Finite Group Databasebeta

← #28335596

Subgroups — #28335596

Order 13696.

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

normal subgroups only

This group has 214 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#20293338C211yesmaximal, characteristic
2#20293006C211yesmaximal, characteristic, complemented
3#20292923C211yesmaximal, characteristic, complemented
4#20292918C211yesmaximal, characteristic
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#333×C107C211yesmaximal, characteristic, complemented
8C2×#13133808—22no—
9#16812109C2211yescharacteristic
10#16812100—21no—
11#16812089—21no—
12#16812085—22no—
13C4×Dic214C2211yescharacteristic
14C4×D428C2211yescharacteristic
15#16812071—21no—
16C4⋊C4×D107—21no—
17#16812043—21no—
18#16812041—21no—
19#16812034—21no—
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×C4×D107—22no—
30#13133808—44no—
31C2×C107⋊C8—21no—
32C4⋊Dic107—22no—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107D411yescharacteristic
36C4⋊Dic107D411yescharacteristic
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39(C2×D107)⋊C4—22no—
40(C2×D107)⋊C4—21no—
41C4×Dic107—21no—
42C2×Q8×C107D412yes—
43C2×C8×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C42×C107C2311yescharacteristic
48C22×D107—21no—
49Q8×C107—42no—
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—
59D428Q1612yescomplemented
60C4×D107—41no—
61C4×D107—41no—
62C4×D107—82no—
63Dic214—21no—
64Dic214—21no—
65Dic214—42no—
66C8×C107—41no—
67C8×C107—41no—
68C107⋊C8—22no—
69C22×C107#12011yescharacteristic
70C2×D107—22no—
71C2×D107—41no—
72C2×D107—41no—
73C4×C107—21no—
74C4×C107—42no—
75C4×C107C2×Q1611yescharacteristic
76C4×C107#13511yescharacteristic
77C4×C107#13711yescharacteristic
78C4×C107#13711yescharacteristic
79Dic107—41no—
80Dic107—41no—
81Dic107—41no—
82Dic107—81no—
83C2×C107#45111yescharacteristic
84C2×C107#45211yescharacteristic
85C2×C107#45311yescharacteristic
86D107—42no—
87#1632—1071nomaximal, Sylow
88C107#163211yescharacteristic, complemented, Sylow
89#554—1071no—
90#494—1071no—
91#477—1071no—
92#476—1071no—
93#333D10711yescharacteristic, complemented
94Q8⋊C8—1071no—
95D4⋊C8—1071no—
96C2×Q16—2142no—
97C4⋊Q8—1071no—
98C4⋊Q8C2×D10711yescharacteristic
99#125—2141no—
100C22⋊Q8—2141no—
101C22⋊Q8—2142no—
102C4×Q8—1071no—
103C4×D4—1071no—
104C4×D4—2141no—
105C2×C4⋊C4—2141no—
106#107—2141no—
107#106—2141no—
108C4⋊C8C2×D10711yescharacteristic
109C4⋊C8C2×D10711yescharacteristic
110#103—2141no—
111#102—2141no—
112C4×C8—1071no—
113C2×Q8—1071no—
114C2×Q8—2141no—
115C2×Q8D42812yes—
116C2×D4—1071no—
117C22×C4—2142no—
118C22×C4—2141no—
119Q16—4284no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2141no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—1071no—
126C4⋊C4—1071no—
127C4⋊C4—2142no—
128C4⋊C4—2142no—
129C4⋊C4—2141no—
130C4⋊C4—2141no—
131C4⋊C4—2141no—
132C22⋊C4—2142no—
133C22⋊C4—2141no—
134C42—2141no—
135C42C22×D10711yescharacteristic
136C23—2141no—
137Q8—42no—
138Q8—4282no—
139Q8—2141no—
140Q8—2141no—
141D4—1072no—
142D4—2141no—
143C2×C4—22no—
144C2×C4—8562no—
145C2×C4—4281no—
146C2×C4—4281no—
147C2×C4—4281no—
148C2×C4—2141no—
149C2×C4—2141no—
150C2×C4—2141no—
151C2×C4D4×D10711yescharacteristic
152C2×C4D4×D10711yescharacteristic
153C2×C4C2×D42811yescharacteristic
154C8—41no—
155C8—41no—
156C8—2142no—
157C22—4281no—
158C22—4281no—
159C22—2142no—
160C22#1681207911yescharacteristic
161C4—21no—
162C4—42no—
163C4—8561no—
164C4—4281no—
165C4—4281no—
166C4—4281no—
167C4Q16×D10711yescharacteristic
168C4#1681213011yescharacteristic
169C4#1681213811yescharacteristic
170C4#1681214111yescharacteristic
171C2—4282no—
172C2#2029278411yescharacteristic
173C2#2029279211yescharacteristic
174C2#2029279711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#333×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#333D107
hypocenterC107#1632
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20293006C2semidirect
3. #20292923C2semidirect
5. #20292357C2semidirect
7. #333×C107C2semidirect
59. D428Q16semidirect
88. C107#1632semidirect
93. #333D107semidirect

Maximal subgroups

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

subgroupindex
#202933382
#202930062
#202929232
#202929182
#202923572
#202923542
#333×C1072
#1632107