Finite Group Databasebeta

← #28337269

Subgroups — #28337269

Order 13696.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16812098C211yesmaximal, characteristic
2C2×#16812033C211yesmaximal, characteristic, complemented
3C2×#16812031C211yesmaximal, characteristic
4C2×#16812029C211yesmaximal, characteristic
5C2×C8×Dic107C211yesmaximal, characteristic, complemented
6#20292492C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C22×C107⋊C8C2211yescharacteristic
10#16812098—22no—
11#16812098C414yescomplemented
12C2×C4⋊Dic107C2211yescharacteristic
13C2×(C2×D107)⋊C4—22no—
14C2×C4×Dic107C2211yescharacteristic
15#16812033—22no—
16#16812031—22no—
17#16812029—22no—
18C8×Dic107—22no—
19C107⋊#95—22no—
20C22×Q8×C107C2211yescharacteristic
21C22×C8×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23#103×C107—22no—
24C23×D107—21no—
25C2×D428—21no—
26C2×D428—21no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428C2×C412yes—
31C22×Dic107—21no—
32C22×Dic107D412yes—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C2×C107⋊C8D412yes—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38(C2×D107)⋊C4—28no—
39C4×Dic107C2×C412yes—
40C4×Dic107D412yes—
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×C107C2311yescharacteristic
50C2×C8×C107—21no—
51C2×C8×C107—21no—
52C2×C8×C107D412yes—
53C4⋊C4×C107—21no—
54C4⋊C4×C107—21no—
55C23×C107C2×D411yescharacteristic
56C22×D107—22no—
57C22×D107—41no—
58C22×D107—41no—
59C22×D107—42no—
60C22×D107—42no—
61Q8×C107—22no—
62Q8×C107—22no—
63Q8×C107—41no—
64Q8×C107—41no—
65C2×C4×C107—22no—
66C2×C4×C107—41no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107—42no—
70C2×C4×C107C22×C411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107#4111yescharacteristic
75C2×C4×C107#4111yescharacteristic
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—41no—
79C2×Dic107—42no—
80C2×Dic107C22⋊C414yes—
81D428—22no—
82D428—22no—
83D428—41no—
84D428—41no—
85C8×C107—22no—
86C107⋊C8—22no—
87C22×C107C2×C22⋊C411yescharacteristic
88C22×C107#12411yescharacteristic
89C22×C107#12711yescharacteristic
90C22×C107#13512yes—
91C22×C107#13512yes—
92C2×D107—42no—
93C2×D107—44no—
94C2×D107—82no—
95C2×D107—81no—
96C2×D107—81no—
97C4×C107—42no—
98C4×C107—81no—
99C4×C107C4×D411yescharacteristic
100C4×C107C4×D411yescharacteristic
101C4×C107#12111yescharacteristic
102C4×C107#12111yescharacteristic
103Dic107—24no—
104Dic107—81no—
105C2×C107#39011yescharacteristic
106C2×C107#41612yes—
107C2×C107#48812yes—
108C2×C107#49511yescharacteristic
109C2×C107#49511yescharacteristic
110D107—82no—
111#1958—1071nomaximal, Sylow
112C107#195811yescharacteristic, complemented, Sylow
113C2×#124—1071no—
114C2×#103—1071no—
115C2×#103D10711yescharacteristic, complemented
116C2×#102—1071no—
117C2×#102—1071no—
118C2×C4×C8—1071no—
119#382—1071no—
120C22×Q8C2×D10711yescharacteristic
121C22×D4—1071no—
122C22×C8—1071no—
123C22×C8C2×D10711yescharacteristic
124#124—1074no—
125#124—2142no—
126C2×C4⋊C4—1071no—
127C2×C4⋊C4C2×D10711yescharacteristic
128C2×C22⋊C4—2142no—
129C2×C42—1071no—
130#103—22no—
131#103—2142no—
132#102—2142no—
133#102—2142no—
134C4×C8—2142no—
135#95—2142no—
136C24—2141no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C4×D10712yes—
143C2×D4—1072no—
144C2×D4—2141no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4—2141no—
148C2×D4—2141no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1072no—
152C22×C4—2141no—
153C22×C4C22×D10711yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—1072no—
157C2×C8—2141no—
158C2×C8—2141no—
159C2×C8C22⋊D10712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—2141no—
163C4⋊C4—2141no—
164C22⋊C4—2148no—
165C42—1072no—
166C42—1072no—
167C23—4281no—
168C23—4281no—
169C23—4282no—
170C23—4282no—
171C23—2142no—
172C23D4×D10711yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177D4—4281no—
178D4—4281no—
179D4—2142no—
180D4—2142no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—1074no—
187C2×C4—4281no—
188C2×C4—4282no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4C2×C22⋊D10711yescharacteristic
195C2×C4#1313381711yescharacteristic
196C2×C4#1313381811yescharacteristic
197C8—22no—
198C8—2142no—
199C22—8562no—
200C22—8561no—
201C22—8561no—
202C22—4284no—
203C22—4282no—
204C22C22⋊C4×D10711yescharacteristic
205C22#1681209511yescharacteristic
206C22#1681210511yescharacteristic
207C22#1681213012yes—
208C22#1681213212yes—
209C4—42no—
210C4—81no—
211C4—8561no—
212C4—2144no—
213C4(C22⋊D107)⋊C411yescharacteristic
214C4C4×C22⋊D10711yescharacteristic
215C4#1681208111yescharacteristic
216C4#1681208211yescharacteristic
217C2—8562no—
218C2#2029252711yescharacteristic
219C2#2029262212yes—
220C2#2029298312yes—
221C2#2029300811yescharacteristic
222C2#2029301211yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC22×D428C22
cosocleC22×C4×C107C23
hypercenterC2×#103D107
hypocenterC107#1958
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16812033C2semidirect
5. C2×C8×Dic107C2semidirect
6. #20292492C2semidirect
7. C2×#103×C107C2semidirect
11. #16812098C4semidirect
112. C107#1958semidirect
115. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
C2×#168120332
C2×#168120312
C2×#168120292
C2×C8×Dic1072
#202924922
C2×#103×C1072
#1958107