Finite Group Databasebeta

← #28335499

Subgroups — #28335499

Order 13696.

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

normal subgroups only

This group has 246 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#20293204C211yesmaximal, characteristic
2#20292946C211yesmaximal, characteristic, complemented
3#20292943C211yesmaximal, characteristic, complemented
4#20292567C211yesmaximal, characteristic, complemented
5#20292360C211yesmaximal, characteristic, complemented
6#20292352C211yesmaximal, characteristic, complemented
7#327×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9#16812087—21no—
10#16812086—21no—
11#16812083C2211yescharacteristic
12#16812082—21no—
13#16812081—21no—
14#16812081—21no—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×Dic107⋊C4C2211yescharacteristic
17#16812041—21no—
18#16812040—21no—
19#16812031—21no—
20#16812030—21no—
21C22⋊(C107⋊C8)C2211yescharacteristic
22C107⋊#95—21no—
23C107⋊#95—21no—
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27#13133817—42no—
28#13133817—42no—
29C2×Dic214—21no—
30C2×D428D411yescharacteristic
31C2×C22⋊D107—21no—
32C2×C22⋊D107—21no—
33C2×C22⋊D107—21no—
34C22×Dic107—21no—
35C22×Dic107—21no—
36C22×Dic107D411yescharacteristic
37C22×Dic107D411yescharacteristic
38C2×C4×D107—21no—
39C2×C107⋊C8—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107D411yescharacteristic
43Dic107⋊C4—21no—
44Dic107⋊C4—21no—
45Dic107⋊C4—21no—
46C22⋊Dic107—21no—
47(C2×D107)⋊C4—21no—
48(C2×D107)⋊C4—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—
59D4×C107—41no—
60D4×C107—41no—
61D4×C107—41no—
62C2×C4×C107—21no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C22⋊D107—42no—
70C22⋊D107—41no—
71C22⋊D107—41no—
72C22⋊D107—81no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—42no—
78C2×Dic107—41no—
79C2×Dic107—42no—
80C2×Dic107—41no—
81C2×Dic107SD1612yes—
82D428—41no—
83C4×D107—42no—
84Dic214—42no—
85C8×C107—41no—
86C107⋊C8—41no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107—41no—
92C22×C107#12011yescharacteristic
93C22×C107C2×SD1611yescharacteristic
94C22×C107AGL(1,ℤ/8)11yescharacteristic
95C2×D107—41no—
96C2×D107—41no—
97C2×D107—41no—
98C4×C107—21no—
99C4×C107—21no—
100C4×C107—41no—
101C4×C107—81no—
102Dic107—22no—
103Dic107—41no—
104Dic107—41no—
105Dic107—81no—
106C2×C107—21no—
107C2×C107—21no—
108C2×C107—81no—
109C2×C107#45011yescharacteristic
110C2×C107#45411yescharacteristic
111C2×C107#45811yescharacteristic
112D107—81no—
113#1616—1071nomaximal, Sylow
114C107#161611yescharacteristic, complemented, Sylow
115#537—1071no—
116#481—1071no—
117#481—1071no—
118#400—1071no—
119#327—1071no—
120#327D10711yescharacteristic, complemented
121#324—1071no—
122C2×#41—2141no—
123C22⋊Q8—2141no—
124C22⋊Q8—2141no—
125#121—1071no—
126#121—2141no—
127#121—2141no—
128#121—2141no—
129#121C2×D10711yescharacteristic
130C2×C4⋊C4—1071no—
131C2×C4⋊C4—1071no—
132#106—2141no—
133#106—2141no—
134#102—2141no—
135#102—2141no—
136C22⋊C8—1071no—
137C22⋊C8C2×D10711yescharacteristic
138#95—2141no—
139#95—2141no—
140#95C2×D10711yescharacteristic
141#41—4282no—
142#41—4282no—
143C2×Q8—2141no—
144C2×D4—21no—
145C2×D4—1071no—
146C2×D4—2141no—
147C2×D4—2141no—
148C2×D4—2141no—
149C2×D4D42811yescharacteristic
150C22×C4—21no—
151C22×C4—1071no—
152C22×C4—1071no—
153C22×C4—2141no—
154C22×C4—2141no—
155C22×C4—2141no—
156C22×C4C22×D10711yescharacteristic
157C2×C8—21no—
158C2×C8—2141no—
159C4⋊C4—1071no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C4⋊C4—2141no—
163C4⋊C4—2141no—
164C4⋊C4—2141no—
165C4⋊C4D42811yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2141no—
168C22⋊C4—2141no—
169C22⋊C4—2141no—
170C23—21no—
171C23—2141no—
172C23D4×D10711yescharacteristic
173Q8—4282no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—8561no—
178D4—4281no—
179D4—4281no—
180D4—4281no—
181D4—4282no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1072no—
188C2×C4—4282no—
189C2×C4—4282no—
190C2×C4—4281no—
191C2×C4—4282no—
192C2×C4—4281no—
193C2×C4—2141no—
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—4281no—
207C22—4281no—
208C22—4281no—
209C22#1681207911yescharacteristic
210C22SD16×D10711yescharacteristic
211C22#1681213311yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—8561no—
217C4—4281no—
218C4—4281no—
219C4—2142no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—8561no—
224C2#2029277911yescharacteristic
225C2#2029280111yescharacteristic
226C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107#120
prime radical#20293204C2
hypercenter#327D107
hypocenterC107#1616
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. #20292946C2semidirect
3. #20292943C2semidirect
4. #20292567C2semidirect
5. #20292360C2semidirect
6. #20292352C2semidirect
7. #327×C107C2semidirect
114. C107#1616semidirect
120. #327D107semidirect

Maximal subgroups

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

subgroupindex
#202932042
#202929462
#202929432
#202925672
#202923602
#202923522
#327×C1072
#1616107