Finite Group Databasebeta

← #28372637

Subgroups — #28372637

Order 13952.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13182012C211yesmaximal, characteristic, complemented
2#20295533C212yesmaximal, complemented
3#20295281C212yesmaximal, complemented
4#20295276C212yesmaximal, complemented
5#20295269C214yesmaximal, complemented
6#20295064C211yesmaximal, characteristic, complemented
7#20295022C212yesmaximal, complemented
8C2×#135×C109C211yesmaximal, characteristic, complemented
9C22×C4×D109C2211yescharacteristic
10C2×#13182012—24no—
11C2×#13182012C2212yes—
12C2×#13182012C2212yes—
13C22×#11724784C2211yescharacteristic
14#16812680—22no—
15#16812675C2214yes—
16#16812670C2214yes—
17#16812667—22no—
18C4×C22⋊D109—24no—
19#16812658C2212yes—
20#16812633C2212yescomplemented
21#16812631C2212yescomplemented
22#16812623C2214yescomplemented
23#16812620C2214yescomplemented
24C2×#41×C109C2212yes—
25#135×C109—24no—
26C2×Q16×C109C2211yescharacteristic, complemented
27C2×SD16×C109C2212yescomplemented
28C2×D8×C109C2211yescharacteristic, complemented
29C22×C8×C109C2211yescharacteristic
30C23×D109D411yescharacteristic
31C2×C22⋊D109—24no—
32C22×Dic109D411yescharacteristic
33C2×C4×D109—22no—
34C2×C4×D109—22no—
35C2×C4×D109D412yes—
36C2×C4×D109C2312yes—
37#13182012—24no—
38#13182012—24no—
39C2×#11724784—22no—
40C2×#11724784C2312yes—
41C4⋊Dic109C2314yes—
42#13182008—24no—
43C22⋊Dic109—22no—
44C22⋊Dic109—24no—
45#13182006—24no—
46C4×Dic109—22no—
47#41×C109—24no—
48#41×C109—24no—
49C2×Q8×C109C2312yes—
50C2×D4×C109—22no—
51C2×D4×C109C2312yes—
52C22×C4×C109—22no—
53C22×C4×C109C2311yescharacteristic
54Q16×C109—22no—
55SD16×C109—24no—
56D8×C109—22no—
57C2×C8×C109D414yescomplemented
58C2×C8×C109C2311yescharacteristic
59C2×C8×C109C2311yescharacteristic
60C23×C109—22no—
61C23×C109C2×D411yescharacteristic
62C22×D109—22no—
63C22×D109—42no—
64C22×D109C2×D412yes—
65Q8×C109—22no—
66Q8×C109—22no—
67D4×C109—22no—
68D4×C109—22no—
69D4×C109—44no—
70C2×C4×C109—22no—
71C2×C4×C109—44no—
72C2×C4×C109C2×D411yescharacteristic
73C2×C4×C109C2×D412yes—
74C2×C4×C109#4112yes—
75C2×C4×C109C2411yescharacteristic
76C22⋊D109—48no—
77C2×Dic109—22no—
78C2×Dic109—24no—
79C2×Dic109C2×D412yes—
80C4×D109—24no—
81C4×D109—22no—
82C4×D109—22no—
83C8×C109C2×D412yes—
84C8×C109C2×D412yes—
85#11724784—22no—
86C22×C109—22no—
87C22×C109—42no—
88C22×C109—44no—
89C22×C109#12112yes—
90C22×C109C22×D411yescharacteristic
91C2×D109—22no—
92C2×D109—24no—
93C2×D109—44no—
94C4×C109—42no—
95C4×C109#12112yes—
96C4×C109C22×D411yescharacteristic
97C4×C109C2×#4111yescharacteristic
98Dic109—22no—
99Dic109—44no—
100C2×C109—22no—
101C2×C109—82no—
102C2×C109C2×#12111yescharacteristic
103C2×C109#57512yes—
104D109—42no—
105#3054—1091nomaximal, Sylow
106C109#305411yescharacteristic, complemented, Sylow
107C2×#135D10911yescharacteristic, complemented
108C22×M16—1091no—
109#525—1092no—
110#470—1092no—
111#469—1092no—
112#468—1094no—
113#427—1091no—
114#416—1092no—
115C2×#41C2×D10912yes—
116C23×C4—1091no—
117#135—24no—
118C2×Q16C2×D10911yescharacteristic, complemented
119C2×SD16C2×D10912yescomplemented
120C2×D8C2×D10911yescharacteristic, complemented
121C2×M16—1092no—
122C2×M16—1092no—
123C2×M16—2184no—
124C22×C8—1091no—
125C22×C8C2×D10911yescharacteristic
126#123—2182no—
127C22⋊Q8—1094no—
128#121—1094no—
129#120—2182no—
130C4×D4—2184no—
131#117—1092no—
132#107—1092no—
133#106—1092no—
134#103—1094no—
135#102—1094no—
136C24—1091no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D10912yes—
140C2×D4—22no—
141C2×D4—2184no—
142C2×D4C22×D10912yes—
143C22×C4—22no—
144C22×C4—1091no—
145C22×C4—1092no—
146C22×C4—1092no—
147C22×C4—2182no—
148C22×C4—2182no—
149C22×C4C22×D10911yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—2184no—
154M16—2184no—
155C2×C8—1092no—
156C2×C8—2182no—
157C2×C8C22⋊D10914yescomplemented
158C2×C8C22×D10911yescharacteristic
159C2×C8C22×D10911yescharacteristic
160C4⋊C4—1094no—
161C4⋊C4—2184no—
162C22⋊C4—2184no—
163C22⋊C4—2184no—
164C22⋊C4—2182no—
165C42—2182no—
166C23—22no—
167C23—1092no—
168C23—4362no—
169C23—2182no—
170C23D4×D10911yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—4368no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—1092no—
180C2×C4—2184no—
181C2×C4—2182no—
182C2×C4—2184no—
183C2×C4—2182no—
184C2×C4—2182no—
185C2×C4D4×D10911yescharacteristic
186C2×C4C2×C22⋊D10912yes—
187C2×C4#1318202812yes—
188C2×C4C23×D10911yescharacteristic
189C8—2182no—
190C8C2×C22⋊D10912yes—
191C8C2×C22⋊D10912yes—
192C22—22no—
193C22—42no—
194C22—44no—
195C22—4364no—
196C22—2182no—
197C22—2184no—
198C22#1681267012yes—
199C22C2×D4×D10911yescharacteristic
200C4—42no—
201C4—4364no—
202C4—2182no—
203C4#1681267012yes—
204C4C22×C22⋊D10911yescharacteristic
205C4C2×#1318202811yescharacteristic
206C2—22no—
207C2—82no—
208C2—4362no—
209C2C2×#1681267011yescharacteristic
210C2#2029582312yes—
subgroupG/H
centerC2×C4D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#135×C109C2
socleC22×C109C22×D4
hypercenterC2×#135D109
hypocenterC109#3054
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#13182012C2semidirect
2. #20295533C2semidirect
3. #20295281C2semidirect
4. #20295276C2semidirect
5. #20295269C2semidirect
6. #20295064C2semidirect
7. #20295022C2semidirect
8. C2×#135×C109C2semidirect
20. #16812633C22semidirect
21. #16812631C22semidirect
22. #16812623C22semidirect
23. #16812620C22semidirect
26. C2×Q16×C109C22semidirect
27. C2×SD16×C109C22semidirect
28. C2×D8×C109C22semidirect
57. C2×C8×C109D4semidirect
106. C109#3054semidirect
107. C2×#135D109semidirect
118. C2×Q16C2×D109semidirect
119. C2×SD16C2×D109semidirect
120. C2×D8C2×D109semidirect
157. C2×C8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C22×#131820122
#202955332
#202952812
#202952762
#202952692
#202950642
#202950222
C2×#135×C1092
#3054109