Finite Group Databasebeta

← #28337008

Subgroups — #28337008

Order 13696.

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

normal subgroups only

This group has 340 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#20293191C211yesmaximal, characteristic
2#20292611C211yesmaximal, characteristic, complemented
3C2×#16812025C212yesmaximal, complemented
4#20292399C212yesmaximal, complemented
5#568×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7C2×#13133802C2212yes—
8#16812105—21no—
9#16812098—21no—
10#16812083—22no—
11#16812077C414yes—
12#16812068C2211yescharacteristic
13#16812028—21no—
14#16812026—21no—
15#16812025—24no—
16C2×#41×C107C2211yescharacteristic
17#131×C107—24no—
18C2×M16×C107—22no—
19C2×M16×C107C2212yes—
20C23×D107C2×C412yes—
21C2×D428—21no—
22C2×D428—21no—
23C2×D428—22no—
24C2×D428—22no—
25C2×C22⋊D107—24no—
26#13133802—22no—
27#13133802—22no—
28C2×C107⋊C8—22no—
29C4⋊Dic107—21no—
30C22⋊Dic107C2×C412yes—
31(C2×D107)⋊C4—24no—
32C4×Dic107—21no—
33#41×C107—22no—
34#41×C107—22no—
35C2×Q8×C107D411yescharacteristic
36C2×D4×C107C2×C412yes—
37C2×D4×C107D411yescharacteristic
38C22×C4×C107D412yes—
39C22×C4×C107C2311yescharacteristic
40M16×C107—24no—
41M16×C107—22no—
42M16×C107D414yescomplemented
43C2×C8×C107—24no—
44C2×C8×C107—22no—
45C23×C107C22⋊C412yes—
46C23×C107C22×C411yescharacteristic
47C22×D107—22no—
48C22×D107—24no—
49C22×D107—82no—
50Q8×C107—22no—
51D4×C107—24no—
52D4×C107—22no—
53C2×C4×C107—22no—
54C2×C4×C107—22no—
55C2×C4×C107C22⋊C412yes—
56C2×C4×C107C2×D412yes—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107#4112yes—
60C22⋊D107—84no—
61C2×Dic107—22no—
62D428—41no—
63D428—42no—
64D428—41no—
65C8×C107—24no—
66C107⋊C8—42no—
67C22×C107—22no—
68C22×C107—41no—
69C22×C107—42no—
70C22×C107C2×C22⋊C411yescharacteristic
71C22×C107C4×D412yes—
72C2×D107—44no—
73C2×D107—42no—
74C2×D107—84no—
75C4×C107—22no—
76C4×C107#12011yescharacteristic
77C4×C107#12112yes—
78C4×C107#12311yescharacteristic
79Dic107—42no—
80C2×C107—21no—
81C2×C107—22no—
82C2×C107—42no—
83C2×C107#38611yescharacteristic
84D107—82no—
85#1914—1071nomaximal, Sylow
86C107#191411yescharacteristic, complemented, Sylow
87#568D10711yescharacteristic, complemented
88#535—1071no—
89#413—1071no—
90C2×#100—1072no—
91#344—1072no—
92C2×#41C2×D10711yescharacteristic
93C22×D4—1071no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1072no—
97C2×M16C2×D10712yes—
98#127—2141no—
99#124—2141no—
100#121—2142no—
101#120—1074no—
102#117—1071no—
103#101—2141no—
104#100—2141no—
105#100—2144no—
106C24—1072no—
107#41—22no—
108#41—22no—
109C2×Q8D42811yescharacteristic
110C2×D4—2144no—
111C2×D4—2142no—
112C2×D4—2141no—
113C2×D4—2141no—
114C2×D4—2142no—
115C2×D4C4×D10712yes—
116C2×D4D42811yescharacteristic
117C22×C4C22⋊D10712yes—
118C22×C4C22×D10711yescharacteristic
119M16—24no—
120M16—22no—
121M16—2142no—
122M16—2142no—
123M16C22⋊D10714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—2142no—
127C4⋊C4—2141no—
128C22⋊C4—1072no—
129C22⋊C4—2144no—
130C42—2141no—
131C23—8562no—
132C23—2144no—
133C23—2142no—
134C23(C2×D107)⋊C412yes—
135C23C2×C4×D10711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—8564no—
140D4—4281no—
141D4—4282no—
142D4—4281no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—2142no—
146C2×C4(C2×D107)⋊C412yes—
147C2×C4C2×C22⋊D10712yes—
148C2×C4C2×C22⋊D10711yescharacteristic
149C2×C4C2×D42811yescharacteristic
150C2×C4#1313381812yes—
151C8—24no—
152C8—4282no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—8564no—
157C22—4284no—
158C22—4282no—
159C22C2×(C2×D107)⋊C411yescharacteristic
160C22C4×C22⋊D10712yes—
161C4—22no—
162C4—4282no—
163C4#1681207811yescharacteristic
164C4#1681208312yes—
165C4#1681209211yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—8562no—
170C2#2029250611yescharacteristic
subgroupG/H
centerC2#20292506
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#568×C107C2
socleC2×C107#386
prime radical#20293191C2
cosocleC22×C4×C107C23
hypercenter#568D107
hypocenterC107#1914
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20292611C2semidirect
3. C2×#16812025C2semidirect
4. #20292399C2semidirect
5. #568×C107C2semidirect
42. M16×C107D4semidirect
86. C107#1914semidirect
87. #568D107semidirect
123. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931912
#202926112
C2×#168120252
#202923992
#568×C1072
#1914107