Finite Group Databasebeta

← #28373429

Subgroups — #28373429

Order 13952.

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

normal subgroups only

This group has 392 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#20295575C211yesmaximal, characteristic
2#125×D109C211yesmaximal, characteristic, complemented
3#20295393C212yesmaximal, complemented
4#20295389C212yesmaximal, complemented
5#20295383C211yesmaximal, characteristic, complemented
6#20295381C211yesmaximal, characteristic, complemented
7#20295380C211yesmaximal, characteristic, complemented
8#20295026C212yesmaximal, complemented
9#20295024C212yesmaximal, complemented
10C4×#13182012C211yesmaximal, characteristic, complemented
11#488×C109C211yesmaximal, characteristic, complemented
12C2×#13182030C2211yescharacteristic
13C2×#13182028C2211yescharacteristic
14C2×#13182012C2212yes—
15#16812697C2211yescharacteristic
16#16812695C2211yescharacteristic
17#16812691C2211yescharacteristic
18#16812690C2212yes—
19#16812689—22no—
20#16812687C2211yescharacteristic
21#16812672—22no—
22#16812659—22no—
23C4⋊C4×D109—22no—
24C4⋊C4×D109C2212yes—
25C42×D109C2211yescharacteristic
26#16812624C2212yes—
27#16812623C2212yes—
28#16812622C2212yes—
29#16812621C2212yes—
30#16812620C2212yes—
31#16812619C2212yes—
32#16812609C2212yes—
33C4×#11724784C2211yescharacteristic
34#125×C109C2211yescharacteristic
35#124×C109C2211yescharacteristic
36#103×C109C2212yes—
37#102×C109C2212yes—
38C4×C8×C109C2211yescharacteristic
39#13182030—24no—
40#13182028—24no—
41C2×Dic218C2311yescharacteristic
42C2×D436—21no—
43C2×D436C2311yescharacteristic
44C2×C22⋊D109—21no—
45C2×C22⋊D109—22no—
46C22×Dic109—21no—
47C2×C4×D109—22no—
48C2×C4×D109—21no—
49C2×C4×D109D412yes—
50C2×C4×D109C2311yescharacteristic
51#13182012—24no—
52C2×#11724784C2312yes—
53C4⋊Dic109—22no—
54C4⋊Dic109C2312yes—
55#13182008—22no—
56#13182008—22no—
57#13182008—22no—
58#13182008—22no—
59#13182006—22no—
60C4×Dic109D412yes—
61C4×Dic109C2311yescharacteristic
62C2×Q8×C109C2311yescharacteristic
63C2×D4×C109C2311yescharacteristic
64C2×C8×C109C2312yes—
65C4⋊C4×C109—22no—
66C4⋊C4×C109C2312yes—
67C22⋊C4×C109—22no—
68C42×C109C2311yescharacteristic
69C23×C109—21no—
70C22×D109—21no—
71C22×D109C2×D411yescharacteristic
72Q8×C109—22no—
73D4×C109—22no—
74C2×C4×C109—21no—
75C2×C4×C109—22no—
76C2×C4×C109C2×D412yes—
77C2×C4×C109C2411yescharacteristic
78C22⋊D109—44no—
79C22⋊D109—42no—
80C2×Dic109—21no—
81C2×Dic109—22no—
82C2×Dic109—42no—
83C2×Dic109C2×D412yes—
84C2×Dic109C2×D411yescharacteristic
85D436—22no—
86D436—42no—
87C4×D109—42no—
88C4×D109—44no—
89C4×D109—42no—
90C4×D109#4114yes—
91Dic218—22no—
92C8×C109—22no—
93#11724784—22no—
94C22×C109—41no—
95C22×C109—42no—
96C22×C109C22×D411yescharacteristic
97C2×D109—22no—
98C2×D109—41no—
99C2×D109—42no—
100C2×D109#12412yes—
101C4×C109—22no—
102C4×C109—41no—
103C4×C109—42no—
104C4×C109C2×#4112yes—
105Dic109—22no—
106Dic109—41no—
107Dic109—42no—
108Dic109#12412yes—
109C2×C109—81no—
110C2×C109C2×#12411yescharacteristic
111C2×C109#57512yes—
112D109—22no—
113D109—81no—
114#3142—1091nomaximal, Sylow
115C109#314211yescharacteristic, complemented, Sylow
116#532—1091no—
117C2×#125—1091no—
118#491—1092no—
119#490—1092no—
120#488—1091no—
121#488—1091no—
122#488—1091no—
123#488D10911yescharacteristic, complemented
124#417—1092no—
125#417—1092no—
126C4×M16—1091no—
127C2×#41—1091no—
128C2×#41—1091no—
129C2×M16—1092no—
130C4⋊Q8—1091no—
131#127—1091no—
132#125—1092no—
133#125—1091no—
134#125—2182no—
135#125C2×D10911yescharacteristic
136#124—1091no—
137#124C2×D10911yescharacteristic
138#121—2182no—
139C4×D4—2182no—
140C2×C4⋊C4—1092no—
141C2×C4⋊C4—2182no—
142C2×C42—1091no—
143#103—1092no—
144#103—1092no—
145#103—1092no—
146#103C2×D10912yes—
147#102—1092no—
148#102—1092no—
149#102—1092no—
150#102C2×D10912yes—
151#97—1092no—
152C4×C8—1091no—
153C4×C8C2×D10911yescharacteristic
154#41—2184no—
155#41—2184no—
156C2×Q8—1091no—
157C2×Q8C22×D10911yescharacteristic
158C2×D4—1091no—
159C2×D4—2181no—
160C2×D4—2182no—
161C2×D4—2181no—
162C2×D4C22×D10911yescharacteristic
163C22×C4—1091no—
164C22×C4—1092no—
165C22×C4—2181no—
166C22×C4—2182no—
167C22×C4—2181no—
168M16—2184no—
169C2×C8—1092no—
170C2×C8C22×D10912yes—
171C4⋊C4—22no—
172C4⋊C4—1092no—
173C4⋊C4—2182no—
174C4⋊C4—2182no—
175C4⋊C4—2182no—
176C4⋊C4—2182no—
177C4⋊C4—2182no—
178C4⋊C4C22×D10912yes—
179C22⋊C4—22no—
180C22⋊C4—2182no—
181C42—1092no—
182C42—1091no—
183C42C22×D10911yescharacteristic
184C23—21no—
185C23—1091no—
186C23—2181no—
187Q8—22no—
188Q8—2182no—
189D4—22no—
190D4—4362no—
191D4—4362no—
192D4—4364no—
193D4—2182no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—1091no—
197C2×C4—1094no—
198C2×C4—1092no—
199C2×C4—4362no—
200C2×C4—4362no—
201C2×C4—4364no—
202C2×C4—4362no—
203C2×C4—2181no—
204C2×C4—2182no—
205C2×C4D4×D10912yes—
206C2×C4C23×D10911yescharacteristic
207C8—22no—
208C8—2182no—
209C22—41no—
210C22—42no—
211C22—1092no—
212C22—4361no—
213C22—4362no—
214C22—2182no—
215C22C2×D4×D10911yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1092no—
220C4—4361no—
221C4—4362no—
222C4—2182no—
223C4#41×D10912yes—
224C2—81no—
225C2—8721no—
226C2—2182no—
227C2#124×D10911yescharacteristic
228C2#2029582312yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#488×C109C2
socleC22×C109C22×D4
prime radical#20295575C2
hypercenter#488D109
hypocenterC109#3142
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D109C2semidirect
3. #20295393C2semidirect
4. #20295389C2semidirect
5. #20295383C2semidirect
6. #20295381C2semidirect
7. #20295380C2semidirect
8. #20295026C2semidirect
9. #20295024C2semidirect
10. C4×#13182012C2semidirect
11. #488×C109C2semidirect
115. C109#3142semidirect
123. #488D109semidirect

Maximal subgroups

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

subgroupindex
#202955752
#125×D1092
#202953932
#202953892
#202953832
#202953812
#202953802
#202950262
#202950242
C4×#131820122
#488×C1092
#3142109