Finite Group Databasebeta

← #28362949

Subgroups — #28362949

Order 13952.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13182029C211yesmaximal, characteristic
2C2×#16812613C214yesmaximal, complemented
3#20294889C211yesmaximal, characteristic, complemented
4#381×C109C211yesmaximal, characteristic, complemented
5C2×#13182029—28no—
6C2×#13182029—22no—
7C2×#13182029—22no—
8C22×Dic218C411yescharacteristic, complemented
9C22×D436C411yescharacteristic, complemented
10C22×C22⋊D109C2212yes—
11C22×C4×D109C412yescomplemented
12C2×C22⋊Dic109C2212yes—
13#16812613—28no—
14#16812606—22no—
15C23×C4×C109C2211yescharacteristic
16C2×C22⋊C4×C109C2212yes—
17#95×C109—22no—
18C23×D109C2×C412yes—
19#13182029—216no—
20#13182029—216no—
21C2×Dic218—22no—
22C2×Dic218—24no—
23C2×D436—24no—
24C2×D436—22no—
25C2×C22⋊D109—24no—
26C2×C22⋊D109—22no—
27C2×C22⋊D109—22no—
28C2×C22⋊D109D418yes—
29C22×Dic109—22no—
30C22×Dic109C2×C412yes—
31C2×C4×D109—24no—
32C2×C4×D109—24no—
33C2×C4×D109—24no—
34C22⋊Dic109—24no—
35C24×C109C2311yescharacteristic
36C22×C4×C109—24no—
37C22×C4×C109—22no—
38C22×C4×C109C2×C411yescharacteristic
39C22×C4×C109C2×C411yescharacteristic
40C22×C4×C109D412yes—
41C22×C4×C109D412yes—
42C22⋊C4×C109—24no—
43C23×C109—42no—
44C23×C109C22×C411yescharacteristic
45C23×C109C2×D414yes—
46C23×C109C2×D411yescharacteristic
47C23×C109C2×D411yescharacteristic
48C22×D109—22no—
49C22×D109—22no—
50C22×D109—44no—
51C22×D109C22⋊C414yes—
52C2×C4×C109—22no—
53C2×C4×C109—22no—
54C2×C4×C109—24no—
55C2×C4×C109—24no—
56C2×C4×C109—44no—
57C2×C4×C109—42no—
58C2×C4×C109C22⋊C412yes—
59C2×C4×C109C22⋊C412yes—
60C22⋊D109—28no—
61C22⋊D109—28no—
62C2×Dic109—22no—
63C2×Dic109—22no—
64C2×Dic109—44no—
65C2×Dic109—42no—
66C2×Dic109C22⋊C414yes—
67D436—28no—
68C4×D109—216no—
69Dic218—28no—
70C22×C109—22no—
71C22×C109—22no—
72C22×C109—22no—
73C22×C109—44no—
74C22×C109C2×C22⋊C412yes—
75C22×C109C2×C22⋊C411yescharacteristic
76C22×C109#12012yes—
77C22×C109#12012yes—
78C2×D109—24no—
79C2×D109—44no—
80C2×D109—44no—
81C2×D109—44no—
82C4×C109—22no—
83C4×C109—22no—
84C4×C109—82no—
85Dic109—24no—
86Dic109—82no—
87C2×C109—22no—
88C2×C109—22no—
89C2×C109—22no—
90C2×C109#37911yescharacteristic
91C2×C109#41012yes—
92D109—44no—
93#1782—1091nomaximal, Sylow
94C109#178211yescharacteristic, complemented, Sylow
95C22×#41—1091no—
96C2×#99—1094no—
97#381—1091no—
98#381D10911yescharacteristic, complemented
99C2×#41—2188no—
100C2×#41—2182no—
101C2×#41—2182no—
102C22×Q8—1091no—
103C22×D4—1092no—
104C22×D4—1091no—
105C23×C4—1092no—
106C23×C4C2×D10911yescharacteristic
107C2×C22⋊C4—1092no—
108C2×C22⋊C4C2×D10912yes—
109#99—2188no—
110#95—22no—
111#95—2182no—
112C24—1092no—
113C24C22×D10911yescharacteristic
114#41—21816no—
115#41—21816no—
116C2×Q8—2184no—
117C2×Q8—2182no—
118C2×D4—1098no—
119C2×D4—2184no—
120C2×D4—2184no—
121C2×D4—2182no—
122C2×D4—2182no—
123C2×D4—2182no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—1092no—
127C22×C4—2184no—
128C22×C4—2184no—
129C22×C4—2182no—
130C22×C4—2184no—
131C22×C4C4×D10911yescharacteristic
132C22×C4C4×D10911yescharacteristic
133C22×C4D43612yes—
134C22×C4C22⋊D10912yes—
135C22⋊C4—24no—
136C22⋊C4—2184no—
137C23—42no—
138C23—1094no—
139C23—4364no—
140C23—2182no—
141C23—2182no—
142C23C2×C4×D10911yescharacteristic
143C23D4×D10914yes—
144C23C2×C22⋊D10911yescharacteristic
145C23C2×D43611yescharacteristic
146Q8—2188no—
147D4—2188no—
148D4—2188no—
149D4—2188no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—1094no—
157C2×C4—4364no—
158C2×C4—4362no—
159C2×C4—21816no—
160C2×C4—2182no—
161C2×C4—2182no—
162C2×C4#1318200612yes—
163C2×C4#1318200612yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—4364no—
169C22—4364no—
170C22—4364no—
171C22—2184no—
172C22C22⋊C4×D10912yes—
173C22C2×#1318200611yescharacteristic
174C22#1681266712yes—
175C22#1681266912yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—8722no—
180C4—2184no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—4364no—
185C2#2029488111yescharacteristic
186C2#2029499912yes—
subgroupG/H
centerC22C2×#13182006
commutatorC23×C109C22×C4
FrattiniC24C22×D109
Fitting#381×C109C2
socleC22×C109C2×C22⋊C4
prime radicalC22×#13182029C2
cosocleC24×C109C23
hypercenter#381D109
hypocenterC109#1782
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#16812613C2semidirect
3. #20294889C2semidirect
4. #381×C109C2semidirect
8. C22×Dic218C4semidirect
9. C22×D436C4semidirect
11. C22×C4×D109C4semidirect
94. C109#1782semidirect
98. #381D109semidirect

Maximal subgroups

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

subgroupindex
C22×#131820292
C2×#168126132
#202948892
#381×C1092
#1782109