Finite Group Databasebeta

← #28378937

Subgroups — #28378937

Order 13952.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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
1D109⋊(C22⋊C8)C211yesmaximal, characteristic
2#20295968C211yesmaximal, characteristic
3#20295899C212yesmaximal, complemented
4#20295899C212yesmaximal, complemented
5#117×D109C211yesmaximal, characteristic
6C2×D109⋊C8C2212yes—
7C2×C4⋊#89137C2212yes—
8C2×C22⋊#89137—21no—
9#16812756—21no—
10C22⋊#11724786—21no—
11(C2×#89137)⋊C4—21no—
12C22×C4×D109C2211yescharacteristic
13#16812658C411yescharacteristic, complemented
14#16812657C411yescharacteristic, complemented
15#16812657C411yescharacteristic, complemented
16#16812656—21no—
17#16812655—21no—
18C4⋊C4×D109C2211yescharacteristic
19C4⋊C4×D109C2211yescharacteristic
20C22⋊C4×D109—21no—
21C42×D109—21no—
22#117×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24D109⋊C8—24no—
25C2×#11724786—22no—
26C4⋊#89137—24no—
27C22⋊#89137—42no—
28C23×D109D411yescharacteristic
29C22×Dic109D411yescharacteristic
30C2×C4×D109—24no—
31C2×C4×D109—21no—
32C2×C4×D109—21no—
33C2×C4×D109C2×C411yescharacteristic
34C2×C4×D109D411yescharacteristic
35C2×C4×D109D411yescharacteristic
36C2×C4×D109C2311yescharacteristic
37C4⋊Dic109C2×C411yescharacteristic
38C4⋊Dic109C2×C411yescharacteristic
39#13182008—21no—
40#13182008—21no—
41C22⋊Dic109—21no—
42#13182006—21no—
43#13182006—21no—
44C4×Dic109—21no—
45C4×Dic109—21no—
46C4×Dic109—21no—
47C22×C4×C109C2×C411yescharacteristic
48C4⋊C4×C109C2×C411yescharacteristic
49C4⋊C4×C109C2×C411yescharacteristic
50C22⋊C4×C109—21no—
51C42×C109—21no—
52C23×C109C22⋊C411yescharacteristic
53C22×D109—22no—
54C22×D109—22no—
55C22×D109—21no—
56C22×D109—21no—
57C22×D109C22⋊C411yescharacteristic
58C22×D109C2×D411yescharacteristic
59C2×#89137—42no—
60C2×#89137—42no—
61C2×#89137—42no—
62C2×C4×C109—22no—
63C2×C4×C109—21no—
64C2×C4×C109—21no—
65C2×C4×C109C22⋊C411yescharacteristic
66C2×C4×C109C22×C411yescharacteristic
67C2×Dic109—22no—
68C2×Dic109—21no—
69C2×Dic109—21no—
70C2×Dic109C22⋊C411yescharacteristic
71C2×Dic109C2×D411yescharacteristic
72C4×D109—22no—
73C4×D109—22no—
74C4×D109—22no—
75C4×D109—42no—
76C4×D109—41no—
77C4×D109—41no—
78C4×D109#4114yes—
79#11724786—42no—
80C22×C109—21no—
81C22×C109—21no—
82C22×C109—21no—
83C22×C109C2×C22⋊C411yescharacteristic
84C2×D109—21no—
85C2×D109—21no—
86C2×D109—21no—
87C2×D109—42no—
88C2×D109—42no—
89C2×D109#12312yes—
90C2×D109#13512yes—
91C2×D109AGL(1,ℤ/8)11yescharacteristic
92C2×D109#13711yescharacteristic
93#89137—82no—
94C4×C109—21no—
95C4×C109—41no—
96C4×C109—41no—
97C4×C109#11712yes—
98Dic109—21no—
99Dic109—41no—
100Dic109—41no—
101Dic109#12312yes—
102C2×C109—41no—
103C2×C109#38111yescharacteristic
104C2×C109#41611yescharacteristic
105C2×C109#41711yescharacteristic
106D109—41no—
107D109#48212yes—
108D109#48512yes—
109#1825—1091nomaximal, Sylow
110C109#182511yescharacteristic, complemented, Sylow
111C2×#117—1091no—
112C2×C22⋊C8—1091no—
113#380—1091no—
114#340—1092no—
115#340—1092no—
116C23×C4—1091no—
117C22×C8—1092no—
118#117—1091no—
119#117—1091no—
120#117—1091no—
121#117—2181no—
122#117—2181no—
123#117#8913711yescharacteristic, complemented
124C2×C4⋊C4—1091no—
125C2×C4⋊C4—1091no—
126C2×C4⋊C4—1092no—
127C2×C22⋊C4—2181no—
128C2×C22⋊C4—2181no—
129C2×C42—2181no—
130C22⋊C8—2181no—
131C22⋊C8—2181no—
132#95—2181no—
133C24—1091no—
134C22×C4—1091no—
135C22×C4—1091no—
136C22×C4—1091no—
137C22×C4—1091no—
138C22×C4—1091no—
139C22×C4—2184no—
140C22×C4—2181no—
141C22×C4—2181no—
142C22×C4—2182no—
143C22×C4C2×#8913711yescharacteristic
144C2×C8—2182no—
145C2×C8—2184no—
146C4⋊C4—1091no—
147C4⋊C4—1091no—
148C4⋊C4—2181no—
149C4⋊C4—2181no—
150C4⋊C4—2184no—
151C4⋊C4C2×#8913711yescharacteristic
152C4⋊C4C2×#8913711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—4362no—
155C22⋊C4—2181no—
156C22⋊C4—2181no—
157C22⋊C4—2181no—
158C42—21no—
159C42—2181no—
160C42—2181no—
161C42—2181no—
162C23—1091no—
163C23—1091no—
164C23—2181no—
165C23—2182no—
166C23—2182no—
167C23—2181no—
168C23C22⋊#8913711yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—1091no—
173C2×C4—1094no—
174C2×C4—1091no—
175C2×C4—4362no—
176C2×C4—4361no—
177C2×C4—4361no—
178C2×C4—4362no—
179C2×C4—4362no—
180C2×C4—4362no—
181C2×C4—2182no—
182C2×C4—2182no—
183C2×C4—2182no—
184C2×C4—2181no—
185C2×C4—2181no—
186C2×C4—2182no—
187C2×C4C22⋊#8913711yescharacteristic
188C2×C4C22×#8913711yescharacteristic
189C8—4362no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—1092no—
194C22—1092no—
195C22—1091no—
196C22—1091no—
197C22—4362no—
198C22—4362no—
199C22—2181no—
200C22—2181no—
201C22—2181no—
202C22C2×C22⋊#8913711yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—8722no—
207C4—1092no—
208C4—4361no—
209C4—4361no—
210C4—2181no—
211C4#1681277812yes—
212C2—41no—
213C2—1092no—
214C2—1092no—
215C2—4361no—
216C2#2029596911yescharacteristic
217C2#2029601611yescharacteristic
218C2#2029601711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#117×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC2×C4×D109C23
hypercenter#117#89137
hypocenterC109#1825
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20295899C2semidirect
4. #20295899C2semidirect
13. #16812658C4semidirect
14. #16812657C4semidirect
15. #16812657C4semidirect
22. #117×C109C4semidirect
110. C109#1825semidirect
123. #117#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
#202959682
#202958992
#117×D1092
#1825109