Finite Group Databasebeta

← #28335511

Subgroups — #28335511

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#20293215C211yesmaximal, characteristic
2#20292963C211yesmaximal, characteristic, complemented
3#20292946C211yesmaximal, characteristic, complemented
4#20292563C211yesmaximal, characteristic, complemented
5#20292365C211yesmaximal, characteristic, complemented
6#20292362C211yesmaximal, characteristic, complemented
7#323×C107C211yesmaximal, characteristic, complemented
8#16812102—21no—
9#16812095—21no—
10#16812091—21no—
11#16812088C2211yescharacteristic
12#16812083C2211yescharacteristic
13#16812082—21no—
14#16812078—21no—
15C4×C22⋊D107—21no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4—21no—
18C2×C22⋊Dic107—21no—
19#16812043—21no—
20#16812041—21no—
21#16812034—21no—
22#16812031—21no—
23C107⋊#95C2211yescharacteristic
24C2×C22⋊C4×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26C22⋊C8×C107C2211yescharacteristic
27C2×Dic214D411yescharacteristic
28C2×D428D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C2×C22⋊D107—21no—
32C22×Dic107—21no—
33C22×Dic107—21no—
34C2×C4×D107—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107D411yescharacteristic
38C4⋊Dic107D411yescharacteristic
39Dic107⋊C4—21no—
40Dic107⋊C4—21no—
41Dic107⋊C4—42no—
42C22⋊Dic107—21no—
43C22⋊Dic107—21no—
44C22⋊Dic107—21no—
45C22⋊Dic107—42no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—21no—
48C4×Dic107—21no—
49C24×C107D411yescharacteristic
50C22×C4×C107D411yescharacteristic
51C22×C4×C107C2311yescharacteristic
52C2×C8×C107—21no—
53C2×C8×C107—21no—
54C22⋊C4×C107—21no—
55C22⋊C4×C107—21no—
56C23×C107—21no—
57C23×C107—21no—
58C23×C107—21no—
59C23×C107—41no—
60C23×C107—41no—
61C23×C107C2×D411yescharacteristic
62C22×D107—21no—
63C2×C4×C107—21no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C22⋊D107—41no—
69C22⋊D107—41no—
70C22⋊D107—81no—
71C22⋊D107—81no—
72C2×Dic107—22no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81D428—41no—
82C4×D107—81no—
83Dic214—41no—
84C8×C107—41no—
85C8×C107—41no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107—42no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107—41no—
92C22×C107—81no—
93C22×C107#12011yescharacteristic
94C22×C107#13511yescharacteristic
95C22×C107#13711yescharacteristic
96C2×D107—41no—
97C2×D107—41no—
98C2×D107—41no—
99C4×C107—21no—
100C4×C107—21no—
101C4×C107—41no—
102Dic107—42no—
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#327—1071no—
122#323D10711yescharacteristic, complemented
123#126—2141no—
124#124—2141no—
125#123—2141no—
126C22⋊Q8—1071no—
127#121—1071no—
128#121—2141no—
129#120—2141no—
130C4×D4—2141no—
131C2×C4⋊C4—1071no—
132C2×C4⋊C4—2141no—
133C2×C22⋊C4—2141no—
134C2×C22⋊C4C2×D10711yescharacteristic
135#107—2141no—
136#106—2141no—
137#103—2141no—
138#102—2141no—
139C22⋊C8C2×D10711yescharacteristic
140C22⋊C8C2×D10711yescharacteristic
141#95—1071no—
142C24D42811yescharacteristic
143C2×Q8—1071no—
144C2×D4—1071no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4—2141no—
148C22×C4—2141no—
149C22×C4—2141no—
150C22×C4—2141no—
151C22×C4D42811yescharacteristic
152C22×C4C22×D10711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—1071no—
156C4⋊C4—1071no—
157C4⋊C4—4282no—
158C4⋊C4—2141no—
159C4⋊C4—2141no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—4282no—
165C22⋊C4—2141no—
166C22⋊C4—2141no—
167C22⋊C4—2141no—
168C22⋊C4—2141no—
169C22⋊C4—2141no—
170C42—2141no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—2141no—
177C23C2×D42811yescharacteristic
178Q8—4281no—
179D4—8561no—
180D4—8561no—
181D4—4281no—
182D4—4281no—
183D4—4281no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8561no—
188C2×C4—4281no—
189C2×C4—4281no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4—4281no—
193C2×C4—2142no—
194C2×C4—2141no—
195C2×C4—2141no—
196C2×C4—2141no—
197C2×C4D4×D10711yescharacteristic
198C2×C4D4×D10711yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—4281no—
209C22—4281no—
210C22—4281no—
211C22#1681207911yescharacteristic
212C22#1681213111yescharacteristic
213C22#1681214211yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—8561no—
218C4—4282no—
219C4—4281no—
220C4—4281no—
221C4—4281no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—8561no—
226C2#2029279111yescharacteristic
227C2#2029279711yescharacteristic
228C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107#120
prime radical#20293215C2
hypercenter#323D107
hypocenterC107#1618
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292963C2semidirect
3. #20292946C2semidirect
4. #20292563C2semidirect
5. #20292365C2semidirect
6. #20292362C2semidirect
7. #323×C107C2semidirect
115. C107#1618semidirect
122. #323D107semidirect

Maximal subgroups

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

subgroupindex
#202932152
#202929632
#202929462
#202925632
#202923652
#202923622
#323×C1072
#1618107