Finite Group Databasebeta

← #28371509

Subgroups — #28371509

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×#16812745C211yesmaximal, characteristic
2#20295792C212yesmaximal, complemented
3#20295010C211yesmaximal, characteristic, complemented
4#20295009C214yesmaximal, complemented
5C2×#16812615C212yesmaximal, complemented
6#100×D109C214yesmaximal, complemented
7#413×C109C211yesmaximal, characteristic, complemented
8#16812745—24no—
9#16812745—24no—
10C2×#13182030C2212yes—
11C2×#13182029C412yes—
12C2×#13182029C2211yescharacteristic
13C22×D436C412yes—
14C22×D436C2211yescharacteristic
15C2×D4×D109C414yes—
16C2×D4×D109C2212yes—
17#16812707—24no—
18#16812705—24no—
19C2×#13182013C2212yescomplemented
20M16×D109C2218yescomplemented
21#16812618C2212yescomplemented
22#16812616C2212yescomplemented
23#16812615C2218yescomplemented
24C2×#41×C109C2211yescharacteristic
25C2×M16×C109C2212yescomplemented
26#101×C109C2212yescomplemented
27#100×C109C2212yescomplemented
28C23×D109C2×C412yes—
29C23×D109C2×C414yes—
30#13182030—28no—
31#13182029—24no—
32#13182029—24no—
33#13182029D414yes—
34#13182029D414yes—
35C2×Dic218C2×C411yescharacteristic
36C2×D436—22no—
37C2×D436—22no—
38C2×D436—24no—
39C2×D436—24no—
40C2×D436C2×C414yes—
41C2×D436C2×C412yes—
42C2×D436C2×C411yescharacteristic
43C2×D436C2312yes—
44C2×C22⋊D109C2×C414yes—
45C2×C22⋊D109C2×C412yes—
46D4×D109—216no—
47D4×D109—28no—
48C2×C4×D109C2×C414yes—
49C2×C4×D109C2312yes—
50#13182013—22no—
51#13182013C2314yes—
52#13182012—28no—
53C2×#11724784—22no—
54C8×D109—28no—
55#41×C109—22no—
56#41×C109—22no—
57C2×Q8×C109C2311yescharacteristic
58C2×D4×C109C2×C412yes—
59C2×D4×C109C2311yescharacteristic
60C22×C4×C109C2×C412yes—
61C22×C4×C109C2311yescharacteristic
62M16×C109—22no—
63M16×C109C2314yes—
64C2×C8×C109—22no—
65C23×C109C22×C412yes—
66C23×C109C22×C411yescharacteristic
67C22×D109—24no—
68C22×D109—28no—
69C22×D109—84no—
70C22×D109—82no—
71C22×D109C22×C414yes—
72C22×D109C22×C412yes—
73Q8×C109—22no—
74D4×C109—24no—
75D4×C109—22no—
76C2×C4×C109—22no—
77C2×C4×C109—22no—
78C2×C4×C109C22×C412yes—
79C2×C4×C109C22×C411yescharacteristic
80C2×C4×C109C2×D412yes—
81C2×C4×C109C2×D412yes—
82C2×C4×C109C2411yescharacteristic
83C22⋊D109—28no—
84C22⋊D109C22⋊C418yes—
85C2×Dic109C22×C412yes—
86D436—28no—
87D436—22no—
88D436—22no—
89D436—24no—
90D436C22⋊C412yes—
91D436C22⋊C412yes—
92C4×D109—28no—
93C4×D109C2×D418yes—
94Dic218C22⋊C412yes—
95Dic218C22⋊C412yes—
96C8×C109—24no—
97#11724784—24no—
98C22×C109—22no—
99C22×C109—41no—
100C22×C109—42no—
101C22×C109C2×C22⋊C412yes—
102C22×C109C23×C411yescharacteristic
103C2×D109—24no—
104C2×D109—42no—
105C2×D109—44no—
106C2×D109—88no—
107C2×D109—84no—
108C2×D109C2×C22⋊C414yes—
109C4×C109—22no—
110C4×C109C2×C22⋊C411yescharacteristic
111C4×C109C2×C22⋊C411yescharacteristic
112C4×C109C22×D412yes—
113Dic109C2×C22⋊C414yes—
114C2×C109—21no—
115C2×C109—22no—
116C2×C109—42no—
117C2×C109C22×C22⋊C411yescharacteristic
118D109—24no—
119D109—44no—
120#2888—1091nomaximal, Sylow
121C109#288811yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1091no—
123#568—1092no—
124#413—1091no—
125#413—1094no—
126#413D10911yescharacteristic, complemented
127C2×#100—1092no—
128C2×#100—1094no—
129H(5,𝔽2)—2184no—
130H(5,𝔽2)—2184no—
131C2×#41—1092no—
132C2×#41—1092no—
133C2×#41—1091no—
134C2×#41C2×D10911yescharacteristic
135C22×D4—1094no—
136C22×D4—1092no—
137C22×D4—1092no—
138C22×D4—1091no—
139#131—2184no—
140#131—2184no—
141C2×M16—1092no—
142C2×M16—1098no—
143C2×M16C2×D10912yescomplemented
144#101—1092no—
145#101C2×D10912yescomplemented
146#100—1092no—
147#100—1098no—
148#100C2×D10912yescomplemented
149C24—1094no—
150C24—1092no—
151#41—22no—
152#41—22no—
153#41—1094no—
154#41—1094no—
155#41—2184no—
156#41—2184no—
157#41—2188no—
158C2×Q8—1091no—
159C2×Q8C22×D10911yescharacteristic
160C2×D4—1092no—
161C2×D4—1094no—
162C2×D4—1091no—
163C2×D4—1094no—
164C2×D4—1092no—
165C2×D4—1092no—
166C2×D4—2184no—
167C2×D4—2184no—
168C2×D4—2188no—
169C2×D4—21816no—
170C2×D4—2182no—
171C2×D4—2182no—
172C2×D4C4×D10912yes—
173C2×D4C22×D10911yescharacteristic
174C22×C4—1092no—
175C22×C4—1094no—
176C22×C4C4×D10912yes—
177C22×C4C22×D10911yescharacteristic
178M16—22no—
179M16—1094no—
180M16—2188no—
181M16—2182no—
182M16C22×D10914yes—
183C2×C8—22no—
184C2×C8—2188no—
185C2×C8—2182no—
186C23—8722no—
187C23—8724no—
188C23—1092no—
189C23—1094no—
190C23—2188no—
191C23—2184no—
192C23C2×C4×D10912yes—
193C23C2×C4×D10911yescharacteristic
194Q8—22no—
195Q8—1092no—
196Q8—1092no—
197D4—24no—
198D4—22no—
199D4—1098no—
200D4—1092no—
201D4—1092no—
202D4—2188no—
203D4—2188no—
204D4—2182no—
205D4—2182no—
206D4—2184no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—1092no—
210C2×C4—1098no—
211C2×C4—2188no—
212C2×C4C2×C4×D10912yes—
213C2×C4C2×C4×D10911yescharacteristic
214C2×C4D4×D10912yes—
215C2×C4D4×D10912yes—
216C2×C4C23×D10911yescharacteristic
217C8—24no—
218C8—2184no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—8724no—
223C22—8728no—
224C22—1094no—
225C22—4364no—
226C22—4362no—
227C22—2184no—
228C22C22⋊C4×D10912yes—
229C22C22×C4×D10911yescharacteristic
230C4—22no—
231C4—1094no—
232C4C22⋊C4×D10911yescharacteristic
233C4C22⋊C4×D10911yescharacteristic
234C4C2×D4×D10912yes—
235C2—21no—
236C2—22no—
237C2—42no—
238C2—4364no—
239C2—2184no—
240C2C2×C22⋊C4×D10911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D109
commutatorC22×C109C23×C4
FrattiniC2×C4C23×D109
Fitting#413×C109C2
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812745C2
cosocleC2×C4×C109C24
hypercenter#413D109
hypocenterC109#2888
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20295792C2semidirect
3. #20295010C2semidirect
4. #20295009C2semidirect
5. C2×#16812615C2semidirect
6. #100×D109C2semidirect
7. #413×C109C2semidirect
19. C2×#13182013C22semidirect
20. M16×D109C22semidirect
21. #16812618C22semidirect
22. #16812616C22semidirect
23. #16812615C22semidirect
25. C2×M16×C109C22semidirect
26. #101×C109C22semidirect
27. #100×C109C22semidirect
121. C109#2888semidirect
126. #413D109semidirect
143. C2×M16C2×D109semidirect
145. #101C2×D109semidirect
148. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127452
#202957922
#202950102
#202950092
C2×#168126152
#100×D1092
#413×C1092
#2888109