Finite Group Databasebeta

← #28335510

Subgroups — #28335510

Order 13696.

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

normal subgroups only

This group has 240 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#20292963C211yesmaximal, characteristic, complemented
3#20292943C211yesmaximal, characteristic, complemented
4#20292564C211yesmaximal, characteristic, complemented
5#20292363C211yesmaximal, characteristic, complemented
6#20292350C211yesmaximal, characteristic, complemented
7#327×C107C211yesmaximal, characteristic, complemented
8#16812104—21no—
9#16812095—21no—
10#16812093—21no—
11#16812088C2211yescharacteristic
12#16812081—21no—
13#16812077—21no—
14D4×Dic107—21no—
15C2×C4⋊Dic107—21no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×(C2×D107)⋊C4—21no—
18C2×(C2×D107)⋊C4C2211yescharacteristic
19#16812043—21no—
20#16812040—21no—
21#16812034—21no—
22#16812030—21no—
23C22⋊(C107⋊C8)C2211yescharacteristic
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28C2×Dic214D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C22×Dic107—21no—
32C22×Dic107—21no—
33C22×Dic107D411yescharacteristic
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107—42no—
39C4⋊Dic107D411yescharacteristic
40Dic107⋊C4—21no—
41C22⋊Dic107—21no—
42C22⋊Dic107—21no—
43(C2×D107)⋊C4—21no—
44(C2×D107)⋊C4—21no—
45(C2×D107)⋊C4—21no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—42no—
48C4×Dic107—21no—
49C2×D4×C107—21no—
50C2×D4×C107D411yescharacteristic
51C22×C4×C107—21no—
52C22×C4×C107C2311yescharacteristic
53C2×C8×C107—21no—
54C4⋊C4×C107D411yescharacteristic
55C22⋊C4×C107—21no—
56C23×C107—21no—
57C23×C107C2×D411yescharacteristic
58C22×D107—21no—
59C22×D107—21no—
60C22×D107—21no—
61C22×D107—41no—
62C22×D107—41no—
63D4×C107—41no—
64D4×C107—41no—
65D4×C107—41no—
66C2×C4×C107—22no—
67C2×C4×C107—21no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C22⋊D107—81no—
74C22⋊D107—81no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107—41no—
82C2×Dic107—81no—
83Dic214—41no—
84C8×C107—41no—
85C107⋊C8—41no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107—41no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107#12011yescharacteristic
92C22×C107#13511yescharacteristic
93C22×C107#13711yescharacteristic
94C2×D107—42no—
95C2×D107—41no—
96C2×D107—41no—
97C2×D107—41no—
98C2×D107—81no—
99C4×C107—21no—
100C4×C107—21no—
101C4×C107—42no—
102C4×C107—41no—
103Dic107—41no—
104Dic107—41no—
105Dic107—41no—
106Dic107—81no—
107C2×C107—21no—
108C2×C107—21no—
109C2×C107—81no—
110C2×C107#45211yescharacteristic
111C2×C107#45311yescharacteristic
112C2×C107#45811yescharacteristic
113D107—81no—
114#1618—1071nomaximal, Sylow
115C107#161811yescharacteristic, complemented, Sylow
116#538—1071no—
117#484—1071no—
118#481—1071no—
119#399—1071no—
120#328—1071no—
121#327D10711yescharacteristic, complemented
122#323—1071no—
123#126—2141no—
124#124—2141no—
125#123—2141no—
126C22⋊Q8—1071no—
127#121—2141no—
128#121C2×D10711yescharacteristic
129#120—2141no—
130C4×D4—2141no—
131C2×C4⋊C4—1071no—
132C2×C4⋊C4—2141no—
133C2×C22⋊C4—1071no—
134C2×C22⋊C4—2141no—
135#107—2141no—
136#106—2141no—
137#103—2141no—
138#102—2141no—
139C22⋊C8—1071no—
140C22⋊C8C2×D10711yescharacteristic
141#95C2×D10711yescharacteristic
142C24—1071no—
143C2×Q8—1071no—
144C2×D4—21no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4D42811yescharacteristic
148C22×C4—21no—
149C22×C4—1071no—
150C22×C4—2141no—
151C22×C4—2141no—
152C22×C4C22×D10711yescharacteristic
153C2×C8—21no—
154C2×C8—2141no—
155C4⋊C4—1071no—
156C4⋊C4—4282no—
157C4⋊C4—2141no—
158C4⋊C4—2141no—
159C4⋊C4—2141no—
160C4⋊C4—2141no—
161C4⋊C4D42811yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—4282no—
164C22⋊C4—2141no—
165C22⋊C4—2141no—
166C22⋊C4—2141no—
167C22⋊C4—2141no—
168C22⋊C4—2141no—
169C22⋊C4—2141no—
170C42—2141no—
171C23—21no—
172C23—4281no—
173C23—4281no—
174C23—2141no—
175C23—2141no—
176C23—2141no—
177C23D4×D10711yescharacteristic
178Q8—4281no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—8561no—
183D4—8561no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8561no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4—4281no—
193C2×C4—4281no—
194C2×C4—2141no—
195C2×C4—2141no—
196C2×C4—2141no—
197C2×C4D4×D10711yescharacteristic
198C2×C4C2×D42811yescharacteristic
199C8—41no—
200C8—4281no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—8561no—
207C22—4282no—
208C22—4281no—
209C22—4281no—
210C22—4281no—
211C22#1681207911yescharacteristic
212C22#1681212811yescharacteristic
213C22#1681213811yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—8561no—
219C4—4281no—
220C4—4281no—
221C4—4281no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—8561no—
226C2#2029279011yescharacteristic
227C2#2029279611yescharacteristic
228C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107#120
prime radical#20293214C2
hypercenter#327D107
hypocenterC107#1618
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20292963C2semidirect
3. #20292943C2semidirect
4. #20292564C2semidirect
5. #20292363C2semidirect
6. #20292350C2semidirect
7. #327×C107C2semidirect
115. C107#1618semidirect
121. #327D107semidirect

Maximal subgroups

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

subgroupindex
#202932142
#202929632
#202929432
#202925642
#202923632
#202923502
#327×C1072
#1618107