Finite Group Databasebeta

← #28335438

Subgroups — #28335438

Order 13696.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293214C211yesmaximal, characteristic
2#20292783C211yesmaximal, characteristic, complemented
3#20292538C211yesmaximal, characteristic, complemented
4#20292365C211yesmaximal, characteristic, complemented
5#20292361C211yesmaximal, characteristic, complemented
6#20292350C211yesmaximal, characteristic, complemented
7#448×C107C211yesmaximal, characteristic, complemented
8C2×#13133808—21no—
9#16812104—21no—
10#16812095—21no—
11#16812093—21no—
12#16812088C2211yescharacteristic
13#16812081—21no—
14#16812077—21no—
15D4×Dic107—21no—
16C2×(C2×D107)⋊C4—21no—
17C2×(C2×D107)⋊C4C2211yescharacteristic
18#16812033—21no—
19C22⋊(C107⋊C8)C2211yescharacteristic
20C107⋊#95—21no—
21C107⋊#95C2211yescharacteristic
22C22×Q8×C107C2211yescharacteristic
23C2×SD16×C107—21no—
24#121×C107C2211yescharacteristic
25#103×C107—21no—
26C22⋊C8×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28C2×Dic214D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C22×Dic107—21no—
32C22×Dic107—21no—
33C22×Dic107D411yescharacteristic
34#13133808—42no—
35C2×C107⋊C8—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107D411yescharacteristic
38Dic107⋊C4—21no—
39C22⋊Dic107—21no—
40C22⋊Dic107—21no—
41(C2×D107)⋊C4—21no—
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44(C2×D107)⋊C4—21no—
45(C2×D107)⋊C4—42no—
46C4×Dic107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—42no—
50C2×D4×C107—21no—
51C2×D4×C107D411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54SD16×C107—42no—
55C2×C8×C107—21no—
56C4⋊C4×C107D411yescharacteristic
57C22⋊C4×C107—21no—
58C23×C107—21no—
59C23×C107C2×D411yescharacteristic
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—21no—
63C22×D107—41no—
64C22×D107—41no—
65Q8×C107—22no—
66Q8×C107—22no—
67Q8×C107—41no—
68Q8×C107—41no—
69D4×C107—41no—
70D4×C107—41no—
71D4×C107—41no—
72C2×C4×C107—22no—
73C2×C4×C107—21no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C22⋊D107—81no—
80C22⋊D107—81no—
81C2×Dic107—21no—
82C2×Dic107—21no—
83C2×Dic107—21no—
84C2×Dic107—41no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88C2×Dic107—81no—
89Dic214—41no—
90C8×C107—41no—
91C107⋊C8—41no—
92C22×C107—21no—
93C22×C107—21no—
94C22×C107—41no—
95C22×C107—41no—
96C22×C107—41no—
97C22×C107#12011yescharacteristic
98C22×C107#13511yescharacteristic
99C22×C107#13711yescharacteristic
100C2×D107—42no—
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C2×D107—81no—
105C4×C107—21no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—42no—
109Dic107—41no—
110Dic107—41no—
111Dic107—41no—
112Dic107—81no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—81no—
116C2×C107#45211yescharacteristic
117C2×C107#45511yescharacteristic
118C2×C107B(4,𝔽2)11yescharacteristic
119D107—81no—
120#1606—1071nomaximal, Sylow
121C107#160611yescharacteristic, complemented, Sylow
122#538—1071no—
123#451—1071no—
124#448D10711yescharacteristic, complemented
125#393—1071no—
126#328—1071no—
127#327—1071no—
128#323—1071no—
129C22×Q8C2×D10711yescharacteristic
130C2×Q16—2141no—
131C2×SD16—21no—
132#126—2141no—
133#124—2141no—
134#123—2141no—
135C22⋊Q8—1071no—
136#121—2141no—
137#121C2×D10711yescharacteristic
138#120—2141no—
139C4×D4—2141no—
140C2×C22⋊C4—1071no—
141C2×C22⋊C4—2141no—
142#103—21no—
143#103—2141no—
144C22⋊C8—1071no—
145C22⋊C8C2×D10711yescharacteristic
146#95—1071no—
147#95—2141no—
148C24—1071no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1071no—
153C2×D4—21no—
154C2×D4—2141no—
155C2×D4—2141no—
156C2×D4C22⋊D10711yescharacteristic
157C22×C4—21no—
158C22×C4—1071no—
159C22×C4—2141no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—4282no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—2141no—
166C4⋊C4—1071no—
167C4⋊C4—2141no—
168C4⋊C4—2141no—
169C4⋊C4C22⋊D10711yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—4282no—
172C22⋊C4—2141no—
173C22⋊C4—2141no—
174C22⋊C4—2141no—
175C22⋊C4—2141no—
176C22⋊C4—2141no—
177C22⋊C4—2141no—
178C42—2141no—
179C23—21no—
180C23—4281no—
181C23—4281no—
182C23—2141no—
183C23—2141no—
184C23—2141no—
185C23D4×D10711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—4281no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—8561no—
195D4—8561no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—8561no—
202C2×C4—4281no—
203C2×C4—4281no—
204C2×C4—4281no—
205C2×C4—4281no—
206C2×C4—2141no—
207C2×C4—2141no—
208C2×C4—2141no—
209C2×C4D4×D10711yescharacteristic
210C2×C4C2×C22⋊D10711yescharacteristic
211C8—41no—
212C8—4281no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—8561no—
219C22—4282no—
220C22—4281no—
221C22—4281no—
222C22—4281no—
223C22#1681207711yescharacteristic
224C22#1681213011yescharacteristic
225C22#1681213811yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—8561no—
231C4—4281no—
232C4—4281no—
233C4—4281no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—8561no—
238C2#2029278611yescharacteristic
239C2#2029280311yescharacteristic
240C2#2029281311yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#448×C107C2
socleC22×C107#120
prime radical#20293214C2
hypercenter#448D107
hypocenterC107#1606
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #20292783C2semidirect
3. #20292538C2semidirect
4. #20292365C2semidirect
5. #20292361C2semidirect
6. #20292350C2semidirect
7. #448×C107C2semidirect
121. C107#1606semidirect
124. #448D107semidirect

Maximal subgroups

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

subgroupindex
#202932142
#202927832
#202925382
#202923652
#202923612
#202923502
#448×C1072
#1606107