Finite Group Databasebeta

← #28371357

Subgroups — #28371357

Order 13952.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812683C211yesmaximal, characteristic
2C22×C4×Dic109C211yesmaximal, characteristic, complemented
3#20294916C212yesmaximal, complemented
4#20294915C212yesmaximal, complemented
5#20294913C212yesmaximal, complemented
6#20294912C212yesmaximal, complemented
7#20294905C211yesmaximal, characteristic
8#20294903C211yesmaximal, characteristic
9#20294898C211yesmaximal, characteristic, complemented
10#20294887C211yesmaximal, characteristic, complemented
11#385×C109C211yesmaximal, characteristic, complemented
12C22×C22⋊D109C2211yescharacteristic
13C23×Dic109C2211yescharacteristic
14C23×Dic109C2211yescharacteristic
15#16812683—24no—
16C2×C4⋊Dic109C2211yescharacteristic
17C2×C4⋊Dic109C2211yescharacteristic
18C2×#13182008C2212yes—
19C2×C22⋊Dic109C2211yescharacteristic
20C2×#13182006C2211yescharacteristic
21C2×#13182006C2212yes—
22C2×#13182006C2212yes—
23C2×#13182006C2211yescharacteristic
24C2×C4×Dic109—24no—
25C2×C4×Dic109C2212yes—
26C2×C4×Dic109C2212yes—
27#16812606C2212yes—
28#16812606C2212yes—
29#16812606C2212yes—
30#16812606C2212yes—
31#16812606C2212yes—
32C23×C4×C109C2211yescharacteristic
33C2×C4⋊C4×C109C2212yes—
34C2×C22⋊C4×C109C2211yescharacteristic
35C2×C22⋊C4×C109C2211yescharacteristic
36#95×C109C2212yes—
37C23×D109C2311yescharacteristic
38C2×C22⋊D109—22no—
39C2×C22⋊D109—22no—
40C2×C22⋊D109—22no—
41C22×Dic109—22no—
42C22×Dic109—22no—
43C22×Dic109—22no—
44C22×Dic109—24no—
45C22×Dic109D414yes—
46C22×Dic109C2312yes—
47C22×Dic109C2311yescharacteristic
48C22×Dic109C2311yescharacteristic
49C22×Dic109C2311yescharacteristic
50C22×Dic109C2312yes—
51C4⋊Dic109—22no—
52C4⋊Dic109—22no—
53#13182008—24no—
54C22⋊Dic109—22no—
55#13182006—24no—
56#13182006—22no—
57#13182006—22no—
58#13182006—24no—
59C4×Dic109—24no—
60C4×Dic109—24no—
61C24×C109C2311yescharacteristic
62C22×C4×C109—22no—
63C22×C4×C109—24no—
64C22×C4×C109C2311yescharacteristic
65C22×C4×C109C2312yes—
66C22×C4×C109C2311yescharacteristic
67C22×C4×C109C2312yes—
68C4⋊C4×C109—24no—
69C22⋊C4×C109—22no—
70C22⋊C4×C109—22no—
71C23×C109—22no—
72C23×C109—22no—
73C23×C109C2×D412yes—
74C23×C109#4112yes—
75C23×C109#4112yes—
76C23×C109C2411yescharacteristic
77C22×D109—21no—
78C22×D109—22no—
79C22×D109—22no—
80C22×D109—21no—
81C22×D109—21no—
82C2×C4×C109—28no—
83C2×C4×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109—21no—
86C2×C4×C109—21no—
87C2×C4×C109—22no—
88C2×C4×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109#4114yes—
93C22⋊D109—44no—
94C2×Dic109—28no—
95C2×Dic109—28no—
96C2×Dic109—22no—
97C2×Dic109—22no—
98C2×Dic109—21no—
99C2×Dic109—21no—
100C2×Dic109—21no—
101C2×Dic109—21no—
102C2×Dic109—21no—
103C2×Dic109—22no—
104C2×Dic109—22no—
105C2×Dic109—22no—
106C2×Dic109—21no—
107C2×Dic109—21no—
108C2×Dic109C2×D412yes—
109C2×Dic109C2×D412yes—
110C2×Dic109#4114yes—
111C22×C109—22no—
112C22×C109—22no—
113C22×C109—24no—
114C22×C109—24no—
115C22×C109#12314yes—
116C22×C109C22×D411yescharacteristic
117C22×C109C2×#4112yes—
118C22×C109C2×#4112yes—
119C22×C109C2×#4111yescharacteristic
120C22×C109C2×#4111yescharacteristic
121C2×D109—41no—
122C2×D109—41no—
123C2×D109—41no—
124C2×D109—42no—
125C2×D109—42no—
126C4×C109—24no—
127C4×C109—41no—
128C4×C109—41no—
129C4×C109—42no—
130Dic109—22no—
131Dic109—22no—
132Dic109—24no—
133Dic109—41no—
134Dic109—42no—
135C2×C109—24no—
136C2×C109C2×#12311yescharacteristic
137C2×C109#52511yescharacteristic
138C2×C109#52511yescharacteristic
139C2×C109#52912yes—
140C2×C109#52912yes—
141D109—81no—
142#2853—1091nomaximal, Sylow
143C109#285311yescharacteristic, complemented, Sylow
144C2×#123—1091no—
145C22×C42—1091no—
146#388—1092no—
147#388—1092no—
148#387—1092no—
149#387—1092no—
150#386—1091no—
151#386—1091no—
152#385—1091no—
153#385D10911yescharacteristic, complemented
154#381—1091no—
155C22×D4—1091no—
156C23×C4—1091no—
157C23×C4—1091no—
158C23×C4C2×D10911yescharacteristic
159#123—2184no—
160C2×C4⋊C4—1091no—
161C2×C4⋊C4—1091no—
162C2×C4⋊C4—1092no—
163C2×C4⋊C4C2×D10912yes—
164C2×C22⋊C4—1091no—
165C2×C22⋊C4—1091no—
166C2×C22⋊C4—1091no—
167C2×C22⋊C4—1092no—
168C2×C22⋊C4—1092no—
169C2×C22⋊C4C2×D10911yescharacteristic
170C2×C22⋊C4C2×D10911yescharacteristic
171C2×C42—1092no—
172C2×C42—1092no—
173C2×C42—2184no—
174#95—1092no—
175#95—1092no—
176#95—1092no—
177#95—1092no—
178#95—1092no—
179#95C2×D10912yes—
180C24—1091no—
181C24C22×D10911yescharacteristic
182C2×D4—2182no—
183C2×D4—2182no—
184C2×D4—2182no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—1094no—
188C22×C4—1091no—
189C22×C4—1091no—
190C22×C4—1092no—
191C22×C4—1092no—
192C22×C4—1091no—
193C22×C4—2184no—
194C22×C4—2182no—
195C22×C4—2182no—
196C22×C4—2182no—
197C22×C4C22×D10912yes—
198C22×C4C22×D10911yescharacteristic
199C22×C4C22×D10911yescharacteristic
200C22×C4C22×D10912yes—
201C4⋊C4—24no—
202C4⋊C4—2184no—
203C4⋊C4—2182no—
204C4⋊C4—2182no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2184no—
208C22⋊C4—2182no—
209C22⋊C4—2182no—
210C22⋊C4—2182no—
211C22⋊C4—2184no—
212C42—2184no—
213C42—2184no—
214C23—22no—
215C23—22no—
216C23—2181no—
217C23—2182no—
218C23—2182no—
219C23—2181no—
220C23—2181no—
221C23D4×D10912yes—
222C23#1318202912yes—
223C23#1318203012yes—
224C23C23×D10911yescharacteristic
225D4—4364no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—1092no—
237C2×C4—1092no—
238C2×C4—1094no—
239C2×C4—2181no—
240C2×C4—2182no—
241C2×C4—2182no—
242C2×C4—2181no—
243C2×C4—2182no—
244C2×C4—2181no—
245C2×C4—2188no—
246C2×C4—2188no—
247C2×C4—2182no—
248C2×C4—2182no—
249C2×C4—2181no—
250C2×C4—2181no—
251C2×C4—2181no—
252C2×C4—2181no—
253C2×C4#1318202814yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—4361no—
259C22—4361no—
260C22—4361no—
261C22—4362no—
262C22—4362no—
263C22#1681268414yes—
264C22C2×D4×D10911yescharacteristic
265C22#41×D10912yes—
266C22C2×#1318202812yes—
267C22C2×#1318202911yescharacteristic
268C22C2×#1318203011yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—4361no—
274C4—4362no—
275C4—2182no—
276C4—2182no—
277C4—2184no—
278C2—24no—
279C2—8721no—
280C2C2×#1681268411yescharacteristic
281C2#2029553211yescharacteristic
282C2#2029553411yescharacteristic
283C2#2029555612yes—
284C2#2029556212yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#385×C109C2
prime radicalC22×C22⋊D109C22
hypercenter#385D109
hypocenterC109#2853
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic109C2semidirect
3. #20294916C2semidirect
4. #20294915C2semidirect
5. #20294913C2semidirect
6. #20294912C2semidirect
9. #20294898C2semidirect
10. #20294887C2semidirect
11. #385×C109C2semidirect
143. C109#2853semidirect
153. #385D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126832
C22×C4×Dic1092
#202949162
#202949152
#202949132
#202949122
#202949052
#202949032
#202948982
#202948872
#385×C1092
#2853109