Finite Group Databasebeta

← #28335725

Subgroups — #28335725

Order 13696.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293292C211yesmaximal, characteristic
2#20293023C211yesmaximal, characteristic, complemented
3#20292913C211yesmaximal, characteristic, complemented
4#20292833C211yesmaximal, characteristic
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#332×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107—21no—
9C2×#13133807—21no—
10#16812109C2211yescharacteristic
11#16812108—21no—
12#16812087—22no—
13#16812085—21no—
14C4×Dic214C2211yescharacteristic
15C4×D428C2211yescharacteristic
16#16812071—21no—
17#16812041—21no—
18#16812040—22no—
19#16812034—21no—
20#16812034—21no—
21C4×C107⋊C8C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214D411yescharacteristic
28Q8×D107—42no—
29Q8×D107—42no—
30C2×D428D411yescharacteristic
31C2×C4×D107—22no—
32C2×C4×D107—21no—
33#13133807—42no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107D411yescharacteristic
38C4⋊Dic107D411yescharacteristic
39Dic107⋊C4—22no—
40Dic107⋊C4—21no—
41(C2×D107)⋊C4—22no—
42(C2×D107)⋊C4—21no—
43C4×Dic107—21no—
44C2×Q8×C107—21no—
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C4⋊C4×C107—21no—
48C4⋊C4×C107D412yes—
49C42×C107C2311yescharacteristic
50C22×D107—21no—
51Q8×C107—41no—
52Q8×C107—41no—
53C2×C4×C107—22no—
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×Dic107—21no—
58C2×Dic107—21no—
59C2×Dic107—21no—
60C2×Dic107—41no—
61D428—21no—
62D428SD1612yes—
63C4×D107—44no—
64C4×D107—41no—
65C4×D107—41no—
66Dic214—21no—
67Dic214—44no—
68Dic214—41no—
69Dic214—41no—
70Dic214SD1612yescomplemented
71C8×C107—41no—
72C8×C107—41no—
73C107⋊C8—22no—
74C22×C107#12011yescharacteristic
75C2×D107—22no—
76C2×D107—41no—
77C2×D107—41no—
78C4×C107—21no—
79C4×C107—42no—
80C4×C107C2×SD1611yescharacteristic
81C4×C107C2×SD1611yescharacteristic
82C4×C107AGL(1,ℤ/8)11yescharacteristic
83C4×C107#13711yescharacteristic
84Dic107—22no—
85Dic107—41no—
86Dic107—41no—
87Dic107—81no—
88C2×C107#44811yescharacteristic
89C2×C107#45011yescharacteristic
90C2×C107#45611yescharacteristic
91D107—42no—
92#1651—1071nomaximal, Sylow
93C107#165111yescharacteristic, complemented, Sylow
94D4×Q8—1071no—
95#498—1071no—
96#475—1071no—
97#461—1071no—
98#332D10711yescharacteristic, complemented
99Q8⋊C8—1071no—
100D4⋊C8—1071no—
101C22×Q8—2141no—
102C2×SD16—2141no—
103C4⋊Q8—1071no—
104C4⋊Q8—2141no—
105C4⋊Q8C2×D10711yescharacteristic
106C22⋊Q8—2141no—
107C22⋊Q8—2142no—
108C4×Q8—1071no—
109C4×D4—1071no—
110C4×D4—2141no—
111#106—2141no—
112#106—2142no—
113C4⋊C8C2×D10711yescharacteristic
114C4⋊C8C2×D10711yescharacteristic
115#103—2141no—
116#103—2141no—
117C4×C8—1071no—
118C2×Q8—21no—
119C2×Q8—1071no—
120C2×Q8—4282no—
121C2×Q8—4282no—
122C2×Q8—2142no—
123C2×Q8—2141no—
124C2×D4—1071no—
125C22×C4—2142no—
126C22×C4—2141no—
127SD16—4282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2141no—
131C4⋊C4—21no—
132C4⋊C4—1071no—
133C4⋊C4—1071no—
134C4⋊C4—2142no—
135C4⋊C4—2141no—
136C4⋊C4—2141no—
137C4⋊C4—2141no—
138C4⋊C4D42812yes—
139C22⋊C4—2142no—
140C22⋊C4—2141no—
141C42—2141no—
142C42C22×D10711yescharacteristic
143C23—2141no—
144Q8—41no—
145Q8—41no—
146Q8—1072no—
147Q8—4284no—
148Q8—4281no—
149Q8—4281no—
150Q8—2141no—
151D4—1072no—
152D4—2141no—
153C2×C4—22no—
154C2×C4—4284no—
155C2×C4—4281no—
156C2×C4—4281no—
157C2×C4—4281no—
158C2×C4—2141no—
159C2×C4—2141no—
160C2×C4—2141no—
161C2×C4D4×D10711yescharacteristic
162C2×C4D4×D10711yescharacteristic
163C2×C4C2×D42811yescharacteristic
164C8—41no—
165C8—41no—
166C8—2142no—
167C22—4281no—
168C22—4281no—
169C22—2142no—
170C22#1681207911yescharacteristic
171C4—21no—
172C4—42no—
173C4—8561no—
174C4—4281no—
175C4—4281no—
176C4—2142no—
177C4SD16×D10711yescharacteristic
178C4SD16×D10711yescharacteristic
179C4#1681213311yescharacteristic
180C4#1681214111yescharacteristic
181C2—4282no—
182C2#2029276711yescharacteristic
183C2#2029277911yescharacteristic
184C2#2029281211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#332×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#332D107
hypocenterC107#1651
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. #20293023C2semidirect
3. #20292913C2semidirect
5. #20292357C2semidirect
7. #332×C107C2semidirect
70. Dic214SD16semidirect
93. C107#1651semidirect
98. #332D107semidirect

Maximal subgroups

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

subgroupindex
#202932922
#202930232
#202929132
#202928332
#202923572
#202923542
#332×C1072
#1651107