Finite Group Databasebeta

← #28337596

Subgroups — #28337596

Order 13696.

326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16812108C211yesmaximal, characteristic, complemented
2C2×#16812031C211yesmaximal, characteristic
3C2×#16812029C211yesmaximal, characteristic
4#20292528C211yesmaximal, characteristic
5#20292389C211yesmaximal, characteristic, complemented
6#20292381C211yesmaximal, characteristic, complemented
7#340×C107C211yesmaximal, characteristic, complemented
8C22×Dic214—21no—
9C22×D428C2211yescharacteristic
10C22×C107⋊C8C2211yescharacteristic
11#16812108—22no—
12#16812108—22no—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4—21no—
15C2×(C2×D107)⋊C4—21no—
16C2×(C2×D107)⋊C4—21no—
17C2×C4×Dic107C2211yescharacteristic
18#16812031—22no—
19#16812029—22no—
20C22×C8×C107C2211yescharacteristic
21C2×C4⋊C4×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23C23×D107—21no—
24C2×Dic214—24no—
25C2×Dic214—42no—
26C2×Dic214—42no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428Q812yescomplemented
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C22×Dic107D411yescharacteristic
36C2×C107⋊C8—21no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C4⋊Dic107—21no—
40C4⋊Dic107D412yes—
41Dic107⋊C4—24no—
42(C2×D107)⋊C4—42no—
43(C2×D107)⋊C4—42no—
44C4×Dic107—21no—
45C4×Dic107—21no—
46C22×C4×C107—21no—
47C22×C4×C107—21no—
48C22×C4×C107C2311yescharacteristic
49C2×C8×C107—21no—
50C2×C8×C107—21no—
51C2×C8×C107—21no—
52C4⋊C4×C107—21no—
53C4⋊C4×C107—21no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107D412yes—
56C23×C107C2×D411yescharacteristic
57C22×D107—22no—
58C22×D107—41no—
59C22×D107—41no—
60C22×D107—41no—
61C22×D107—41no—
62C22×D107—41no—
63C22×D107—41no—
64C2×C4×C107—22no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×Q811yescharacteristic
73C2×C4×C107#4111yescharacteristic
74C2×C4×C107#4111yescharacteristic
75C2×C4×C107#4111yescharacteristic
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—21no—
79C2×Dic107—21no—
80C2×Dic107—22no—
81C2×Dic107—41no—
82C2×Dic107—41no—
83C2×Dic107D812yes—
84C2×Dic107SD1612yes—
85D428—22no—
86D428—22no—
87D428—41no—
88D428—41no—
89Dic214—48no—
90C8×C107—41no—
91C107⋊C8—41no—
92C22×C107#12011yescharacteristic
93C22×C107C22⋊Q811yescharacteristic
94C22×C107#12411yescharacteristic
95C22×C107C2×D811yescharacteristic
96C22×C107C2×SD1611yescharacteristic
97C22×C107AGL(1,ℤ/8)11yescharacteristic
98C22×C107#13711yescharacteristic
99C2×D107—42no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—81no—
103C2×D107—81no—
104C2×D107—81no—
105C2×D107—81no—
106C4×C107—42no—
107C4×C107—81no—
108C4×C107C22⋊Q811yescharacteristic
109C4×C107C22⋊Q811yescharacteristic
110C4×C107#12411yescharacteristic
111C4×C107#12411yescharacteristic
112Dic107—22no—
113Dic107—22no—
114Dic107—42no—
115C2×C107#39311yescharacteristic
116C2×C107#44711yescharacteristic
117C2×C107#44811yescharacteristic
118C2×C107#47411yescharacteristic
119C2×C107#47611yescharacteristic
120C2×C107#48611yescharacteristic
121C2×C107#48911yescharacteristic
122D107—82no—
123#2023—1071nomaximal, Sylow
124C107#202311yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—1071no—
126C2×#102—1071no—
127C2×#102—1071no—
128#390—1071no—
129#340—1071no—
130#340D10711yescharacteristic, complemented
131#336—1071no—
132C22×Q8—2141no—
133C22×D4—1071no—
134C22×C8—1071no—
135C22×C8C2×D10711yescharacteristic
136C4⋊Q8—2142no—
137C4⋊Q8—2142no—
138C2×C4⋊C4—1071no—
139C2×C4⋊C4—2141no—
140C2×C4⋊C4C2×D10711yescharacteristic
141C2×C4⋊C4C2×D10711yescharacteristic
142C2×C22⋊C4—2141no—
143C2×C22⋊C4—2141no—
144C2×C42—1071no—
145#102—2142no—
146#102—2142no—
147C24—2141no—
148C2×Q8—4282no—
149C2×Q8—4282no—
150C2×Q8—2144no—
151C2×D4—1072no—
152C2×D4—2141no—
153C2×D4—2141no—
154C2×D4—2141no—
155C2×D4—2141no—
156C2×D4—2141no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—1071no—
160C22×C4—1071no—
161C22×C4—2141no—
162C22×C4C22×D10711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2141no—
167C2×C8—2141no—
168C2×C8—2141no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1072no—
173C4⋊C4—2144no—
174C4⋊C4—2141no—
175C4⋊C4D42812yes—
176C22⋊C4—4282no—
177C22⋊C4—4282no—
178C42—2141no—
179C42—2141no—
180C23—4281no—
181C23—4281no—
182C23—4281no—
183C23—4281no—
184C23—4281no—
185C23—4281no—
186C23—2142no—
187C23D4×D10711yescharacteristic
188Q8—4288no—
189D4—4281no—
190D4—4281no—
191D4—2142no—
192D4—2142no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1072no—
200C2×C4—1072no—
201C2×C4—4281no—
202C2×C4—4281no—
203C2×C4—2142no—
204C2×C4—2141no—
205C2×C4—2141no—
206C2×C4—2141no—
207C2×C4—2141no—
208C2×C4D4×D10711yescharacteristic
209C2×C4C2×D42811yescharacteristic
210C2×C4Q8×D10711yescharacteristic
211C2×C4#1313381711yescharacteristic
212C2×C4#1313381811yescharacteristic
213C2×C4#1313381811yescharacteristic
214C8—41no—
215C8—4281no—
216C22—8561no—
217C22—8561no—
218C22—8561no—
219C22—8561no—
220C22—4282no—
221C22—4282no—
222C22—4282no—
223C22#1681207911yescharacteristic
224C22#1681208711yescharacteristic
225C22#1681209511yescharacteristic
226C22D8×D10711yescharacteristic
227C22SD16×D10711yescharacteristic
228C22#1681213611yescharacteristic
229C22#1681214111yescharacteristic
230C4—42no—
231C4—81no—
232C4—4282no—
233C4—2142no—
234C4—2142no—
235C4#1681208711yescharacteristic
236C4#1681208911yescharacteristic
237C4#1681209511yescharacteristic
238C4#1681209711yescharacteristic
239C2—8562no—
240C2#2029254011yescharacteristic
241C2#2029276211yescharacteristic
242C2#2029276711yescharacteristic
243C2#2029290611yescharacteristic
244C2#2029291611yescharacteristic
245C2#2029297211yescharacteristic
246C2#2029298811yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socleC23×C107C2×D4
prime radicalC22×D428C22
hypercenter#340D107
hypocenterC107#2023
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
1. C2×#16812108C2semidirect
5. #20292389C2semidirect
6. #20292381C2semidirect
7. #340×C107C2semidirect
32. C2×D428Q8semidirect
124. C107#2023semidirect
130. #340D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121082
C2×#168120312
C2×#168120292
#202925282
#202923892
#202923812
#340×C1072
#2023107