Finite Group Databasebeta

← #28336693

Subgroups — #28336693

Order 13696.

286 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#20293108C211yesmaximal, characteristic
2#20292667C211yesmaximal, characteristic, complemented
3C2×#16812037C211yesmaximal, characteristic
4#20292626C211yesmaximal, characteristic
5#20292396C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7#382×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10C2×#13133802C2211yescharacteristic
11#16812074—21no—
12#16812071—21no—
13C4×C22⋊D107—22no—
14#16812068C2211yescharacteristic
15#16812067—21no—
16C4⋊C4×D107—21no—
17#16812042—21no—
18#16812040—21no—
19#16812037C414yescomplemented
20#16812032—21no—
21#16812029—21no—
22C2×C4⋊C4×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24C2×C42×C107C2211yescharacteristic
25#95×C107—22no—
26#13133818—21no—
27#13133818—21no—
28#13133818C2×C412yes—
29#13133818Q812yes—
30C2×Dic214D411yescharacteristic
31C2×D428D411yescharacteristic
32C2×C22⋊D107—21no—
33C2×C4×D107—21no—
34C2×C4×D107—22no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38#13133802C2×C412yescomplemented
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C4⋊Dic107D411yescharacteristic
42C4⋊Dic107D411yescharacteristic
43Dic107⋊C4—22no—
44C22⋊Dic107—21no—
45(C2×D107)⋊C4—22no—
46C4×Dic107—22no—
47C4×Dic107—21no—
48C22×C4×C107—21no—
49C22×C4×C107—21no—
50C22×C4×C107D412yes—
51C22×C4×C107C2311yescharacteristic
52C4⋊C4×C107—21no—
53C4⋊C4×C107—21no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C42×C107—21no—
57C42×C107C2×C412yescomplemented
58C23×C107C2×D411yescharacteristic
59C22×D107—21no—
60C2×C4×C107—22no—
61C2×C4×C107—22no—
62C2×C4×C107—41no—
63C2×C4×C107—41no—
64C2×C4×C107—42no—
65C2×C4×C107—42no—
66C2×C4×C107—41no—
67C2×C4×C107C22×C411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×Q811yescharacteristic
71C2×C4×C107#4111yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C22⋊D107—22no—
74C22⋊D107—22no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78D428—21no—
79D428C4⋊C412yes—
80C4×D107—22no—
81C4×D107—22no—
82C4×D107—82no—
83Dic214—21no—
84Dic214C4⋊C412yes—
85C107⋊C8—22no—
86C107⋊C8—41no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107C4×D411yescharacteristic
90C22×C107#12011yescharacteristic
91C22×C107C22⋊Q811yescharacteristic
92C2×D107—22no—
93C2×D107—41no—
94C2×D107—41no—
95C4×C107—42no—
96C4×C107—42no—
97C4×C107—81no—
98C4×C107C2×C4⋊C411yescharacteristic
99C4×C107C4×D411yescharacteristic
100C4×C107C22⋊Q811yescharacteristic
101C4×C107#12311yescharacteristic
102Dic107—22no—
103Dic107—41no—
104Dic107—41no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107#38511yescharacteristic
108C2×C107#45411yescharacteristic
109C2×C107#45511yescharacteristic
110D107—42no—
111#1860—1071nomaximal, Sylow
112C107#186011yescharacteristic, complemented, Sylow
113#520—1071no—
114#428—1071no—
115C2×#104—1071no—
116#417—1071no—
117#382D10711yescharacteristic, complemented
118#343—1071no—
119#339—1071no—
120C2×#41—1071no—
121C2×M16—1071no—
122C2×M16—1071no—
123C4×Q8—2141no—
124C4×D4—2141no—
125C4×D4—2142no—
126#117—1071no—
127#117—2141no—
128C2×C4⋊C4—2141no—
129C2×C4⋊C4C2×D10711yescharacteristic
130C2×C4⋊C4C2×D10711yescharacteristic
131C2×C42C2×D10711yescharacteristic
132#107—2141no—
133#106—2141no—
134#104—1074no—
135#103—2141no—
136#102—2141no—
137#95—22no—
138#41—1072no—
139#41—1072no—
140#41—2141no—
141#41—2141no—
142C2×Q8—1071no—
143C2×D4—1071no—
144C2×D4—2141no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—2142no—
148C22×C4—2141no—
149C22×C4C22⋊D10712yes—
150C22×C4C22×D10711yescharacteristic
151M16—1072no—
152M16—2141no—
153M16—2141no—
154M16—2141no—
155C2×C8—2141no—
156C2×C8—2141no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1071no—
162C4⋊C4—1071no—
163C4⋊C4—2142no—
164C22⋊C4—2142no—
165C22⋊C4—2141no—
166C42—21no—
167C42—2142no—
168C42—2141no—
169C42C4×D10712yescomplemented
170C23—2141no—
171C23C2×C22⋊D10711yescharacteristic
172Q8—1072no—
173Q8—2141no—
174D4—1072no—
175D4—2142no—
176D4—2142no—
177D4—2141no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—8562no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4—2141no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4Q8×D10711yescharacteristic
195C2×C4#1313381811yescharacteristic
196C2×C4#1313381911yescharacteristic
197C8—4281no—
198C8—2142no—
199C22—21no—
200C22—21no—
201C22—4281no—
202C22—4281no—
203C22—2142no—
204C22C4×C22⋊D10711yescharacteristic
205C22#1681207711yescharacteristic
206C22#1681208511yescharacteristic
207C4—42no—
208C4—42no—
209C4—81no—
210C4—4281no—
211C4—4281no—
212C4—2142no—
213C4C4⋊C4×D10711yescharacteristic
214C4#1681207111yescharacteristic
215C4#1681208711yescharacteristic
216C4#1681209411yescharacteristic
217C2—21no—
218C2—21no—
219C2—4282no—
220C2#2029249811yescharacteristic
221C2#2029279911yescharacteristic
222C2#2029280411yescharacteristic
subgroupG/H
centerC22#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#382×C107C2
socleC22×C107#120
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenter#382D107
hypocenterC107#1860
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #20292667C2semidirect
5. #20292396C2semidirect
6. #20292387C2semidirect
7. #382×C107C2semidirect
19. #16812037C4semidirect
38. #13133802C2×C4semidirect
57. C42×C107C2×C4semidirect
112. C107#1860semidirect
117. #382D107semidirect
169. C42C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202931082
#202926672
C2×#168120372
#202926262
#202923962
#202923872
#382×C1072
#1860107