C109C2×D411yescharacteristic
85C22×D109—22no—
86C22×D109—28no—
87C22×D109—24no—
88C22×D109—84no—
89C22×D109—82no—
90C22×D109C2×D412yes—
91Q8×C109—21no—
92Q8×C109—22no—
93Q8×C109—22no—
94Q8×C109—42no—
95Q8×C109C2×D412yes—
96D4×C109—24no—
97D4×C109—21no—
98D4×C109C2×D412yes—
99C2×C4×C109—21no—
100C2×C4×C109—24no—
101C2×C4×C109—22no—
102C2×C4×C109—42no—
103C2×C4×C109C2×D411yescharacteristic
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2411yescharacteristic
107C22⋊D109—28no—
108C22⋊D109—24no—
109C2×Dic109—22no—
110C2×Dic109C2×D412yes—
111D436—28no—
112D436—22no—
113D436—24no—
114D436—22no—
115D436C2×D414yes—
116C4×D109—28no—
117C4×D109—22no—
118C4×D109—22no—
119Dic218—22no—
120C8×C109—22no—
121#11724784—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—42no—
125C22×C109#12012yes—
126C22×C109C22×D411yescharacteristic
127C2×D109—22no—
128C2×D109—24no—
129C2×D109—24no—
130C2×D109—44no—
131C2×D109—84no—
132C2×D109—88no—
133C4×C109—22no—
134C4×C109—42no—
135C4×C109#12012yes—
136C4×C109C22×D411yescharacteristic
137C4×C109C22×D411yescharacteristic
138Dic109—22no—
139Dic109—42no—
140C2×C109—22no—
141C2×C109—42no—
142C2×C109C2×#12011yescharacteristic
143C2×C109#57712yes—
144D109—42no—
145D109—44no—
146#3009—1091nomaximal, Sylow
147C109#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1091no—
149C2×#137D10911yescharacteristic, complemented
150C22×SD16—1091no—
151#536—1091no—
152#452—1094no—
153#450—1094no—
154#418—1091no—
155#416—1091no—
156#408—1091no—
157H(5,𝔽2)—2184no—
158H(5,𝔽2)—2184no—
159C2×#41—1091no—
160C2×#41—1092no—
161C2×#41—2181no—
162C2×#41C2×D10911yescharacteristic
163C22×Q8C2×D10911yescharacteristic, complemented
164C22×D4—1092no—
165C22×D4—1091no—
166C22×D4—2181no—
167C22×D4—2182no—
168#137—24no—
169C2×Q16C2×D10912yescomplemented
170C2×SD16—1094no—
171C2×SD16—2184no—
172C2×SD16C2×D10912yescomplemented
173C2×M16C2×D10911yescharacteristic, complemented
174C22×C8—1091no—
175C4⋊Q8—2181no—
176#124—2181no—
177#123—2182no—
178C22⋊Q8—1094no—
179#117—1091no—
180#103—1092no—
181#102—1092no—
182#102—1094no—
183C22⋊C8—1094no—
184C24—2182no—
185C24—2181no—
186#41—22no—
187#41—2188no—
188#41—2184no—
189#41—2184no—
190#41—2182no—
191#41—2182no—
192#41C22⋊D10914yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1091no—
197C2×Q8C22×D10912yes—
198C2×Q8C22×D10911yescharacteristic
199C2×D4—21no—
200C2×D4—1092no—
201C2×D4—1094no—
202C2×D4—1091no—
203C2×D4—1092no—
204C2×D4—2182no—
205C2×D4—2188no—
206C2×D4—2182no—
207C2×D4—2184no—
208C2×D4—21816no—
209C2×D4—2184no—
210C2×D4—2181no—
211C2×D4—2182no—
212C2×D4C22×D10911yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—1092no—
216C22×C4—2182no—
217C22×C4C22×D10911yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2188no—
221M16—22no—
222C2×C8—1092no—
223C2×C8—2182no—
224C2×C8C22×D10912yes—
225C4⋊C4—1092no—
226C4⋊C4—2184no—
227C22⋊C4—2184no—
228C22⋊C4—2181no—
229C42—2181no—
230C23—21no—
231C23—8722no—
232C23—8724no—
233C23—1092no—
234C23—2188no—
235C23—2184no—
236C23—2182no—
237C23D4×D10911yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2182no—
243Q8C2×C22⋊D10912yes—
244D4—24no—
245D4—21no—
246D4—1094no—
247D4—2188no—
248D4—2188no—
249D4—2184no—
250D4—2184no—
251D4—2182no—
252D4—2182no—
253D4C2×C22⋊D10912yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—1092no—
259C2×C4—2188no—
260C2×C4—2182no—
261C2×C4—2182no—
262C2×C4—2182no—
263C2×C4D4×D10911yescharacteristic
264C2×C4D4×D10912yes—
265C2×C4C2×C22⋊D10912yes—
266C2×C4C23×D10911yescharacteristic
267C8—22no—
268C8—2182no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—8724no—
273C22—8728no—
274C22—4364no—
275C22—2184no—
276C22—2182no—
277C22—2184no—
278C22#1681266712yes—
279C22C2×D4×D10911yescharacteristic
280C4—22no—
281C4—42no—
282C4—4362no—
283C4—2182no—
284C4#1681266712yes—
285C4C2×D4×D10911yescharacteristic
286C4C22×C22⋊D10911yescharacteristic
287C2—22no—
288C2—42no—
289C2—4364no—
290C2—4362no—
291C2C2×#1681266711yescharacteristic
292C2#2029583512yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#137×C109C2
socleC22×C109C22×D4
prime radicalC2×#16812745C2
hypercenterC2×#137D109
hypocenterC109#3009
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#13182016C2semidirect
3. #20295600C2semidirect
4. #20295187C2semidirect
5. #20295175C2semidirect
6. #20295032C2semidirect
7. #20295022C2semidirect
8. #20294994C2semidirect
9. C2×#137×C109C2semidirect
20. C2×#13182016C22semidirect
21. C22×#11724784C22semidirect
25. #16812675C22semidirect
26. #16812658C22semidirect
27. #16812623C22semidirect
28. #16812621C22semidirect
29. #16812620C22semidirect
30. #16812611C22semidirect
32. C22×Q8×C109C22semidirect
34. C2×Q16×C109C22semidirect
35. C2×SD16×C109C22semidirect
36. C2×M16×C109C22semidirect
147. C109#3009semidirect
149. C2×#137D109semidirect
163. C22×Q8C2×D109semidirect
169. C2×Q16C2×D109semidirect
172. C2×SD16C2×D109semidirect
173. C2×M16C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127452
C22×#131820162
#202956002
#202951872
#202951752
#202950322
#202950222
#202949942
C2×#137×C1092
#3009109