Finite Group Databasebeta

← #28335419

Subgroups — #28335419

Order 13696.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#20293252C211yesmaximal, characteristic
2#20292785C211yesmaximal, characteristic, complemented
3#20292782C211yesmaximal, characteristic, complemented
4#20292540C211yesmaximal, characteristic
5#20292363C211yesmaximal, characteristic, complemented
6#20292350C211yesmaximal, characteristic
7#328×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C2×Dic428—21no—
10C2×#13133808—21no—
11#16812089—21no—
12#16812088C2211yescharacteristic
13#16812087—21no—
14#16812087—21no—
15#16812086—21no—
16#16812085—21no—
17C2×(C2×D107)⋊C4—21no—
18C2×(C2×D107)⋊C4C2211yescharacteristic
19C22⋊C4×D107—21no—
20#16812034—21no—
21#16812032—21no—
22C22⋊(C107⋊C8)C2211yescharacteristic
23C107⋊#95—21no—
24C22⋊Q8×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—42no—
32C2×Dic214D411yescharacteristic
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C2×C4×D107—21no—
36C2×C4×D107—21no—
37Dic428—42no—
38#13133808—42no—
39C2×C107⋊C8—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107D411yescharacteristic
42Dic107⋊C4—21no—
43Dic107⋊C4—21no—
44Dic107⋊C4—21no—
45C22⋊Dic107—21no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—21no—
48(C2×D107)⋊C4—21no—
49(C2×D107)⋊C4—21no—
50(C2×D107)⋊C4—42no—
51C2×Q8×C107D411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—21no—
55C4⋊C4×C107—21no—
56C4⋊C4×C107D411yescharacteristic
57C22⋊C4×C107—21no—
58C23×C107C2×D411yescharacteristic
59C22×D107—21no—
60C22×D107—21no—
61C22×D107—41no—
62C22×D107—41no—
63C22×D107Q1612yescomplemented
64Q8×C107—41no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×Dic107—22no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—41no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C4×D107—81no—
81C4×D107—81no—
82Dic214—22no—
83Dic214—22no—
84Dic214—41no—
85Dic214—41no—
86Dic214—41no—
87Dic214—81no—
88C8×C107—41no—
89C107⋊C8—41no—
90C22×C107—21no—
91C22×C107—21no—
92C22×C107#12011yescharacteristic
93C22×C107C2×Q1611yescharacteristic
94C22×C107#13711yescharacteristic
95C2×D107—22no—
96C2×D107—41no—
97C2×D107—41no—
98C2×D107—42no—
99C2×D107—81no—
100C4×C107—21no—
101C4×C107—21no—
102C4×C107—41no—
103C4×C107—41no—
104C4×C107—81no—
105Dic107—41no—
106Dic107—41no—
107Dic107—42no—
108Dic107—41no—
109C2×C107—21no—
110C2×C107—21no—
111C2×C107#45111yescharacteristic
112C2×C107#45511yescharacteristic
113C2×C107B(4,𝔽2)11yescharacteristic
114D107—42no—
115#1602—1071nomaximal, Sylow
116C107#160211yescharacteristic, complemented, Sylow
117#543—1071no—
118#451—1071no—
119#451—1071no—
120#393—1071no—
121#328—1071no—
122#328D10711yescharacteristic, complemented
123#323—1071no—
124C22×Q8—1071no—
125C2×Q16—2141no—
126C2×Q16—2141no—
127C22⋊Q8—1071no—
128C22⋊Q8—2141no—
129C22⋊Q8—2141no—
130C22⋊Q8—2141no—
131C22⋊Q8—2141no—
132C22⋊Q8—2141no—
133C22⋊Q8C2×D10711yescharacteristic
134C2×C22⋊C4—1071no—
135C2×C22⋊C4—2141no—
136C2×C22⋊C4—2141no—
137#103—2141no—
138#103—2141no—
139C22⋊C8—1071no—
140C22⋊C8C2×D10711yescharacteristic
141#95—2141no—
142#95C2×D10711yescharacteristic
143C24—1071no—
144C2×Q8—1071no—
145C2×Q8—4282no—
146C2×Q8—2141no—
147C2×Q8—2141no—
148C2×Q8—2141no—
149C2×Q8D42811yescharacteristic
150C22×C4—21no—
151C22×C4—1071no—
152C22×C4—2141no—
153C22×C4—2141no—
154C22×C4—2141no—
155C22×C4C22×D10711yescharacteristic
156Q16—4282no—
157Q16—4282no—
158C2×C8—21no—
159C2×C8—2141no—
160C4⋊C4—21no—
161C4⋊C4—1071no—
162C4⋊C4—2141no—
163C4⋊C4—2141no—
164C4⋊C4—2141no—
165C4⋊C4—2141no—
166C4⋊C4D42811yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—4282no—
169C22⋊C4—2141no—
170C22⋊C4—2141no—
171C22⋊C4—2141no—
172C22⋊C4—2141no—
173C22⋊C4—2141no—
174C23—1072no—
175C23—4281no—
176C23—4281no—
177C23—2141no—
178C23—2141no—
179C23D4×D10711yescharacteristic
180Q8—41no—
181Q8—8561no—
182Q8—4281no—
183Q8—4281no—
184Q8—4281no—
185Q8—2142no—
186Q8—2142no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8561no—
194C2×C4—8561no—
195C2×C4—4281no—
196C2×C4—4281no—
197C2×C4—4281no—
198C2×C4—2142no—
199C2×C4—2141no—
200C2×C4—2141no—
201C2×C4—2141no—
202C2×C4D4×D10711yescharacteristic
203C2×C4C2×D42811yescharacteristic
204C8—41no—
205C8—4281no—
206C22—21no—
207C22—21no—
208C22—8561no—
209C22—4282no—
210C22—4281no—
211C22—4281no—
212C22—2142no—
213C22#1681207911yescharacteristic
214C22Q16×D10711yescharacteristic
215C22#1681213811yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—4281no—
222C4—4281no—
223C4—4281no—
224C4—4282no—
225C2—21no—
226C2—21no—
227C2—4282no—
228C2#2029278411yescharacteristic
229C2#2029280611yescharacteristic
230C2#2029281511yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#328×C107C2
socleC22×C107#120
prime radicalC23×D107D4
hypercenter#328D107
hypocenterC107#1602
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20292785C2semidirect
3. #20292782C2semidirect
5. #20292363C2semidirect
7. #328×C107C2semidirect
63. C22×D107Q16semidirect
116. C107#1602semidirect
122. #328D107semidirect

Maximal subgroups

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

subgroupindex
#202932522
#202927852
#202927822
#202925402
#202923632
#202923502
#328×C1072
#1602107