Finite Group Databasebeta

← #28371353

Subgroups — #28371353

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×#16812680C211yesmaximal, characteristic
2C22×C4×Dic109C211yesmaximal, characteristic, complemented
3#20294920C212yesmaximal, complemented
4#20294919C212yesmaximal, complemented
5#20294909C212yesmaximal, complemented
6#20294908C212yesmaximal, complemented
7#20294906C211yesmaximal, characteristic
8#20294901C211yesmaximal, characteristic, complemented
9#20294899C211yesmaximal, characteristic, complemented
10#20294887C211yesmaximal, characteristic, complemented
11#386×C109C211yesmaximal, characteristic
12C23×Dic109C2211yescharacteristic
13C23×Dic109C2211yescharacteristic
14#16812680—24no—
15C2×C4⋊Dic109C2212yes—
16C2×#13182008C2212yes—
17C2×#13182008C2211yescharacteristic
18C2×#13182008C2211yescharacteristic
19C2×C22⋊Dic109C2211yescharacteristic
20C2×C22⋊Dic109C2212yes—
21C2×C22⋊Dic109C2211yescharacteristic
22C2×C22⋊Dic109C2211yescharacteristic
23C2×C22⋊Dic109C2211yescharacteristic
24C2×C4×Dic109—24no—
25C2×C4×Dic109C2212yes—
26C2×C4×Dic109C2212yes—
27#16812606C2212yes—
28#16812606C2212yes—
29#16812606C2212yes—
30#16812606C2212yes—
31#16812606C2212yes—
32C22×D4×C109C2211yescharacteristic
33C23×C4×C109C2211yescharacteristic
34C2×C22⋊C4×C109C2212yes—
35C2×C22⋊C4×C109C2211yescharacteristic
36#95×C109C2212yes—
37C22×Dic109—22no—
38C22×Dic109—22no—
39C22×Dic109—22no—
40C22×Dic109—24no—
41C22×Dic109D414yes—
42C22×Dic109C2312yes—
43C22×Dic109C2311yescharacteristic
44C22×Dic109C2311yescharacteristic
45C22×Dic109C2311yescharacteristic
46C22×Dic109C2311yescharacteristic
47C22×Dic109C2312yes—
48C4⋊Dic109—24no—
49#13182008—24no—
50#13182008—22no—
51#13182008—22no—
52C22⋊Dic109—22no—
53C22⋊Dic109—22no—
54C22⋊Dic109—22no—
55C22⋊Dic109—22no—
56C22⋊Dic109—24no—
57C4×Dic109—24no—
58C4×Dic109—24no—
59C24×C109C2311yescharacteristic
60C24×C109C2311yescharacteristic
61C2×D4×C109—22no—
62C2×D4×C109—22no—
63C2×D4×C109—22no—
64C22×C4×C109—22no—
65C22×C4×C109—24no—
66C22×C4×C109C2311yescharacteristic
67C22×C4×C109C2312yes—
68C22×C4×C109C2312yes—
69C22⋊C4×C109—22no—
70C22⋊C4×C109—24no—
71C23×C109—22no—
72C23×C109—22no—
73C23×C109—21no—
74C23×C109—22no—
75C23×C109—21no—
76C23×C109—21no—
77C23×C109—22no—
78C23×C109C2×D412yes—
79C23×C109#4112yes—
80C23×C109#4112yes—
81C23×C109C2411yescharacteristic
82D4×C109—44no—
83C2×C4×C109—28no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—21no—
87C2×C4×C109—22no—
88C2×C4×C109—22no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109#4114yes—
93C2×Dic109—28no—
94C2×Dic109—28no—
95C2×Dic109—22no—
96C2×Dic109—22no—
97C2×Dic109—21no—
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—22no—
107C2×Dic109—22no—
108C2×Dic109C2×D412yes—
109C2×Dic109C2×D412yes—
110C2×Dic109#4114yes—
111C22×C109—22no—
112C22×C109—22no—
113C22×C109—24no—
114C22×C109—24no—
115C22×C109—41no—
116C22×C109—41no—
117C22×C109—41no—
118C22×C109—42no—
119C22×C109—42no—
120C22×C109#12314yes—
121C22×C109C22×D411yescharacteristic
122C22×C109C2×#4112yes—
123C22×C109C2×#4111yescharacteristic
124C22×C109C2×#4112yes—
125C22×C109C2×#4111yescharacteristic
126C4×C109—24no—
127C4×C109—41no—
128C4×C109—42no—
129Dic109—22no—
130Dic109—22no—
131Dic109—24no—
132Dic109—41no—
133Dic109—41no—
134Dic109—42no—
135C2×C109—24no—
136C2×C109—81no—
137C2×C109C2×#12311yescharacteristic
138C2×C109#52511yescharacteristic
139C2×C109#52511yescharacteristic
140C2×C109#52912yes—
141C2×C109#52912yes—
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#386D10911yescharacteristic
152#385—1091no—
153#385—1091no—
154#381—1091no—
155C22×D4C2×D10911yescharacteristic
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⋊C4—1092no—
164C2×C22⋊C4—1091no—
165C2×C22⋊C4—1091no—
166C2×C22⋊C4—1091no—
167C2×C22⋊C4—1091no—
168C2×C22⋊C4—1092no—
169C2×C22⋊C4C2×D10911yescharacteristic
170C2×C22⋊C4C2×D10912yes—
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—
180C24C22×D10911yescharacteristic
181C24C22×D10911yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
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—1091no—
194C22×C4—2184no—
195C22×C4—2182no—
196C22×C4—2182no—
197C22×C4—2182no—
198C22×C4C22×D10912yes—
199C22×C4C22×D10911yescharacteristic
200C22×C4C22×D10912yes—
201C4⋊C4—2182no—
202C4⋊C4—2182no—
203C4⋊C4—2184no—
204C4⋊C4—2184no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—2184no—
208C22⋊C4—2182no—
209C22⋊C4—2182no—
210C22⋊C4—2182no—
211C22⋊C4—2182no—
212C42—2184no—
213C42—2184no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D10912yes—
222C23#1318202812yes—
223C23#1318202912yes—
224C23C23×D10911yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—1092no—
236C2×C4—1092no—
237C2×C4—1094no—
238C2×C4—2181no—
239C2×C4—2181no—
240C2×C4—2182no—
241C2×C4—2182no—
242C2×C4—2182no—
243C2×C4—2182no—
244C2×C4—2182no—
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—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1681268114yes—
264C22C2×D4×D10911yescharacteristic
265C22#41×D10912yes—
266C22C2×#1318202811yescharacteristic
267C22C2×#1318202812yes—
268C22C2×#1318202911yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—4361no—
273C4—4361no—
274C4—4362no—
275C4—2182no—
276C4—2182no—
277C4—2184no—
278C2—24no—
279C2—81no—
280C2C2×#1681268111yescharacteristic
281C2#2029553211yescharacteristic
282C2#2029553411yescharacteristic
283C2#2029555612yes—
284C2#2029555712yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#386×C109C2
prime radicalC22×D4×C109C22
hypercenter#386D109
hypocenterC109#2853
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic109C2semidirect
3. #20294920C2semidirect
4. #20294919C2semidirect
5. #20294909C2semidirect
6. #20294908C2semidirect
8. #20294901C2semidirect
9. #20294899C2semidirect
10. #20294887C2semidirect
143. C109#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#168126802
C22×C4×Dic1092
#202949202
#202949192
#202949092
#202949082
#202949062
#202949012
#202948992
#202948872
#386×C1092
#2853109