Finite Group Databasebeta

← #28335642

Subgroups — #28335642

Order 13696.

308 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D428C211yesmaximal, characteristic
2#20292974C211yesmaximal, characteristic, complemented
3#20292908C211yesmaximal, characteristic, complemented
4#20292832C211yesmaximal, characteristic, complemented
5#20292370C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C22×D428—21no—
9C2×D4×D107—21no—
10C2×#13133806—21no—
11#16812109C2211yescharacteristic, complemented
12#16812106C2211yescharacteristic
13#16812084—21no—
14#16812083—21no—
15#16812079—22no—
16C4×D428C2211yescharacteristic
17#16812043—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic, complemented
19#16812032—21no—
20#16812031—21no—
21#16812031—22no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic, complemented
24C4×C8×C107C2211yescharacteristic, complemented
25C23×D107—22no—
26C2×Dic214—21no—
27C2×D428—22no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—42no—
31C2×D428—42no—
32C2×D428D411yescharacteristic
33C2×D428D412yes—
34C2×C22⋊D107—22no—
35D4×D107—42no—
36D4×D107—42no—
37C2×C4×D107—21no—
38#13133806—42no—
39C2×C107⋊C8—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107D411yescharacteristic
43(C2×D107)⋊C4—22no—
44(C2×D107)⋊C4—21no—
45C2×D4×C107D411yescharacteristic
46C22×C4×C107—21no—
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C4⋊C4×C107D411yescharacteristic
50C22⋊C4×C107—21no—
51C42×C107C2311yescharacteristic
52C23×C107—21no—
53C22×D107—21no—
54C22×D107—22no—
55C22×D107—44no—
56C22×D107—44no—
57C22×D107—82no—
58D4×C107—21no—
59D4×C107D812yes—
60C2×C4×C107—21no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C22⋊D107—44no—
67C2×Dic107—21no—
68C2×Dic107—41no—
69D428—21no—
70D428—42no—
71D428—44no—
72D428—41no—
73D428—41no—
74D428—41no—
75D428—41no—
76D428SD1612yescomplemented
77C4×D107—41no—
78C4×D107—41no—
79Dic214—41no—
80Dic214—41no—
81C8×C107—41no—
82C8×C107—41no—
83C107⋊C8—41no—
84C22×C107—22no—
85C22×C107—41no—
86C22×C107—41no—
87C22×C107#12011yescharacteristic
88C2×D107—22no—
89C2×D107—41no—
90C2×D107—41no—
91C2×D107—42no—
92C2×D107—42no—
93C2×D107—42no—
94C2×D107—88no—
95C4×C107—21no—
96C4×C107—41no—
97C4×C107C2×D811yescharacteristic
98C4×C107C2×SD1611yescharacteristic
99C4×C107AGL(1,ℤ/8)11yescharacteristic
100C4×C107AGL(1,ℤ/8)11yescharacteristic
101Dic107—41no—
102Dic107—81no—
103C2×C107—42no—
104C2×C107#44711yescharacteristic
105C2×C107#45011yescharacteristic
106C2×C107#45511yescharacteristic
107D107—42no—
108D107—82no—
109#1639—1071nomaximal, Sylow
110C107#163911yescharacteristic, complemented, Sylow
111D42—1071no—
112#486—1071no—
113#474—1071no—
114#461—1071no—
115#331—1071no—
116D4⋊C8—1071no—
117D4⋊C8D10711yescharacteristic, complemented
118C22×D4—2141no—
119C22×D4—2141no—
120C2×SD16—2141no—
121C4⋊Q8—1071no—
122#127—1071no—
123#121—2141no—
124#121—2141no—
125#120—2142no—
126C4×D4—1071no—
127C4×D4C2×D10711yescharacteristic
128#107—2141no—
129C4⋊C8—1071no—
130C4⋊C8C2×D10711yescharacteristic, complemented
131#103—2141no—
132#102—2141no—
133#102—2142no—
134C4×C8C2×D10711yescharacteristic, complemented
135C24—2142no—
136C2×Q8—2141no—
137C2×D4—1071no—
138C2×D4—1072no—
139C2×D4—4282no—
140C2×D4—4282no—
141C2×D4—4282no—
142C2×D4—4282no—
143C2×D4—2142no—
144C2×D4—2142no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4D42811yescharacteristic
148C22×C4—21no—
149C22×C4—2141no—
150SD16—4282no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2141no—
154C4⋊C4—1071no—
155C4⋊C4—2141no—
156C4⋊C4—2141no—
157C4⋊C4D42811yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—2142no—
160C22⋊C4—2141no—
161C42C22×D10711yescharacteristic
162C23—21no—
163C23—8562no—
164C23—4284no—
165C23—4284no—
166C23—2142no—
167C23—2141no—
168Q8—4281no—
169Q8—4281no—
170D4—21no—
171D4—1072no—
172D4—4282no—
173D4—4284no—
174D4—4284no—
175D4—4281no—
176D4—4281no—
177D4—4281no—
178D4—4281no—
179D4—2141no—
180D4D85612yes—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—4281no—
185C2×C4—4281no—
186C2×C4—4281no—
187C2×C4—2141no—
188C2×C4D4×D10711yescharacteristic
189C2×C4D4×D10711yescharacteristic
190C2×C4C2×D42811yescharacteristic
191C8—41no—
192C8—41no—
193C8—4281no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—8568no—
198C22—4282no—
199C22—4282no—
200C22—4282no—
201C22—4281no—
202C22—4281no—
203C22—2142no—
204C22#1681207911yescharacteristic
205C4—21no—
206C4—41no—
207C4—8561no—
208C4—4281no—
209C4C2×D85611yescharacteristic
210C4SD16×D10711yescharacteristic
211C4#1681213311yescharacteristic
212C4#1681213711yescharacteristic
213C2—42no—
214C2—8562no—
215C2—4282no—
216C2#2029276311yescharacteristic
217C2#2029277911yescharacteristic
218C2#2029280611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radicalD4×D428C2
hypercenterD4⋊C8D107
hypocenterC107#1639
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20292974C2semidirect
3. #20292908C2semidirect
4. #20292832C2semidirect
5. #20292370C2semidirect
6. #20292356C2semidirect
7. D4⋊C8×C107C2semidirect
11. #16812109C22semidirect
18. C4⋊(C107⋊C8)C22semidirect
23. C4⋊C8×C107C22semidirect
24. C4×C8×C107C22semidirect
76. D428SD16semidirect
110. C107#1639semidirect
117. D4⋊C8D107semidirect
130. C4⋊C8C2×D107semidirect
134. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
D4×D4282
#202929742
#202929082
#202928322
#202923702
#202923562
D4⋊C8×C1072
#1639107