Finite Group Databasebeta

← #28372396

Subgroups — #28372396

Order 13952.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#20295824C212yesmaximal, complemented
2#20295600C211yesmaximal, characteristic, complemented
3#20295209C214yesmaximal, complemented
4#20295199C214yesmaximal, complemented
5#20295043C212yesmaximal, complemented
6C2×#16812618C211yesmaximal, characteristic, complemented
7C2×2−1+4×C109C211yesmaximal, characteristic, complemented
8C2×#13182029C2211yescharacteristic
9#16812730—22no—
10#16812729C2214yescomplemented
11#16812725—22no—
12#16812724C2214yescomplemented
13#16812719—24no—
14C2×#13182013C2212yes—
15#16812697C2212yescomplemented
16#16812688C2212yescomplemented
17#16812682—22no—
18#16812675—22no—
19#16812658C2211yescharacteristic
20#16812626C2218yescomplemented
21#16812618C2214yescomplemented
222−1+4×C109—24no—
232−1+4×C109—22no—
242−1+4×C109C2214yes—
25C2×#41×C109—22no—
26C2×#41×C109—22no—
27C2×#41×C109C2212yes—
28C22×Q8×C109—22no—
29C22×Q8×C109C2211yescharacteristic
30#13182029—22no—
31#13182029—22no—
32C2×Dic218C2311yescharacteristic
33C2×D436C2311yescharacteristic
34C2×C22⋊D109—21no—
35C2×C4×D109—21no—
36#13182019—24no—
37#13182017—24no—
38#13182016—24no—
39#13182014—24no—
40#13182013—22no—
41#13182013C2314yes—
42C2×#11724784—22no—
43C4⋊Dic109—21no—
44#13182008—24no—
45C22⋊Dic109—21no—
46#13182006—24no—
47C4×Dic109C2312yes—
48#41×C109—28no—
49#41×C109—28no—
50#41×C109—28no—
51#41×C109—28no—
52#41×C109—22no—
53#41×C109—22no—
54#41×C109D414yes—
55#41×C109C2314yes—
56C2×Q8×C109—22no—
57C2×Q8×C109—28no—
58C2×Q8×C109—28no—
59C2×Q8×C109—22no—
60C2×Q8×C109—22no—
61C2×Q8×C109D412yes—
62C2×Q8×C109D412yes—
63C2×Q8×C109C2312yes—
64C2×D4×C109—22no—
65C2×D4×C109—22no—
66C2×D4×C109D412yes—
67C22×C4×C109—24no—
68C22×C4×C109—22no—
69C22×C4×C109D412yes—
70C22×C4×C109C2311yescharacteristic
71C23×C109—22no—
72C23×C109C2×D411yescharacteristic
73C22×D109—21no—
74Q8×C109—28no—
75Q8×C109—22no—
76Q8×C109—24no—
77Q8×C109—24no—
78Q8×C109C2×D414yes—
79D4×C109—28no—
80D4×C109—24no—
81D4×C109—24no—
82D4×C109—22no—
83D4×C109C2×D414yes—
84C2×C4×C109—216no—
85C2×C4×C109—22no—
86C2×C4×C109—22no—
87C2×C4×C109—22no—
88C2×C4×C109—24no—
89C2×C4×C109—24no—
90C2×C4×C109C2×D414yes—
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2×D412yes—
93C2×C4×C109C2×D412yes—
94C2×C4×C109C2411yescharacteristic
95C22⋊D109—41no—
96C22⋊D109—41no—
97C2×Dic109—22no—
98C2×Dic109—21no—
99D436—22no—
100C4×D109—42no—
101Dic218—22no—
102#11724784—24no—
103C22×C109—24no—
104C22×C109—41no—
105C22×C109—44no—
106C22×C109#12011yescharacteristic
107C22×C109#12011yescharacteristic
108C22×C109C22×D411yescharacteristic
109C2×D109—42no—
110C2×D109—41no—
111C4×C109—24no—
112C4×C109—24no—
113C4×C109#12012yes—
114C4×C109C22×D412yes—
115Dic109—42no—
116Dic109—41no—
117C2×C109—21no—
118C2×C109—21no—
119C2×C109—21no—
120C2×C109—44no—
121C2×C109C2×#12011yescharacteristic
122D109—81no—
123#3020—1091nomaximal, Sylow
124C109#302011yescharacteristic, complemented, Sylow
125C2×2−1+4D10911yescharacteristic, complemented
126#575—1092no—
127#536—1091no—
128#456—1094no—
129#454—1094no—
130#421—1092no—
131C2×#101—1091no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D10914yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1091no—
138C2×#41C2×D10912yes—
139C22×Q8—22no—
140C22×Q8C2×D10911yescharacteristic
141#137—1094no—
142#137—2182no—
143AGL(1,ℤ/8)—1094no—
144AGL(1,ℤ/8)—2182no—
145#135—2184no—
146C2×M16—1092no—
147C4⋊Q8—1092no—
148#124—1092no—
149#123—2182no—
150C22⋊Q8—2182no—
151#117—1091no—
152#104—1098no—
153#101—1094no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—2182no—
161#41—2182no—
162#41C22⋊D10914yes—
163#41C22×D10914yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1091no—
170C2×Q8C22⋊D10912yes—
171C2×Q8C22⋊D10912yes—
172C2×Q8C22×D10912yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—1091no—
176C2×D4—2181no—
177C2×D4C22⋊D10912yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—2181no—
181C22×C4C22⋊D10912yes—
182C22×C4C22×D10911yescharacteristic
183Q16—2184no—
184SD16—2184no—
185SD16—2184no—
186D8—2184no—
187M16—1094no—
188M16—2182no—
189C2×C8—2182no—
190C4⋊C4—2184no—
191C4⋊C4—2181no—
192C22⋊C4—2181no—
193C22⋊C4—2184no—
194C42—1092no—
195C23—22no—
196C23—2181no—
197C23C2×C22⋊D10911yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—2182no—
203Q8C2×C22⋊D10914yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—4361no—
209D4—4361no—
210D4—2182no—
211D4C2×C22⋊D10914yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—4362no—
219C2×C4—2182no—
220C2×C4—2181no—
221C2×C4C2×C22⋊D10914yes—
222C2×C4C2×C22⋊D10911yescharacteristic
223C2×C4C2×C22⋊D10912yes—
224C2×C4C2×C22⋊D10912yes—
225C2×C4C23×D10911yescharacteristic
226C8—2184no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—4362no—
231C22—4361no—
232C22#1681266811yescharacteristic
233C22#1681266811yescharacteristic
234C22C22×C22⋊D10911yescharacteristic
235C4—24no—
236C4—24no—
237C4—4362no—
238C4—4361no—
239C4#1681266812yes—
240C4C22×C22⋊D10912yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—8721no—
246C2C2×#1681266811yescharacteristic
subgroupG/H
centerC2C2×#16812668
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×2−1+4×C109C2
socleC2×C109C2×#120
hypercenterC2×2−1+4D109
hypocenterC109#3020
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20295824C2semidirect
2. #20295600C2semidirect
3. #20295209C2semidirect
4. #20295199C2semidirect
5. #20295043C2semidirect
6. C2×#16812618C2semidirect
7. C2×2−1+4×C109C2semidirect
10. #16812729C22semidirect
12. #16812724C22semidirect
15. #16812697C22semidirect
16. #16812688C22semidirect
20. #16812626C22semidirect
21. #16812618C22semidirect
124. C109#3020semidirect
125. C2×2−1+4D109semidirect

Maximal subgroups

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

subgroupindex
#202958242
#202956002
#202952092
#202951992
#202950432
C2×#168126182
C2×2−1+4×C1092
#3020109