Finite Group Databasebeta

← #28338108

Subgroups — #28338108

Order 13696.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#20293186C211yesmaximal, characteristic
2C2×#16812024C211yesmaximal, characteristic, complemented
3#20292418C212yesmaximal, complemented
4#20292415C212yesmaximal, complemented
5C2×#100×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107C411yescharacteristic, complemented
7#16812092—21no—
8#16812090C411yescharacteristic, complemented
9#16812090C2211yescharacteristic
10#16812081—21no—
11#16812081—21no—
12#16812078—21no—
13#16812077C411yescharacteristic, complemented
14#16812077C2211yescharacteristic
15C2×C22⋊Dic107—21no—
16C2×C22⋊Dic107C411yescharacteristic, complemented
17#16812024—21no—
18#16812024C2212yes—
19C22×D4×C107C2211yescharacteristic
20C2×M16×C107—21no—
21#100×C107—21no—
22#100×C107C2212yes—
23C23×D107C2×C411yescharacteristic, complemented
24C2×C22⋊D107—21no—
25C2×C22⋊D107—21no—
26C2×C22⋊D107—21no—
27C2×C22⋊D107—21no—
28C2×C22⋊D107—41no—
29C2×C22⋊D107—41no—
30C22×Dic107—21no—
31C22×Dic107—41no—
32C22×Dic107C2×C411yescharacteristic, complemented
33Dic107⋊C4—21no—
34Dic107⋊C4—21no—
35C22⋊Dic107—21no—
36C22⋊Dic107—21no—
37C22⋊Dic107—21no—
38C22⋊Dic107—22no—
39C22⋊Dic107—41no—
40C22⋊Dic107C2×C411yescharacteristic, complemented
41C22⋊Dic107C2×C411yescharacteristic, complemented
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44C24×C107C2×C411yescharacteristic
45C24×C107D411yescharacteristic
46C2×D4×C107—41no—
47C2×D4×C107—41no—
48C2×D4×C107C2×C411yescharacteristic
49C2×D4×C107D411yescharacteristic
50C2×D4×C107D411yescharacteristic
51C2×D4×C107C2311yescharacteristic
52C22×C4×C107D411yescharacteristic
53M16×C107—22no—
54M16×C107—41no—
55C2×C8×C107—41no—
56C23×C107—21no—
57C23×C107—21no—
58C23×C107—21no—
59C23×C107—21no—
60C23×C107—81no—
61C23×C107—81no—
62C23×C107C22⋊C411yescharacteristic
63C23×C107C22⋊C411yescharacteristic
64C23×C107C22⋊C411yescharacteristic
65C23×C107C22×C411yescharacteristic
66C23×C107C2×D411yescharacteristic
67C22×D107—21no—
68C22×D107—41no—
69C22×D107—81no—
70D4×C107—41no—
71D4×C107—41no—
72D4×C107—41no—
73D4×C107—41no—
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107C22⋊C411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C22⋊D107—42no—
79C22⋊D107—41no—
80C22⋊D107—41no—
81C22⋊D107—81no—
82C22⋊D107—81no—
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—41no—
87C2×Dic107—42no—
88C2×Dic107—81no—
89C2×Dic107—82no—
90C8×C107—42no—
91C22×C107—21no—
92C22×C107—21no—
93C22×C107—21no—
94C22×C107—21no—
95C22×C107—41no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107—41no—
99C22×C107—41no—
100C22×C107—42no—
101C22×C107—81no—
102C22×C107—81no—
103C22×C107—81no—
104C22×C107#9911yescharacteristic
105C22×C107#9911yescharacteristic
106C22×C107C2×C22⋊C411yescharacteristic
107C2×D107—41no—
108C2×D107—41no—
109C2×D107—41no—
110C2×D107—82no—
111C4×C107—21no—
112C4×C107—21no—
113Dic107—41no—
114Dic107—41no—
115Dic107—41no—
116Dic107—82no—
117C2×C107—21no—
118C2×C107—21no—
119C2×C107—21no—
120C2×C107—41no—
121C2×C107—41no—
122C2×C107—41no—
123C2×C107—41no—
124C2×C107C2×#9911yescharacteristic
125D107—81no—
126#2121—1071nomaximal, Sylow
127C107#212111yescharacteristic, complemented, Sylow
128#534—1071no—
129C2×#100D10711yescharacteristic, complemented
130C2×#99—1071no—
131#352—1072no—
132AΣL(1,𝔽16)—1072no—
133C22×D4—1071no—
134C22×D4C2×D10711yescharacteristic
135C2×M16—21no—
136#123—1071no—
137#123—1071no—
138#123—2141no—
139#121—2141no—
140#121—2141no—
141#120—1071no—
142#120—1071no—
143#120—2141no—
144C2×C22⋊C4—1071no—
145C2×C22⋊C4—2141no—
146#100—21no—
147#100C2×D10712yes—
148#99—1072no—
149#99—2141no—
150C24—1071no—
151C24C4×D10711yescharacteristic
152C24D42811yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—4281no—
156C2×D4—4281no—
157C2×D4—2141no—
158C2×D4—2141no—
159C2×D4—2141no—
160C2×D4—2141no—
161C2×D4C4×D10711yescharacteristic
162C2×D4D42811yescharacteristic
163C2×D4C22⋊D10711yescharacteristic
164C2×D4C22×D10711yescharacteristic
165C22×C4—1071no—
166C22×C4—4281no—
167C22×C4—2141no—
168C22×C4C22⋊D10711yescharacteristic
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C4⋊C4—2141no—
173C4⋊C4—2141no—
174C22⋊C4—1071no—
175C22⋊C4—1071no—
176C22⋊C4—4281no—
177C22⋊C4—2142no—
178C22⋊C4—2141no—
179C22⋊C4—2141no—
180C22⋊C4—2141no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—8561no—
190C23—4281no—
191C23—2141no—
192C23(C2×D107)⋊C411yescharacteristic
193C23(C2×D107)⋊C411yescharacteristic
194C23(C2×D107)⋊C411yescharacteristic
195C23C2×C4×D10711yescharacteristic
196C23C2×D42811yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—8561no—
202D4—8561no—
203D4—4282no—
204D4—4281no—
205D4—4281no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—8562no—
209C2×C4—8561no—
210C2×C4—4282no—
211C2×C4—4281no—
212C2×C4—2141no—
213C2×C4—2141no—
214C2×C4—2141no—
215C2×C4(C2×D107)⋊C411yescharacteristic
216C2×C4C2×C22⋊D10711yescharacteristic
217C8—42no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—8562no—
232C22—4281no—
233C22—4281no—
234C22—4281no—
235C22#1681202311yescharacteristic
236C22#1681202311yescharacteristic
237C22C2×(C2×D107)⋊C411yescharacteristic
238C4—21no—
239C4—21no—
240C4—8562no—
241C4—4281no—
242C4—4281no—
243C4—4281no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—8561no—
252C2C2×#1681202311yescharacteristic
subgroupG/H
centerC2C2×#16812023
commutatorC23×C107C22×C4
FrattiniC2×D4C22×D107
FittingC2×#100×C107C2
socleC2×C107C2×#99
prime radical#20293186C2
cosocleC2×D4×C107C23
hypercenterC2×#100D107
hypocenterC107#2121
LCSG ▷ C23×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16812024C2semidirect
3. #20292418C2semidirect
4. #20292415C2semidirect
5. C2×#100×C107C2semidirect
6. C22×C22⋊D107C4semidirect
8. #16812090C4semidirect
13. #16812077C4semidirect
16. C2×C22⋊Dic107C4semidirect
23. C23×D107C2×C4semidirect
32. C22×Dic107C2×C4semidirect
40. C22⋊Dic107C2×C4semidirect
41. C22⋊Dic107C2×C4semidirect
127. C107#2121semidirect
129. C2×#100D107semidirect

Maximal subgroups

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

subgroupindex
#202931862
C2×#168120242
#202924182
#202924152
C2×#100×C1072
#2121107