Finite Group Databasebeta

← #28335634

Subgroups — #28335634

Order 13696.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293351C211yesmaximal, characteristic
2#20292979C211yesmaximal, characteristic, complemented
3#20292923C211yesmaximal, characteristic, complemented
4#20292918C211yesmaximal, characteristic
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#332×C107C211yesmaximal, characteristic, complemented
8#16812109C2211yescharacteristic
9#16812104—22no—
10#16812098—21no—
11#16812089—21no—
12Q8×Dic107—21no—
13C4×Dic214C2211yescharacteristic
14C4×D428C2211yescharacteristic
15#16812067—21no—
16#16812043—21no—
17#16812041—21no—
18#16812034—21no—
19#16812032—22no—
20#16812031—21no—
21C4×C107⋊C8C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×D428D411yescharacteristic
28C2×C4×D107—21no—
29C2×C107⋊C8—21no—
30C4⋊Dic107—22no—
31C4⋊Dic107—21no—
32C4⋊Dic107—21no—
33C4⋊Dic107D411yescharacteristic
34C4⋊Dic107D411yescharacteristic
35Dic107⋊C4—21no—
36(C2×D107)⋊C4—22no—
37(C2×D107)⋊C4—22no—
38(C2×D107)⋊C4—21no—
39C4×Dic107—22no—
40C4×Dic107—21no—
41C2×Q8×C107—21no—
42C2×C8×C107—21no—
43C2×C8×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D412yes—
46C42×C107C2311yescharacteristic
47C22×D107—21no—
48Q8×C107—41no—
49Q8×C107—41no—
50C2×C4×C107—22no—
51C2×C4×C107C2×D411yescharacteristic
52C2×C4×C107C2×D411yescharacteristic
53C2×C4×C107C2×D411yescharacteristic
54C2×Dic107—21no—
55C2×Dic107—21no—
56C2×Dic107—21no—
57C2×Dic107—41no—
58D428—21no—
59D428—21no—
60C4×D107—41no—
61C4×D107—41no—
62Dic214—21no—
63Dic214—21no—
64Dic214—42no—
65C8×C107—41no—
66C8×C107—41no—
67C107⋊C8—41no—
68C22×C107#12011yescharacteristic
69C2×D107—41no—
70C2×D107—41no—
71C2×D107—41no—
72C4×C107—21no—
73C4×C107—42no—
74C4×C107#13511yescharacteristic
75C4×C107#13511yescharacteristic
76C4×C107#13711yescharacteristic
77C4×C107#13711yescharacteristic
78Dic107—41no—
79Dic107—41no—
80Dic107—41no—
81Dic107—81no—
82C2×C107#45211yescharacteristic
83C2×C107#45211yescharacteristic
84C2×C107#45411yescharacteristic
85D107—81no—
86#1638—1071nomaximal, Sylow
87C107#163811yescharacteristic, complemented, Sylow
88#556—1071no—
89#487—1071no—
90#477—1071no—
91#476—1071no—
92#332D10711yescharacteristic, complemented
93Q8⋊C8—1071no—
94D4⋊C8—1071no—
95C4⋊Q8—1071no—
96C4⋊Q8C2×D10711yescharacteristic
97#126—2142no—
98#124—2141no—
99C22⋊Q8—2141no—
100C4×Q8—1071no—
101C4×Q8—2141no—
102C4×D4—1071no—
103#117—2141no—
104#107—2141no—
105#106—2141no—
106C4⋊C8C2×D10711yescharacteristic
107C4⋊C8C2×D10711yescharacteristic
108#103—2141no—
109#103—2142no—
110#102—2141no—
111C4×C8—1071no—
112C2×Q8—21no—
113C2×Q8—1071no—
114C2×Q8—2141no—
115C2×D4—1071no—
116C22×C4—2141no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—2141no—
120C4⋊C4—21no—
121C4⋊C4—1071no—
122C4⋊C4—1071no—
123C4⋊C4—2142no—
124C4⋊C4—2141no—
125C4⋊C4—2141no—
126C4⋊C4—2141no—
127C4⋊C4D42812yes—
128C22⋊C4—2142no—
129C22⋊C4—2142no—
130C22⋊C4—2141no—
131C42—2142no—
132C42—2141no—
133C42C22×D10711yescharacteristic
134C23—2141no—
135Q8—41no—
136Q8—41no—
137Q8—4282no—
138Q8—2141no—
139Q8—2141no—
140D4—2141no—
141D4—2141no—
142C2×C4—22no—
143C2×C4—4281no—
144C2×C4—4281no—
145C2×C4—4281no—
146C2×C4—2141no—
147C2×C4—2141no—
148C2×C4—2141no—
149C2×C4D4×D10711yescharacteristic
150C2×C4D4×D10711yescharacteristic
151C2×C4C2×D42811yescharacteristic
152C8—41no—
153C8—41no—
154C8—4281no—
155C22—4281no—
156C22—4281no—
157C22—4281no—
158C22#1681207911yescharacteristic
159C4—21no—
160C4—42no—
161C4—8561no—
162C4—4281no—
163C4—4281no—
164C4—4281no—
165C4#1681212811yescharacteristic
166C4#1681213011yescharacteristic
167C4#1681213811yescharacteristic
168C4#1681214111yescharacteristic
169C2—8561no—
170C2#2029279011yescharacteristic
171C2#2029279211yescharacteristic
172C2#2029280211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#332×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#332D107
hypocenterC107#1638
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20292979C2semidirect
3. #20292923C2semidirect
5. #20292357C2semidirect
7. #332×C107C2semidirect
87. C107#1638semidirect
92. #332D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202929792
#202929232
#202929182
#202923572
#202923542
#332×C1072
#1638107