Finite Group Databasebeta

← #28366321

Subgroups — #28366321

Order 13952.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16812664C211yesmaximal, characteristic, complemented
2C2×#16812661C211yesmaximal, characteristic
3#20294964C211yesmaximal, characteristic, complemented
4#20294955C212yesmaximal, complemented
5#20294944C211yesmaximal, characteristic
6#20294943C211yesmaximal, characteristic
7#20294939C212yesmaximal, complemented
8#20294873C212yesmaximal, complemented
9#20294873C212yesmaximal, complemented
10#20294871C211yesmaximal, characteristic, complemented
11#400×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C22×D436C2211yescharacteristic
14C22×C4×D109C2212yes—
15#16812664—24no—
16#16812661—24no—
17C2×#13182008C2211yescharacteristic
18C2×#13182008C2211yescharacteristic
19C2×#13182006C2212yes—
20C2×#13182006C2212yes—
21C2×#13182006C2212yes—
22C2×#13182006C2212yes—
23C2×#13182006C2212yes—
24C2×C4×Dic109C2211yescharacteristic
25C2×C4×Dic109C2211yescharacteristic
26C2×C4×Dic109C2212yes—
27C2×C4×Dic109C2211yescharacteristic
28#16812606C2212yes—
29#16812606C2212yes—
30#16812606C2211yescharacteristic
31#16812606C2212yes—
32C2×C4⋊C4×C109C2211yescharacteristic
33C2×C4⋊C4×C109C2211yescharacteristic
34#95×C109C2212yes—
35#95×C109C2211yescharacteristic
36#95×C109C2212yes—
37C23×D109C2312yes—
38C2×Dic218—22no—
39C2×Dic218—22no—
40C2×Dic218—22no—
41C2×D436—24no—
42C2×D436—22no—
43C22×Dic109C2312yes—
44C22×Dic109C2311yescharacteristic
45C22×Dic109C2311yescharacteristic
46C22×Dic109C2311yescharacteristic
47C22×Dic109C2311yescharacteristic
48C2×C4×D109—24no—
49C2×C4×D109—24no—
50C2×C4×D109—24no—
51#13182008—22no—
52#13182008—22no—
53#13182006—24no—
54#13182006—24no—
55#13182006—24no—
56#13182006—24no—
57#13182006—24no—
58C4×Dic109—22no—
59C4×Dic109—22no—
60C4×Dic109—24no—
61C4×Dic109D414yes—
62C22×C4×C109C2312yes—
63C22×C4×C109C2312yes—
64C22×C4×C109C2311yescharacteristic
65C22×C4×C109C2311yescharacteristic
66C22×C4×C109C2311yescharacteristic
67C4⋊C4×C109—22no—
68C4⋊C4×C109—22no—
69C23×C109C2411yescharacteristic
70C22×D109—22no—
71C22×D109—22no—
72C22×D109—22no—
73C22×D109—22no—
74C22×D109—22no—
75C22×D109—22no—
76C22×D109#4114yes—
77C2×C4×C109—21no—
78C2×C4×C109—22no—
79C2×C4×C109—21no—
80C2×C4×C109—21no—
81C2×C4×C109—22no—
82C2×C4×C109—22no—
83C2×C4×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—22no—
87C2×C4×C109—22no—
88C2×C4×C109C2×D412yes—
89C2×C4×C109#4112yes—
90C2×Dic109—21no—
91C2×Dic109—21no—
92C2×Dic109—21no—
93C2×Dic109—21no—
94C2×Dic109—21no—
95C2×Dic109—22no—
96C2×Dic109—22no—
97C2×Dic109—22no—
98C2×Dic109—22no—
99C2×Dic109C2×D412yes—
100C2×Dic109C2×D412yes—
101C2×Dic109#4112yes—
102C2×Dic109#4114yes—
103D436—28no—
104C4×D109—48no—
105Dic218—28no—
106C22×C109C22×D411yescharacteristic
107C22×C109C2×#4112yes—
108C22×C109C2×#4111yescharacteristic
109C22×C109C2×#4112yes—
110C22×C109C2×#4111yescharacteristic
111C2×D109—24no—
112C2×D109—24no—
113C2×D109—24no—
114C2×D109—42no—
115C2×D109—42no—
116C2×D109—42no—
117C2×D109—42no—
118C4×C109—22no—
119C4×C109—22no—
120C4×C109—41no—
121C4×C109—42no—
122C4×C109—42no—
123Dic109—22no—
124Dic109—22no—
125Dic109—41no—
126Dic109—42no—
127Dic109#12414yes—
128C2×C109#52511yescharacteristic
129C2×C109#52511yescharacteristic
130C2×C109C2×#12411yescharacteristic
131C2×C109#52912yes—
132C2×C109#52912yes—
133D109—44no—
134#2398—1091nomaximal, Sylow
135C109#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—1091no—
137C2×C4×D4—1091no—
138#400—1091no—
139#400D10911yescharacteristic, complemented
140#397—1092no—
141#394—1091no—
142#394—1091no—
143#393—1092no—
144C4×C22⋊C4—1092no—
145C4×C22⋊C4—1092no—
146#376—1091no—
147C22×Q8—1091no—
148C22×D4—1091no—
149C23×C4—1092no—
150C4×Q8—2184no—
151C4×D4—2184no—
152C2×C4⋊C4—1091no—
153C2×C4⋊C4—1091no—
154C2×C4⋊C4C2×D10911yescharacteristic
155C2×C4⋊C4C2×D10911yescharacteristic
156C2×C22⋊C4—1092no—
157C2×C22⋊C4—1092no—
158C2×C22⋊C4—1092no—
159C2×C22⋊C4—1092no—
160C2×C22⋊C4—1092no—
161C2×C42—1091no—
162C2×C42—1091no—
163C2×C42—1091no—
164C2×C42—1092no—
165#95—1092no—
166#95—1091no—
167#95—1092no—
168#95—1092no—
169#95C2×D10911yescharacteristic
170#95C2×D10912yes—
171#95C2×D10912yes—
172C24—1092no—
173C2×Q8—2182no—
174C2×Q8—2182no—
175C2×Q8—2182no—
176C2×D4—2184no—
177C2×D4—2182no—
178C22×C4—1091no—
179C22×C4—1091no—
180C22×C4—1091no—
181C22×C4—1092no—
182C22×C4—1091no—
183C22×C4—2184no—
184C22×C4—2184no—
185C22×C4—2184no—
186C22×C4C22×D10912yes—
187C22×C4C22×D10912yes—
188C22×C4C22×D10911yescharacteristic
189C22×C4C22×D10911yescharacteristic
190C22×C4C22×D10911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2182no—
194C4⋊C4—2182no—
195C22⋊C4—2184no—
196C22⋊C4—2184no—
197C22⋊C4—2184no—
198C22⋊C4—2184no—
199C22⋊C4—2184no—
200C42—1094no—
201C42—2184no—
202C42—2182no—
203C42—2182no—
204C23—1094no—
205C23—2182no—
206C23—2182no—
207C23—2182no—
208C23—2182no—
209C23—2182no—
210C23—2182no—
211C23C23×D10911yescharacteristic
212Q8—2188no—
213D4—2188no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—1092no—
226C2×C4—1094no—
227C2×C4—1092no—
228C2×C4—1092no—
229C2×C4—4368no—
230C2×C4—2181no—
231C2×C4—2182no—
232C2×C4—2182no—
233C2×C4—2182no—
234C2×C4—2182no—
235C2×C4—2181no—
236C2×C4—2181no—
237C2×C4—2181no—
238C2×C4—2181no—
239C2×C4D4×D10912yes—
240C2×C4#1318203012yes—
241C22—4362no—
242C22—4362no—
243C22—4362no—
244C22—4362no—
245C22—2184no—
246C22—2184no—
247C22—2184no—
248C22C2×D4×D10911yescharacteristic
249C22#41×D10912yes—
250C22#41×D10911yescharacteristic
251C22#41×D10912yes—
252C22C2×#1318203011yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—1094no—
259C4—4361no—
260C4—4362no—
261C4—2182no—
262C4—2182no—
263C2—4364no—
264C2#2029553411yescharacteristic
265C2#2029553911yescharacteristic
266C2#124×D10911yescharacteristic
267C2#2029555912yes—
268C2#2029556112yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#400×C109C2
prime radicalC22×D436C22
hypercenter#400D109
hypocenterC109#2398
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16812664C2semidirect
3. #20294964C2semidirect
4. #20294955C2semidirect
7. #20294939C2semidirect
8. #20294873C2semidirect
9. #20294873C2semidirect
10. #20294871C2semidirect
11. #400×C109C2semidirect
135. C109#2398semidirect
139. #400D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126642
C2×#168126612
#202949642
#202949552
#202949442
#202949432
#202949392
#202948732
#202948712
#400×C1092
#2398109