Finite Group Databasebeta

← #28365596

Subgroups — #28365596

Order 13952.

924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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
2C2×D4×Dic109C211yesmaximal, characteristic, complemented
3C2×#16812661C211yesmaximal, characteristic, complemented
4C2×#16812659C214yesmaximal, complemented
5#20294928C211yesmaximal, characteristic, complemented
6#20294926C214yesmaximal, complemented
7#20294895C211yesmaximal, characteristic, complemented
8C42×Dic109C211yesmaximal, characteristic, complemented
9#390×C109C211yesmaximal, characteristic, complemented
10C22×D436C2211yescharacteristic
11C22×C22⋊D109C2214yes—
12C23×Dic109C2212yes—
13C22×C4×D109C2212yes—
14#16812695C418yescomplemented
15D4×Dic109—24no—
16#16812661—24no—
17#16812659—216no—
18C2×C4⋊Dic109C2211yescharacteristic
19C2×#13182008C2214yes—
20C2×C22⋊Dic109C2212yes—
21C2×#13182006C2212yes—
22C2×#13182006C2214yes—
23C2×C4×Dic109C2212yes—
24C2×C4×Dic109C2211yescharacteristic
25C2×C4×Dic109C2212yes—
26C2×C4×Dic109C2211yescharacteristic
27C22×D4×C109C2211yescharacteristic
28C2×C4⋊C4×C109C2211yescharacteristic
29C2×C22⋊C4×C109C2214yes—
30C2×C42×C109C2211yescharacteristic
31C23×D109C2312yes—
32C2×D436—22no—
33C2×D436—22no—
34C2×D436C2×C414yescomplemented
35C2×C22⋊D109—28no—
36C2×C22⋊D109—28no—
37C2×C22⋊D109C2×C4116yescomplemented
38C22×Dic109—28no—
39C22×Dic109—24no—
40C22×Dic109C2312yes—
41C22×Dic109C2311yescharacteristic
42C22×Dic109C2312yes—
43C22×Dic109C2311yescharacteristic
44C2×C4×D109—28no—
45C2×C4×D109—24no—
46C4⋊Dic109—22no—
47#13182008—28no—
48C22⋊Dic109—24no—
49#13182006—28no—
50#13182006—24no—
51C4×Dic109—24no—
52C4×Dic109C2×C414yescomplemented
53C4×Dic109D418yescomplemented
54C4×Dic109D414yescomplemented
55C24×C109C2312yes—
56C2×D4×C109—22no—
57C2×D4×C109—22no—
58C2×D4×C109C2×C414yescomplemented
59C22×C4×C109C2312yes—
60C22×C4×C109C2312yes—
61C22×C4×C109C2311yescharacteristic
62C4⋊C4×C109—22no—
63C22⋊C4×C109—28no—
64C42×C109—22no—
65C23×C109—24no—
66C23×C109—22no—
67C23×C109—22no—
68C23×C109—24no—
69C23×C109C22×C414yes—
70C23×C109C2411yescharacteristic
71C22×D109—24no—
72C22×D109—22no—
73C22×D109—24no—
74C22×D109—22no—
75C22×D109C22×C414yes—
76D4×C109—28no—
77C2×C4×C109—22no—
78C2×C4×C109—22no—
79C2×C4×C109—24no—
80C2×C4×C109—22no—
81C2×C4×C109C22×C412yes—
82C2×C4×C109C2×D414yes—
83C2×C4×C109C2×D412yes—
84C2×C4×C109#4112yes—
85C22⋊D109—232no—
86C2×Dic109—22no—
87C2×Dic109—24no—
88C2×Dic109—22no—
89C2×Dic109—48no—
90C2×Dic109C22×C414yes—
91C2×Dic109C2×D412yes—
92C2×Dic109C2×D414yes—
93C2×Dic109C2×D414yes—
94C2×Dic109C2×D412yes—
95C2×Dic109#4114yes—
96D436—28no—
97C4×D109—48no—
98C22×C109—28no—
99C22×C109—24no—
100C22×C109—42no—
101C22×C109—44no—
102C22×C109—42no—
103C22×C109C23×C411yescharacteristic
104C22×C109C22×D412yes—
105C22×C109C22×D411yescharacteristic
106C22×C109C2×#4112yes—
107C22×C109C2×#4111yescharacteristic
108C2×D109—24no—
109C2×D109—28no—
110C2×D109—42no—
111C2×D109—44no—
112C2×D109—42no—
113C4×C109—24no—
114C4×C109—42no—
115C4×C109C4×D414yescomplemented
116Dic109—22no—
117Dic109—42no—
118Dic109C4×D418yescomplemented
119Dic109#12714yescomplemented
120C2×C109—44no—
121C2×C109C2×C4×D412yes—
122C2×C109C2×C4×D411yescharacteristic
123C2×C109C2×#12711yescharacteristic
124C2×C109#53212yes—
125C2×C109#53211yescharacteristic
126D109—44no—
127C4×#127—1091nomaximal, Sylow
128C109C4×#12711yescharacteristic, complemented, Sylow
129C2×#127—1091no—
130C2×C4×D4—1091no—
131C2×C4×D4—1091no—
132C2×C4×D4—1094no—
133#390—1094no—
134#390—1091no—
135#390D10911yescharacteristic, complemented
136#384—1091no—
137C43—1091no—
138C22×D4—1094no—
139C22×D4—1091no—
140C22×D4C2×D10911yescharacteristic
141C23×C4—1092no—
142C23×C4—1092no—
143#127—1098no—
144C4×D4—21816no—
145C4×D4—2184no—
146C4×D4—2184no—
147C2×C4⋊C4—1094no—
148C2×C4⋊C4—1091no—
149C2×C4⋊C4C2×D10911yescharacteristic
150C2×C22⋊C4—1092no—
151C2×C22⋊C4—1092no—
152C2×C22⋊C4—1094no—
153C2×C22⋊C4C2×D10914yes—
154C2×C42—1092no—
155C2×C42—1091no—
156C2×C42—1091no—
157C2×C42—1092no—
158C2×C42C2×D10911yescharacteristic
159C24—1092no—
160C24C22×D10912yes—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—10916no—
164C2×D4—1094no—
165C2×D4—2188no—
166C2×D4—2188no—
167C2×D4—2182no—
168C2×D4—2182no—
169C2×D4C4×D10914yescomplemented
170C22×C4—1092no—
171C22×C4—1091no—
172C22×C4—1091no—
173C22×C4—1092no—
174C22×C4—2188no—
175C22×C4—2184no—
176C22×C4—2188no—
177C22×C4—2184no—
178C22×C4C22×D10912yes—
179C22×C4C22×D10912yes—
180C22×C4C22×D10911yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—2188no—
183C4⋊C4—2182no—
184C22⋊C4—28no—
185C22⋊C4—2184no—
186C22⋊C4—2184no—
187C22⋊C4—2188no—
188C42—22no—
189C42—1094no—
190C42—1098no—
191C42—1094no—
192C42—2184no—
193C23—24no—
194C23—22no—
195C23—24no—
196C23—22no—
197C23—1094no—
198C23—2184no—
199C23—2182no—
200C23—2184no—
201C23—2182no—
202C23C2×C4×D10914yes—
203C23C23×D10911yescharacteristic
204D4—28no—
205D4—21832no—
206D4—2188no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—1094no—
212C2×C4—1094no—
213C2×C4—1094no—
214C2×C4—1092no—
215C2×C4—1094no—
216C2×C4—1092no—
217C2×C4—4368no—
218C2×C4—4368no—
219C2×C4—2184no—
220C2×C4—2182no—
221C2×C4—2182no—
222C2×C4C2×C4×D10912yes—
223C2×C4D4×D10914yes—
224C2×C4D4×D10912yes—
225C2×C4#1318202812yes—
226C22—28no—
227C22—24no—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—4362no—
232C22—4364no—
233C22—4362no—
234C22—2188no—
235C22—2184no—
236C22C22×C4×D10911yescharacteristic
237C22C2×D4×D10912yes—
238C22C2×D4×D10911yescharacteristic
239C22#41×D10912yes—
240C22C2×#1318202811yescharacteristic
241C4—24no—
242C4—42no—
243C4—1098no—
244C4—1094no—
245C4—4362no—
246C4—2182no—
247C4#1681265914yescomplemented
248C2—44no—
249C2—4364no—
250C2C4×D4×D10912yes—
251C2C2×#1681265911yescharacteristic
252C2#127×D10911yescharacteristic
253C2#2029557112yes—
254C2#2029557511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#390×C109C2
socle, cosocleC23×C109C24
prime radicalC2×#16812695C2
hypercenter#390D109
hypocenterC109C4×#127
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic109C2semidirect
3. C2×#16812661C2semidirect
4. C2×#16812659C2semidirect
5. #20294928C2semidirect
6. #20294926C2semidirect
7. #20294895C2semidirect
8. C42×Dic109C2semidirect
9. #390×C109C2semidirect
14. #16812695C4semidirect
34. C2×D436C2×C4semidirect
37. C2×C22⋊D109C2×C4semidirect
52. C4×Dic109C2×C4semidirect
53. C4×Dic109D4semidirect
54. C4×Dic109D4semidirect
58. C2×D4×C109C2×C4semidirect
115. C4×C109C4×D4semidirect
118. Dic109C4×D4semidirect
119. Dic109#127semidirect
128. C109C4×#127semidirect
135. #390D109semidirect
169. C2×D4C4×D109semidirect
247. C4#16812659semidirect

Maximal subgroups

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

subgroupindex
C2×#168126952
C2×D4×Dic1092
C2×#168126612
C2×#168126592
#202949282
#202949262
#202948952
C42×Dic1092
#390×C1092
C4×#127109