Finite Group Databasebeta

← #28373206

Subgroups — #28373206

Order 13952.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20295562C211yesmaximal, characteristic
2C2×#16812664C211yesmaximal, characteristic, complemented
3#20295344C211yesmaximal, characteristic
4#20295332C212yesmaximal, complemented
5#20295251C212yesmaximal, complemented
6#20295181C211yesmaximal, characteristic, complemented
7#20295167C211yesmaximal, characteristic
8#20295130C212yesmaximal, complemented
9#20295125C212yesmaximal, complemented
10#20295080C211yesmaximal, characteristic, complemented
11#484×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C2×#13182019C2212yes—
14C2×#13182018C2212yes—
15#16812694—21no—
16#16812690C2211yescharacteristic
17#16812688C2211yescharacteristic
18#16812683—21no—
19#16812673C2211yescharacteristic
20Q8×Dic109C2211yescharacteristic
21#16812664—21no—
22#16812664—21no—
23#16812664C2212yes—
24#16812664C2212yes—
25#16812661C2211yescharacteristic
26#16812659—21no—
27#16812657—21no—
28C2×#13182008—21no—
29C2×C4×Dic109—21no—
30C2×C4×Dic109C2211yescharacteristic
31#16812630C2212yes—
32#16812628C2212yes—
33#16812624C2212yes—
34#16812622C2212yes—
35#16812619C2212yes—
36C22⋊#11724784C2211yescharacteristic
37#16812609C2212yes—
38C22⋊Q8×C109C2211yescharacteristic
39C2×C4⋊C4×C109C2211yescharacteristic
40#107×C109C2212yes—
41#103×C109C2212yes—
42C22⋊C8×C109C2211yescharacteristic
43C2×Dic218—21no—
44C2×Dic218—24no—
45C2×Dic218C2312yes—
46C2×D436C2311yescharacteristic
47C2×C22⋊D109—21no—
48C22×Dic109—21no—
49C22×Dic109D411yescharacteristic
50C22×Dic109D411yescharacteristic
51C2×C4×D109—21no—
52#13182019—24no—
53#13182018—24no—
54C2×#11724784C2312yes—
55C4⋊Dic109—21no—
56C4⋊Dic109—21no—
57C4⋊Dic109C2311yescharacteristic
58#13182008—22no—
59#13182008—22no—
60#13182008—21no—
61#13182006—21no—
62#13182006—21no—
63#13182006—21no—
64#13182006—21no—
65C4×Dic109—22no—
66C4×Dic109—22no—
67C4×Dic109—21no—
68C4×Dic109D411yescharacteristic
69C4×Dic109D411yescharacteristic
70C4×Dic109C2311yescharacteristic
71C4×Dic109C2311yescharacteristic
72C2×Q8×C109C2311yescharacteristic
73C22×C4×C109—21no—
74C22×C4×C109C2311yescharacteristic
75C2×C8×C109C2312yes—
76C4⋊C4×C109—21no—
77C4⋊C4×C109—21no—
78C4⋊C4×C109C2312yes—
79C4⋊C4×C109C2311yescharacteristic
80C22⋊C4×C109—21no—
81C23×C109C2×D411yescharacteristic
82C22×D109—21no—
83Q8×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109—21no—
86C2×C4×C109—21no—
87C2×C4×C109—22no—
88C2×C4×C109—42no—
89C2×C4×C109C2×D411yescharacteristic
90C2×C4×C109C2411yescharacteristic
91C22⋊D109—42no—
92C2×Dic109—24no—
93C2×Dic109—22no—
94C2×Dic109—22no—
95C2×Dic109—21no—
96C2×Dic109—41no—
97C2×Dic109C2×D411yescharacteristic
98C2×Dic109C2×D411yescharacteristic
99C2×Dic109C2×D411yescharacteristic
100C2×Dic109C2×D411yescharacteristic
101D436—22no—
102C4×D109—42no—
103Dic218—24no—
104Dic218—22no—
105Dic218#4114yes—
106C8×C109—22no—
107#11724784—22no—
108C22×C109—21no—
109C22×C109—21no—
110C22×C109#13712yes—
111C22×C109C22×D411yescharacteristic
112C2×D109—41no—
113C2×D109—41no—
114C2×D109—41no—
115C4×C109—21no—
116C4×C109—41no—
117C4×C109—41no—
118C4×C109—42no—
119C4×C109C2×#4112yes—
120Dic109—24no—
121Dic109—21no—
122Dic109—21no—
123Dic109—21no—
124Dic109—21no—
125Dic109—41no—
126C2×C109—22no—
127C2×C109#52511yescharacteristic
128C2×C109C2×#13711yescharacteristic
129C2×C109#57511yescharacteristic
130D109—81no—
131#3117—1091nomaximal, Sylow
132C109#311711yescharacteristic, complemented, Sylow
133#529—1091no—
134C2×C4×Q8—1091no—
135#484D10911yescharacteristic, complemented
136#481—1091no—
137#479—1092no—
138#464—1092no—
139#451—1091no—
140#448—1091no—
141#441—1092no—
142#440—1092no—
143#432—1091no—
144C22×Q8—1091no—
145C2×Q16—1092no—
146C2×SD16—1092no—
147#126—2181no—
148#125—1091no—
149#124—1091no—
150#123—2181no—
151C22⋊Q8C2×D10911yescharacteristic
152#121—1091no—
153C4×Q8—1091no—
154C4×Q8—1092no—
155C4×Q8—1092no—
156C4×Q8—2181no—
157C4×Q8—2181no—
158C4×D4—1091no—
159C4×D4—2181no—
160#117—2181no—
161C2×C4⋊C4—2181no—
162C2×C4⋊C4C2×D10911yescharacteristic
163C2×C42—1091no—
164C2×C42—2181no—
165#107C2×D10912yes—
166#106—1092no—
167C4⋊C8—1092no—
168#103—1092no—
169#103—1092no—
170#103C2×D10912yes—
171#102—1092no—
172C22⋊C8—1091no—
173C22⋊C8C2×D10911yescharacteristic
174#97—1092no—
175C2×Q8—1092no—
176C2×Q8—2184no—
177C2×Q8—2181no—
178C2×Q8C22×D10911yescharacteristic
179C2×D4—1091no—
180C2×D4—2181no—
181C22×C4—21no—
182C22×C4—1091no—
183C22×C4—1091no—
184C22×C4—2181no—
185C22×C4—2181no—
186C22×C4C22×D10911yescharacteristic
187Q16—2184no—
188SD16—2184no—
189C2×C8—1092no—
190C2×C8C22×D10912yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—1091no—
194C4⋊C4—2182no—
195C4⋊C4—2182no—
196C4⋊C4—2181no—
197C4⋊C4—2181no—
198C4⋊C4—2181no—
199C4⋊C4C22×D10912yes—
200C4⋊C4C22×D10911yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2181no—
203C22⋊C4—2181no—
204C22⋊C4—2181no—
205C22⋊C4—2181no—
206C42—1091no—
207C42—1091no—
208C42—1091no—
209C42—1091no—
210C42—2182no—
211C42—2182no—
212C42—2181no—
213C23—2181no—
214C23D4×D10911yescharacteristic
215Q8—22no—
216Q8—1094no—
217Q8—2184no—
218Q8—2182no—
219D4—4362no—
220D4—2182no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—1091no—
227C2×C4—1091no—
228C2×C4—1091no—
229C2×C4—1091no—
230C2×C4—4362no—
231C2×C4—4361no—
232C2×C4—2181no—
233C2×C4—2184no—
234C2×C4—2182no—
235C2×C4—2182no—
236C2×C4D4×D10911yescharacteristic
237C2×C4C23×D10911yescharacteristic
238C8—22no—
239C8—2182no—
240C22—21no—
241C22—21no—
242C22—4361no—
243C22—4361no—
244C22—4361no—
245C22#1681273112yes—
246C22C2×D4×D10911yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—4361no—
252C4—2184no—
253C4—2181no—
254C4—2181no—
255C4—2181no—
256C4—2181no—
257C4#41×D10912yes—
258C2—22no—
259C2—8721no—
260C2#2029553411yescharacteristic
261C2C2×#1681273111yescharacteristic
262C2#2029582211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#484×C109C2
socleC22×C109C22×D4
prime radical#16812673C22
hypercenter#484D109
hypocenterC109#3117
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#16812664C2semidirect
4. #20295332C2semidirect
5. #20295251C2semidirect
6. #20295181C2semidirect
8. #20295130C2semidirect
9. #20295125C2semidirect
10. #20295080C2semidirect
11. #484×C109C2semidirect
132. C109#3117semidirect
135. #484D109semidirect

Maximal subgroups

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

subgroupindex
#202955622
C2×#168126642
#202953442
#202953322
#202952512
#202951812
#202951672
#202951302
#202951252
#202950802
#484×C1092
#3117109