Finite Group Databasebeta

← #28364063

Subgroups — #28364063

Order 13952.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812732C211yesmaximal, characteristic, complemented
2#20295593C211yesmaximal, characteristic
3C2×#16812686C211yesmaximal, characteristic, complemented
4C2×#16812625C211yesmaximal, characteristic, complemented
5#20295036C211yesmaximal, characteristic, complemented
6#20294787C211yesmaximal, characteristic, complemented
7C2×#100×C109C211yesmaximal, characteristic, complemented
8C2×#13182029C2211yescharacteristic
9C22×Dic218C2211yescharacteristic
10#16812732—22no—
11#16812732—22no—
12C2×Dic436—21no—
13C2×#13182018—21no—
14#16812695—21no—
15#16812686—21no—
16#16812686—22no—
17#16812686—22no—
18#16812670—22no—
19#16812668—22no—
20#16812658C2211yescharacteristic
21C2×C22⋊Dic109—22no—
22C2×C4×Dic109C2211yescharacteristic
23#16812625—22no—
24#16812624—22no—
25C22×D4×C109C2211yescharacteristic
26C2×M16×C109C2211yescharacteristic
27C2×M16×C109C2211yescharacteristic
28#100×C109—22no—
29#13182029—21no—
30#13182029—21no—
31#13182029—21no—
32#13182029—21no—
33C2×Dic218—21no—
34C2×Dic218—21no—
35C2×Dic218—21no—
36C2×Dic218—21no—
37C2×Dic218—21no—
38C2×Dic218D411yescharacteristic
39C2×Dic218D412yescomplemented
40C2×D436D411yescharacteristic
41C2×C22⋊D109—21no—
42C2×C22⋊D109—24no—
43C22×Dic109—21no—
44C22×Dic109—21no—
45C2×C4×D109—21no—
46Dic436—24no—
47#13182018—24no—
48C4⋊Dic109D412yes—
49C22⋊Dic109—28no—
50C22⋊Dic109—24no—
51C22⋊Dic109—21no—
52C4×Dic109—21no—
53C4×Dic109—21no—
54C4×Dic109D412yescomplemented
55C24×C109D412yes—
56C2×D4×C109—21no—
57C2×D4×C109—21no—
58C2×D4×C109—21no—
59C2×D4×C109—21no—
60C2×D4×C109—21no—
61C2×D4×C109—21no—
62C22×C4×C109C2311yescharacteristic
63M16×C109—21no—
64M16×C109—21no—
65M16×C109—21no—
66M16×C109D412yescomplemented
67C2×C8×C109—21no—
68C2×C8×C109—21no—
69C23×C109—22no—
70C23×C109—22no—
71C23×C109—22no—
72C23×C109—44no—
73C23×C109C2×D411yescharacteristic
74C22×D109—21no—
75D4×C109—41no—
76D4×C109—41no—
77D4×C109—41no—
78D4×C109—41no—
79C2×C4×C109C2×D411yescharacteristic
80C2×C4×C109C2×D411yescharacteristic
81C2×C4×C109C2×D411yescharacteristic
82C2×C4×C109C2×D411yescharacteristic
83C2×C4×C109C2×D411yescharacteristic
84C2×C4×C109#4111yescharacteristic
85C22⋊D109—41no—
86C22⋊D109—41no—
87C22⋊D109—84no—
88C2×Dic109—22no—
89C2×Dic109—22no—
90C2×Dic109—21no—
91C2×Dic109—22no—
92C2×Dic109—42no—
93C2×Dic109—41no—
94C2×Dic109—41no—
95D436—21no—
96D436—21no—
97C4×D109—41no—
98C4×D109—41no—
99Dic218—22no—
100Dic218—21no—
101Dic218—22no—
102Dic218—21no—
103Dic218—41no—
104Dic218—41no—
105C8×C109—22no—
106C8×C109—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#12011yescharacteristic
118C4×C109#12111yescharacteristic
119C4×C109#12111yescharacteristic
120C4×C109#12711yescharacteristic
121Dic109—42no—
122Dic109—42no—
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×#124—1091no—
136C2×#104—1091no—
137#419—1091no—
138C2×#100D10911yescharacteristic, complemented
139#339—1091no—
140C2×#41—1091no—
141C22×Q8—1091no—
142C22×D4C2×D10911yescharacteristic
143#137—2182no—
144#137—2182no—
145C2×Q16—2181no—
146C2×SD16—2181no—
147C2×M16C2×D10911yescharacteristic
148C2×M16C2×D10911yescharacteristic
149#127—2181no—
150#124—2182no—
151#124—2182no—
152#124—2181no—
153#121—2182no—
154#120—2182no—
155#117—1091no—
156C2×C22⋊C4—2182no—
157C2×C42—1091no—
158#104—2182no—
159#103—2182no—
160#100—22no—
161C24D43612yes—
162#41—2181no—
163#41—2181no—
164#41—2181no—
165#41—2181no—
166C2×Q8—1091no—
167C2×Q8—1092no—
168C2×Q8—2181no—
169C2×Q8—2181no—
170C2×Q8—2181no—
171C2×Q8—2181no—
172C2×Q8—2181no—
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—2181no—
183C22×C4—2181no—
184C22×C4—2181no—
185C22×C4C22×D10911yescharacteristic
186Q16—2184no—
187SD16—2184no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16C22⋊D10912yescomplemented
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—1092no—
195C22⋊C4—2188no—
196C22⋊C4—2184no—
197C22⋊C4—2181no—
198C42—1092no—
199C42—2181no—
200C42—2181no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—2181no—
206C23C2×D43611yescharacteristic
207Q8—4361no—
208Q8—4361no—
209Q8—2182no—
210Q8—2181no—
211Q8—2182no—
212Q8—2181no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—8724no—
218D4—4361no—
219D4—4361no—
220D4—2181no—
221D4—2181no—
222C2×C4—4362no—
223C2×C4—4361no—
224C2×C4—4361no—
225C2×C4—4361no—
226C2×C4—4361no—
227C2×C4—2182no—
228C2×C4—2182no—
229C2×C4—2182no—
230C2×C4—2181no—
231C2×C4D4×D10911yescharacteristic
232C2×C4D4×D10911yescharacteristic
233C2×C4D4×D10911yescharacteristic
234C2×C4D4×D10911yescharacteristic
235C2×C4C2×C22⋊D10911yescharacteristic
236C2×C4#1318202911yescharacteristic
237C8—22no—
238C8—41no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—4361no—
245C22—4362no—
246C22#1681266911yescharacteristic
247C22#1681266911yescharacteristic
248C22#1681267311yescharacteristic
249C4—4362no—
250C4—4361no—
251C4—4362no—
252C4—4362no—
253C4#1681266711yescharacteristic
254C4#1681267211yescharacteristic
255C4#1681267211yescharacteristic
256C4#1681269511yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—8721no—
261C2#2029493511yescharacteristic
262C2#2029520612yes—
subgroupG/H
centerC22#16812669
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
FittingC2×#100×C109C2
socleC22×C109#120
prime radical#20295593C2
hypercenterC2×#100D109
hypocenterC109#2004
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
3. C2×#16812686C2semidirect
4. C2×#16812625C2semidirect
5. #20295036C2semidirect
6. #20294787C2semidirect
7. C2×#100×C109C2semidirect
39. C2×Dic218D4semidirect
54. C4×Dic109D4semidirect
66. M16×C109D4semidirect
132. C109#2004semidirect
138. C2×#100D109semidirect
191. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
#202955932
C2×#168126862
C2×#168126252
#202950362
#202947872
C2×#100×C1092
#2004109