Finite Group Databasebeta

← #28337667

Subgroups — #28337667

Order 13696.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#16812089C211yesmaximal, characteristic
2C2×#16812042C211yesmaximal, characteristic, complemented
3C2×#16812034C211yesmaximal, characteristic, complemented
4C2×#16812032C211yesmaximal, characteristic, complemented
5C2×(C2×D107)⋊C8C211yesmaximal, characteristic
6#20292484C211yesmaximal, characteristic
7#340×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×C4×D107C2211yescharacteristic
10C22×C107⋊C8C2211yescharacteristic
11#16812089—22no—
12#16812089—22no—
13C2×C4⋊Dic107—21no—
14C2×C4⋊Dic107C2211yescharacteristic
15C2×(C2×D107)⋊C4—21no—
16C2×(C2×D107)⋊C4—21no—
17#16812042—22no—
18#16812034—22no—
19#16812032—22no—
20(C2×D107)⋊C8—22no—
21C107⋊#95—21no—
22C22×C8×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C23×D107D411yescharacteristic
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214D412yes—
32C22×Dic107—21no—
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C2×C4×D107—21no—
36C2×C4×D107—21no—
37C2×C4×D107—21no—
38C2×C4×D107—21no—
39C2×C4×D107—21no—
40C2×C4×D107—21no—
41C2×C107⋊C8—21no—
42C2×C107⋊C8—21no—
43C2×C107⋊C8D412yes—
44C4⋊Dic107—24no—
45C4⋊Dic107—21no—
46C4⋊Dic107—21no—
47(C2×D107)⋊C4—24no—
48(C2×D107)⋊C4—42no—
49C22×C4×C107—21no—
50C22×C4×C107—21no—
51C22×C4×C107C2311yescharacteristic
52C2×C8×C107—21no—
53C2×C8×C107—21no—
54C2×C8×C107—21no—
55C4⋊C4×C107—21no—
56C4⋊C4×C107—21no—
57C4⋊C4×C107—21no—
58C4⋊C4×C107D412yes—
59C23×C107C2×D411yescharacteristic
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—21no—
63C22×D107—21no—
64C22×D107—21no—
65C22×D107—21no—
66C22×D107Q1612yes—
67C2×C4×C107—22no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107#4111yescharacteristic
77C2×C4×C107#4111yescharacteristic
78C2×C4×C107#4111yescharacteristic
79C2×Dic107—22no—
80C2×Dic107—21no—
81C2×Dic107—21no—
82C2×Dic107—21no—
83C2×Dic107—41no—
84C2×Dic107—41no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88C4×D107—41no—
89C4×D107—41no—
90C4×D107—41no—
91C4×D107—41no—
92Dic214—22no—
93Dic214—22no—
94Dic214—41no—
95Dic214—41no—
96C8×C107—41no—
97C107⋊C8—22no—
98C22×C107#12011yescharacteristic
99C22×C107#12111yescharacteristic
100C22×C107#12311yescharacteristic
101C22×C107C2×Q1611yescharacteristic
102C22×C107#13511yescharacteristic
103C22×C107AGL(1,ℤ/8)11yescharacteristic
104C22×C107#13711yescharacteristic
105C2×D107—22no—
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—41no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C4×C107—42no—
113C4×C107—81no—
114C4×C107#12111yescharacteristic
115C4×C107#12111yescharacteristic
116C4×C107#12311yescharacteristic
117C4×C107#12411yescharacteristic
118Dic107—42no—
119Dic107—41no—
120Dic107—81no—
121C2×C107#39411yescharacteristic
122C2×C107#44911yescharacteristic
123C2×C107#45111yescharacteristic
124C2×C107#46711yescharacteristic
125C2×C107#46811yescharacteristic
126C2×C107#48411yescharacteristic
127C2×C107#48511yescharacteristic
128D107—42no—
129#2036—1071nomaximal, Sylow
130C107#203611yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—1071no—
132C2×#107—1071no—
133C2×#103—1071no—
134C2×#103—1071no—
135C2×C22⋊C8—1071no—
136#380—1071no—
137#340D10711yescharacteristic, complemented
138C22×Q8—1071no—
139C23×C4—1071no—
140C22×C8—1071no—
141C22×C8C2×D10711yescharacteristic
142C22⋊Q8—2142no—
143C22⋊Q8—2142no—
144C2×C4⋊C4—1071no—
145C2×C4⋊C4—2141no—
146C2×C4⋊C4C2×D10711yescharacteristic
147C2×C4⋊C4C2×D10711yescharacteristic
148C2×C22⋊C4—2141no—
149C2×C22⋊C4—2141no—
150#107—2142no—
151#103—2142no—
152#103—2142no—
153C22⋊C8—2142no—
154#95—2141no—
155C24—1071no—
156C2×Q8—1072no—
157C2×Q8—2141no—
158C2×Q8—2141no—
159C2×Q8—2141no—
160C2×Q8—2141no—
161C2×Q8—2141no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1071no—
165C22×C4—2141no—
166C22×C4—2141no—
167C22×C4—2141no—
168C22×C4—2141no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1072no—
178C2×C8—2141no—
179C2×C8—2141no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—2144no—
184C4⋊C4—2141no—
185C4⋊C4—2141no—
186C4⋊C4D42812yes—
187C22⋊C4—4282no—
188C22⋊C4—2144no—
189C23—1072no—
190C23—2141no—
191C23—2141no—
192C23—2141no—
193C23—2141no—
194C23—2141no—
195C23—2141no—
196C23D4×D10711yescharacteristic
197Q8—4281no—
198Q8—4281no—
199Q8—2142no—
200Q8—2142no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—4281no—
208C2×C4—4281no—
209C2×C4—4281no—
210C2×C4—4281no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—4281no—
216C2×C4—2142no—
217C2×C4—2141no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4D4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C2×C4C2×D42811yescharacteristic
223C2×C4#1313381811yescharacteristic
224C2×C4#1313381811yescharacteristic
225C2×C4#1313381911yescharacteristic
226C8—41no—
227C8—2142no—
228C22—4281no—
229C22—4281no—
230C22—4281no—
231C22—4281no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22#1681207911yescharacteristic
236C22#1681208211yescharacteristic
237C22#1681209411yescharacteristic
238C22Q16×D10711yescharacteristic
239C22#1681213011yescharacteristic
240C22#1681213311yescharacteristic
241C22#1681213811yescharacteristic
242C4—42no—
243C4—81no—
244C4—8561no—
245C4—4281no—
246C4—4282no—
247C4#1681208211yescharacteristic
248C4#1681208411yescharacteristic
249C4#1681209411yescharacteristic
250C4#1681209711yescharacteristic
251C2—4282no—
252C2#2029254511yescharacteristic
253C2#2029277311yescharacteristic
254C2#2029278411yescharacteristic
255C2#2029286711yescharacteristic
256C2#2029287211yescharacteristic
257C2#2029296411yescharacteristic
258C2#2029297111yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
hypercenter#340D107
hypocenterC107#2036
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16812042C2semidirect
3. C2×#16812034C2semidirect
4. C2×#16812032C2semidirect
7. #340×C107C2semidirect
130. C107#2036semidirect
137. #340D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×#168120422
C2×#168120342
C2×#168120322
C2×(C2×D107)⋊C82
#202924842
#340×C1072
#2036107