Finite Group Databasebeta

← #28338421

Subgroups — #28338421

Order 13696.

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

normal subgroups only

This group has 220 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#20293428C211yesmaximal, characteristic, complemented
2#20293065C211yesmaximal, characteristic, complemented
3C2×#16812050C211yesmaximal, characteristic, complemented
4#20292898C211yesmaximal, characteristic, complemented
5#20292433C211yesmaximal, characteristic, complemented
6#20292412C211yesmaximal, characteristic, complemented
7#361×C107C211yesmaximal, characteristic, complemented
8#16812154—21no—
9#16812137—21no—
10#16812135—21no—
11#16812133—22no—
12#16812131C2211yescharacteristic
13#16812128—21no—
14C2×D856C2211yescharacteristic, complemented
15D8×D107—21no—
16#16812115C2211yescharacteristic
17#16812054—21no—
18D1712—21no—
19#16812050—22no—
20C2×C107⋊C16C2211yescharacteristic, complemented
21#16812025—21no—
22C2×D8×C107C2211yescharacteristic
23M32×C107C2211yescharacteristic, complemented
24#108×C107C2211yescharacteristic, complemented
25#13133818—21no—
26#13133818D411yescharacteristic
27#13133817—22no—
28#13133817—41no—
29C2×D428—21no—
30C2×C22⋊D107—42no—
31D4×D107—22no—
32D4×D107—41no—
33Dic428D411yescharacteristic
34#13133807—21no—
35#13133807—41no—
36#13133806—22no—
37D856—21no—
38D856—21no—
39D856—41no—
40D856D411yescharacteristic
41#13133803—22no—
42#13133802D411yescharacteristic
43#13133801—21no—
44C8×D107—21no—
45C107⋊C16—21no—
46C2×D4×C107—21no—
47D8×C107—21no—
48D8×C107D412yes—
49M16×C107—21no—
50C2×C8×C107C2311yescharacteristic
51C16×C107—21no—
52C23×C107—41no—
53C22×D107—42no—
54C22×D107—41no—
55D4×C107—22no—
56D4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C22⋊D107—21no—
59C22⋊D107—42no—
60C22⋊D107—42no—
61C22⋊D107—41no—
62C2×Dic107—42no—
63D428—21no—
64D428—41no—
65D428—41no—
66D428D811yescharacteristic
67C4×D107—21no—
68C4×D107—41no—
69Dic214—21no—
70Dic214D811yescharacteristic
71C8×C107—41no—
72C8×C107C2×D411yescharacteristic
73C8×C107C2×D411yescharacteristic
74C107⋊C8—21no—
75C22×C107—42no—
76C22×C107—161no—
77C22×C107AGL(1,ℤ/8)11yescharacteristic
78C2×D107—21no—
79C2×D107—41no—
80C2×D107—81no—
81C2×D107—162no—
82C2×D107—161no—
83C4×C107#12011yescharacteristic
84C4×C107C2×D811yescharacteristic
85Dic107—21no—
86Dic107—41no—
87C2×C107—21no—
88C2×C107—82no—
89C2×C107#44711yescharacteristic
90D107—41no—
91D107—81no—
92D107—161no—
93#2192—1071nomaximal, Sylow
94C107#219211yescharacteristic, complemented, Sylow
95#576—1071no—
96#509—1071no—
97C2×D16—1071no—
98#472—1071no—
99#361D10711yescharacteristic, complemented
100#359—1071no—
101#350—1071no—
102H(5,𝔽2)—2141no—
103AGL(1,ℤ/8)—2141no—
104AGL(1,ℤ/8)—2141no—
105AGL(1,ℤ/8)—2142no—
106#135—1071no—
107#135—2141no—
108C2×D8—1071no—
109C2×D8—2141no—
110C2×D8C2×D10711yescharacteristic
111#131—1071no—
112SD32—2141no—
113D16—2141no—
114D16—2142no—
115M32C2×D10711yescharacteristic, complemented
116C2×C16—1071no—
117#108C2×D10711yescharacteristic, complemented
118#100—2141no—
119#41—1071no—
120#41—4281no—
121#41—2142no—
122#41—2141no—
123C2×D4—21no—
124C2×D4—4281no—
125C2×D4—4282no—
126C2×D4—2142no—
127C2×D4—2141no—
128Q16—1071no—
129SD16—4281no—
130SD16—2142no—
131SD16—2141no—
132D8—21no—
133D8—1071no—
134D8—4281no—
135D8—2142no—
136D8—2141no—
137D8—2141no—
138D8D42812yes—
139M16—21no—
140M16—1071no—
141M16—2141no—
142C2×C8—2141no—
143C2×C8C22×D10711yescharacteristic
144C16—21no—
145C16—2141no—
146C23—41no—
147C23—4282no—
148C23—4281no—
149Q8—1071no—
150Q8—2141no—
151D4—22no—
152D4—41no—
153D4—1071no—
154D4—4282no—
155D4—4281no—
156D4—4282no—
157D4—4281no—
158D4—4281no—
159D4—2141no—
160D4—2141no—
161C2×C4—4282no—
162C2×C4—4281no—
163C2×C4—2141no—
164C2×C4D4×D10711yescharacteristic
165C8—41no—
166C8—2141no—
167C8D4×D10711yescharacteristic
168C8C2×D42811yescharacteristic
169C22—42no—
170C22—161no—
171C22—8561no—
172C22—4281no—
173C22—17121no—
174C22—17122no—
175C22—2141no—
176C22#1681213611yescharacteristic
177C4—4281no—
178C4—2141no—
179C4#1681207911yescharacteristic
180C4D8×D10711yescharacteristic
181C2—21no—
182C2—82no—
183C2—8561no—
184C2—4281no—
185C2—17121no—
186C2#2029276211yescharacteristic
subgroupG/H
centerC2#20292762
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#361×C107C2
socleC2×C107#447
hypercenter#361D107
hypocenterC107#2192
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #20293428C2semidirect
2. #20293065C2semidirect
3. C2×#16812050C2semidirect
4. #20292898C2semidirect
5. #20292433C2semidirect
6. #20292412C2semidirect
7. #361×C107C2semidirect
14. C2×D856C22semidirect
20. C2×C107⋊C16C22semidirect
23. M32×C107C22semidirect
24. #108×C107C22semidirect
94. C107#2192semidirect
99. #361D107semidirect
115. M32C2×D107semidirect
117. #108C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934282
#202930652
C2×#168120502
#202928982
#202924332
#202924122
#361×C1072
#2192107