Finite Group Databasebeta

← #28364058

Subgroups — #28364058

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
1#20295590C211yesmaximal, characteristic
2C2×#16812688C211yesmaximal, characteristic, complemented
3C2×#16812626C211yesmaximal, characteristic, complemented
4#20295035C211yesmaximal, characteristic, complemented
5C2×#16812615C211yesmaximal, characteristic, complemented
6#20294787C211yesmaximal, characteristic, complemented
7C2×#137×C109C211yesmaximal, characteristic, complemented
8C22×D436C2211yescharacteristic
9C2×#13182013C2211yescharacteristic
10#16812695—21no—
11#16812688—22no—
12#16812688—21no—
13#16812688—22no—
14#16812673—22no—
15#16812667—22no—
16#16812658C2211yescharacteristic
17C2×#13182006—22no—
18C2×C4×Dic109C2211yescharacteristic
19#16812626—22no—
20#16812623—22no—
21#16812615—22no—
22C2×#41×C109C2211yescharacteristic
23C22×Q8×C109C2211yescharacteristic
24#137×C109—22no—
25#137×C109—22no—
26C2×Q16×C109—21no—
27C2×SD16×C109—21no—
28C2×M16×C109C2211yescharacteristic
29C23×D109D412yes—
30C2×D436—21no—
31C2×D436—21no—
32C2×D436—21no—
33C2×D436—21no—
34C2×D436—21no—
35C2×D436—21no—
36C2×C22⋊D109—24no—
37C22×Dic109—21no—
38#13182013—21no—
39#13182013—21no—
40C2×#11724784—21no—
41C4⋊Dic109D412yes—
42C22⋊Dic109—21no—
43#13182006—28no—
44#13182006—24no—
45C4×Dic109—21no—
46C4×Dic109—21no—
47C4×Dic109D412yescomplemented
48#41×C109—21no—
49#41×C109—21no—
50#41×C109—21no—
51#41×C109—21no—
52C2×Q8×C109—21no—
53C2×Q8×C109—21no—
54C2×Q8×C109—21no—
55C2×Q8×C109—21no—
56C2×Q8×C109—21no—
57C2×Q8×C109D411yescharacteristic
58C2×Q8×C109D412yescomplemented
59C2×D4×C109—21no—
60C2×D4×C109D411yescharacteristic
61C22×C4×C109—21no—
62C22×C4×C109—21no—
63C22×C4×C109C2311yescharacteristic
64Q16×C109—24no—
65SD16×C109—24no—
66M16×C109—21no—
67M16×C109D412yescomplemented
68C2×C8×C109—21no—
69C23×C109—21no—
70C23×C109C2×D411yescharacteristic
71C22×D109—22no—
72C22×D109—22no—
73C22×D109—22no—
74C22×D109—44no—
75Q8×C109—22no—
76Q8×C109—21no—
77Q8×C109—22no—
78Q8×C109—21no—
79Q8×C109—41no—
80Q8×C109—41no—
81D4×C109—21no—
82D4×C109—21no—
83D4×C109—41no—
84D4×C109—41no—
85C2×C4×C109—21no—
86C2×C4×C109—22no—
87C2×C4×C109—41no—
88C2×C4×C109—41no—
89C2×C4×C109—41no—
90C2×C4×C109—41no—
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109#4111yescharacteristic
97C22⋊D109—84no—
98C2×Dic109—22no—
99C2×Dic109—22no—
100C2×Dic109—42no—
101D436—41no—
102D436—41no—
103D436—41no—
104D436—41no—
105C8×C109—22no—
106#11724784—41no—
107C22×C109—22no—
108C22×C109—41no—
109C22×C109—42no—
110C22×C109#12011yescharacteristic
111C22×C109#12011yescharacteristic
112C22×C109#12111yescharacteristic
113C2×D109—44no—
114C2×D109—42no—
115C2×D109—44no—
116C2×D109—82no—
117C4×C109—41no—
118C4×C109—42no—
119C4×C109#12011yescharacteristic
120C4×C109#12111yescharacteristic
121C4×C109#12111yescharacteristic
122C4×C109#12711yescharacteristic
123Dic109—42no—
124Dic109—42no—
125C2×C109—21no—
126C2×C109—21no—
127C2×C109—81no—
128C2×C109#39211yescharacteristic
129C2×C109#45512yes—
130D109—82no—
131#2004—1091nomaximal, Sylow
132C109#200411yescharacteristic, complemented, Sylow
133C2×#137D10911yescharacteristic, complemented
134#535—1091no—
135C2×#124—1091no—
136C2×#104—1091no—
137#419—1091no—
138C2×#100—1091no—
139#339—1091no—
140C2×#41C2×D10911yescharacteristic
141C22×Q8C2×D10911yescharacteristic
142C22×D4—1091no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1091no—
148C2×M16C2×D10911yescharacteristic
149#127—2181no—
150#124—2182no—
151#124—2181no—
152#124—2182no—
153#121—2182no—
154#120—2182no—
155#117—1091no—
156C2×C22⋊C4—2182no—
157C2×C42—1091no—
158#104—2182no—
159#103—2182no—
160#100—2182no—
161C24—1092no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D10911yescharacteristic
172C2×Q8C22⋊D10912yescomplemented
173C2×D4—21no—
174C2×D4—2184no—
175C2×D4—2181no—
176C2×D4—2181no—
177C2×D4—2181no—
178C2×D4—2181no—
179C2×D4—2181no—
180C2×D4—2181no—
181C2×D4C22⋊D10911yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2181no—
185C22×C4C22×D10911yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—2181no—
190M16—2181no—
191M16C22⋊D10912yescomplemented
192C2×C8—21no—
193C2×C8—2181no—
194C4⋊C4—1092no—
195C22⋊C4—2188no—
196C22⋊C4—2184no—
197C22⋊C4—2181no—
198C42—1092no—
199C42—2181no—
200C42—2181no—
201C23—21no—
202C23—4364no—
203C23—2182no—
204C23—2182no—
205C23—2182no—
206C23D4×D10911yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—8724no—
218D4—4361no—
219D4—4361no—
220D4—4361no—
221D4—4361no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4362no—
229C2×C4—2182no—
230C2×C4—2182no—
231C2×C4D4×D10911yescharacteristic
232C2×C4D4×D10911yescharacteristic
233C2×C4C2×C22⋊D10911yescharacteristic
234C2×C4C2×C22⋊D10911yescharacteristic
235C2×C4C2×C22⋊D10911yescharacteristic
236C2×C4#1318202811yescharacteristic
237C8—22no—
238C8—4361no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8722no—
243C22—4364no—
244C22—4362no—
245C22—4364no—
246C22#1681266711yescharacteristic
247C22#1681266711yescharacteristic
248C22#1681267011yescharacteristic
249C4—41no—
250C4—42no—
251C4—4362no—
252C4—4362no—
253C4#1681266811yescharacteristic
254C4#1681267111yescharacteristic
255C4#1681267111yescharacteristic
256C4#1681269511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—8722no—
261C2#2029493411yescharacteristic
262C2#2029520212yes—
subgroupG/H
centerC22#16812667
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
FittingC2×#137×C109C2
socleC22×C109#120
prime radical#20295590C2
hypercenterC2×#137D109
hypocenterC109#2004
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16812688C2semidirect
3. C2×#16812626C2semidirect
4. #20295035C2semidirect
5. C2×#16812615C2semidirect
6. #20294787C2semidirect
7. C2×#137×C109C2semidirect
47. C4×Dic109D4semidirect
58. C2×Q8×C109D4semidirect
67. M16×C109D4semidirect
132. C109#2004semidirect
133. C2×#137D109semidirect
172. C2×Q8C22⋊D109semidirect
191. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955902
C2×#168126882
C2×#168126262
#202950352
C2×#168126152
#202947872
C2×#137×C1092
#2004109