Finite Group Databasebeta

← #28338422

Subgroups — #28338422

Order 13696.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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#20293441C211yesmaximal, characteristic, complemented
2C2×#16812053C211yesmaximal, characteristic, complemented
3#20292894C211yesmaximal, characteristic, complemented
4#20292444C211yesmaximal, characteristic, complemented
5#20292434C211yesmaximal, characteristic, complemented
6#20292412C211yesmaximal, characteristic, complemented
7#509×C107C211yesmaximal, characteristic, complemented
8#16812156—21no—
9#16812139—21no—
10#16812138—22no—
11#16812135—21no—
12#16812128—21no—
13C2×Dic428C2211yescharacteristic
14Q16×D107—21no—
15#16812115C2211yescharacteristic
16#16812053—22no—
17C2×C107⋊C16C2211yescharacteristic
18#16812045C2211yescharacteristic
19#16812026—21no—
20#135×C107C2211yescharacteristic
21C2×D8×C107C2211yescharacteristic
22SD32×C107—21no—
23D16×C107—21no—
24M32×C107C2211yescharacteristic
25#13133818—41no—
26#13133818D411yescharacteristic
27#13133817—22no—
28#13133817—42no—
29C2×Dic214—21no—
30C2×Dic214—41no—
31Q8×D107—22no—
32Dic428—21no—
33Dic428D412yes—
34#13133808—22no—
35#13133806—22no—
36#13133806—41no—
37#13133803—41no—
38#13133802—21no—
39#13133802D411yescharacteristic
40#13133801—21no—
41C8×D107—21no—
42C107⋊C16—21no—
43#41×C107—21no—
44C2×D4×C107—21no—
45Q16×C107D411yescharacteristic
46SD16×C107—21no—
47D8×C107—21no—
48D8×C107—21no—
49D8×C107D411yescharacteristic
50C2×C8×C107C2311yescharacteristic
51C16×C107—21no—
52C23×C107—41no—
53Q8×C107—21no—
54D4×C107—21no—
55D4×C107—41no—
56D4×C107—41no—
57D4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107C2×D411yescharacteristic
60C22⋊D107—21no—
61C22⋊D107—42no—
62C2×Dic107—42no—
63C2×Dic107—41no—
64D428D811yescharacteristic
65C4×D107—21no—
66C4×D107—42no—
67Dic214—22no—
68Dic214—42no—
69Dic214—41no—
70Dic214D811yescharacteristic
71C8×C107C2×D411yescharacteristic
72C8×C107C2×D411yescharacteristic
73C107⋊C8—21no—
74C107⋊C8—41no—
75C22×C107—41no—
76C22×C107—81no—
77C22×C107—161no—
78C22×C107AGL(1,ℤ/8)11yescharacteristic
79C2×D107—21no—
80C4×C107—41no—
81C4×C107#12011yescharacteristic
82C4×C107C2×D811yescharacteristic
83Dic107—21no—
84Dic107—42no—
85C2×C107—21no—
86C2×C107—81no—
87C2×C107—161no—
88C2×C107#44711yescharacteristic
89D107—41no—
90#2193—1071nomaximal, Sylow
91C107#219311yescharacteristic, complemented, Sylow
92#578—1071no—
93#509D10711yescharacteristic, complemented
94C2×SD32—1071no—
95#472—1071no—
96#362—1071no—
97#359—1071no—
98#350—1071no—
992−1+4—2141no—
100#137—2141no—
101#137—2142no—
102AGL(1,ℤ/8)—2141no—
103#135—2141no—
104#135C2×D10711yescharacteristic
105C2×Q16—1071no—
106C2×Q16—2141no—
107C2×D8C2×D10711yescharacteristic
108#131—1071no—
109SD32—21no—
110SD32—2142no—
111D16—21no—
112M32C2×D10711yescharacteristic
113C2×C16—1071no—
114#108—1071no—
115#100—2141no—
116#41—21no—
117#41—1071no—
118#41—4281no—
119#41—4282no—
120#41—2142no—
121C2×Q8—4281no—
122C2×Q8—2142no—
123C2×Q8—2141no—
124C2×D4—21no—
125Q16—1072no—
126Q16—2142no—
127Q16—2141no—
128Q16C22⋊D10711yescharacteristic
129SD16—21no—
130SD16—4281no—
131SD16—2142no—
132D8—21no—
133D8—21no—
134D8—4281no—
135D8C22⋊D10711yescharacteristic
136M16—1071no—
137M16—2141no—
138M16—2141no—
139C2×C8—2141no—
140C2×C8C22×D10711yescharacteristic
141C16—21no—
142C16—2141no—
143C23—41no—
144Q8—21no—
145Q8—1071no—
146Q8—4282no—
147Q8—4281no—
148Q8—2142no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—1071no—
154D4—4282no—
155D4—2141no—
156C2×C4—41no—
157C2×C4—4282no—
158C2×C4—4281no—
159C2×C4—4282no—
160C2×C4—2141no—
161C2×C4D4×D10711yescharacteristic
162C8—4281no—
163C8—2141no—
164C8D4×D10711yescharacteristic
165C8C2×C22⋊D10711yescharacteristic
166C22—41no—
167C22—81no—
168C22—161no—
169C22—2141no—
170C22#1681213311yescharacteristic
171C4—41no—
172C4—4282no—
173C4—2141no—
174C4#1681207711yescharacteristic
175C4D8×D10711yescharacteristic
176C2—21no—
177C2—81no—
178C2—161no—
179C2—4281no—
180C2#2029275911yescharacteristic
subgroupG/H
centerC2#20292759
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#509×C107C2
socleC2×C107#447
hypercenter#509D107
hypocenterC107#2193
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20293441C2semidirect
2. C2×#16812053C2semidirect
3. #20292894C2semidirect
4. #20292444C2semidirect
5. #20292434C2semidirect
6. #20292412C2semidirect
7. #509×C107C2semidirect
91. C107#2193semidirect
93. #509D107semidirect

Maximal subgroups

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

subgroupindex
#202934412
C2×#168120532
#202928942
#202924442
#202924342
#202924122
#509×C1072
#2193107