Finite Group Databasebeta

← #28363415

Subgroups — #28363415

Order 13952.

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

normal subgroups only

This group has 332 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#20295572C211yesmaximal, characteristic
2#20295042C212yesmaximal, complemented
3#20295000C211yesmaximal, characteristic, complemented
4#20294787C212yesmaximal, complemented
5#568×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2211yescharacteristic
7#16812697C411yescharacteristic, complemented
8#16812695C411yescharacteristic, complemented
9#16812688C411yescharacteristic, complemented
10#16812686C411yescharacteristic, complemented
11#16812671—22no—
12C4×C22⋊D109—22no—
13#16812658C2212yes—
14C2×C4×Dic109C2211yescharacteristic
15#16812625—24no—
16#16812614—22no—
17C2×#41×C109C2211yescharacteristic
18#131×C109—24no—
19C2×M16×C109—22no—
20C2×M16×C109C2212yes—
21#13182029—22no—
22#13182029—22no—
23C2×Dic218C2×C411yescharacteristic
24C2×D436C2×C411yescharacteristic
25C2×C22⋊D109—22no—
26C2×C22⋊D109—21no—
27C22×Dic109—21no—
28C2×C4×D109—21no—
29C4⋊Dic109—22no—
30#13182008—22no—
31C22⋊Dic109—22no—
32C22⋊Dic109—22no—
33#13182006—22no—
34C4×Dic109—21no—
35C4×Dic109—22no—
36C4×Dic109C2×C411yescharacteristic
37C4×Dic109C2×C411yescharacteristic
38#41×C109—22no—
39#41×C109—22no—
40C2×Q8×C109C2×C411yescharacteristic
41C2×D4×C109C2×C411yescharacteristic
42C2×D4×C109D412yes—
43C22×C4×C109D412yes—
44C22×C4×C109C2311yescharacteristic
45M16×C109—24no—
46M16×C109—22no—
47M16×C109D414yescomplemented
48C2×C8×C109—24no—
49C2×C8×C109—22no—
50C23×C109C22⋊C412yes—
51C23×C109C2×D411yescharacteristic
52C22×D109—21no—
53Q8×C109—22no—
54D4×C109—24no—
55D4×C109—22no—
56C2×C4×C109—22no—
57C2×C4×C109—22no—
58C2×C4×C109C22⋊C412yes—
59C2×C4×C109C22×C411yescharacteristic
60C2×C4×C109C2×D412yes—
61C2×C4×C109C2×D411yescharacteristic
62C2×C4×C109#4112yes—
63C22⋊D109—44no—
64C22⋊D109—42no—
65C2×Dic109—21no—
66C2×Dic109—21no—
67C2×Dic109—21no—
68C2×Dic109—42no—
69C2×Dic109—42no—
70D436—22no—
71C4×D109—41no—
72C4×D109—41no—
73Dic218—22no—
74C8×C109—24no—
75C22×C109—22no—
76C22×C109—41no—
77C22×C109—42no—
78C22×C109C2×C22⋊C411yescharacteristic
79C22×C109#12112yes—
80C2×D109—42no—
81C2×D109—41no—
82C4×C109—22no—
83C4×C109C4×D412yes—
84C4×C109#12011yescharacteristic
85C4×C109#12311yescharacteristic
86Dic109—22no—
87Dic109—41no—
88Dic109—41no—
89Dic109—82no—
90C2×C109—21no—
91C2×C109—22no—
92C2×C109—42no—
93C2×C109#38611yescharacteristic
94D109—81no—
95#1884—1091nomaximal, Sylow
96C109#188411yescharacteristic, complemented, Sylow
97#568D10911yescharacteristic, complemented
98#532—1091no—
99#421—1092no—
100#410—1091no—
101#339—1092no—
102C2×#41—1091no—
103C2×#41C2×D10911yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D10912yes—
107C4⋊Q8—1091no—
108#127—1091no—
109#124—1091no—
110#124—1091no—
111#121—2182no—
112C4×D4—2182no—
113#117—1092no—
114C2×C42—1091no—
115#104—2184no—
116#99—2182no—
117#41—22no—
118#41—22no—
119#41—2182no—
120#41—2182no—
121C2×Q8—1091no—
122C2×Q8C4×D10911yescharacteristic
123C2×D4—1091no—
124C2×D4—2182no—
125C2×D4—2181no—
126C2×D4C4×D10911yescharacteristic
127C2×D4D43612yes—
128C22×C4—2181no—
129C22×C4—2181no—
130C22×C4C22⋊D10912yes—
131C22×C4C22×D10911yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D10914yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—2182no—
138C4⋊C4—2182no—
139C22⋊C4—2182no—
140C22⋊C4—2182no—
141C22⋊C4—2182no—
142C42—1091no—
143C42—1091no—
144C42—2182no—
145C42—2181no—
146C23—2181no—
147C23#1318200612yes—
148C23C2×D43611yescharacteristic
149Q8—22no—
150Q8—2182no—
151D4—24no—
152D4—22no—
153D4—4362no—
154D4—4364no—
155D4—2182no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—4362no—
159C2×C4—4362no—
160C2×C4—4361no—
161C2×C4—4361no—
162C2×C4—2181no—
163C2×C4—2181no—
164C2×C4—2181no—
165C2×C4#1318200612yes—
166C2×C4C2×C4×D10911yescharacteristic
167C2×C4C2×C22⋊D10912yes—
168C2×C4C2×C22⋊D10911yescharacteristic
169C2×C4#1318202912yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—4362no—
175C22—4361no—
176C22C2×#1318200611yescharacteristic
177C22#1681267312yes—
178C4—22no—
179C4—8722no—
180C4—4361no—
181C4—4361no—
182C4—2182no—
183C4C4×C22⋊D10912yes—
184C4#1681266811yescharacteristic
185C4#1681268211yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—8721no—
190C2#2029490511yescharacteristic
subgroupG/H
centerC4#16812668
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#568×C109C2
socleC2×C109#386
prime radical#20295572C2
cosocleC22×C4×C109C23
hypercenter#568D109
hypocenterC109#1884
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20295042C2semidirect
3. #20295000C2semidirect
4. #20294787C2semidirect
5. #568×C109C2semidirect
7. #16812697C4semidirect
8. #16812695C4semidirect
9. #16812688C4semidirect
10. #16812686C4semidirect
47. M16×C109D4semidirect
96. C109#1884semidirect
97. #568D109semidirect
134. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955722
#202950422
#202950002
#202947872
#568×C1092
#1884109