Finite Group Databasebeta

← #28338416

Subgroups — #28338416

Order 13696.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293432C211yesmaximal, characteristic, complemented
2#20293063C211yesmaximal, characteristic, complemented
3C2×#16812050C211yesmaximal, characteristic, complemented
4#20292439C211yesmaximal, characteristic, complemented
5#20292433C211yesmaximal, characteristic, complemented
6#20292411C211yesmaximal, characteristic, complemented
7#472×C107C211yesmaximal, characteristic, complemented
8#16812155—21no—
9#16812137—22no—
10#16812137—21no—
11#16812131—21no—
12#16812131C2211yescharacteristic
13C2×D856—21no—
14C2×D856C2211yescharacteristic
15#16812053—21no—
16#16812050—22no—
17#16812050—21no—
18#16812049C2211yescharacteristic, complemented
19C2×C107⋊C16C2211yescharacteristic, complemented
20AGL(1,ℤ/8)×C107—21no—
21C2×D8×C107C2211yescharacteristic, complemented
22#131×C107C2211yescharacteristic
23#108×C107C2211yescharacteristic, complemented
24#100×C107—21no—
25#13133819—41no—
26#13133818—22no—
27#13133818—21no—
28C2×D428—22no—
29C2×D428—21no—
30D4×D107—42no—
31D4×D107—41no—
32Dic428D411yescharacteristic
33#13133807—22no—
34#13133807—21no—
35D856—22no—
36D856—21no—
37D856D411yescharacteristic
38D856D412yes—
39C107⋊C16—21no—
40C107⋊C16—21no—
41#41×C107D411yescharacteristic
42C2×D4×C107—21no—
43SD16×C107—41no—
44D8×C107—21no—
45D8×C107—21no—
46D8×C107—41no—
47M16×C107—21no—
48M16×C107—21no—
49M16×C107D411yescharacteristic
50C2×C8×C107—21no—
51C2×C8×C107C2311yescharacteristic
52C23×C107—41no—
53C22×D107—42no—
54C22×D107—41no—
55Q8×C107D811yescharacteristic
56D4×C107—21no—
57D4×C107—41no—
58D4×C107—41no—
59D4×C107D811yescharacteristic
60C2×C4×C107—21no—
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—42no—
63C22⋊D107—41no—
64D428—21no—
65D428—22no—
66D428—42no—
67D428—41no—
68C4×D107—42no—
69C4×D107—41no—
70Dic214—21no—
71C8×C107—21no—
72C8×C107—41no—
73C8×C107C2×D411yescharacteristic
74C8×C107C2×D411yescharacteristic
75C22×C107—21no—
76C22×C107—81no—
77C22×C107—161no—
78C22×C107AGL(1,ℤ/8)11yescharacteristic
79C2×D107—42no—
80C2×D107—41no—
81C2×D107—161no—
82C2×D107—162no—
83C4×C107—21no—
84C4×C107#12011yescharacteristic
85C4×C107C2×D811yescharacteristic
86Dic107—41no—
87C2×C107—21no—
88C2×C107—41no—
89C2×C107—161no—
90C2×C107#44711yescharacteristic
91D107—82no—
92D107—81no—
93#2192—1071nomaximal, Sylow
94C107#219211yescharacteristic, complemented, Sylow
95#576—1071no—
96#509—1071no—
97C2×D16—1071no—
98#472D10711yescharacteristic, complemented
99#361—1071no—
100#359—1071no—
101#350—1071no—
102H(5,𝔽2)—2141no—
103AGL(1,ℤ/8)—21no—
104AGL(1,ℤ/8)—2142no—
105AGL(1,ℤ/8)—2141no—
106#135—1071no—
107#135—2141no—
108C2×D8—1071no—
109C2×D8—2141no—
110C2×D8C2×D10711yescharacteristic, complemented
111#131C2×D10711yescharacteristic
112SD32—2141no—
113D16—2142no—
114D16—2141no—
115M32—1071no—
116C2×C16—1071no—
117#108C2×D10711yescharacteristic, complemented
118#100—21no—
119#41—4281no—
120#41—2142no—
121#41—2141no—
122#41C22⋊D10711yescharacteristic
123C2×D4—21no—
124C2×D4—4282no—
125C2×D4—4281no—
126C2×D4—2142no—
127C2×D4—2141no—
128Q16—1071no—
129SD16—41no—
130SD16—2142no—
131SD16—2141no—
132D8—21no—
133D8—21no—
134D8—41no—
135D8—1072no—
136D8—1071no—
137D8—2142no—
138D8—2141no—
139M16—21no—
140M16—21no—
141M16C22⋊D10711yescharacteristic
142C2×C8—21no—
143C2×C8C22×D10711yescharacteristic
144C16—2141no—
145C16—2141no—
146C23—41no—
147C23—4282no—
148C23—4281no—
149Q8—2141no—
150Q8#1313380311yescharacteristic
151D4—21no—
152D4—41no—
153D4—41no—
154D4—4282no—
155D4—4282no—
156D4—4281no—
157D4—4281no—
158D4—2142no—
159D4—2141no—
160D4#1313380311yescharacteristic
161C2×C4—21no—
162C2×C4—4282no—
163C2×C4—4281no—
164C2×C4C2×C22⋊D10711yescharacteristic
165C8—21no—
166C8—41no—
167C8D4×D10711yescharacteristic
168C8D4×D10711yescharacteristic
169C22—21no—
170C22—81no—
171C22—161no—
172C22—4282no—
173C22—4281no—
174C22—17121no—
175C22—17122no—
176C22#1681213411yescharacteristic
177C4—21no—
178C4—4281no—
179C4#1681207711yescharacteristic
180C4C2×#1313380311yescharacteristic
181C2—21no—
182C2—41no—
183C2—161no—
184C2—8562no—
185C2—8561no—
186C2#2029276011yescharacteristic
subgroupG/H
centerC2#20292760
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#472×C107C2
socleC2×C107#447
hypercenter#472D107
hypocenterC107#2192
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20293432C2semidirect
2. #20293063C2semidirect
3. C2×#16812050C2semidirect
4. #20292439C2semidirect
5. #20292433C2semidirect
6. #20292411C2semidirect
7. #472×C107C2semidirect
18. #16812049C22semidirect
19. C2×C107⋊C16C22semidirect
21. C2×D8×C107C22semidirect
23. #108×C107C22semidirect
94. C107#2192semidirect
98. #472D107semidirect
110. C2×D8C2×D107semidirect
117. #108C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934322
#202930632
C2×#168120502
#202924392
#202924332
#202924112
#472×C1072
#2192107