Finite Group Databasebeta

← #28373204

Subgroups — #28373204

Order 13952.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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×Q8×Dic109C211yesmaximal, characteristic, complemented
3#20295359C211yesmaximal, characteristic, complemented
4#20295345C211yesmaximal, characteristic
5#20295331C212yesmaximal, complemented
6#20295250C212yesmaximal, complemented
7#20295165C211yesmaximal, characteristic
8#20295133C212yesmaximal, complemented
9#20295129C212yesmaximal, complemented
10#20295080C211yesmaximal, characteristic, complemented
11#451×C109C211yesmaximal, characteristic, complemented
12C2×#13182016C2212yes—
13#16812694—21no—
14#16812690C2211yescharacteristic
15#16812688C2211yescharacteristic
16#16812683—21no—
17#16812673C2211yescharacteristic
18Q8×Dic109—21no—
19Q8×Dic109—21no—
20Q8×Dic109C2212yes—
21Q8×Dic109C2212yes—
22Q8×Dic109C2211yescharacteristic
23#16812661C2211yescharacteristic
24#16812659—21no—
25#16812657—21no—
26C2×C4⋊Dic109—21no—
27C2×C4⋊Dic109C2211yescharacteristic
28C2×C4×Dic109—21no—
29C2×C4×Dic109C2211yescharacteristic
30#16812632C2212yes—
31#16812631C2212yes—
32#16812628C2212yes—
33#16812623C2212yes—
34#16812623C2212yes—
35#16812621C2212yes—
36C22⋊#11724784C2211yescharacteristic
37#16812609C2212yes—
38C22×Q8×C109C2211yescharacteristic
39C2×Q16×C109C2212yes—
40C22⋊Q8×C109C2211yescharacteristic
41#103×C109C2212yes—
42C22⋊C8×C109C2211yescharacteristic
43C2×D436C2311yescharacteristic
44C2×C22⋊D109—21no—
45C22×Dic109—21no—
46C22×Dic109D411yescharacteristic
47C22×Dic109D411yescharacteristic
48C2×C4×D109—21no—
49#13182016—24no—
50C2×#11724784C2312yes—
51C4⋊Dic109—21no—
52C4⋊Dic109—22no—
53C4⋊Dic109—22no—
54C4⋊Dic109—21no—
55C4⋊Dic109—21no—
56C4⋊Dic109C2312yes—
57C4⋊Dic109C2311yescharacteristic
58#13182008—21no—
59#13182006—21no—
60#13182006—21no—
61#13182006—21no—
62#13182006—21no—
63C4×Dic109—22no—
64C4×Dic109—22no—
65C4×Dic109—21no—
66C4×Dic109D411yescharacteristic
67C4×Dic109D411yescharacteristic
68C4×Dic109C2311yescharacteristic
69C4×Dic109C2311yescharacteristic
70C2×Q8×C109—24no—
71C2×Q8×C109—21no—
72C2×Q8×C109C2311yescharacteristic
73C2×Q8×C109C2312yes—
74C22×C4×C109—21no—
75C22×C4×C109C2311yescharacteristic
76Q16×C109—24no—
77C2×C8×C109C2312yes—
78C4⋊C4×C109—21no—
79C4⋊C4×C109C2311yescharacteristic
80C22⋊C4×C109—21no—
81C23×C109C2×D411yescharacteristic
82C22×D109—21no—
83Q8×C109—24no—
84Q8×C109—22no—
85Q8×C109—22no—
86Q8×C109#4114yes—
87C2×C4×C109—24no—
88C2×C4×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—21no—
91C2×C4×C109—21no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—42no—
95C2×Dic109—22no—
96C2×Dic109—22no—
97C2×Dic109—21no—
98C2×Dic109—41no—
99C2×Dic109—42no—
100C2×Dic109C2×D411yescharacteristic
101C2×Dic109C2×D411yescharacteristic
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104D436—22no—
105C4×D109—42no—
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—24no—
116C4×C109—21no—
117C4×C109—41no—
118C4×C109—41no—
119C4×C109C2×#4112yes—
120Dic109—21no—
121Dic109—21no—
122Dic109—21no—
123Dic109—21no—
124Dic109—41no—
125Dic109—42no—
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#484—1091no—
136#481—1091no—
137#479—1092no—
138#464—1092no—
139#451D10911yescharacteristic, complemented
140#448—1091no—
141#441—1092no—
142#440—1092no—
143#432—1091no—
144C22×Q8C2×D10911yescharacteristic
145C2×Q16C2×D10912yes—
146C2×SD16—1092no—
147#126—2181no—
148#125—1091no—
149#124—1091no—
150#123—2181no—
151C22⋊Q8C2×D10911yescharacteristic
152#121—1091no—
153C4×Q8—1092no—
154C4×Q8—1092no—
155C4×Q8—1091no—
156C4×Q8—2181no—
157C4×Q8—2181no—
158C4×D4—1091no—
159C4×D4—2181no—
160#117—2181no—
161C2×C4⋊C4—1091no—
162C2×C4⋊C4—2181no—
163C2×C42—1091no—
164C2×C42—2181no—
165#107—1092no—
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—24no—
176C2×Q8—21no—
177C2×Q8C22×D10911yescharacteristic
178C2×Q8C22×D10912yes—
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—24no—
188SD16—2184no—
189C2×C8—1092no—
190C2×C8C22×D10912yes—
191C4⋊C4—21no—
192C4⋊C4—1092no—
193C4⋊C4—1091no—
194C4⋊C4—2181no—
195C4⋊C4—2182no—
196C4⋊C4—2182no—
197C4⋊C4—2181no—
198C4⋊C4—2181no—
199C4⋊C4—2181no—
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—24no—
216Q8—22no—
217Q8—22no—
218Q8#1318202814yes—
219D4—4362no—
220D4—2182no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—1091no—
227C2×C4—1091no—
228C2×C4—1091no—
229C2×C4—1091no—
230C2×C4—4362no—
231C2×C4—4362no—
232C2×C4—4361no—
233C2×C4—2182no—
234C2×C4—2181no—
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—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—4361no—
252C4—4362no—
253C4—2181no—
254C4—2181no—
255C4—2181no—
256C4—2181no—
257C4C2×#1318202812yes—
258C2—22no—
259C2—8721no—
260C2#2029553211yescharacteristic
261C2C2×#1681273111yescharacteristic
262C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#451×C109C2
socleC22×C109C22×D4
prime radical#16812673C22
hypercenter#451D109
hypocenterC109#3117
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic109C2semidirect
3. #20295359C2semidirect
5. #20295331C2semidirect
6. #20295250C2semidirect
8. #20295133C2semidirect
9. #20295129C2semidirect
10. #20295080C2semidirect
11. #451×C109C2semidirect
132. C109#3117semidirect
139. #451D109semidirect

Maximal subgroups

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

subgroupindex
#202955622
C2×Q8×Dic1092
#202953592
#202953452
#202953312
#202952502
#202951652
#202951332
#202951292
#202950802
#451×C1092
#3117109