Finite Group Databasebeta

← #28337321

Subgroups — #28337321

Order 13696.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#13133808C211yesmaximal, characteristic
2C2×#16812042C211yesmaximal, characteristic
3C2×#16812033C211yesmaximal, characteristic
4C2×#16812032C211yesmaximal, characteristic
5C2×C4×C107⋊C8C211yesmaximal, characteristic
6#20292533C211yesmaximal, characteristic
7#391×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C2×#13133808—22no—
10C2×#13133808—22no—
11C2×#13133808C414yescomplemented
12C22×C107⋊C8C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15#16812042—22no—
16#16812033—22no—
17#16812032—22no—
18C4×C107⋊C8—22no—
19C107⋊#95—22no—
20C22×Q8×C107C2211yescharacteristic
21C2×C4⋊C4×C107C2211yescharacteristic
22C2×C42×C107C2211yescharacteristic
23#95×C107—22no—
24C2×Dic214—21no—
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214C2×C412yes—
30C22×Dic107—21no—
31C22×Dic107—21no—
32#13133808—28no—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C2×C107⋊C8C2×C412yes—
36C2×C107⋊C8D412yes—
37C2×C107⋊C8D412yes—
38C2×C107⋊C8D412yes—
39C4⋊Dic107—21no—
40C4⋊Dic107—21no—
41C2×Q8×C107—21no—
42C2×Q8×C107—21no—
43C2×Q8×C107—21no—
44C2×Q8×C107—21no—
45C2×Q8×C107—21no—
46C2×Q8×C107C2×C412yes—
47C22×C4×C107—21no—
48C22×C4×C107—21no—
49C22×C4×C107D412yes—
50C22×C4×C107C2311yescharacteristic
51C4⋊C4×C107—21no—
52C4⋊C4×C107—21no—
53C42×C107—21no—
54C42×C107—21no—
55C23×C107C2×D411yescharacteristic
56Q8×C107—22no—
57Q8×C107—22no—
58Q8×C107—41no—
59Q8×C107—41no—
60C2×C4×C107—22no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—42no—
67C2×C4×C107Q1614yescomplemented
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C2×C4×C107#4111yescharacteristic
74C2×Dic107—22no—
75C2×Dic107—41no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78C2×Dic107—42no—
79Dic214—22no—
80Dic214—22no—
81Dic214—41no—
82Dic214—41no—
83C107⋊C8—22no—
84C107⋊C8C22⋊C414yes—
85C22×C107C4×D411yescharacteristic
86C22×C107#12111yescharacteristic
87C22×C107#12711yescharacteristic
88C22×C107C2×Q1612yes—
89C22×C107#13512yes—
90C4×C107—24no—
91C4×C107—42no—
92C4×C107—81no—
93C4×C107C2×C22⋊C411yescharacteristic
94C4×C107C4×D411yescharacteristic
95C4×C107#12111yescharacteristic
96C4×C107#12411yescharacteristic
97Dic107—42no—
98Dic107—81no—
99C2×C107#39011yescharacteristic
100C2×C107C4×Q1612yes—
101C2×C107#46712yes—
102C2×C107#49411yescharacteristic
103C2×C107#49511yescharacteristic
104#1969—1071nomaximal, Sylow
105C107#196911yescharacteristic, complemented, Sylow
106C22×Q16—1071no—
107C2×#107—1071no—
108C2×#103—1071no—
109C2×#103—1071no—
110C2×C4×C8—1071no—
111#391—1071no—
112#391D10711yescharacteristic
113C22×Q8—1071no—
114C22×Q8C2×D10711yescharacteristic
115C2×Q16—1074no—
116C2×Q16—2142no—
117C2×Q16—2142no—
118C22×C8—1071no—
119C22×C8—1071no—
120C2×C4⋊C4—1071no—
121C2×C4⋊C4C2×D10711yescharacteristic
122C2×C42C2×D10711yescharacteristic
123#107—2142no—
124#103—2142no—
125#103—2142no—
126C4×C8—2142no—
127#95—22no—
128#95—2142no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—1072no—
135C2×Q8—2141no—
136C2×Q8—2141no—
137C2×Q8—2141no—
138C2×Q8—2141no—
139C2×Q8—2141no—
140C2×Q8C4×D10712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—2141no—
144C22×C4—2141no—
145C22×C4C22⋊D10712yes—
146C22×C4C22×D10711yescharacteristic
147Q16—2148no—
148C2×C8—1072no—
149C2×C8—1072no—
150C2×C8—1072no—
151C2×C8—1072no—
152C2×C8—2141no—
153C2×C8—2141no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2141no—
157C4⋊C4—2141no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D10711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—4281no—
166Q8—4281no—
167Q8—2142no—
168Q8—2142no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—4281no—
177C2×C4—4282no—
178C2×C4—4281no—
179C2×C4—4281no—
180C2×C4—2142no—
181C2×C4#1313380814yescomplemented
182C2×C4C2×C4×D10711yescharacteristic
183C2×C4D4×D10711yescharacteristic
184C2×C4D4×D10711yescharacteristic
185C2×C4D4×D10711yescharacteristic
186C2×C4#1313381711yescharacteristic
187C2×C4#1313381811yescharacteristic
188C8—1074no—
189C8—2142no—
190C22C4×C22⋊D10711yescharacteristic
191C22#1681208111yescharacteristic
192C22#1681210511yescharacteristic
193C22C2×#1313380812yes—
194C22#1681212912yes—
195C4—24no—
196C4—42no—
197C4—81no—
198C4—8561no—
199C4—4282no—
200C4C22⋊C4×D10711yescharacteristic
201C4(C22⋊D107)⋊C411yescharacteristic
202C4#1681208211yescharacteristic
203C4#1681209511yescharacteristic
204C2#2029252711yescharacteristic
205C2C4×#1313380812yes—
206C2#2029286612yes—
207C2#2029300611yescharacteristic
208C2#2029300911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#391×C107C2
socle, prime radicalC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenter#391D107
hypocenterC107#1969
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#13133808C4semidirect
67. C2×C4×C107Q16semidirect
105. C107#1969semidirect
181. C2×C4#13133808semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×#168120422
C2×#168120332
C2×#168120322
C2×C4×C107⋊C82
#202925332
#391×C1072
#1969107