Finite Group Databasebeta

← #28336992

Subgroups — #28336992

Order 13696.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293196C211yesmaximal, characteristic
2#20293101C211yesmaximal, characteristic, complemented
3#20292645C211yesmaximal, characteristic, complemented
4#20292643C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×D428C2211yescharacteristic
10C2×#13133802C2211yescharacteristic
11#16812106C412yescomplemented
12#16812097C412yescomplemented
13#16812084—22no—
14#16812079—22no—
15C4×D428—22no—
16#16812071—22no—
17#16812068C2211yescharacteristic
18C22⋊C4×D107—22no—
19#16812037—22no—
20#16812035—22no—
21#16812025—22no—
22C2×M16×C107C2211yescharacteristic
23#117×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25C23×D107D412yes—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30C2×Dic214C2×C411yescharacteristic, complemented
31C2×D428—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—24no—
36C2×D428C2×C412yes—
37C2×D428C2×C411yescharacteristic, complemented
38C2×D428D412yes—
39C2×C22⋊D107—21no—
40C2×C4×D107—21no—
41C2×C4×D107—24no—
42#13133802—21no—
43#13133802—21no—
44C2×C107⋊C8—21no—
45C4⋊Dic107—21no—
46C22⋊Dic107—21no—
47(C2×D107)⋊C4—24no—
48(C2×D107)⋊C4—24no—
49C4×Dic107D412yescomplemented
50C22×C4×C107C2311yescharacteristic
51M16×C107—21no—
52M16×C107—21no—
53C2×C8×C107—21no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C22⋊C4×C107—21no—
57C22⋊C4×C107—21no—
58C42×C107C2×C412yescomplemented
59C42×C107D412yescomplemented
60C23×C107C2×D411yescharacteristic
61C22×D107—22no—
62C22×D107—22no—
63C22×D107—21no—
64C22×D107—82no—
65C22×D107C22⋊C414yes—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C2×C4×C107#4111yescharacteristic
74C22⋊D107—41no—
75C22⋊D107—41no—
76C2×Dic107—21no—
77C2×Dic107—22no—
78D428—22no—
79D428—22no—
80D428—21no—
81D428—21no—
82D428—22no—
83D428—22no—
84D428—84no—
85C4×D107—41no—
86C4×D107—41no—
87C4×D107—84no—
88Dic214—21no—
89Dic214—21no—
90C8×C107—41no—
91C107⋊C8—41no—
92C22×C107—41no—
93C22×C107C2×C22⋊C411yescharacteristic
94C22×C107#12011yescharacteristic
95C22×C107#12311yescharacteristic
96C2×D107—24no—
97C2×D107—44no—
98C2×D107—44no—
99C2×D107—42no—
100C2×D107—41no—
101C2×D107—82no—
102C4×C107—42no—
103C4×C107—42no—
104C4×C107C4×D411yescharacteristic
105C4×C107C4×D411yescharacteristic
106C4×C107#12111yescharacteristic
107C4×C107#12111yescharacteristic
108Dic107—42no—
109Dic107—41no—
110C2×C107—21no—
111C2×C107—21no—
112C2×C107—21no—
113C2×C107#38611yescharacteristic
114D107—44no—
115D107—81no—
116#1911—1071nomaximal, Sylow
117C107#191111yescharacteristic, complemented, Sylow
118#535—1071no—
119#518—1071no—
120#421—1071no—
121#421—1071no—
122C2×#100—1071no—
123H(3,ℤ/4)—1071no—
124H(3,ℤ/4)D10711yescharacteristic, complemented
125C2×#41—1071no—
126C22×D4—1071no—
127C2×M16—1071no—
128C2×M16C2×D10711yescharacteristic
129#127—1072no—
130#124—1072no—
131#121—2142no—
132#120—2142no—
133C4×D4—2142no—
134C4×D4—2142no—
135#117—1071no—
136#117C2×D10711yescharacteristic
137#117C2×D10711yescharacteristic
138C2×C22⋊C4—2142no—
139#104—2142no—
140#104—2142no—
141#100—2142no—
142C24—1072no—
143#41—2141no—
144#41—2141no—
145#41—2141no—
146#41—2141no—
147C2×Q8—1071no—
148C2×D4—1072no—
149C2×D4—1071no—
150C2×D4—1072no—
151C2×D4—2144no—
152C2×D4—2141no—
153C2×D4—2141no—
154C2×D4—2141no—
155C2×D4—2141no—
156C2×D4—2141no—
157C22×C4—2144no—
158C22×C4—2141no—
159C22×C4C22×D10711yescharacteristic
160M16—21no—
161M16—21no—
162M16—2141no—
163M16—2141no—
164C2×C8—21no—
165C2×C8—2141no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2141no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—2144no—
172C22⋊C4—2144no—
173C22⋊C4—2141no—
174C42—1072no—
175C42C4×D10712yescomplemented
176C42D42812yescomplemented
177C23—8562no—
178C23—1074no—
179C23—2142no—
180C23—2141no—
181C23—2142no—
182C23D4×D10711yescharacteristic
183Q8—2141no—
184Q8—2141no—
185D4—8564no—
186D4—4281no—
187D4—4281no—
188D4—2141no—
189D4—2142no—
190D4—2141no—
191D4—2142no—
192D4—2142no—
193D4—2142no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—8564no—
197C2×C4—4281no—
198C2×C4—4281no—
199C2×C4—2141no—
200C2×C4—2142no—
201C2×C4C2×C4×D10711yescharacteristic
202C2×C4D4×D10711yescharacteristic
203C2×C4D4×D10711yescharacteristic
204C2×C4C2×D42811yescharacteristic
205C2×C4#1313381811yescharacteristic
206C2×C4#1313381911yescharacteristic
207C8—41no—
208C8—4281no—
209C22—41no—
210C22—8562no—
211C22—4282no—
212C22—4284no—
213C22—4284no—
214C22—4281no—
215C22—2144no—
216C22C22⋊C4×D10711yescharacteristic
217C22#1681207911yescharacteristic
218C22#1681209411yescharacteristic
219C4—42no—
220C4—42no—
221C4—4282no—
222C4—4281no—
223C4#1681207111yescharacteristic
224C4C4×D42811yescharacteristic
225C4#1681208211yescharacteristic
226C4#1681208411yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—8561no—
231C2—4284no—
232C2#2029250511yescharacteristic
subgroupG/H
centerC2#20292505
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingH(3,ℤ/4)×C107C2
socleC2×C107#386
prime radical#20293196C2
cosocleC22×C4×C107C23
hypercenterH(3,ℤ/4)D107
hypocenterC107#1911
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #20293101C2semidirect
3. #20292645C2semidirect
4. #20292643C2semidirect
5. C2×#16812025C2semidirect
6. #20292383C2semidirect
7. H(3,ℤ/4)×C107C2semidirect
11. #16812106C4semidirect
12. #16812097C4semidirect
30. C2×Dic214C2×C4semidirect
37. C2×D428C2×C4semidirect
49. C4×Dic107D4semidirect
58. C42×C107C2×C4semidirect
59. C42×C107D4semidirect
117. C107#1911semidirect
124. H(3,ℤ/4)D107semidirect
175. C42C4×D107semidirect
176. C42D428semidirect

Maximal subgroups

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

subgroupindex
#202931962
#202931012
#202926452
#202926432
C2×#168120252
#202923832
H(3,ℤ/4)×C1072
#1911107