Finite Group Databasebeta

← #28371376

Subgroups — #28371376

Order 13952.

924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 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×#16812695C211yesmaximal, characteristic, complemented
2C2×#16812674C211yesmaximal, characteristic, complemented
3C2×#16812672C214yesmaximal, complemented
4C2×#16812670C211yesmaximal, characteristic, complemented
5C2×C42×D109C211yesmaximal, characteristic, complemented
6#20294928C211yesmaximal, characteristic, complemented
7#20294926C214yesmaximal, complemented
8#20294895C211yesmaximal, characteristic, complemented
9#390×C109C211yesmaximal, characteristic, complemented
10C22×D436C2211yescharacteristic
11C22×D436C2212yes—
12C22×C22⋊D109C2214yes—
13C22×C22⋊D109C2212yes—
14C22×C4×D109C2211yescharacteristic
15C22×C4×D109C2212yes—
16#16812695—24no—
17#16812674—24no—
18#16812672—216no—
19#16812670—24no—
20C2×C4⋊Dic109C2211yescharacteristic
21C2×#13182008C2214yes—
22C2×C22⋊Dic109C2212yes—
23C2×#13182006C2212yes—
24C2×#13182006C2214yes—
25C2×C4×Dic109C2212yes—
26C2×C4×Dic109C2211yescharacteristic
27C42×D109—24no—
28C22×D4×C109C2211yescharacteristic
29C2×C4⋊C4×C109C2211yescharacteristic
30C2×C22⋊C4×C109C2214yes—
31C2×C42×C109C2211yescharacteristic
32C23×D109C2312yes—
33C23×D109C2311yescharacteristic
34C2×D436—28no—
35C2×D436—24no—
36C2×D436—22no—
37C2×D436—22no—
38C2×D436—22no—
39C2×C22⋊D109—24no—
40C2×C22⋊D109—28no—
41C2×C22⋊D109—28no—
42C2×C22⋊D109—28no—
43C2×C22⋊D109—28no—
44C22×Dic109C2312yes—
45C22×Dic109C2311yescharacteristic
46C22×Dic109C2312yes—
47C2×C4×D109—22no—
48C2×C4×D109—22no—
49C2×C4×D109—24no—
50C2×C4×D109—24no—
51C2×C4×D109D414yescomplemented
52C2×C4×D109D418yescomplemented
53C4⋊Dic109—22no—
54#13182008—28no—
55C22⋊Dic109—24no—
56#13182006—28no—
57#13182006—24no—
58C4×Dic109—24no—
59C4×Dic109D414yescomplemented
60C24×C109C2312yes—
61C2×D4×C109—22no—
62C2×D4×C109—22no—
63C2×D4×C109—22no—
64C22×C4×C109C2312yes—
65C22×C4×C109C2312yes—
66C22×C4×C109C2311yescharacteristic
67C4⋊C4×C109—22no—
68C22⋊C4×C109—28no—
69C42×C109—22no—
70C23×C109—24no—
71C23×C109—22no—
72C23×C109—22no—
73C23×C109—22no—
74C23×C109—24no—
75C23×C109C2411yescharacteristic
76C22×D109—21no—
77C22×D109—22no—
78C22×D109—21no—
79C22×D109—24no—
80C22×D109—22no—
81C22×D109—22no—
82C22×D109—24no—
83C22×D109—22no—
84C22×D109C2×D412yes—
85C22×D109C2×D414yes—
86D4×C109—44no—
87C2×C4×C109—22no—
88C2×C4×C109—22no—
89C2×C4×C109—24no—
90C2×C4×C109—22no—
91C2×C4×C109C2×D414yes—
92C2×C4×C109C2×D412yes—
93C2×C4×C109C2×D412yes—
94C2×C4×C109#4112yes—
95C22⋊D109—48no—
96C22⋊D109—416no—
97C2×Dic109—21no—
98C2×Dic109—21no—
99C2×Dic109—24no—
100C2×Dic109—22no—
101C2×Dic109C2×D412yes—
102C2×Dic109C2×D414yes—
103C2×Dic109C2×D414yes—
104C2×Dic109#4114yes—
105D436—48no—
106D436—44no—
107C4×D109—28no—
108C4×D109—216no—
109C22×C109—44no—
110C22×C109—42no—
111C22×C109—42no—
112C22×C109—44no—
113C22×C109—42no—
114C22×C109C22×D411yescharacteristic
115C22×C109C22×D412yes—
116C22×C109C22×D411yescharacteristic
117C22×C109C2×#4112yes—
118C22×C109C2×#4111yescharacteristic
119C2×D109—24no—
120C2×D109—22no—
121C2×D109—24no—
122C2×D109—22no—
123C2×D109—42no—
124C2×D109—44no—
125C2×D109—42no—
126C2×D109—44no—
127C2×D109—42no—
128C2×D109#12714yescomplemented
129C4×C109—24no—
130C4×C109—42no—
131C4×C109#12114yescomplemented
132Dic109—22no—
133Dic109—42no—
134Dic109#12118yescomplemented
135C2×C109—82no—
136C2×C109C2×#12112yes—
137C2×C109C2×#12111yescharacteristic
138C2×C109C2×#12711yescharacteristic
139C2×C109#53212yes—
140C2×C109#53211yescharacteristic
141D109—24no—
142D109—82no—
143#2856—1091nomaximal, Sylow
144C109#285611yescharacteristic, complemented, Sylow
145C2×#127—1091no—
146C2×#121—1091no—
147C2×#121—1091no—
148C2×#121—1094no—
149C22×C42—1091no—
150#390—1094no—
151#390—1091no—
152#390D10911yescharacteristic, complemented
153#384—1091no—
154C22×D4—1092no—
155C22×D4—1094no—
156C22×D4—1092no—
157C22×D4—1091no—
158C22×D4C2×D10911yescharacteristic
159C23×C4—1092no—
160C23×C4—1091no—
161#127—2184no—
162#121—21816no—
163#121—2184no—
164#121—2184no—
165C2×C4⋊C4—1094no—
166C2×C4⋊C4—1091no—
167C2×C4⋊C4C2×D10911yescharacteristic
168C2×C22⋊C4—1092no—
169C2×C22⋊C4—1092no—
170C2×C22⋊C4—1094no—
171C2×C22⋊C4C2×D10914yes—
172C2×C42—1091no—
173C2×C42—1092no—
174C2×C42—2184no—
175C2×C42C2×D10911yescharacteristic
176C24—1091no—
177C24—1092no—
178C24C22×D10912yes—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2188no—
183C2×D4—2184no—
184C2×D4—2188no—
185C2×D4—2188no—
186C2×D4—2188no—
187C2×D4—2188no—
188C2×D4—2184no—
189C2×D4—2182no—
190C2×D4—2182no—
191C2×D4—2182no—
192C22×C4—1092no—
193C22×C4—1091no—
194C22×C4—1098no—
195C22×C4—1094no—
196C22×C4—1092no—
197C22×C4—2184no—
198C22×C4—2184no—
199C22×C4—2182no—
200C22×C4—2182no—
201C22×C4C22×D10912yes—
202C22×C4C22×D10912yes—
203C22×C4C22×D10911yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—2188no—
206C4⋊C4—2182no—
207C22⋊C4—28no—
208C22⋊C4—2184no—
209C22⋊C4—2184no—
210C22⋊C4—2188no—
211C42—22no—
212C42—1094no—
213C42—2184no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—1094no—
220C23—1092no—
221C23—2184no—
222C23—2182no—
223C23—2182no—
224C23—2184no—
225C23—2182no—
226C23—2181no—
227C23—2182no—
228C23—2181no—
229C23C23×D10911yescharacteristic
230D4—44no—
231D4—4368no—
232D4—43616no—
233D4—4368no—
234D4—4364no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—1094no—
240C2×C4—1094no—
241C2×C4—1094no—
242C2×C4—1092no—
243C2×C4—21816no—
244C2×C4—2184no—
245C2×C4—2182no—
246C2×C4—2181no—
247C2×C4—2181no—
248C2×C4—2188no—
249C2×C4D4×D10914yes—
250C2×C4D4×D10912yes—
251C2×C4D4×D10912yes—
252C2×C4#1318202912yes—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—1094no—
259C22—4364no—
260C22—4362no—
261C22—4362no—
262C22—4364no—
263C22—4362no—
264C22—2184no—
265C22—2182no—
266C22—2184no—
267C22—2182no—
268C22C2×D4×D10911yescharacteristic
269C22C2×D4×D10912yes—
270C22C2×D4×D10911yescharacteristic
271C22#41×D10912yes—
272C22C2×#1318202911yescharacteristic
273C4—24no—
274C4—42no—
275C4—1098no—
276C4—4362no—
277C4—2182no—
278C4#1681267214yescomplemented
279C2—82no—
280C2—8722no—
281C2—2184no—
282C2#121×D10912yes—
283C2C2×#1681267211yescharacteristic
284C2#127×D10911yescharacteristic
285C2#2029557312yes—
286C2#2029557511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#390×C109C2
hypercenter#390D109
hypocenterC109#2856
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812695C2semidirect
2. C2×#16812674C2semidirect
3. C2×#16812672C2semidirect
4. C2×#16812670C2semidirect
5. C2×C42×D109C2semidirect
6. #20294928C2semidirect
7. #20294926C2semidirect
8. #20294895C2semidirect
9. #390×C109C2semidirect
51. C2×C4×D109D4semidirect
52. C2×C4×D109D4semidirect
59. C4×Dic109D4semidirect
128. C2×D109#127semidirect
131. C4×C109#121semidirect
134. Dic109#121semidirect
144. C109#2856semidirect
152. #390D109semidirect
278. C4#16812672semidirect

Maximal subgroups

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

subgroupindex
C2×#168126952
C2×#168126742
C2×#168126722
C2×#168126702
C2×C42×D1092
#202949282
#202949262
#202948952
#390×C1092
#2856109