Finite Group Databasebeta

← #28379010

Subgroups — #28379010

Order 13952.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#13182039C211yesmaximal, characteristic, complemented
2C2×#16812756C211yesmaximal, characteristic, complemented
3C2×C22⋊#11724786C212yesmaximal, complemented
4#20295892C212yesmaximal, complemented
5C2×#16812659C211yesmaximal, characteristic
6C2×#13182039—24no—
7C2×#13182039—22no—
8C22×#11724786C2212yes—
9C22×#11724786C2212yes—
10#16812756—22no—
11C22⋊#11724786—24no—
12C22×C22⋊D109C411yescharacteristic
13C23×Dic109C2211yescharacteristic
14C22×C4×D109C2211yescharacteristic
15#16812659—22no—
16#16812659—22no—
17C2×#13182008C411yescharacteristic
18C2×#13182006C411yescharacteristic
19C2×C4×Dic109C2211yescharacteristic
20C2×C22⋊C4×C109C411yescharacteristic
21#13182039—28no—
22C2×#11724786—22no—
23C2×#11724786—22no—
24C2×#11724786—22no—
25C2×#11724786—22no—
26C2×#11724786—22no—
27C2×#11724786D414yescomplemented
28C23×D109C2×C411yescharacteristic
29C2×C22⋊D109—22no—
30C2×C22⋊D109—22no—
31C2×C22⋊D109—22no—
32C22×Dic109—21no—
33C22×Dic109—21no—
34C22×Dic109—22no—
35C22×Dic109—22no—
36C22×Dic109C2×C411yescharacteristic
37C22×Dic109C2×C411yescharacteristic
38C22×Dic109C2311yescharacteristic
39C2×C4×D109—22no—
40C2×C4×D109—22no—
41C2×C4×D109D412yes—
42C2×C4×D109D412yes—
43#13182008—22no—
44#13182006—22no—
45C4×Dic109—21no—
46C4×Dic109—21no—
47C24×C109C2×C411yescharacteristic
48C22×C4×C109C2×C411yescharacteristic
49C22×C4×C109C2×C411yescharacteristic
50C22⋊C4×C109—22no—
51C23×C109—21no—
52C23×C109—22no—
53C23×C109—21no—
54C23×C109—22no—
55C23×C109M1612yescomplemented
56C23×C109C22×C411yescharacteristic
57C22×D109—22no—
58C22×D109—21no—
59C22×D109—21no—
60C22×D109—22no—
61C22×D109C22⋊C412yes—
62C2×C4×C109—22no—
63C2×C4×C109—21no—
64C2×C4×C109—22no—
65C2×C4×C109C22⋊C412yes—
66C22⋊D109—28no—
67C2×Dic109—21no—
68C2×Dic109—21no—
69C2×Dic109—21no—
70C2×Dic109—22no—
71C2×Dic109—41no—
72C2×Dic109—42no—
73C2×Dic109—41no—
74C2×Dic109M1612yescomplemented
75C2×Dic109C2×D412yes—
76C2×Dic109C2×D411yescharacteristic
77C2×Dic109C2×D411yescharacteristic
78C2×Dic109#4112yes—
79C4×D109—22no—
80C4×D109—24no—
81C4×D109—22no—
82#11724786—24no—
83#11724786—42no—
84C22×C109—22no—
85C22×C109—24no—
86C22×C109—41no—
87C22×C109—41no—
88C22×C109—42no—
89C22×C109C2×C22⋊C411yescharacteristic
90C22×C109C4×D412yes—
91C22×C109C2×M1611yescharacteristic
92C22×C109C2×M1611yescharacteristic
93C22×C109#13112yes—
94C2×D109—22no—
95C2×D109—22no—
96C2×D109—22no—
97C2×D109—42no—
98C2×D109—42no—
99C4×C109—22no—
100C4×C109—41no—
101Dic109—22no—
102Dic109—41no—
103Dic109#12011yescharacteristic
104Dic109#12112yes—
105Dic109#12311yescharacteristic
106C2×C109—42no—
107C2×C109#38611yescharacteristic
108C2×C109#40711yescharacteristic
109C2×C109#40811yescharacteristic
110C2×C109#43512yes—
111C2×C109#43612yes—
112D109—42no—
113#1870—1091nomaximal, Sylow
114C109#187011yescharacteristic, complemented, Sylow
115C22×M16—1091no—
116C2×C4×D4—1091no—
117C2×C22⋊C8—1092no—
118C2×C22⋊C8—1091no—
119#336—1092no—
120C22×D4—1091no—
121C23×C4—1091no—
122C23×C4—1091no—
123C2×M16—2184no—
124C2×M16—2182no—
125C22×C8—1092no—
126C22×C8—1092no—
127C4×D4—2182no—
128C4×D4—2182no—
129C2×C4⋊C4—1091no—
130C2×C22⋊C4—1091no—
131C2×C22⋊C4#8913711yescharacteristic
132C2×C42—1091no—
133C22⋊C8—2184no—
134C22⋊C8—2182no—
135C24—1091no—
136C24C2×#8913711yescharacteristic
137C2×D4—2182no—
138C2×D4—2182no—
139C2×D4—2182no—
140C22×C4—1091no—
141C22×C4—1091no—
142C22×C4—1092no—
143C22×C4—1092no—
144C22×C4—1091no—
145C22×C4—2182no—
146C22×C4—2182no—
147C22×C4—2182no—
148C22×C4—2181no—
149C22×C4—2181no—
150C22×C4—2182no—
151C22×C4C2×#8913711yescharacteristic
152C22×C4C2×#8913711yescharacteristic
153M16—2188no—
154C2×C8—1094no—
155C2×C8—2182no—
156C2×C8—2182no—
157C2×C8—2182no—
158C2×C8—2182no—
159C2×C8—2182no—
160C4⋊C4—2182no—
161C22⋊C4—22no—
162C22⋊C4—2182no—
163C42—2181no—
164C42—2181no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—1092no—
170C23—2182no—
171C23—2181no—
172C23—2181no—
173C23—2182no—
174C23#1318203912yescomplemented
175C23C22×#8913711yescharacteristic
176D4—2188no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—1091no—
181C2×C4—1091no—
182C2×C4—1092no—
183C2×C4—1092no—
184C2×C4—1092no—
185C2×C4—4362no—
186C2×C4—4361no—
187C2×C4—4361no—
188C2×C4—2182no—
189C2×C4—2184no—
190C2×C4—2182no—
191C2×C4—2181no—
192C2×C4—2182no—
193C2×C4—2181no—
194C2×C4—2181no—
195C2×C4C22⋊#8913712yes—
196C8—4362no—
197C8—2184no—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—4362no—
204C22—4362no—
205C22—2182no—
206C22—2182no—
207C22—2182no—
208C22C2×C22⋊#8913711yescharacteristic
209C22D4×#8913712yes—
210C22D109⋊M1611yescharacteristic
211C22C2×#1318203911yescharacteristic
212C22#1681278612yes—
213C4—22no—
214C4—41no—
215C4—1092no—
216C4—1091no—
217C4—1091no—
218C4—4361no—
219C4—2182no—
220C2—42no—
221C2—4362no—
222C2#2029597411yescharacteristic
223C2#2029599711yescharacteristic
224C2#2029600011yescharacteristic
225C2#2029605512yes—
226C2#2029605912yes—
subgroupG/H
center, FrattiniC23C22×#89137
commutator, socleC23×C109C22×C4
FittingC2×C22⋊C4×C109C4
prime radicalC22×C22⋊D109C4
cosocleC22×Dic109C23
hypercenterC2×C22⋊C4#89137
hypocenterC109#1870
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13182039C2semidirect
2. C2×#16812756C2semidirect
3. C2×C22⋊#11724786C2semidirect
4. #20295892C2semidirect
27. C2×#11724786D4semidirect
55. C23×C109M16semidirect
74. C2×Dic109M16semidirect
114. C109#1870semidirect
174. C23#13182039semidirect

Maximal subgroups

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

subgroupindex
C22×#131820392
C2×#168127562
C2×C22⋊#117247862
#202958922
C2×#168126592
#1870109