Finite Group Databasebeta

← #28366317

Subgroups — #28366317

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×#16812659C211yesmaximal, characteristic
3#20294967C212yesmaximal, complemented
4#20294952C211yesmaximal, characteristic, complemented
5#20294941C212yesmaximal
6#20294939C211yesmaximal, characteristic, complemented
7#20294938C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic109C212yesmaximal, complemented
9#20294873C212yesmaximal, complemented
10#20294871C211yesmaximal, characteristic, complemented
11#397×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C22×C22⋊D109C2211yescharacteristic
14C23×Dic109C2211yescharacteristic
15C22×C4×D109C2211yescharacteristic
16#16812664—24no—
17#16812659—24no—
18C2×#13182008C2212yes—
19C2×#13182008C2211yescharacteristic
20C2×C22⋊Dic109C2212yes—
21C2×#13182006C2212yes—
22C2×#13182006C2211yescharacteristic
23C2×#13182006C2212yes—
24C2×C4×Dic109C2212yes—
25C2×C4×Dic109C2212yes—
26C2×C4×Dic109C2211yescharacteristic
27#16812606C2211yescharacteristic
28#16812606C2212yes—
29#16812606C2212yes—
30#16812606C2212yes—
31#16812606C2212yes—
32C2×C4⋊C4×C109C2211yescharacteristic
33C2×C22⋊C4×C109C2211yescharacteristic
34C2×C22⋊C4×C109C2212yes—
35#95×C109C2211yescharacteristic
36#95×C109C2212yes—
37C23×D109C2311yescharacteristic
38C2×Dic218—22no—
39C2×Dic218—24no—
40C2×C22⋊D109—22no—
41C2×C22⋊D109—22no—
42C2×C22⋊D109—22no—
43C22×Dic109—22no—
44C22×Dic109—24no—
45C22×Dic109C2312yes—
46C22×Dic109C2311yescharacteristic
47C22×Dic109C2311yescharacteristic
48C22×Dic109C2312yes—
49C22×Dic109C2311yescharacteristic
50C2×C4×D109—22no—
51C2×C4×D109—24no—
52#13182008—22no—
53#13182008—24no—
54C22⋊Dic109—24no—
55#13182006—24no—
56#13182006—22no—
57#13182006—24no—
58C4×Dic109—24no—
59C4×Dic109—24no—
60C4×Dic109D414yes—
61C24×C109C2311yescharacteristic
62C22×C4×C109C2312yes—
63C22×C4×C109C2311yescharacteristic
64C22×C4×C109C2312yes—
65C22×C4×C109C2311yescharacteristic
66C4⋊C4×C109—22no—
67C22⋊C4×C109—24no—
68C22⋊C4×C109—22no—
69C23×C109—21no—
70C23×C109—22no—
71C23×C109—22no—
72C23×C109—21no—
73C23×C109#4112yes—
74C23×C109C2411yescharacteristic
75C22×D109—21no—
76C22×D109—22no—
77C22×D109—22no—
78C22×D109—21no—
79C22×D109#4112yes—
80C2×C4×C109—22no—
81C2×C4×C109—21no—
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×C109—22no—
89C2×C4×C109C2×D412yes—
90C22⋊D109—28no—
91C2×Dic109—22no—
92C2×Dic109—21no—
93C2×Dic109—21no—
94C2×Dic109—22no—
95C2×Dic109—21no—
96C2×Dic109—22no—
97C2×Dic109—22no—
98C2×Dic109—22no—
99C2×Dic109—22no—
100C2×Dic109—44no—
101C2×Dic109C2×D412yes—
102C2×Dic109C2×D412yes—
103C2×Dic109#4114yes—
104C2×Dic109#4114yes—
105C4×D109—44no—
106Dic218—28no—
107C22×C109—22no—
108C22×C109—24no—
109C22×C109—41no—
110C22×C109—41no—
111C22×C109—42no—
112C22×C109C22×D411yescharacteristic
113C22×C109C2×#4111yescharacteristic
114C22×C109C2×#4112yes—
115C22×C109C2×#4112yes—
116C22×C109C2×#4111yescharacteristic
117C2×D109—22no—
118C2×D109—24no—
119C2×D109—41no—
120C2×D109—41no—
121C2×D109—42no—
122C4×C109—22no—
123C4×C109—42no—
124C4×C109—41no—
125C4×C109—42no—
126Dic109—24no—
127Dic109—22no—
128Dic109—41no—
129Dic109—42no—
130Dic109#12414yes—
131C2×C109—42no—
132C2×C109#52511yescharacteristic
133C2×C109#52511yescharacteristic
134C2×C109C2×#12411yescharacteristic
135C2×C109#52912yes—
136C2×C109#52912yes—
137D109—42no—
138#2398—1091nomaximal, Sylow
139C109#239811yescharacteristic, complemented, Sylow
140C2×C4×Q8—1091no—
141C2×C4×D4—1091no—
142#400—1092no—
143#397—1091no—
144#397D10911yescharacteristic, complemented
145#394—1092no—
146#393—1091no—
147#393—1091no—
148C4×C22⋊C4—1092no—
149C4×C22⋊C4—1092no—
150#376—1091no—
151C22×Q8—1091no—
152C22×D4—1091no—
153C23×C4—1091no—
154C23×C4—1091no—
155C4×Q8—2184no—
156C4×D4—2184no—
157C2×C4⋊C4—1091no—
158C2×C4⋊C4—1092no—
159C2×C4⋊C4C2×D10911yescharacteristic
160C2×C22⋊C4—1092no—
161C2×C22⋊C4—1091no—
162C2×C22⋊C4—1092no—
163C2×C22⋊C4—1092no—
164C2×C22⋊C4C2×D10911yescharacteristic
165C2×C22⋊C4C2×D10912yes—
166C2×C42—1091no—
167C2×C42—1092no—
168C2×C42—1092no—
169#95—1091no—
170#95—1092no—
171#95—1092no—
172#95—1092no—
173#95—1092no—
174#95C2×D10911yescharacteristic
175#95C2×D10912yes—
176C24—1091no—
177C24C22×D10911yescharacteristic
178C2×Q8—2184no—
179C2×Q8—2182no—
180C2×D4—2182no—
181C2×D4—2182no—
182C2×D4—2182no—
183C22×C4—1091no—
184C22×C4—1092no—
185C22×C4—1091no—
186C22×C4—1092no—
187C22×C4—1091no—
188C22×C4—2182no—
189C22×C4—2184no—
190C22×C4—2182no—
191C22×C4—2184no—
192C22×C4C22×D10912yes—
193C22×C4C22×D10911yescharacteristic
194C22×C4C22×D10912yes—
195C22×C4C22×D10911yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—2182no—
198C4⋊C4—2184no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—2184no—
202C22⋊C4—2184no—
203C22⋊C4—2184no—
204C22⋊C4—2182no—
205C42—1094no—
206C42—2184no—
207C42—2184no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—1092no—
213C23—2181no—
214C23—2182no—
215C23—2182no—
216C23—2181no—
217C23#1318202812yes—
218C23C23×D10911yescharacteristic
219Q8—2188no—
220D4—2188no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—1092no—
231C2×C4—1094no—
232C2×C4—1092no—
233C2×C4—1094no—
234C2×C4—4364no—
235C2×C4—4364no—
236C2×C4—2181no—
237C2×C4—2182no—
238C2×C4—2182no—
239C2×C4—2182no—
240C2×C4—2182no—
241C2×C4—2182no—
242C2×C4—2181no—
243C2×C4—2182no—
244C2×C4—2181no—
245C2×C4D4×D10912yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—4361no—
252C22—4361no—
253C22—4362no—
254C22—2182no—
255C22—2184no—
256C22C2×D4×D10911yescharacteristic
257C22#41×D10912yes—
258C22#41×D10911yescharacteristic
259C22#41×D10912yes—
260C22C2×#1318202811yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—1094no—
266C4—4361no—
267C4—4362no—
268C4—2184no—
269C4—2182no—
270C2—42no—
271C2—4362no—
272C2#2029553411yescharacteristic
273C2#2029553611yescharacteristic
274C2#124×D10911yescharacteristic
275C2#2029555312yes—
276C2#2029555912yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#397×C109C2
prime radicalC22×C22⋊D109C22
hypercenter#397D109
hypocenterC109#2398
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16812664C2semidirect
3. #20294967C2semidirect
4. #20294952C2semidirect
6. #20294939C2semidirect
7. #20294938C2semidirect
8. C22⋊C4×Dic109C2semidirect
9. #20294873C2semidirect
10. #20294871C2semidirect
11. #397×C109C2semidirect
139. C109#2398semidirect
144. #397D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126642
C2×#168126592
#202949672
#202949522
#202949412
#202949392
#202949382
C22⋊C4×Dic1092
#202948732
#202948712
#397×C1092
#2398109