Finite Group Databasebeta

← #28366269

Subgroups — #28366269

Order 13952.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16812661C211yesmaximal, characteristic, complemented
2C2×#16812659C212yesmaximal, complemented
3C2×#16812655C211yesmaximal, characteristic, complemented
4#20294957C211yesmaximal, characteristic, complemented
5#20294950C212yesmaximal, complemented
6#20294944C211yesmaximal, characteristic, complemented
7#20294941C212yesmaximal, complemented
8#20294935C211yesmaximal, characteristic
9#20294877C211yesmaximal, characteristic, complemented
10#20294873C212yesmaximal, complemented
11#397×C109C211yesmaximal, characteristic, complemented
12C22×D436C2211yescharacteristic
13C22×C22⋊D109C2212yes—
14C23×Dic109C2211yescharacteristic
15C22×C4×D109C2212yes—
16#16812661—24no—
17#16812659—28no—
18#16812655—24no—
19C2×#13182008C2212yes—
20C2×#13182008C2211yescharacteristic
21C2×#13182008C2211yescharacteristic
22C2×#13182008C2212yes—
23C2×C22⋊Dic109C2211yescharacteristic
24C2×#13182006C2212yes—
25C2×#13182006C2212yes—
26C2×#13182006C2212yes—
27C2×#13182006C2212yes—
28C2×C4×Dic109C2211yescharacteristic
29C2×C4×Dic109C2212yes—
30C2×C4×Dic109C2211yescharacteristic
31#16812606C2211yescharacteristic
32#16812606C2212yes—
33C2×C4⋊C4×C109C2211yescharacteristic
34C2×C22⋊C4×C109C2211yescharacteristic
35C2×C22⋊C4×C109C2212yes—
36#95×C109C2211yescharacteristic
37#95×C109C2212yes—
38C23×D109C2312yes—
39C2×D436—24no—
40C2×D436—22no—
41C2×C22⋊D109—24no—
42C2×C22⋊D109—24no—
43C2×C22⋊D109—24no—
44C22×Dic109—22no—
45C22×Dic109—22no—
46C22×Dic109D414yescomplemented
47C22×Dic109C2311yescharacteristic
48C22×Dic109C2311yescharacteristic
49C22×Dic109C2311yescharacteristic
50C22×Dic109C2312yes—
51C22×Dic109C2311yescharacteristic
52C2×C4×D109—24no—
53C2×C4×D109—24no—
54C2×C4×D109—24no—
55#13182008—22no—
56#13182008—22no—
57#13182008—24no—
58#13182008—24no—
59C22⋊Dic109—22no—
60#13182006—24no—
61#13182006—24no—
62#13182006—24no—
63#13182006—24no—
64C4×Dic109—22no—
65C4×Dic109—24no—
66C4×Dic109D414yescomplemented
67C24×C109C2311yescharacteristic
68C22×C4×C109C2312yes—
69C22×C4×C109C2311yescharacteristic
70C22×C4×C109C2312yes—
71C22×C4×C109C2311yescharacteristic
72C4⋊C4×C109—22no—
73C22⋊C4×C109—24no—
74C22⋊C4×C109—22no—
75C23×C109—21no—
76C23×C109—21no—
77C23×C109—22no—
78C23×C109—22no—
79C23×C109C2×D412yes—
80C23×C109C2411yescharacteristic
81C22×D109—22no—
82C22×D109—22no—
83C22×D109—22no—
84C22×D109—22no—
85C22×D109—22no—
86C22×D109—22no—
87C22×D109#4114yes—
88C2×C4×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109—22no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109C2×D412yes—
98C22⋊D109—216no—
99C2×Dic109—22no—
100C2×Dic109—21no—
101C2×Dic109—22no—
102C2×Dic109—21no—
103C2×Dic109—22no—
104C2×Dic109—28no—
105C2×Dic109—21no—
106C2×Dic109—22no—
107C2×Dic109C2×D412yes—
108C2×Dic109C2×D412yes—
109C2×Dic109C2×D412yes—
110C2×Dic109C2×D412yes—
111C2×Dic109#4114yes—
112C2×Dic109#4112yes—
113D436—28no—
114C4×D109—48no—
115C22×C109—22no—
116C22×C109—24no—
117C22×C109—41no—
118C22×C109—42no—
119C22×C109—41no—
120C22×C109C22×D411yescharacteristic
121C22×C109C22×D411yescharacteristic
122C22×C109C2×#4111yescharacteristic
123C22×C109C2×#4112yes—
124C22×C109C2×#4112yes—
125C2×D109—24no—
126C2×D109—24no—
127C2×D109—24no—
128C2×D109—42no—
129C2×D109—42no—
130C2×D109—42no—
131C2×D109—42no—
132C4×C109—22no—
133C4×C109—42no—
134C4×C109—42no—
135C4×C109—41no—
136Dic109—22no—
137Dic109—24no—
138Dic109—22no—
139Dic109—41no—
140Dic109#12114yes—
141C2×C109—42no—
142C2×C109C2×#12111yescharacteristic
143C2×C109#52512yes—
144C2×C109#52511yescharacteristic
145C2×C109#52912yes—
146C2×C109#53211yescharacteristic
147D109—44no—
148#2392—1091nomaximal, Sylow
149C109#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1091no—
151C2×C4×D4—1092no—
152C2×#117—1091no—
153#398—1091no—
154#397D10911yescharacteristic, complemented
155#396—1092no—
156#394—1092no—
157#394—1091no—
158#392—1091no—
159C4×C4⋊C4—1091no—
160C4×C22⋊C4—1092no—
161C22×D4—1092no—
162C22×D4—1091no—
163C23×C4—1091no—
164C23×C4—1092no—
165C4×D4—2184no—
166C4×D4—2188no—
167#117—2184no—
168C2×C4⋊C4—1091no—
169C2×C4⋊C4—1091no—
170C2×C4⋊C4—1092no—
171C2×C4⋊C4—1092no—
172C2×C4⋊C4C2×D10911yescharacteristic
173C2×C22⋊C4—1091no—
174C2×C22⋊C4—1092no—
175C2×C22⋊C4—1092no—
176C2×C22⋊C4—1092no—
177C2×C22⋊C4—1092no—
178C2×C22⋊C4C2×D10911yescharacteristic
179C2×C22⋊C4C2×D10912yes—
180C2×C42—1091no—
181C2×C42—1091no—
182C2×C42—1092no—
183#95—1092no—
184#95—1091no—
185#95C2×D10912yes—
186#95C2×D10911yescharacteristic
187C24—1092no—
188C24C22×D10911yescharacteristic
189C2×D4—2184no—
190C2×D4—2184no—
191C2×D4—2184no—
192C2×D4—2184no—
193C2×D4—2182no—
194C22×C4—1091no—
195C22×C4—1094no—
196C22×C4—1092no—
197C22×C4—1091no—
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—2184no—
201C22×C4—2184no—
202C22×C4—2184no—
203C22×C4—2182no—
204C22×C4—2182no—
205C22×C4C22×D10912yes—
206C22×C4C22×D10912yes—
207C22×C4C22×D10911yescharacteristic
208C22×C4C22×D10911yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—2184no—
211C4⋊C4—2182no—
212C4⋊C4—2184no—
213C4⋊C4—2182no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—2184no—
217C22⋊C4—2184no—
218C22⋊C4—2184no—
219C22⋊C4—2184no—
220C22⋊C4—2182no—
221C42—1094no—
222C42—2182no—
223C42—2184no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—1094no—
229C23—2182no—
230C23—2182no—
231C23—2182no—
232C23—2182no—
233C23—2182no—
234C23—2182no—
235C23D4×D10912yes—
236C23C23×D10911yescharacteristic
237D4—2188no—
238D4—21816no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—1092no—
249C2×C4—1092no—
250C2×C4—1092no—
251C2×C4—1092no—
252C2×C4—1092no—
253C2×C4—1094no—
254C2×C4—4368no—
255C2×C4—2188no—
256C2×C4—2181no—
257C2×C4—2182no—
258C2×C4—2182no—
259C2×C4—2181no—
260C2×C4—2182no—
261C2×C4—2181no—
262C2×C4—2182no—
263C2×C4D4×D10912yes—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—4362no—
270C22—4362no—
271C22—4362no—
272C22—4362no—
273C22—2184no—
274C22—2184no—
275C22—2184no—
276C22C2×D4×D10911yescharacteristic
277C22C2×D4×D10911yescharacteristic
278C22#41×D10911yescharacteristic
279C22#41×D10912yes—
280C22#41×D10912yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—1094no—
286C4—4361no—
287C4—2182no—
288C4—2184no—
289C4—2182no—
290C2—42no—
291C2—4364no—
292C2#121×D10911yescharacteristic
293C2#2029553411yescharacteristic
294C2#2029553412yes—
295C2#2029555912yes—
296C2#2029557511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#397×C109C2
prime radical#20294935C2
hypercenter#397D109
hypocenterC109#2392
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16812661C2semidirect
2. C2×#16812659C2semidirect
3. C2×#16812655C2semidirect
4. #20294957C2semidirect
5. #20294950C2semidirect
6. #20294944C2semidirect
7. #20294941C2semidirect
9. #20294877C2semidirect
10. #20294873C2semidirect
11. #397×C109C2semidirect
46. C22×Dic109D4semidirect
66. C4×Dic109D4semidirect
149. C109#2392semidirect
154. #397D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126612
C2×#168126592
C2×#168126552
#202949572
#202949502
#202949442
#202949412
#202949352
#202948772
#202948732
#397×C1092
#2392109