Finite Group Databasebeta

← #28363444

Subgroups — #28363444

Order 13952.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16812683C211yesmaximal, characteristic
2#20295499C211yesmaximal, characteristic
3C2×#16812613C211yesmaximal, characteristic
4#20294887C211yesmaximal, characteristic, complemented
5#20294887C211yesmaximal, characteristic, complemented
6#20294792C211yesmaximal, characteristic, complemented
7#342×C109C211yesmaximal, characteristic, complemented
8C22×C22⋊D109C2211yescharacteristic
9C23×Dic109C2211yescharacteristic
10#16812683—22no—
11#16812683C414yescomplemented
12C4×C22⋊D109—22no—
13#16812659—22no—
14#16812655—21no—
15C2×C4⋊Dic109—21no—
16C2×C22⋊Dic109C2211yescharacteristic
17C2×C22⋊Dic109C2211yescharacteristic
18C2×#13182006—21no—
19C22⋊C4×D109—21no—
20#16812613—22no—
21#16812606—22no—
22#16812606—22no—
23C2×C22⋊C4×C109C2211yescharacteristic
24C2×C22⋊C4×C109C2211yescharacteristic
25C2×C22⋊C4×C109C2211yescharacteristic
26C23×D109D411yescharacteristic
27C2×C22⋊D109—21no—
28C2×C22⋊D109—21no—
29C2×C22⋊D109—21no—
30C2×C22⋊D109—21no—
31C2×C22⋊D109C2×C412yes—
32C2×C22⋊D109D412yes—
33C22×Dic109—21no—
34C22×Dic109—21no—
35C22×Dic109—21no—
36C22×Dic109—21no—
37C22×Dic109—21no—
38C22×Dic109—21no—
39C22×Dic109—21no—
40C22×Dic109C2×C412yescomplemented
41C22×Dic109D411yescharacteristic
42C22×Dic109D412yes—
43C2×C4×D109—22no—
44C4⋊Dic109—24no—
45#13182008—22no—
46C22⋊Dic109—21no—
47C22⋊Dic109—21no—
48C22⋊Dic109—21no—
49C22⋊Dic109—21no—
50#13182006—22no—
51#13182006—24no—
52C4×Dic109—22no—
53C24×C109C2311yescharacteristic
54C22×C4×C109—21no—
55C22×C4×C109—21no—
56C22×C4×C109D412yes—
57C22⋊C4×C109—21no—
58C22⋊C4×C109—21no—
59C22⋊C4×C109—21no—
60C22⋊C4×C109—21no—
61C22⋊C4×C109—21no—
62C22⋊C4×C109C2×C412yescomplemented
63C23×C109—42no—
64C23×C109C22×C411yescharacteristic
65C23×C109C2×D411yescharacteristic
66C23×C109C2×D411yescharacteristic
67C23×C109C2×D411yescharacteristic
68C23×C109C2×D411yescharacteristic
69C23×C109#4111yescharacteristic
70C23×C109#4111yescharacteristic
71C22×D109—21no—
72C22×D109—21no—
73C22×D109—42no—
74C22×D109C22⋊C412yes—
75C2×C4×C109—22no—
76C2×C4×C109—22no—
77C2×C4×C109—42no—
78C2×C4×C109—42no—
79C2×C4×C109—42no—
80C2×C4×C109—41no—
81C22⋊D109—22no—
82C22⋊D109—22no—
83C22⋊D109—22no—
84C22⋊D109—22no—
85C2×Dic109—21no—
86C2×Dic109—21no—
87C2×Dic109—22no—
88C2×Dic109—22no—
89C2×Dic109—22no—
90C2×Dic109—22no—
91C2×Dic109—41no—
92C2×Dic109—41no—
93C2×Dic109—41no—
94C2×Dic109—41no—
95C2×Dic109—42no—
96C2×Dic109—42no—
97C2×Dic109—41no—
98C2×Dic109C22⋊C412yes—
99C4×D109—82no—
100C22×C109—22no—
101C22×C109—22no—
102C22×C109—22no—
103C22×C109—41no—
104C22×C109—41no—
105C22×C109—41no—
106C22×C109—41no—
107C22×C109C2×C22⋊C411yescharacteristic
108C22×C109C4×D411yescharacteristic
109C22×C109C4×D411yescharacteristic
110C22×C109#12011yescharacteristic
111C22×C109#12111yescharacteristic
112C22×C109#12111yescharacteristic
113C22×C109#12311yescharacteristic
114C2×D109—22no—
115C2×D109—42no—
116C2×D109—42no—
117C2×D109—81no—
118C4×C109—42no—
119C4×C109—42no—
120C4×C109—81no—
121Dic109—22no—
122Dic109—42no—
123Dic109—42no—
124Dic109—81no—
125C2×C109—21no—
126C2×C109—21no—
127C2×C109—21no—
128C2×C109—21no—
129C2×C109—21no—
130C2×C109—21no—
131C2×C109#38611yescharacteristic
132C2×C109#45812yes—
133D109—42no—
134#1890—1091nomaximal, Sylow
135C109#189011yescharacteristic, complemented, Sylow
136C2×#123—1091no—
137#518—1091no—
138C2×#99—1091no—
139#381—1091no—
140#381—1091no—
141#342—1091no—
142#342D10911yescharacteristic, complemented
143C22×D4—1091no—
144C23×C4—1091no—
145#123—1094no—
146#123—2182no—
147C4×D4—2182no—
148C4×D4—2182no—
149#117—2181no—
150C2×C4⋊C4—2181no—
151C2×C22⋊C4—1091no—
152C2×C22⋊C4—1091no—
153C2×C22⋊C4—2181no—
154C2×C22⋊C4—2181no—
155C2×C22⋊C4C2×D10911yescharacteristic
156C2×C22⋊C4C2×D10911yescharacteristic
157C2×C22⋊C4C2×D10911yescharacteristic
158#99—2182no—
159#95—2182no—
160#95—2182no—
161C24—1091no—
162C24C22×D10911yescharacteristic
163C2×D4—1092no—
164C2×D4—1092no—
165C2×D4—2181no—
166C2×D4—2181no—
167C2×D4—2181no—
168C2×D4—2181no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1091no—
172C22×C4—1092no—
173C22×C4—1092no—
174C22×C4—2182no—
175C22×C4—2181no—
176C22×C4—2181no—
177C22×C4—2181no—
178C22×C4—2181no—
179C22×C4—2181no—
180C22×C4—2181no—
181C22×C4—2181no—
182C22×C4D43612yes—
183C4⋊C4—2184no—
184C4⋊C4—2182no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—2184no—
191C22⋊C4—2182no—
192C22⋊C4—2181no—
193C22⋊C4—2181no—
194C22⋊C4—2181no—
195C22⋊C4—2181no—
196C22⋊C4C4×D10912yescomplemented
197C42—2182no—
198C23—42no—
199C23—1092no—
200C23—4362no—
201C23—2181no—
202C23—2181no—
203C23C2×C4×D10911yescharacteristic
204C23D4×D10911yescharacteristic
205C23D4×D10911yescharacteristic
206C23D4×D10911yescharacteristic
207C23C2×D43611yescharacteristic
208C23#1318202911yescharacteristic
209C23#1318203011yescharacteristic
210D4—2182no—
211D4—2182no—
212D4—2182no—
213D4—2182no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—8722no—
221C2×C4—1092no—
222C2×C4—4362no—
223C2×C4—4361no—
224C2×C4—4361no—
225C2×C4—4361no—
226C2×C4—4361no—
227C2×C4—4361no—
228C2×C4—4362no—
229C2×C4—2182no—
230C2×C4—2182no—
231C2×C4—2182no—
232C2×C4—2182no—
233C2×C4—2181no—
234C2×C4—2181no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—8721no—
243C22—4362no—
244C22—4362no—
245C22—2182no—
246C22C22⋊C4×D10911yescharacteristic
247C22#1681266111yescharacteristic
248C22C4×D43611yescharacteristic
249C22#1681266911yescharacteristic
250C22#1681267211yescharacteristic
251C22#1681267411yescharacteristic
252C22#1681268411yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—8721no—
257C4—4362no—
258C4—4362no—
259C4—2182no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—4362no—
267C2#2029490411yescharacteristic
268C2#2029521712yes—
subgroupG/H
centerC22#16812669
commutatorC23×C109C22×C4
FrattiniC24C22×D109
Fitting#342×C109C2
socleC22×C109#120
prime radicalC22×C22⋊D109C22
cosocleC24×C109C23
hypercenter#342D109
hypocenterC109#1890
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #20294887C2semidirect
5. #20294887C2semidirect
6. #20294792C2semidirect
7. #342×C109C2semidirect
11. #16812683C4semidirect
40. C22×Dic109C2×C4semidirect
62. C22⋊C4×C109C2×C4semidirect
135. C109#1890semidirect
142. #342D109semidirect
196. C22⋊C4C4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126832
#202954992
C2×#168126132
#202948872
#202947922
#342×C1092
#1890109