Finite Group Databasebeta

← #28335589

Subgroups — #28335589

Order 13696.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293297C211yesmaximal, characteristic
2#20292999C211yesmaximal, characteristic, complemented
3#20292923C211yesmaximal, characteristic, complemented
4#20292918C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic, complemented
7#331×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9C2×#13133806—22no—
10#16812109C2211yescharacteristic
11#16812105—21no—
12#16812081—22no—
13#16812080—21no—
14C4×Dic214C2211yescharacteristic
15C4×D428C2211yescharacteristic
16D4×Dic107—21no—
17#16812043—21no—
18#16812041—21no—
19#16812034—21no—
20#16812031—21no—
21C4×C107⋊C8C2211yescharacteristic
22#127×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214—21no—
28C2×Dic214D411yescharacteristic
29C2×D428D411yescharacteristic
30C2×C22⋊D107—22no—
31C2×C22⋊D107—22no—
32C22×Dic107—22no—
33C2×C4×D107—21no—
34#13133806—44no—
35C2×C107⋊C8—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107D411yescharacteristic
39C4⋊Dic107D411yescharacteristic
40Dic107⋊C4—21no—
41C22⋊Dic107—22no—
42(C2×D107)⋊C4—21no—
43C4×Dic107—21no—
44C2×D4×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107D412yes—
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—22no—
51C22×D107—21no—
52D4×C107—42no—
53D4×C107—41no—
54D4×C107—41no—
55D4×C107—41no—
56D4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C22⋊D107—44no—
61C22⋊D107—82no—
62C2×Dic107—21no—
63C2×Dic107—21no—
64C2×Dic107—21no—
65C2×Dic107—44no—
66C2×Dic107—41no—
67D428—21no—
68D428—21no—
69C4×D107—41no—
70C4×D107—41no—
71Dic214—21no—
72Dic214—42no—
73Dic214SD1612yescomplemented
74C8×C107—41no—
75C8×C107—41no—
76C107⋊C8—22no—
77C22×C107—42no—
78C22×C107—42no—
79C22×C107—42no—
80C22×C107#12011yescharacteristic
81C2×D107—41no—
82C2×D107—41no—
83C2×D107—41no—
84C4×C107—21no—
85C4×C107C2×SD1611yescharacteristic
86C4×C107#13511yescharacteristic
87C4×C107AGL(1,ℤ/8)11yescharacteristic
88C4×C107#13711yescharacteristic
89Dic107—22no—
90Dic107—41no—
91Dic107—41no—
92Dic107—81no—
93C2×C107—82no—
94C2×C107#45011yescharacteristic
95C2×C107#45211yescharacteristic
96C2×C107#45311yescharacteristic
97D107—81no—
98#1631—1071nomaximal, Sylow
99C107#163111yescharacteristic, complemented, Sylow
100#550—1071no—
101#492—1071no—
102#477—1071no—
103#476—1071no—
104#331D10711yescharacteristic, complemented
105Q8⋊C8—1071no—
106D4⋊C8—1071no—
107C2×#41—2141no—
108C2×SD16—2142no—
109C4⋊Q8—1071no—
110#127—2141no—
111#127C2×D10711yescharacteristic
112#121—2141no—
113#121—2142no—
114C4×Q8—1071no—
115C4×D4—1071no—
116C4×D4—2141no—
117#107—2141no—
118#106—2141no—
119C4⋊C8C2×D10711yescharacteristic
120C4⋊C8C2×D10711yescharacteristic
121#103—2141no—
122#102—2141no—
123C4×C8—1071no—
124#41—4282no—
125#41—4282no—
126C2×Q8—1071no—
127C2×Q8—2141no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—1071no—
131C2×D4—2142no—
132C2×D4—2142no—
133C2×D4D42812yes—
134C22×C4—2142no—
135C22×C4—2141no—
136SD16—4284no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—2141no—
140C4⋊C4—1071no—
141C4⋊C4—1071no—
142C4⋊C4—2141no—
143C4⋊C4—2141no—
144C4⋊C4—2141no—
145C22⋊C4—2142no—
146C22⋊C4—2141no—
147C42—2141no—
148C42C22×D10711yescharacteristic
149C23—22no—
150C23—2141no—
151Q8—1072no—
152Q8—4282no—
153Q8—2141no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—8562no—
160D4—4284no—
161D4—2141no—
162D4—2141no—
163C2×C4—4281no—
164C2×C4—4284no—
165C2×C4—4281no—
166C2×C4—4281no—
167C2×C4—2141no—
168C2×C4—2141no—
169C2×C4—2141no—
170C2×C4D4×D10711yescharacteristic
171C2×C4D4×D10711yescharacteristic
172C2×C4C2×D42811yescharacteristic
173C8—41no—
174C8—41no—
175C8—2142no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—4281no—
180C22—4281no—
181C22—4281no—
182C22#1681207911yescharacteristic
183C4—21no—
184C4—8561no—
185C4—4281no—
186C4—4281no—
187C4—2142no—
188C4SD16×D10711yescharacteristic
189C4#1681212811yescharacteristic
190C4#1681213311yescharacteristic
191C4#1681213811yescharacteristic
192C2—82no—
193C2—8561no—
194C2#2029277911yescharacteristic
195C2#2029279011yescharacteristic
196C2#2029279711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107#120
prime radical#20293297C2
hypercenter#331D107
hypocenterC107#1631
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20292999C2semidirect
3. #20292923C2semidirect
4. #20292918C2semidirect
5. #20292357C2semidirect
6. #20292354C2semidirect
7. #331×C107C2semidirect
73. Dic214SD16semidirect
99. C107#1631semidirect
104. #331D107semidirect

Maximal subgroups

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

subgroupindex
#202932972
#202929992
#202929232
#202929182
#202923572
#202923542
#331×C1072
#1631107