Finite Group Databasebeta

← #28372510

Subgroups — #28372510

Order 13952.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812724C211yesmaximal, characteristic, complemented
2C2×#16812695C211yesmaximal, characteristic, complemented
3#41×Dic109C211yesmaximal, characteristic, complemented
4#20295224C214yesmaximal, complemented
5#20295219C214yesmaximal, complemented
6#20295049C211yesmaximal, characteristic, complemented
7#20295032C211yesmaximal, characteristic, complemented
8#20295031C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
10C22×D436C2211yescharacteristic
11C22×C22⋊D109—22no—
12#16812724—24no—
13C2×#13182016C2212yescomplemented
14C2×#13182014C2212yescomplemented
15C2×#13182013C2211yescharacteristic
16#16812695—21no—
17#16812695—21no—
18#16812695C2214yes—
19Q8×Dic109C2212yescomplemented
20D4×Dic109—22no—
21D4×Dic109C2212yescomplemented
22#16812658—21no—
23#16812658C2211yescharacteristic
24C2×C4×Dic109—21no—
25C2×C4×Dic109C2211yescharacteristic
26#16812628C2214yescomplemented
27#16812621C2214yescomplemented
28#16812620C2214yescomplemented
29C2×#41×C109C2211yescharacteristic
30C22×D4×C109C2211yescharacteristic
31AGL(1,ℤ/8)×C109—24no—
32C2×SD16×C109C2212yescomplemented
33C2×D8×C109C2212yescomplemented
34C2×M16×C109C2211yescharacteristic
35C23×D109—21no—
36C2×D436—22no—
37C2×D436—22no—
38C2×D436—21no—
39C2×D436C2312yes—
40C2×C22⋊D109—24no—
41C2×C22⋊D109—24no—
42C2×C22⋊D109—48no—
43C22×Dic109—22no—
44C22×Dic109D412yes—
45#13182016—24no—
46#13182014—24no—
47#13182013—22no—
48C2×#11724784C2312yes—
49C4⋊Dic109—22no—
50C4⋊Dic109C2312yes—
51C22⋊Dic109—21no—
52C22⋊Dic109—22no—
53C4×Dic109—21no—
54C4×Dic109—22no—
55C4×Dic109D411yescharacteristic
56C4×Dic109D411yescharacteristic
57C4×Dic109C2312yes—
58C24×C109—21no—
59#41×C109—22no—
60#41×C109D414yescomplemented
61C2×Q8×C109C2311yescharacteristic
62C2×D4×C109—21no—
63C2×D4×C109—22no—
64C2×D4×C109—21no—
65C2×D4×C109—22no—
66C2×D4×C109C2312yes—
67C2×D4×C109C2311yescharacteristic
68C22×C4×C109—21no—
69C22×C4×C109C2311yescharacteristic
70SD16×C109—24no—
71D8×C109—24no—
72M16×C109—22no—
73C2×C8×C109C2312yes—
74C23×C109—21no—
75C23×C109—22no—
76C23×C109—44no—
77C23×C109—42no—
78C23×C109C2×D411yescharacteristic
79C22×D109—22no—
80C22×D109—44no—
81C22×D109—42no—
82Q8×C109—21no—
83Q8×C109C2×D412yescomplemented
84D4×C109—24no—
85D4×C109—22no—
86D4×C109—21no—
87D4×C109—22no—
88D4×C109—42no—
89D4×C109C2×D412yescomplemented
90C2×C4×C109—21no—
91C2×C4×C109—24no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2×D412yes—
94C2×C4×C109#4112yes—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—48no—
97C22⋊D109—48no—
98C2×Dic109—24no—
99C2×Dic109—22no—
100C2×Dic109—44no—
101C2×Dic109C2×D412yes—
102C2×Dic109C2×D412yes—
103D436—22no—
104D436—22no—
105D436—42no—
106C8×C109—22no—
107#11724784—22no—
108C22×C109—22no—
109C22×C109—22no—
110C22×C109—42no—
111C22×C109—42no—
112C22×C109—44no—
113C22×C109—84no—
114C22×C109#12112yes—
115C22×C109C22×D411yescharacteristic
116C2×D109—42no—
117C2×D109—44no—
118C2×D109—84no—
119C4×C109—22no—
120C4×C109#12112yes—
121C4×C109C22×D411yescharacteristic
122C4×C109C2×#4111yescharacteristic
123Dic109—22no—
124Dic109—42no—
125Dic109AGL(1,ℤ/8)14yescomplemented
126C2×C109—22no—
127C2×C109—42no—
128C2×C109—82no—
129C2×C109C2×#12111yescharacteristic
130C2×C109C2×AGL(1,ℤ/8)12yes—
131D109—82no—
132#3038—1091nomaximal, Sylow
133C109#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1091no—
135C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
136C2×#127—1091no—
137C4×#41—1091no—
138#460—1094no—
139#459—1094no—
140#423—1091no—
141#418—1091no—
142#418—1091no—
143C2×#41C2×D10911yescharacteristic
144C22×D4—1091no—
145C22×D4—2182no—
146C22×D4C2×D10911yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2184no—
149C2×SD16—1092no—
150C2×SD16C2×D10912yescomplemented
151C2×D8—1092no—
152C2×D8C2×D10912yescomplemented
153C2×M16—1091no—
154C2×M16C2×D10911yescharacteristic
155#127—1094no—
156#127—2181no—
157#127—2181no—
158C4×Q8—1092no—
159C4×D4—1092no—
160C4×D4—2182no—
161#117—1091no—
162#117—2181no—
163C2×C42—1091no—
164C2×C42—2181no—
165C4⋊C8—1094no—
166#102—1094no—
167#102—1094no—
168C24—21no—
169C24—2181no—
170#41—22no—
171#41C22⋊D10914yescomplemented
172C2×Q8C22×D10911yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—1092no—
178C2×D4—4368no—
179C2×D4—2184no—
180C2×D4—2184no—
181C2×D4—2181no—
182C2×D4—2182no—
183C2×D4—2182no—
184C2×D4C22×D10912yes—
185C2×D4C22×D10911yescharacteristic
186C22×C4—21no—
187C22×C4—1092no—
188C22×C4—2182no—
189C22×C4C22×D10911yescharacteristic
190SD16—24no—
191SD16—2184no—
192D8—24no—
193D8—2184no—
194M16—22no—
195M16—2182no—
196C2×C8—1092no—
197C2×C8C22×D10912yes—
198C4⋊C4—1092no—
199C4⋊C4—2182no—
200C22⋊C4—2182no—
201C22⋊C4—2181no—
202C42—1091no—
203C42—1091no—
204C42—1092no—
205C42—2182no—
206C42—2181no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—4364no—
212C23—4362no—
213C23—2182no—
214C23D4×D10911yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D10912yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—4368no—
223D4—4368no—
224D4—4362no—
225D4—2182no—
226D4—2182no—
227D4C2×C22⋊D10912yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—1092no—
231C2×C4—1092no—
232C2×C4—4364no—
233C2×C4—2184no—
234C2×C4—2182no—
235C2×C4D4×D10911yescharacteristic
236C2×C4C2×C22⋊D10912yes—
237C2×C4#1318202812yes—
238C2×C4C23×D10911yescharacteristic
239C8—22no—
240C8—2182no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—8724no—
248C22—4362no—
249C22—4364no—
250C22#1681267112yes—
251C22C2×D4×D10911yescharacteristic
252C4—22no—
253C4—1094no—
254C4—4362no—
255C4—2182no—
256C4#1681267112yes—
257C4C22×C22⋊D10911yescharacteristic
258C4C2×#1318202811yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—8722no—
263C2C2×#1681267111yescharacteristic
264C2AGL(1,ℤ/8)×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3038
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812724C2semidirect
2. C2×#16812695C2semidirect
3. #41×Dic109C2semidirect
4. #20295224C2semidirect
5. #20295219C2semidirect
6. #20295049C2semidirect
7. #20295032C2semidirect
8. #20295031C2semidirect
9. C2×AGL(1,ℤ/8)×C109C2semidirect
13. C2×#13182016C22semidirect
14. C2×#13182014C22semidirect
19. Q8×Dic109C22semidirect
21. D4×Dic109C22semidirect
26. #16812628C22semidirect
27. #16812621C22semidirect
28. #16812620C22semidirect
32. C2×SD16×C109C22semidirect
33. C2×D8×C109C22semidirect
60. #41×C109D4semidirect
83. Q8×C109C2×D4semidirect
89. D4×C109C2×D4semidirect
125. Dic109AGL(1,ℤ/8)semidirect
133. C109#3038semidirect
135. C2×AGL(1,ℤ/8)D109semidirect
150. C2×SD16C2×D109semidirect
152. C2×D8C2×D109semidirect
171. #41C22⋊D109semidirect
216. Q8C2×C22⋊D109semidirect
227. D4C2×C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127242
C2×#168126952
#41×Dic1092
#202952242
#202952192
#202950492
#202950322
#202950312
C2×AGL(1,ℤ/8)×C1092
#3038109