Finite Group Databasebeta

← #28335613

Subgroups — #28335613

Order 13696.

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

normal subgroups only

This group has 234 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#20293280C211yesmaximal, characteristic
2#20292979C211yesmaximal, characteristic, complemented
3#20292918C211yesmaximal, characteristic, complemented
4#20292918C211yesmaximal, characteristic, complemented
5#20292354C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic, complemented
7#332×C107C211yesmaximal, characteristic, complemented
8C2×#13133819—21no—
9#16812109C2211yescharacteristic
10#16812094—22no—
11#16812085—21no—
12#16812084—21no—
13C4×D428C2211yescharacteristic
14C4×D428C2211yescharacteristic
15C4⋊C4×D107—21no—
16#16812041—21no—
17#16812041—21no—
18#16812032—22no—
19#16812031—21no—
20#16812031—21no—
21C4×C107⋊C8C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25#13133819—42no—
26#13133819—42no—
27C2×Dic214—21no—
28C2×D428—22no—
29C2×D428D411yescharacteristic
30C2×D428D411yescharacteristic
31C2×C4×D107—22no—
32C2×C4×D107—21no—
33C2×C4×D107—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107D411yescharacteristic
38C4⋊Dic107D411yescharacteristic
39Dic107⋊C4—22no—
40(C2×D107)⋊C4—22no—
41(C2×D107)⋊C4—21no—
42(C2×D107)⋊C4—21no—
43C2×Q8×C107—21no—
44C2×C8×C107—21no—
45C2×C8×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107D412yes—
48C42×C107C2311yescharacteristic
49C22×D107—21no—
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—41no—
60D428—21no—
61D428—21no—
62D428—21no—
63D428—44no—
64D428SD1612yes—
65C4×D107—44no—
66C4×D107—41no—
67C4×D107—41no—
68C4×D107—41no—
69C4×D107—41no—
70Dic214—42no—
71C8×C107—41no—
72C8×C107—41no—
73C107⋊C8—41no—
74C22×C107#12011yescharacteristic
75C2×D107—22no—
76C2×D107—41no—
77C2×D107—41no—
78C2×D107—41no—
79C2×D107—41no—
80C2×D107—41no—
81C4×C107—21no—
82C4×C107—42no—
83C4×C107C2×SD1611yescharacteristic
84C4×C107#13511yescharacteristic
85C4×C107AGL(1,ℤ/8)11yescharacteristic
86C4×C107#13711yescharacteristic
87Dic107—41no—
88Dic107—41no—
89Dic107—81no—
90C2×C107#45011yescharacteristic
91C2×C107#45211yescharacteristic
92C2×C107#45411yescharacteristic
93D107—42no—
94D107—81no—
95#1635—1071nomaximal, Sylow
96C107#163511yescharacteristic, complemented, Sylow
97#547—1071no—
98#487—1071no—
99#476—1071no—
100#476—1071no—
101#332D10711yescharacteristic, complemented
102D4⋊C8—1071no—
103D4⋊C8—1071no—
104C2×#41—2141no—
105C4⋊Q8—1071no—
106C4⋊Q8C2×D10711yescharacteristic
107#123—2142no—
108C22⋊Q8—2141no—
109#121—2141no—
110C4×D4—1071no—
111C4×D4—1071no—
112C2×C4⋊C4—2141no—
113#106—2141no—
114#106—2141no—
115C4⋊C8C2×D10711yescharacteristic
116C4⋊C8C2×D10711yescharacteristic
117#103—2142no—
118#102—2141no—
119#102—2141no—
120C4×C8—1071no—
121#41—4282no—
122#41—4282no—
123C2×Q8—21no—
124C2×Q8—2141no—
125C2×D4—1071no—
126C2×D4—1071no—
127C2×D4—2142no—
128C22×C4—2142no—
129C22×C4—2141no—
130C22×C4—2141no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2141no—
134C4⋊C4—21no—
135C4⋊C4—1071no—
136C4⋊C4—1071no—
137C4⋊C4—2142no—
138C4⋊C4—2141no—
139C4⋊C4—2141no—
140C4⋊C4D42812yes—
141C22⋊C4—2142no—
142C22⋊C4—2141no—
143C22⋊C4—2141no—
144C42C22×D10711yescharacteristic
145C23—2141no—
146C23—2141no—
147Q8—41no—
148Q8—41no—
149Q8—4282no—
150D4—1072no—
151D4—4284no—
152D4—2141no—
153D4—2141no—
154D4—2141no—
155C2×C4—22no—
156C2×C4—4284no—
157C2×C4—4281no—
158C2×C4—4281no—
159C2×C4—4281no—
160C2×C4—4281no—
161C2×C4—4281no—
162C2×C4—2141no—
163C2×C4—2141no—
164C2×C4D4×D10711yescharacteristic
165C2×C4D4×D10711yescharacteristic
166C2×C4C2×D42811yescharacteristic
167C8—41no—
168C8—41no—
169C8—4281no—
170C22—4281no—
171C22—4281no—
172C22—4281no—
173C22—4281no—
174C22—4281no—
175C22—2142no—
176C22#1681207911yescharacteristic
177C4—21no—
178C4—42no—
179C4—8561no—
180C4—4281no—
181C4—4281no—
182C4SD16×D10711yescharacteristic
183C4#1681213011yescharacteristic
184C4#1681213311yescharacteristic
185C4#1681214111yescharacteristic
186C2—8561no—
187C2—4282no—
188C2#2029277911yescharacteristic
189C2#2029279211yescharacteristic
190C2#2029280211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#332×C107C2
socleC22×C107#120
prime radical#20293280C2
hypercenter#332D107
hypocenterC107#1635
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. #20292918C2semidirect
4. #20292918C2semidirect
5. #20292354C2semidirect
6. #20292354C2semidirect
7. #332×C107C2semidirect
96. C107#1635semidirect
101. #332D107semidirect

Maximal subgroups

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

subgroupindex
#202932802
#202929792
#202929182
#202923542
#332×C1072
#1635107