Finite Group Databasebeta

← #28362156

Subgroups — #28362156

Order 13952.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#20295696C211yesmaximal, characteristic
2#20295318C211yesmaximal, characteristic, complemented
3#20295313C211yesmaximal, characteristic, complemented
4#20294770C211yesmaximal, characteristic, complemented
5#20294758C211yesmaximal, characteristic, complemented
6#20294755C211yesmaximal, characteristic, complemented
7#492×C109C211yesmaximal, characteristic, complemented
8C2×#13182028—21no—
9#16812695—21no—
10#16812671—22no—
11#16812670—21no—
12C4×Dic218C2211yescharacteristic
13C4×D436C2211yescharacteristic
14D4×Dic109—21no—
15#16812632—21no—
16#16812630—21no—
17C4⋊#11724784C2211yescharacteristic
18C4⋊#11724784C2211yescharacteristic
19#16812622—21no—
20#16812619—21no—
21C2×SD16×C109—22no—
22C4⋊Q8×C109C2211yescharacteristic
23#127×C109C2211yescharacteristic
24C4×C8×C109C2211yescharacteristic
25#13182028—42no—
26#13182028—42no—
27C2×Dic218D411yescharacteristic
28C2×D436D411yescharacteristic
29C2×C22⋊D109—22no—
30C2×C22⋊D109—22no—
31C22×Dic109—22no—
32C2×C4×D109—21no—
33C2×#11724784—21no—
34C2×#11724784—21no—
35C4⋊Dic109D411yescharacteristic
36C4⋊Dic109D411yescharacteristic
37#13182008—21no—
38C22⋊Dic109—22no—
39#13182006—21no—
40C4×Dic109—21no—
41C2×Q8×C109—21no—
42C2×D4×C109—21no—
43C2×D4×C109—21no—
44C2×D4×C109D412yes—
45SD16×C109—44no—
46C2×C8×C109—21no—
47C4⋊C4×C109—21no—
48C4⋊C4×C109—21no—
49C42×C109C2311yescharacteristic
50C23×C109—22no—
51C22×D109—21no—
52Q8×C109—42no—
53D4×C109—42no—
54D4×C109—41no—
55D4×C109—41no—
56D4×C109—41no—
57D4×C109—41no—
58C2×C4×C109—41no—
59C2×C4×C109C2×D411yescharacteristic
60C2×C4×C109C2×D411yescharacteristic
61C2×C4×C109C2×D411yescharacteristic
62C22⋊D109—44no—
63C22⋊D109—82no—
64C2×Dic109—21no—
65C2×Dic109—21no—
66C2×Dic109—21no—
67C2×Dic109—44no—
68D436—21no—
69D436—21no—
70C4×D109—41no—
71C4×D109—41no—
72Dic218—21no—
73Dic218SD1612yescomplemented
74C8×C109—22no—
75#11724784—41no—
76#11724784—41no—
77C22×C109—42no—
78C22×C109—42no—
79C22×C109—42no—
80C22×C109#12011yescharacteristic
81C2×D109—41no—
82C2×D109—41no—
83C2×D109—41no—
84C4×C109—21no—
85C4×C109—81no—
86C4×C109C2×SD1611yescharacteristic
87C4×C109#13511yescharacteristic
88C4×C109AGL(1,ℤ/8)11yescharacteristic
89C4×C109#13711yescharacteristic
90Dic109—22no—
91Dic109—41no—
92Dic109—41no—
93C2×C109—82no—
94C2×C109#45011yescharacteristic
95C2×C109#45211yescharacteristic
96C2×C109#45311yescharacteristic
97D109—81no—
98#1631—1091nomaximal, Sylow
99C109#163111yescharacteristic, complemented, Sylow
100#550—1091no—
101#492D10911yescharacteristic, complemented
102#477—1091no—
103#476—1091no—
104#331—1091no—
105Q8⋊C8—1091no—
106D4⋊C8—1091no—
107C2×#41—2181no—
108C2×SD16—22no—
109C4⋊Q8C2×D10911yescharacteristic
110#127—2181no—
111#127C2×D10911yescharacteristic
112#121—2181no—
113#121—2182no—
114C4×Q8—1091no—
115C4×D4—1091no—
116C4×D4—2181no—
117#107—2181no—
118#106—2181no—
119C4⋊C8—1091no—
120C4⋊C8—1091no—
121#103—2181no—
122#102—2181no—
123C4×C8C2×D10911yescharacteristic
124#41—4362no—
125#41—4362no—
126C2×Q8—21no—
127C2×Q8—1091no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—1091no—
131C2×D4—2182no—
132C2×D4—2182no—
133C2×D4C22⋊D10912yes—
134C22×C4—2182no—
135C22×C4—2181no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—2181no—
139C2×C8—2181no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—1091no—
143C4⋊C4—1091no—
144C4⋊C4—2181no—
145C22⋊C4—2182no—
146C22⋊C4—2181no—
147C42—2181no—
148C42C22×D10911yescharacteristic
149C23—22no—
150C23—2181no—
151Q8—42no—
152Q8—1092no—
153Q8—2181no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—8722no—
160D4—4364no—
161D4—2181no—
162D4—2181no—
163C2×C4—41no—
164C2×C4—4364no—
165C2×C4—4361no—
166C2×C4—4361no—
167C2×C4—2181no—
168C2×C4—2181no—
169C2×C4—2181no—
170C2×C4D4×D10911yescharacteristic
171C2×C4D4×D10911yescharacteristic
172C2×C4C2×C22⋊D10911yescharacteristic
173C8—22no—
174C8—4361no—
175C8—4361no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—4361no—
180C22—4361no—
181C22—4361no—
182C22#1681266711yescharacteristic
183C4—21no—
184C4—81no—
185C4—4361no—
186C4—4361no—
187C4—2182no—
188C4SD16×D10911yescharacteristic
189C4#1681272011yescharacteristic
190C4#1681272611yescharacteristic
191C4#1681272811yescharacteristic
192C2—82no—
193C2—8721no—
194C2#2029517511yescharacteristic
195C2#2029518511yescharacteristic
196C2#2029519311yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#492×C109C2
socleC22×C109#120
prime radical#20295696C2
hypercenter#492D109
hypocenterC109#1631
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20295318C2semidirect
3. #20295313C2semidirect
4. #20294770C2semidirect
5. #20294758C2semidirect
6. #20294755C2semidirect
7. #492×C109C2semidirect
73. Dic218SD16semidirect
99. C109#1631semidirect
101. #492D109semidirect

Maximal subgroups

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

subgroupindex
#202956962
#202953182
#202953132
#202947702
#202947582
#202947552
#492×C1092
#1631109