Finite Group Databasebeta

← #28335752

Subgroups — #28335752

Order 13696.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293371C211yesmaximal, characteristic, complemented
2#20293014C211yesmaximal, characteristic, complemented
3#20292991C211yesmaximal, characteristic, complemented
4#20292368C211yesmaximal, characteristic, complemented
5#20292367C211yesmaximal, characteristic, complemented
6#20292366C211yesmaximal, characteristic, complemented
7#496×C107C211yesmaximal, characteristic, complemented
8C2×#13133806—21no—
9C2×#13133803—21no—
10#16812101C2211yescharacteristic
11#16812097C2211yescharacteristic
12#16812093—21no—
13#16812091—21no—
14#16812086—21no—
15#16812085—21no—
16#16812082—21no—
17#16812081—21no—
18#16812030—21no—
19#16812030—21no—
20#16812020C2211yescharacteristic
21#16812020C2211yescharacteristic
22C2×SD16×C107—21no—
23C2×D8×C107—21no—
24#127×C107C2211yescharacteristic
25#124×C107C2211yescharacteristic
26#97×C107C2211yescharacteristic
27C2×Dic214D411yescharacteristic
28C2×D428D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C22×Dic107—21no—
32C2×C4×D107—21no—
33#13133806—42no—
34#13133803—42no—
35C2×C107⋊C8—21no—
36C2×C107⋊C8—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107D411yescharacteristic
39C4⋊Dic107D411yescharacteristic
40Dic107⋊C4—21no—
41Dic107⋊C4—21no—
42Dic107⋊C4—21no—
43C22⋊Dic107—21no—
44(C2×D107)⋊C4—21no—
45(C2×D107)⋊C4—21no—
46(C2×D107)⋊C4—21no—
47C2×Q8×C107D411yescharacteristic
48C2×D4×C107—21no—
49C2×D4×C107—21no—
50C2×D4×C107—21no—
51C2×D4×C107D411yescharacteristic
52SD16×C107—42no—
53D8×C107—42no—
54C2×C8×C107—21no—
55C22⋊C4×C107—21no—
56C22⋊C4×C107—21no—
57C42×C107C2311yescharacteristic
58C23×C107—21no—
59C23×C107—41no—
60C22×D107—21no—
61Q8×C107—41no—
62D4×C107—41no—
63D4×C107—41no—
64D4×C107—41no—
65D4×C107—42no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—21no—
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C22⋊D107—81no—
73C22⋊D107—81no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—81no—
78D428—41no—
79C4×D107—81no—
80Dic214—41no—
81C8×C107—22no—
82C107⋊C8—22no—
83C107⋊C8—41no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107—41no—
87C22×C107—81no—
88C22×C107—81no—
89C22×C107—81no—
90C22×C107#12011yescharacteristic
91C2×D107—41no—
92C2×D107—41no—
93C2×D107—41no—
94C4×C107—21no—
95C4×C107—21no—
96C4×C107—21no—
97C4×C107—41no—
98Dic107—41no—
99Dic107—41no—
100Dic107—41no—
101C2×C107—81no—
102C2×C107—161no—
103C2×C107#45311yescharacteristic
104C2×C107#45311yescharacteristic
105C2×C107#45411yescharacteristic
106D107—81no—
107#1656—1071nomaximal, Sylow
108C107#165611yescharacteristic, complemented, Sylow
109#562—1071no—
110#496—1071no—
111#496D10711yescharacteristic, complemented
112#490—1071no—
113#330—1071no—
114#329—1071no—
115#329—1071no—
116C2×SD16—21no—
117C2×SD16—2141no—
118C2×D8—21no—
119C2×D8—2141no—
120#127C2×D10711yescharacteristic
121#125—1071no—
122#124—1071no—
123#124C2×D10711yescharacteristic
124#123—2141no—
125#123—2141no—
126C22⋊Q8—2141no—
127C22⋊Q8—2141no—
128#121—2141no—
129#121—2141no—
130#102—2141no—
131#102—2141no—
132#97—1071no—
133#97—1071no—
134#97C2×D10711yescharacteristic
135C2×Q8—1071no—
136C2×Q8C22⋊D10711yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—1071no—
141C2×D4—2141no—
142C2×D4—2141no—
143C2×D4C22⋊D10711yescharacteristic
144C22×C4—2141no—
145C22×C4—2141no—
146SD16—42no—
147SD16—4282no—
148D8—42no—
149D8—4282no—
150C2×C8—21no—
151C2×C8—2141no—
152C2×C8—2141no—
153C4⋊C4—1071no—
154C4⋊C4—1071no—
155C4⋊C4—2141no—
156C4⋊C4—2141no—
157C4⋊C4—2141no—
158C4⋊C4—2141no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—2141no—
162C22⋊C4—2141no—
163C22⋊C4—2141no—
164C22⋊C4—2141no—
165C42C22×D10711yescharacteristic
166C23—21no—
167C23—41no—
168C23—2141no—
169Q8—41no—
170Q8—4281no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—8561no—
178D4—8561no—
179D4—4281no—
180C2×C4—21no—
181C2×C4—8561no—
182C2×C4—8561no—
183C2×C4—2141no—
184C2×C4—2141no—
185C2×C4—2141no—
186C2×C4D4×D10711yescharacteristic
187C2×C4D4×D10711yescharacteristic
188C2×C4C2×C22⋊D10711yescharacteristic
189C8—22no—
190C8—4281no—
191C8—2142no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—4281no—
199C22—4281no—
200C22—4281no—
201C22#1681207711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—4281no—
207C4—4281no—
208C4—4281no—
209C2—81no—
210C2—161no—
211C2—8561no—
212C2#2029279311yescharacteristic
213C2#2029279411yescharacteristic
214C2#2029279811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#496×C107C2
socleC22×C107#120
hypercenter#496D107
hypocenterC107#1656
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20293371C2semidirect
2. #20293014C2semidirect
3. #20292991C2semidirect
4. #20292368C2semidirect
5. #20292367C2semidirect
6. #20292366C2semidirect
7. #496×C107C2semidirect
108. C107#1656semidirect
111. #496D107semidirect

Maximal subgroups

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

subgroupindex
#202933712
#202930142
#202929912
#202923682
#202923672
#202923662
#496×C1072
#1656107