Finite Group Databasebeta

← #28371508

Subgroups — #28371508

Order 13952.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812744C211yesmaximal, characteristic
2#20295792C212yesmaximal, complemented
3#20295008C212yesmaximal, complemented
4#20295007C214yesmaximal, complemented
5C2×#16812616C211yesmaximal, characteristic, complemented
6#100×D109C214yesmaximal, complemented
7C2×#100×C109C211yesmaximal, characteristic, complemented
8#16812744—24no—
9#16812744—24no—
10C2×#13182029C2211yescharacteristic
11C2×#13182029C2211yescharacteristic
12C2×#13182028C412yes—
13C2×#13182028C2212yes—
14C22×C22⋊D109C414yes—
15C2×D4×D109C412yes—
16C2×D4×D109C2212yes—
17#16812707—24no—
18#16812705—24no—
19C2×#13182013C2212yescomplemented
20M16×D109C2218yescomplemented
21#16812617C2214yescomplemented
22#16812616C2214yescomplemented
23#16812615C2214yescomplemented
24C22×D4×C109C2211yescharacteristic
25C2×M16×C109C2212yescomplemented
26#100×C109C2214yescomplemented
27C23×D109C2×C414yes—
28#13182029—22no—
29#13182029—22no—
30#13182029D414yes—
31#13182029D414yes—
32#13182028—28no—
33#13182028—28no—
34C2×Dic218C2×C411yescharacteristic
35C2×Dic218C2311yescharacteristic
36C2×D436C2×C411yescharacteristic
37C2×D436C2311yescharacteristic
38C2×C22⋊D109—28no—
39C2×C22⋊D109—28no—
40C2×C22⋊D109C2×C414yes—
41C2×C22⋊D109C2×C414yes—
42C2×C22⋊D109C2×C412yes—
43C2×C22⋊D109C2×C412yes—
44D4×D109—28no—
45D4×D109—28no—
46C22×Dic109C2×C414yes—
47C2×C4×D109C2×C412yes—
48C2×C4×D109C2312yes—
49#13182013—22no—
50#13182013C2314yes—
51#13182012—28no—
52C2×#11724784—22no—
53C8×D109—28no—
54C24×C109C2×C412yes—
55C2×D4×C109—22no—
56C2×D4×C109—22no—
57C2×D4×C109C2×C412yes—
58C2×D4×C109C2312yes—
59C22×C4×C109C2311yescharacteristic
60M16×C109—22no—
61M16×C109C2314yes—
62C2×C8×C109—22no—
63C23×C109—22no—
64C23×C109—22no—
65C23×C109—82no—
66C23×C109C22×C414yes—
67C23×C109C22×C411yescharacteristic
68C22×D109—28no—
69C22×D109—84no—
70C22×D109C22×C412yes—
71C22×D109C22×C412yes—
72D4×C109—24no—
73D4×C109—24no—
74C2×C4×C109C22×C411yescharacteristic
75C2×C4×C109C2×D412yes—
76C2×C4×C109C2×D412yes—
77C2×C4×C109C2411yescharacteristic
78C22⋊D109—22no—
79C22⋊D109—22no—
80C22⋊D109—28no—
81C22⋊D109—28no—
82C22⋊D109C22⋊C414yes—
83C22⋊D109C22⋊C414yes—
84C2×Dic109—28no—
85C2×Dic109C22×C412yes—
86C2×Dic109C22×C412yes—
87D436—22no—
88D436C22⋊C414yes—
89C4×D109—24no—
90C4×D109C2×D418yes—
91Dic218—22no—
92Dic218C22⋊C414yes—
93C8×C109—24no—
94#11724784—24no—
95C22×C109—24no—
96C22×C109—41no—
97C22×C109—44no—
98C22×C109—82no—
99C22×C109—82no—
100C22×C109C2×C22⋊C411yescharacteristic
101C22×C109C2×C22⋊C411yescharacteristic
102C22×C109C23×C411yescharacteristic
103C2×D109—22no—
104C2×D109—42no—
105C2×D109—42no—
106C2×D109—88no—
107C2×D109C2×C22⋊C414yes—
108C4×C109C2×C22⋊C412yes—
109C4×C109C22×D412yes—
110Dic109—22no—
111Dic109C2×C22⋊C414yes—
112C2×C109—21no—
113C2×C109—21no—
114C2×C109—21no—
115C2×C109—44no—
116C2×C109C22×C22⋊C411yescharacteristic
117D109—24no—
118D109—42no—
119#2888—1091nomaximal, Sylow
120C109#288811yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1091no—
122#568—1092no—
123#413—1092no—
124#413—1094no—
125C2×#100—1091no—
126C2×#100—1094no—
127C2×#100D10911yescharacteristic, complemented
128H(5,𝔽2)—2184no—
129H(5,𝔽2)—2184no—
130C2×#41—1092no—
131C2×#41—1092no—
132C2×#41—1091no—
133C2×#41—1091no—
134C22×D4—1094no—
135C22×D4—1092no—
136C22×D4—1092no—
137C22×D4C2×D10911yescharacteristic
138#131—2184no—
139#131—2184no—
140C2×M16—1092no—
141C2×M16—1098no—
142C2×M16C2×D10912yescomplemented
143#101—1094no—
144#100—1094no—
145#100—1094no—
146#100C2×D10914yescomplemented
147C24—1094no—
148C24C4×D10912yes—
149#41—1094no—
150#41—1094no—
151#41—2188no—
152#41—2188no—
153#41—2182no—
154#41—2182no—
155C2×Q8—1091no—
156C2×Q8—1091no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—1092no—
160C2×D4—1094no—
161C2×D4—1094no—
162C2×D4—1092no—
163C2×D4—1091no—
164C2×D4—1091no—
165C2×D4—2188no—
166C2×D4—2188no—
167C2×D4—2188no—
168C2×D4—2188no—
169C2×D4C4×D10912yes—
170C2×D4C22×D10912yes—
171C22×C4—1092no—
172C22×C4—1094no—
173C22×C4—1092no—
174C22×C4C22×D10911yescharacteristic
175M16—22no—
176M16—1094no—
177M16—2188no—
178M16—2182no—
179M16C22×D10914yes—
180C2×C8—22no—
181C2×C8—2188no—
182C2×C8—2182no—
183C23—22no—
184C23—22no—
185C23—82no—
186C23—8724no—
187C23—1092no—
188C23—1092no—
189C23—2188no—
190C23C2×C4×D10914yes—
191C23C2×C4×D10911yescharacteristic
192Q8—1094no—
193Q8—2182no—
194D4—24no—
195D4—24no—
196D4—1094no—
197D4—1094no—
198D4—1094no—
199D4—2188no—
200D4—2188no—
201D4—2182no—
202D4—2182no—
203D4—2182no—
204C2×C4—1092no—
205C2×C4—1098no—
206C2×C4—1092no—
207C2×C4—2188no—
208C2×C4—2184no—
209C2×C4C2×C4×D10911yescharacteristic
210C2×C4D4×D10912yes—
211C2×C4D4×D10912yes—
212C2×C4C23×D10911yescharacteristic
213C8—24no—
214C8—2184no—
215C22—24no—
216C22—41no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—8728no—
221C22—1094no—
222C22—4362no—
223C22—4362no—
224C22—2182no—
225C22C22⋊C4×D10911yescharacteristic
226C22C22⋊C4×D10911yescharacteristic
227C22C22×C4×D10911yescharacteristic
228C4—1094no—
229C4—2182no—
230C4C22⋊C4×D10912yes—
231C4C2×D4×D10912yes—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—4362no—
237C2—2184no—
238C2C2×C22⋊C4×D10911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D109
commutatorC22×C109C23×C4
FrattiniC2×C4C23×D109
FittingC2×#100×C109C2
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812744C2
cosocleC2×C4×C109C24
hypercenterC2×#100D109
hypocenterC109#2888
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #20295792C2semidirect
3. #20295008C2semidirect
4. #20295007C2semidirect
5. C2×#16812616C2semidirect
6. #100×D109C2semidirect
7. C2×#100×C109C2semidirect
19. C2×#13182013C22semidirect
20. M16×D109C22semidirect
21. #16812617C22semidirect
22. #16812616C22semidirect
23. #16812615C22semidirect
25. C2×M16×C109C22semidirect
26. #100×C109C22semidirect
120. C109#2888semidirect
127. C2×#100D109semidirect
142. C2×M16C2×D109semidirect
146. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127442
#202957922
#202950082
#202950072
C2×#168126162
#100×D1092
C2×#100×C1092
#2888109