Finite Group Databasebeta

← #28346884

Subgroups — #28346884

Order 13696.

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

normal subgroups only

This group has 572 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×#16812130C214yesmaximal, complemented
2#20293178C211yesmaximal, characteristic, complemented
3#20293175C211yesmaximal, characteristic, complemented
4#20293008C214yesmaximal, complemented
5#20293006C211yesmaximal, characteristic, complemented
6#20293003C211yesmaximal, characteristic, complemented
7#20293000C211yesmaximal, characteristic, complemented
8C4×C8×D107C211yesmaximal, characteristic, complemented
9#492×C107C211yesmaximal, characteristic, complemented
10C2×#13133819C2212yes—
11C2×#13133817C2212yes—
12#16812130—216no—
13C2×#13133808C2214yes—
14C2×#13133807C2214yes—
15C2×#13133803C2214yes—
16C2×C8×D107C2212yes—
17#16812109C2211yescharacteristic
18#16812106C2211yescharacteristic
19#16812098C2212yes—
20#16812096C2212yes—
21#16812084—22no—
22#16812080—22no—
23D4×Dic107—22no—
24#16812071—22no—
25C42×D107C2211yescharacteristic
26C4×C107⋊C8C2211yescharacteristic
27C8×Dic107C2212yes—
28C2×SD16×C107C2214yes—
29C4⋊Q8×C107C2211yescharacteristic
30#127×C107C2211yescharacteristic
31C4×C8×C107C2211yescharacteristic
32#13133819—28no—
33#13133817—28no—
34C2×Dic214C2312yes—
35C2×D428—22no—
36C2×D428—22no—
37C2×D428C2312yes—
38C2×C22⋊D107—22no—
39C22×Dic107—22no—
40C2×C4×D107—22no—
41C2×C4×D107D412yes—
42C2×C4×D107C2311yescharacteristic
43#13133808—28no—
44#13133807—28no—
45#13133803—28no—
46C2×C107⋊C8C2312yes—
47C8×D107D418yescomplemented
48C4⋊Dic107—22no—
49C22⋊Dic107—24no—
50(C2×D107)⋊C4—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×D107—22no—
62C22×D107C2×D411yescharacteristic
63Q8×C107—24no—
64D4×C107—24no—
65D4×C107—44no—
66C2×C4×C107—22no—
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2411yescharacteristic
69C22⋊D107—44no—
70C2×Dic107—22no—
71C2×Dic107—44no—
72C2×Dic107C2×D412yes—
73C2×Dic107C2×D411yescharacteristic
74D428—24no—
75D428—44no—
76D428—44no—
77C4×D107—24no—
78C4×D107—44no—
79C4×D107C2×D414yes—
80Dic214—24no—
81C8×C107C2×D414yes—
82C107⋊C8C2×D414yes—
83C22×C107—42no—
84C22×C107—44no—
85C22×C107C22×D411yescharacteristic
86C2×D107—22no—
87C2×D107—42no—
88C2×D107—44no—
89C2×D107#12712yes—
90C4×C107—42no—
91C4×C107#13514yescomplemented
92C4×C107C22×D412yes—
93Dic107—22no—
94Dic107—42no—
95Dic107#12712yes—
96C2×C107—82no—
97C2×C107C2×#12711yescharacteristic
98C2×C107C2×#13512yes—
99D107—22no—
100D107—82no—
101#3147—1071nomaximal, Sylow
102C107#314711yescharacteristic, complemented, Sylow
103C2×#135—1074no—
104#532—1071no—
105#532—1071no—
106#495—1074no—
107#494—1071no—
108#493—1071no—
109#492—1071no—
110#492D10711yescharacteristic, complemented
111C2×C4×C8—1071no—
112C2×#41—1072no—
113C2×#41—1072no—
114#135—21416no—
115C2×Q16—1074no—
116C2×SD16—1074no—
117C2×SD16C2×D10714yes—
118C2×D8—1074no—
119C22×C8—1072no—
120C4⋊Q8—1071no—
121C4⋊Q8C2×D10711yescharacteristic
122#127—1071no—
123#127C2×D10711yescharacteristic
124#124—1072no—
125#124—1072no—
126#121—2142no—
127#121—2142no—
128C4×D4—2142no—
129C4×D4—2142no—
130C2×C42—1071no—
131C4×C8—1072no—
132C4×C8—1071no—
133C4×C8C2×D10711yescharacteristic
134#41—2148no—
135#41—2148no—
136C2×Q8—1072no—
137C2×Q8C22×D10712yes—
138C2×D4—22no—
139C2×D4—1072no—
140C2×D4—2142no—
141C2×D4—2142no—
142C2×D4—2142no—
143C2×D4C22×D10712yes—
144C22×C4—1071no—
145C22×C4—1072no—
146C22×C4—2142no—
147C22×C4—2142no—
148Q16—2148no—
149SD16—28no—
150SD16—2148no—
151D8—2148no—
152C2×C8—1072no—
153C2×C8—1078no—
154C2×C8C22×D10712yes—
155C4⋊C4—22no—
156C4⋊C4—2142no—
157C22⋊C4—2144no—
158C22⋊C4—2144no—
159C42—1072no—
160C42—1071no—
161C42C22×D10711yescharacteristic
162C23—22no—
163C23—1071no—
164C23—2142no—
165Q8—24no—
166Q8—2144no—
167D4—24no—
168D4—44no—
169D4—4284no—
170D4—4284no—
171D4—4284no—
172D4—2144no—
173C2×C4—22no—
174C2×C4—1071no—
175C2×C4—1074no—
176C2×C4—1072no—
177C2×C4—4284no—
178C2×C4—4284no—
179C2×C4—2142no—
180C2×C4—2144no—
181C2×C4D4×D10712yes—
182C2×C4C23×D10711yescharacteristic
183C8—1074no—
184C8D4×D10714yes—
185C22—44no—
186C22—42no—
187C22—1072no—
188C22—4284no—
189C22—4282no—
190C22—2142no—
191C22C2×D4×D10711yescharacteristic
192C4—42no—
193C4—1072no—
194C4—4282no—
195C4—2142no—
196C4#1681213014yescomplemented
197C4C2×D4×D10712yes—
198C2—82no—
199C2—8562no—
200C2—2142no—
201C2#127×D10711yescharacteristic
202C2C2×#1681213012yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
hypercenter#492D107
hypocenterC107#3147
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×#16812130C2semidirect
2. #20293178C2semidirect
3. #20293175C2semidirect
4. #20293008C2semidirect
5. #20293006C2semidirect
6. #20293003C2semidirect
7. #20293000C2semidirect
8. C4×C8×D107C2semidirect
9. #492×C107C2semidirect
47. C8×D107D4semidirect
91. C4×C107#135semidirect
102. C107#3147semidirect
110. #492D107semidirect
196. C4#16812130semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202931782
#202931752
#202930082
#202930062
#202930032
#202930002
C4×C8×D1072
#492×C1072
#3147107