Finite Group Databasebeta

← #28364061

Subgroups — #28364061

Order 13952.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16812730C211yesmaximal, characteristic, complemented
2#20295593C211yesmaximal, characteristic
3C2×#16812627C211yesmaximal, characteristic, complemented
4#20295035C211yesmaximal, characteristic, complemented
5C2×#16812616C211yesmaximal, characteristic, complemented
6#20294786C211yesmaximal, characteristic, complemented
7C2×#124×C109C211yesmaximal, characteristic, complemented
8C2×#13182029C2211yescharacteristic
9#16812730—22no—
10#16812730—22no—
11C2×#13182019—21no—
12C2×#13182016—21no—
13C2×#13182013C2211yescharacteristic
14C2×#13182013C2211yescharacteristic
15#16812695—21no—
16#16812686—21no—
17#16812670—22no—
18#16812668—22no—
19#16812658C2211yescharacteristic
20#16812627—22no—
21#16812623—22no—
22#16812616—22no—
23C22×Q8×C109C2211yescharacteristic
24C22×D4×C109C2211yescharacteristic
25#124×C109—22no—
26#124×C109—22no—
27C2×C22⋊C4×C109—22no—
28C2×C42×C109C2211yescharacteristic
29#13182029—21no—
30#13182029—21no—
31#13182029—21no—
32#13182029—21no—
33C2×Dic218D411yescharacteristic
34C2×D436D411yescharacteristic
35C2×C22⋊D109—21no—
36C2×C22⋊D109—24no—
37C2×C4×D109—21no—
38#13182019—24no—
39#13182016—24no—
40#13182013—21no—
41#13182013—21no—
42#13182013—21no—
43#13182013D412yescomplemented
44C2×#11724784—21no—
45C2×#11724784—21no—
46C4⋊Dic109D412yes—
47C22⋊Dic109—24no—
48C22⋊Dic109—21no—
49C4×Dic109—21no—
50C24×C109D412yes—
51C2×Q8×C109—21no—
52C2×Q8×C109—21no—
53C2×Q8×C109—21no—
54C2×Q8×C109—21no—
55C2×Q8×C109—21no—
56C2×Q8×C109D412yescomplemented
57C2×D4×C109—21no—
58C2×D4×C109—21no—
59C2×D4×C109—21no—
60C2×D4×C109—21no—
61C2×D4×C109—21no—
62C2×D4×C109—21no—
63C22×C4×C109—21no—
64C22×C4×C109—21no—
65C22×C4×C109C2311yescharacteristic
66C22⋊C4×C109—28no—
67C42×C109—21no—
68C42×C109D412yescomplemented
69C23×C109—22no—
70C23×C109—22no—
71C23×C109—22no—
72C23×C109—44no—
73C23×C109C2×D411yescharacteristic
74C22×D109—21no—
75Q8×C109—22no—
76Q8×C109—22no—
77Q8×C109—41no—
78Q8×C109—41no—
79D4×C109—41no—
80D4×C109—41no—
81D4×C109—41no—
82D4×C109—41no—
83C2×C4×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109—42no—
86C2×C4×C109—41no—
87C2×C4×C109—41no—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2×D411yescharacteristic
90C2×C4×C109C2×D411yescharacteristic
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109#4111yescharacteristic
94C22⋊D109—41no—
95C22⋊D109—41no—
96C22⋊D109—84no—
97C2×Dic109—22no—
98C2×Dic109—21no—
99D436—21no—
100D436—21no—
101C4×D109—41no—
102C4×D109—41no—
103Dic218—21no—
104Dic218—21no—
105#11724784—22no—
106#11724784—41no—
107C22×C109—22no—
108C22×C109—44no—
109C22×C109—42no—
110C22×C109—44no—
111C22×C109—82no—
112C22×C109#12011yescharacteristic
113C22×C109#12011yescharacteristic
114C22×C109#12111yescharacteristic
115C2×D109—41no—
116C2×D109—42no—
117C4×C109—42no—
118C4×C109—42no—
119C4×C109#12011yescharacteristic
120C4×C109#12111yescharacteristic
121C4×C109#12111yescharacteristic
122C4×C109#12711yescharacteristic
123Dic109—41no—
124Dic109—42no—
125C2×C109—21no—
126C2×C109—21no—
127C2×C109—82no—
128C2×C109#39211yescharacteristic
129C2×C109#45512yes—
130D109—81no—
131#2004—1091nomaximal, Sylow
132C109#200411yescharacteristic, complemented, Sylow
133C2×#137—1091no—
134#535—1091no—
135C2×#124D10911yescharacteristic, complemented
136C2×#104—1091no—
137#419—1091no—
138C2×#100—1091no—
139#339—1091no—
140C2×#41—1091no—
141C22×Q8C2×D10911yescharacteristic
142C22×D4C2×D10911yescharacteristic
143#137—2182no—
144#137—2182no—
145C2×Q16—2181no—
146C2×SD16—2181no—
147C2×M16—1091no—
148C2×M16—1091no—
149#127—2181no—
150#124—22no—
151#124—22no—
152#124—2181no—
153#121—2182no—
154#120—2182no—
155#117—1091no—
156C2×C22⋊C4—22no—
157C2×C42C2×D10911yescharacteristic
158#104—2182no—
159#103—2182no—
160#100—2182no—
161C24C22⋊D10912yes—
162#41—2181no—
163#41—2181no—
164#41—2181no—
165#41—2181no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1091no—
172C2×Q8C22⋊D10912yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—1091no—
180C2×D4—2184no—
181C2×D4—2181no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2181no—
185C22×C4C22×D10911yescharacteristic
186Q16—2184no—
187SD16—2184no—
188M16—1092no—
189M16—2181no—
190M16—2181no—
191M16—2181no—
192C2×C8—2181no—
193C2×C8—2181no—
194C4⋊C4—1092no—
195C22⋊C4—28no—
196C22⋊C4—2184no—
197C22⋊C4—2181no—
198C42—21no—
199C42—2181no—
200C42C22⋊D10912yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—2181no—
206C23C2×C22⋊D10911yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—2181no—
212Q8—2181no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—8724no—
218D4—4361no—
219D4—4361no—
220D4—2181no—
221D4—2181no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—4361no—
228C2×C4—4361no—
229C2×C4—2182no—
230C2×C4—2181no—
231C2×C4D4×D10911yescharacteristic
232C2×C4D4×D10911yescharacteristic
233C2×C4D4×D10911yescharacteristic
234C2×C4C2×C22⋊D10911yescharacteristic
235C2×C4C2×C22⋊D10911yescharacteristic
236C2×C4#1318202811yescharacteristic
237C8—4361no—
238C8—2182no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—4361no—
245C22—4362no—
246C22#1681266711yescharacteristic
247C22#1681266811yescharacteristic
248C22#1681267111yescharacteristic
249C4—42no—
250C4—42no—
251C4—4361no—
252C4—4362no—
253C4#1681266711yescharacteristic
254C4#1681267011yescharacteristic
255C4#1681267111yescharacteristic
256C4#1681269511yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—8721no—
261C2#2029493411yescharacteristic
262C2#2029520312yes—
subgroupG/H
centerC22#16812667
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
FittingC2×#124×C109C2
socleC22×C109#120
prime radical#20295593C2
hypercenterC2×#124D109
hypocenterC109#2004
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
3. C2×#16812627C2semidirect
4. #20295035C2semidirect
5. C2×#16812616C2semidirect
6. #20294786C2semidirect
7. C2×#124×C109C2semidirect
43. #13182013D4semidirect
56. C2×Q8×C109D4semidirect
68. C42×C109D4semidirect
132. C109#2004semidirect
135. C2×#124D109semidirect
172. C2×Q8C22⋊D109semidirect
200. C42C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
#202955932
C2×#168126272
#202950352
C2×#168126162
#202947862
C2×#124×C1092
#2004109