Finite Group Databasebeta

← #28371294

Subgroups — #28371294

Order 13952.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20294967C212yesmaximal, complemented
2#20294959C212yesmaximal, complemented
3#20294956C212yesmaximal, complemented
4#20294949C212yesmaximal
5#20294946C212yesmaximal, complemented
6#20294938C211yesmaximal, characteristic
7#20294887C211yesmaximal, characteristic
8#20294884C212yesmaximal
9#399×C109C211yesmaximal, characteristic
10C22×Dic218C2211yescharacteristic
11C23×Dic109C2211yescharacteristic
12C2×C4⋊Dic109C2212yes—
13C2×#13182008C2212yes—
14C2×#13182008C2212yes—
15C2×#13182008C2212yes—
16C2×#13182008C2212yes—
17C2×C22⋊Dic109C2212yes—
18C2×C22⋊Dic109C2211yescharacteristic
19#16812606C2211yescharacteristic
20#16812606C2212yes—
21#16812606C2212yes—
22#16812606C2212yes—
23#16812606C2212yes—
24#16812606C2212yes—
25#16812606C2212yes—
26C2×C4⋊C4×C109C2212yes—
27C2×C4⋊C4×C109C2211yescharacteristic
28C2×C22⋊C4×C109C2211yescharacteristic
29C2×C22⋊C4×C109C2212yes—
30#95×C109C2211yescharacteristic
31C2×Dic218—42no—
32C2×Dic218—41no—
33C22×Dic109—41no—
34C22×Dic109—41no—
35C22×Dic109Q814yescomplemented
36C22×Dic109C2312yes—
37C22×Dic109C2312yes—
38C22×Dic109C2312yes—
39C22×Dic109C2312yes—
40C4⋊Dic109—42no—
41#13182008—42no—
42#13182008—42no—
43#13182008—42no—
44#13182008—42no—
45C22⋊Dic109—41no—
46C22⋊Dic109—42no—
47C24×C109C2311yescharacteristic
48C22×C4×C109C2311yescharacteristic
49C22×C4×C109C2312yes—
50C22×C4×C109C2311yescharacteristic
51C22×C4×C109C2312yes—
52C4⋊C4×C109—41no—
53C4⋊C4×C109—42no—
54C22⋊C4×C109—42no—
55C22⋊C4×C109—41no—
56C23×C109—22no—
57C23×C109—21no—
58C23×C109—21no—
59C23×C109—22no—
60C23×C109C2×Q812yes—
61C23×C109C2411yescharacteristic
62C2×C4×C109—22no—
63C2×C4×C109—21no—
64C2×C4×C109—22no—
65C2×C4×C109—21no—
66C2×C4×C109—22no—
67C2×C4×C109—22no—
68C2×C4×C109—22no—
69C2×C4×C109—22no—
70C2×C4×C109—22no—
71C2×C4×C109—22no—
72C2×Dic109—22no—
73C2×Dic109—22no—
74C2×Dic109—22no—
75C2×Dic109—22no—
76C2×Dic109—22no—
77C2×Dic109—22no—
78C2×Dic109—22no—
79C2×Dic109—22no—
80C2×Dic109—22no—
81C2×Dic109—22no—
82C2×Dic109—22no—
83C2×Dic109—44no—
84C2×Dic109C2×Q814yes—
85Dic218—44no—
86C22×C109—22no—
87C22×C109—24no—
88C22×C109—41no—
89C22×C109—42no—
90C22×C109—41no—
91C22×C109C22×Q811yescharacteristic
92C22×C109H(5,𝔽2)11yescharacteristic
93C22×C109H(5,𝔽2)12yes—
94C22×C109H(5,𝔽2)11yescharacteristic
95C22×C1092−1+412yes—
96C4×C109—41no—
97C4×C109—42no—
98C4×C109—42no—
99C4×C109—41no—
100Dic109—24no—
101Dic109—42no—
102Dic109—42no—
103Dic109—42no—
104C2×C109—42no—
105C2×C109#54311yescharacteristic
106C2×C109#54412yes—
107C2×C109#56011yescharacteristic
108C2×C109#56212yes—
109C2×C109#56311yescharacteristic
110#2841—1091nomaximal, Sylow
111C109#284111yescharacteristic, complemented, Sylow
112#400—1092no—
113#399D10911yescharacteristic
114#398—1092no—
115#398—1092no—
116#396—1092no—
117#395—1092no—
118#393—1091no—
119#381—1091no—
120#380—1092no—
121C22×Q8—1091no—
122C23×C4—1091no—
123C2×C4⋊C4—1092no—
124C2×C4⋊C4—1092no—
125C2×C4⋊C4—1092no—
126C2×C4⋊C4—1092no—
127C2×C4⋊C4—1092no—
128C2×C4⋊C4C2×D10912yes—
129C2×C4⋊C4C2×D10911yescharacteristic
130C2×C22⋊C4—1092no—
131C2×C22⋊C4—1091no—
132C2×C22⋊C4C2×D10912yes—
133C2×C22⋊C4C2×D10911yescharacteristic
134#95—1092no—
135#95—1092no—
136#95—1092no—
137#95—1092no—
138#95—1091no—
139#95—1092no—
140#95—1092no—
141#95C2×D10911yescharacteristic
142C24C22×D10911yescharacteristic
143C2×Q8—4361no—
144C2×Q8—4362no—
145C22×C4—1094no—
146C22×C4—1092no—
147C22×C4—1092no—
148C22×C4—1092no—
149C22×C4—1092no—
150C22×C4—4361no—
151C22×C4—4361no—
152C22×C4C22×D10911yescharacteristic
153C22×C4C22×D10911yescharacteristic
154C22×C4C22×D10912yes—
155C22×C4C22×D10912yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—4362no—
159C4⋊C4—4362no—
160C4⋊C4—4362no—
161C4⋊C4—4362no—
162C4⋊C4—4362no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—4362no—
166C22⋊C4—4361no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D10912yes—
172C23C23×D10911yescharacteristic
173Q8—4364no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1094no—
185C2×C4—4364no—
186C2×C4—2182no—
187C2×C4—2182no—
188C2×C4—2182no—
189C2×C4—2182no—
190C2×C4—2182no—
191C2×C4—2182no—
192C2×C4—2182no—
193C2×C4—2182no—
194C2×C4—2182no—
195C2×C4—2182no—
196C2×C4—2182no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D10911yescharacteristic
203C22#1681274412yes—
204C22#1681274411yescharacteristic
205C22#1681274511yescharacteristic
206C22#1681274612yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—4362no—
212C4—4362no—
213C4—4362no—
214C4—2184no—
215C2—42no—
216C2#2029565111yescharacteristic
217C2#2029565512yes—
218C2#2029576311yescharacteristic
219C2#2029577312yes—
220C2#2029577911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#399×C109C2
prime radicalC24×C109C23
hypercenter#399D109
hypocenterC109#2841
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #20294967C2semidirect
2. #20294959C2semidirect
3. #20294956C2semidirect
5. #20294946C2semidirect
35. C22×Dic109Q8semidirect
111. C109#2841semidirect

Maximal subgroups

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

subgroupindex
#202949672
#202949592
#202949562
#202949492
#202949462
#202949382
#202948872
#202948842
#399×C1092
#2841109