Finite Group Databasebeta

← #28340119

Subgroups — #28340119

Order 13696.

852 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×Dic214C211yesmaximal, characteristic, complemented
2C2×#16812097C212yesmaximal, complemented
3C2×#16812083C211yesmaximal, characteristic
4#20292540C214yesmaximal, complemented
5#20292533C212yesmaximal, complemented
6#20292504C214yesmaximal, complemented
7C4×C22⋊C4×C107C211yesmaximal, characteristic, complemented
8C22×Dic214—28no—
9C22×Dic214—24no—
10C22×Dic214C2212yes—
11C22×Dic214C2212yes—
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C23×Dic107C2212yes—
15#16812097—28no—
16#16812083—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×(C2×D107)⋊C4C2218yes—
19C2×(C2×D107)⋊C4C2212yes—
20C107⋊#95C2218yes—
21C23×C4×C107C2211yescharacteristic
22C2×C22⋊C4×C107C2212yes—
23C2×C42×C107C2212yes—
24#95×C107C2212yes—
25C23×D107C2312yes—
26C2×Dic214—232no—
27C2×Dic214—24no—
28C2×Dic214—24no—
29C2×Dic214—24no—
30C2×Dic214—28no—
31C2×Dic214D418yescomplemented
32C2×D428—22no—
33C2×D428—24no—
34C2×C22⋊D107—24no—
35C2×C22⋊D107—28no—
36C22×Dic107—24no—
37C22×Dic107—28no—
38C22×Dic107C2312yes—
39C22×Dic107C2314yes—
40C4⋊Dic107—22no—
41(C2×D107)⋊C4—24no—
42(C2×D107)⋊C4—216no—
43C24×C107C2311yescharacteristic
44C22×C4×C107—24no—
45C22×C4×C107D414yes—
46C22×C4×C107C2314yes—
47C22×C4×C107C2311yescharacteristic
48C22×C4×C107C2311yescharacteristic
49C22⋊C4×C107—24no—
50C42×C107—24no—
51C23×C107—22no—
52C23×C107—22no—
53C23×C107C2×D412yes—
54C23×C107#4114yes—
55C23×C107C2411yescharacteristic
56C22×D107—24no—
57C22×D107—22no—
58C22×D107—24no—
59C22×D107—24no—
60C2×C4×C107—22no—
61C2×C4×C107—28no—
62C2×C4×C107—24no—
63C2×C4×C107—24no—
64C2×C4×C107—24no—
65C2×C4×C107C2×D414yes—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D412yes—
68C22⋊D107—48no—
69C2×Dic107—216no—
70C2×Dic107—24no—
71C2×Dic107—24no—
72C2×Dic107—22no—
73C2×Dic107—24no—
74C2×Dic107C2×D418yes—
75D428—44no—
76Dic214—216no—
77Dic214—216no—
78C22×C107—22no—
79C22×C107—22no—
80C22×C107—28no—
81C22×C107#12414yescomplemented
82C22×C107C22×D412yes—
83C22×C107C22×D411yescharacteristic
84C22×C107C2×#4112yes—
85C22×C1072−1+412yes—
86C2×D107—42no—
87C2×D107—42no—
88C2×D107—42no—
89C2×D107—48no—
90C4×C107—22no—
91C4×C107—44no—
92C4×C107#12014yes—
93Dic107—28no—
94Dic107—42no—
95C2×C107—24no—
96C2×C107C2×#12011yescharacteristic
97C2×C107C2×#12411yescharacteristic
98C2×C107#53611yescharacteristic
99C2×C107#54814yes—
100D107—82no—
101#2436—1071nomaximal, Sylow
102C107#243611yescharacteristic, complemented, Sylow
103C23×Q8—1071no—
104C2×#124—1072no—
105C2×#121—1071no—
106#393—1074no—
107#391—1072no—
108#386—1074no—
109C4×C22⋊C4D10711yescharacteristic, complemented
110C22×Q8—1072no—
111C22×Q8—1072no—
112C22×Q8—2148no—
113C22×Q8—2144no—
114C22×D4—1072no—
115C22×D4—1071no—
116C23×C4—1072no—
117C23×C4C2×D10711yescharacteristic
118#124—2148no—
119#121—2144no—
120C2×C4⋊C4—1071no—
121C2×C22⋊C4—1072no—
122C2×C22⋊C4—1078no—
123C2×C22⋊C4C2×D10712yes—
124C2×C42C2×D10712yes—
125#95—1078no—
126#95C2×D10712yes—
127C24—1072no—
128C24C22×D10711yescharacteristic
129C2×Q8—1078no—
130C2×Q8—2148no—
131C2×Q8—21432no—
132C2×Q8—2144no—
133C2×Q8—2144no—
134C2×Q8—2144no—
135C2×D4—2144no—
136C2×D4—2148no—
137C2×D4—2144no—
138C2×D4—2142no—
139C22×C4—24no—
140C22×C4—1074no—
141C22×C4—1072no—
142C22×C4—2148no—
143C22×C4—2144no—
144C22×C4D42814yes—
145C22×C4C22×D10714yes—
146C22×C4C22×D10711yescharacteristic
147C22×C4C22×D10711yescharacteristic
148C4⋊C4—2142no—
149C22⋊C4—24no—
150C22⋊C4—21416no—
151C22⋊C4—2144no—
152C42—24no—
153C23—22no—
154C23—22no—
155C23—2144no—
156C23—2142no—
157C23—2144no—
158C23—2144no—
159C23C2×D42812yes—
160C23#1313381814yes—
161C23C23×D10711yescharacteristic
162Q8—21416no—
163Q8—21416no—
164D4—4288no—
165D4—4284no—
166C2×C4—22no—
167C2×C4—28no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—1078no—
172C2×C4—2142no—
173C2×C4—2144no—
174C2×C4—21416no—
175C2×C4—2144no—
176C2×C4—2144no—
177C2×C4D4×D10714yes—
178C2×C4C2×D42812yes—
179C2×C4C2×D42812yes—
180C22—22no—
181C22—28no—
182C22—22no—
183C22—4282no—
184C22—4282no—
185C22—4288no—
186C22—4282no—
187C22#1681209714yescomplemented
188C22C2×D4×D10712yes—
189C22C22×D42811yescharacteristic
190C22C2×#1313381812yes—
191C22#1681215612yes—
192C4—22no—
193C4—44no—
194C4—4282no—
195C4—2148no—
196C4#1681207914yes—
197C2—24no—
198C2—8562no—
199C2C2×#1681207911yescharacteristic
200C2C2×#1681209711yescharacteristic
201C2#2029320311yescharacteristic
202C2#2029328614yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC4×C22⋊C4×C107C2
prime radicalC2×#16812083C2
hypercenterC4×C22⋊C4D107
hypocenterC107#2436
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×Dic214C2semidirect
2. C2×#16812097C2semidirect
4. #20292540C2semidirect
5. #20292533C2semidirect
6. #20292504C2semidirect
7. C4×C22⋊C4×C107C2semidirect
31. C2×Dic214D4semidirect
81. C22×C107#124semidirect
102. C107#2436semidirect
109. C4×C22⋊C4D107semidirect
187. C22#16812097semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2142
C2×#168120972
C2×#168120832
#202925402
#202925332
#202925042
C4×C22⋊C4×C1072
#2436107