Finite Group Databasebeta

← #28363478

Subgroups — #28363478

Order 13952.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#20295537C211yesmaximal, characteristic
2C2×#16812658C211yesmaximal, characteristic, complemented
3C2×#16812625C212yesmaximal, complemented
4#20294787C212yesmaximal, complemented
5#407×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2211yescharacteristic
7#16812682—21no—
8#16812678C412yescomplemented
9#16812673C412yescomplemented
10#16812668—21no—
11C4×C22⋊D109—22no—
12#16812658—24no—
13#16812658C2211yescharacteristic
14C2×C4⋊Dic109—21no—
15C2×C22⋊Dic109—21no—
16C2×C4×Dic109C2212yes—
17#16812625—24no—
18C23×C4×C109C2211yescharacteristic
19C2×M16×C109C2212yes—
20C22⋊C8×C109—22no—
21#13182029—22no—
22#13182029—22no—
23C2×Dic218C2×C411yescharacteristic
24C2×D436C2×C411yescharacteristic
25C2×C22⋊D109—22no—
26C2×C22⋊D109—21no—
27C22×Dic109—22no—
28C2×C4×D109—21no—
29C4⋊Dic109—24no—
30C4⋊Dic109C2×C412yes—
31#13182008—22no—
32C22⋊Dic109—24no—
33C22⋊Dic109—22no—
34C22⋊Dic109—21no—
35#13182006—22no—
36C4×Dic109—22no—
37C4×Dic109—21no—
38C4×Dic109D414yescomplemented
39C24×C109D411yescharacteristic
40C22×C4×C109—22no—
41C22×C4×C109—22no—
42C22×C4×C109C2×C412yes—
43C22×C4×C109D411yescharacteristic
44C22×C4×C109D412yes—
45C22×C4×C109C2311yescharacteristic
46M16×C109—22no—
47M16×C109—22no—
48C2×C8×C109—22no—
49C23×C109—21no—
50C23×C109—21no—
51C23×C109—42no—
52C23×C109C22⋊C412yes—
53C23×C109C2×D411yescharacteristic
54C22×D109—21no—
55C2×C4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109—24no—
58C2×C4×C109—24no—
59C2×C4×C109C22⋊C412yes—
60C2×C4×C109C22×C411yescharacteristic
61C2×C4×C109C2×D412yes—
62C2×C4×C109C2×D411yescharacteristic
63C2×C4×C109#4112yes—
64C22⋊D109—44no—
65C22⋊D109—41no—
66C22⋊D109—41no—
67C2×Dic109—22no—
68C2×Dic109—21no—
69C2×Dic109—24no—
70C2×Dic109—44no—
71D436—22no—
72C4×D109—42no—
73Dic218—22no—
74C8×C109—42no—
75C22×C109—22no—
76C22×C109—22no—
77C22×C109—22no—
78C22×C109—22no—
79C22×C109—42no—
80C22×C109—42no—
81C22×C109C2×C22⋊C411yescharacteristic
82C22×C109#12011yescharacteristic
83C22×C109#12311yescharacteristic
84C2×D109—41no—
85C2×D109—42no—
86C4×C109—22no—
87C4×C109C4×D412yes—
88C4×C109#12112yes—
89Dic109—44no—
90Dic109—41no—
91Dic109—42no—
92C2×C109—21no—
93C2×C109—21no—
94C2×C109—42no—
95C2×C109#38611yescharacteristic
96C2×C109#42112yes—
97D109—81no—
98#1897—1091nomaximal, Sylow
99C109#189711yescharacteristic, complemented, Sylow
100#525—1091no—
101C2×#117—1091no—
102C2×#104—1092no—
103#407D10911yescharacteristic, complemented
104#339—1092no—
105C2×#41—1091no—
106C23×C4C2×D10911yescharacteristic
107C2×M16C2×D10912yes—
108#123—2181no—
109C22⋊Q8—1092no—
110#121—1092no—
111#120—2181no—
112C4×D4—2182no—
113#117—1091no—
114#117—2184no—
115C2×C4⋊C4—2181no—
116C2×C22⋊C4—2181no—
117C2×C42—1092no—
118#104—2184no—
119C22⋊C8—22no—
120C24D43611yescharacteristic
121#41—2182no—
122#41—2182no—
123C2×Q8—1091no—
124C2×D4—1091no—
125C2×D4—2182no—
126C2×D4—2181no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—2182no—
130C22×C4—2181no—
131C22×C4C4×D10912yes—
132C22×C4D43611yescharacteristic
133C22×C4C22⋊D10912yes—
134C22×C4C22×D10911yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—1092no—
139C4⋊C4—2184no—
140C4⋊C4—2182no—
141C22⋊C4—2184no—
142C22⋊C4—2182no—
143C22⋊C4—2181no—
144C22⋊C4—2182no—
145C42—1094no—
146C42—2182no—
147C42—2181no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2181no—
152C23#1318200612yes—
153C23C2×D43611yescharacteristic
154Q8—2182no—
155D4—4364no—
156D4—4361no—
157D4—4361no—
158D4—2182no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—4364no—
164C2×C4—4362no—
165C2×C4—2182no—
166C2×C4—2181no—
167C2×C4—2184no—
168C2×C4#1318200612yes—
169C2×C4C2×C4×D10911yescharacteristic
170C2×C4D4×D10912yes—
171C2×C4C2×C22⋊D10911yescharacteristic
172C2×C4#1318202812yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—4362no—
181C22—4361no—
182C22C2×#1318200611yescharacteristic
183C22#1681266911yescharacteristic
184C22#1681268311yescharacteristic
185C4—22no—
186C4—4364no—
187C4—4362no—
188C4—4361no—
189C4#1681265912yes—
190C4#1681267112yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—8721no—
195C2#2029490311yescharacteristic
196C2#2029504212yes—
subgroupG/H
centerC2×C4C2×C22⋊D109
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#407×C109C2
socleC22×C109C2×C22⋊C4
prime radical#20295537C2
cosocleC22×C4×C109C23
hypercenter#407D109
hypocenterC109#1897
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16812658C2semidirect
3. C2×#16812625C2semidirect
4. #20294787C2semidirect
5. #407×C109C2semidirect
8. #16812678C4semidirect
9. #16812673C4semidirect
38. C4×Dic109D4semidirect
99. C109#1897semidirect
103. #407D109semidirect

Maximal subgroups

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

subgroupindex
#202955372
C2×#168126582
C2×#168126252
#202947872
#407×C1092
#1897109