Finite Group Databasebeta

← #28337009

Subgroups — #28337009

Order 13696.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#20293394C211yesmaximal, characteristic, complemented
2#20292611C211yesmaximal, characteristic, complemented
3C2×#16812026C212yesmaximal, complemented
4#20292397C212yesmaximal, complemented
5#535×C107C211yesmaximal, characteristic
6#16812116—24no—
7C2×#13133802—22no—
8C2×#13133802C2212yes—
9C2×#13133802C2212yes—
10#16812028—21no—
11#16812026—24no—
12#16812026—21no—
13C2×#41×C107C2211yescharacteristic
14C22×D4×C107C2211yescharacteristic
15#127×C107—21no—
16#124×C107—21no—
17#121×C107—22no—
18#120×C107C414yes—
19#117×C107C2211yescharacteristic
20#13133802—24no—
21#13133802—22no—
22#13133802—22no—
23#13133802—22no—
24#13133802D414yescomplemented
25C2×C107⋊C8—24no—
26C2×C107⋊C8—22no—
27C2×C107⋊C8—22no—
28C24×C107C2×C412yes—
29#41×C107—22no—
30#41×C107—22no—
31C2×Q8×C107D411yescharacteristic
32C2×D4×C107—24no—
33C2×D4×C107—21no—
34C2×D4×C107—21no—
35C2×D4×C107—22no—
36C2×D4×C107—22no—
37C2×D4×C107C2×C412yes—
38C2×D4×C107D411yescharacteristic
39C22×C4×C107D412yes—
40C22×C4×C107C2311yescharacteristic
41C4⋊C4×C107—21no—
42C22⋊C4×C107—24no—
43C22⋊C4×C107C2×C412yes—
44C42×C107—21no—
45C23×C107—22no—
46C23×C107—24no—
47C23×C107—82no—
48C23×C107C22⋊C412yes—
49C23×C107C22×C411yescharacteristic
50Q8×C107—22no—
51D4×C107—24no—
52D4×C107—22no—
53D4×C107—41no—
54D4×C107—42no—
55D4×C107—41no—
56D4×C107—84no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—22no—
60C2×C4×C107C22⋊C412yes—
61C2×C4×C107C2×D412yes—
62C2×C4×C107C2×D411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107#4112yes—
65C107⋊C8—24no—
66C107⋊C8—42no—
67C22×C107—22no—
68C22×C107—41no—
69C22×C107—42no—
70C22×C107—44no—
71C22×C107—42no—
72C22×C107—84no—
73C22×C107C2×C22⋊C411yescharacteristic
74C22×C107C4×D412yes—
75C4×C107—22no—
76C4×C107—42no—
77C4×C107#12011yescharacteristic
78C4×C107#12112yes—
79C4×C107#12311yescharacteristic
80C2×C107—21no—
81C2×C107—22no—
82C2×C107—42no—
83C2×C107—82no—
84C2×C107#38611yescharacteristic
85#1914—1071nomaximal, Sylow
86C107#191411yescharacteristic, complemented, Sylow
87#568—1071no—
88#535D10711yescharacteristic
89#413—1071no—
90C2×#100—1072no—
91#344—1072no—
92C2×#41C2×D10711yescharacteristic
93C22×D4C2×D10711yescharacteristic
94#131—2144no—
95C2×M16—1072no—
96C2×M16—1072no—
97C2×M16—2142no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic10714yes—
102#117C2×D10711yescharacteristic
103#101—2141no—
104#100—2144no—
105#100—2141no—
106C24C2×Dic10712yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D10711yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic10712yes—
116C2×D4C22⋊D10711yescharacteristic
117C22×C4C22⋊D10712yes—
118C22×C4C22×D10711yescharacteristic
119M16—1074no—
120M16—2142no—
121M16—2142no—
122M16—2144no—
123M16—2142no—
124C2×C8—2142no—
125C2×C8—2144no—
126C2×C8—2142no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic10712yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic10712yes—
135C23C22×Dic10711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic10712yes—
147C2×C4D4×D10712yes—
148C2×C4C2×C22⋊D10711yescharacteristic
149C2×C4C2×C22⋊D10711yescharacteristic
150C2×C4#1313381712yes—
151C8—4282no—
152C8—2144no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic10711yescharacteristic
160C22D4×Dic10712yes—
161C4—22no—
162C4—42no—
163C4#1681207711yescharacteristic
164C4#1681208112yes—
165C4#1681209011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2029250711yescharacteristic
subgroupG/H
centerC2#20292507
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radical#535×C107C2
socleC2×C107#386
cosocleC22×C4×C107C23
hypercenter#535D107
hypocenterC107#1914
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20293394C2semidirect
2. #20292611C2semidirect
3. C2×#16812026C2semidirect
4. #20292397C2semidirect
24. #13133802D4semidirect
86. C107#1914semidirect

Maximal subgroups

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

subgroupindex
#202933942
#202926112
C2×#168120262
#202923972
#535×C1072
#1914107