Finite Group Databasebeta

← #28346915

Subgroups — #28346915

Order 13696.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#16812138C214yesmaximal, complemented
2#20293175C211yesmaximal, characteristic
3C4⋊Q8×D107C211yesmaximal, characteristic, complemented
4#20293018C214yesmaximal, complemented
5#20293007C211yesmaximal, characteristic, complemented
6#20293006C211yesmaximal, characteristic, complemented
7#20292999C211yesmaximal, characteristic, complemented
8C4×#13133801C211yesmaximal, characteristic, complemented
9#492×C107C211yesmaximal, characteristic, complemented
10C2×#13133817C2212yes—
11C2×Q8×D107C2212yes—
12#16812138—216no—
13C2×Dic428C2214yes—
14C2×#13133808C2214yes—
15C2×#13133806C2214yes—
16C2×#13133801C2212yes—
17#16812109C2211yescharacteristic
18#16812109C2211yescharacteristic
19#16812108—22no—
20#16812107C2212yes—
21#16812096C2212yes—
22#16812080—22no—
23D4×Dic107—22no—
24C4⋊C4×D107—22no—
25C42×D107C2211yescharacteristic
26#16812019C2212yes—
27C4×C107⋊C8C2211yescharacteristic
28C2×SD16×C107C2214yes—
29C4⋊Q8×C107C2211yescharacteristic
30#127×C107C2211yescharacteristic
31C4×C8×C107C2211yescharacteristic
32#13133817—28no—
33C2×Dic214—22no—
34C2×Dic214C2312yes—
35C2×Dic214C2312yes—
36Q8×D107—28no—
37C2×C22⋊D107—22no—
38C22×Dic107—22no—
39C2×C4×D107—22no—
40C2×C4×D107D412yes—
41C2×C4×D107C2311yescharacteristic
42Dic428—28no—
43#13133808—28no—
44#13133806—28no—
45#13133801D418yescomplemented
46C2×C107⋊C8C2312yes—
47C4⋊Dic107—22no—
48C4⋊Dic107—22no—
49Dic107⋊C4—24no—
50C22⋊Dic107—24no—
51C4×Dic107D412yes—
52C4×Dic107C2311yescharacteristic
53C2×Q8×C107C2312yes—
54C2×D4×C107—22no—
55C2×D4×C107C2312yes—
56SD16×C107—28no—
57C2×C8×C107C2312yes—
58C4⋊C4×C107—22no—
59C42×C107C2311yescharacteristic
60C23×C107—22no—
61C22×D107C2×D411yescharacteristic
62Q8×C107—24no—
63D4×C107—24no—
64D4×C107—44no—
65C2×C4×C107—22no—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2411yescharacteristic
68C22⋊D107—44no—
69C2×Dic107—22no—
70C2×Dic107—22no—
71C2×Dic107—44no—
72C2×Dic107C2×D412yes—
73C2×Dic107C2×D411yescharacteristic
74C4×D107—24no—
75C4×D107—44no—
76C4×D107C2×D414yes—
77Dic214—24no—
78Dic214—24no—
79Dic214—44no—
80C8×C107C2×D414yes—
81C107⋊C8C2×D414yes—
82C22×C107—42no—
83C22×C107—44no—
84C22×C107C22×D411yescharacteristic
85C2×D107—22no—
86C2×D107#12712yes—
87C4×C107—42no—
88C4×C107#13714yescomplemented
89C4×C107C22×D412yes—
90Dic107—22no—
91Dic107—42no—
92Dic107—42no—
93Dic107#12712yes—
94C2×C107—82no—
95C2×C107C2×#12711yescharacteristic
96C2×C107C2×#13712yes—
97D107—22no—
98#3152—1071nomaximal, Sylow
99C107#315211yescharacteristic, complemented, Sylow
100C2×#137—1074no—
101#532—1071no—
102C2×C4⋊Q8—1071no—
103#497—1074no—
104#494—1071no—
105#494—1071no—
106#492—1071no—
107#492D10711yescharacteristic, complemented
108C4×M16—1071no—
109C2×#41—1072no—
110C22×Q8—1072no—
111#137—21416no—
112C2×Q16—1074no—
113C2×Q16—1074no—
114C2×SD16—1074no—
115C2×SD16C2×D10714yes—
116C2×M16—1072no—
117C4⋊Q8—1072no—
118C4⋊Q8—1071no—
119C4⋊Q8—1071no—
120C4⋊Q8—2142no—
121C4⋊Q8C2×D10711yescharacteristic
122#127C2×D10711yescharacteristic
123#124—1072no—
124#121—2142no—
125C4×D4—2142no—
126C2×C4⋊C4—2142no—
127C2×C42—1071no—
128#97—1072no—
129C4×C8—1071no—
130C4×C8C2×D10711yescharacteristic
131#41—2148no—
132C2×Q8—1072no—
133C2×Q8—1072no—
134C2×Q8—2142no—
135C2×Q8—2148no—
136C2×Q8C22×D10712yes—
137C2×D4—22no—
138C2×D4—2142no—
139C2×D4C22×D10712yes—
140C22×C4—1071no—
141C22×C4—1072no—
142C22×C4—2142no—
143C22×C4—2142no—
144Q16—2148no—
145Q16—2148no—
146SD16—28no—
147SD16—2148no—
148M16—1078no—
149C2×C8—1072no—
150C2×C8C22×D10712yes—
151C4⋊C4—22no—
152C4⋊C4—2144no—
153C4⋊C4—2142no—
154C4⋊C4—2142no—
155C22⋊C4—2144no—
156C42—1072no—
157C42—1071no—
158C42C22×D10711yescharacteristic
159C23—22no—
160C23—1071no—
161Q8—24no—
162Q8—4284no—
163Q8—2144no—
164Q8—2144no—
165D4—24no—
166D4—44no—
167D4—4284no—
168C2×C4—22no—
169C2×C4—1071no—
170C2×C4—1074no—
171C2×C4—1072no—
172C2×C4—4284no—
173C2×C4—4284no—
174C2×C4—2142no—
175C2×C4—2142no—
176C2×C4—2144no—
177C2×C4D4×D10712yes—
178C2×C4C23×D10711yescharacteristic
179C8—1074no—
180C8D4×D10714yes—
181C22—44no—
182C22—42no—
183C22—1072no—
184C22—2142no—
185C22C2×D4×D10711yescharacteristic
186C4—42no—
187C4—1072no—
188C4—4282no—
189C4—4282no—
190C4—2142no—
191C4#1681213814yescomplemented
192C4C2×D4×D10712yes—
193C2—82no—
194C2—2142no—
195C2#127×D10711yescharacteristic
196C2C2×#1681213812yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
prime radical#20293175C2
hypercenter#492D107
hypocenterC107#3152
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
3. C4⋊Q8×D107C2semidirect
4. #20293018C2semidirect
5. #20293007C2semidirect
6. #20293006C2semidirect
7. #20292999C2semidirect
8. C4×#13133801C2semidirect
9. #492×C107C2semidirect
45. #13133801D4semidirect
88. C4×C107#137semidirect
99. C107#3152semidirect
107. #492D107semidirect
191. C4#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
#202931752
C4⋊Q8×D1072
#202930182
#202930072
#202930062
#202929992
C4×#131338012
#492×C1072
#3152107