Finite Group Databasebeta

← #28335529

Subgroups — #28335529

Order 13696.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293298C211yesmaximal, characteristic, complemented
2#20293004C211yesmaximal, characteristic, complemented
3#20292828C211yesmaximal, characteristic, complemented
4#20292822C211yesmaximal, characteristic, complemented
5#20292370C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133818—21no—
9C2×#13133807—21no—
10C2×D856—22no—
11C2×#13133803—21no—
12#16812106C2211yescharacteristic
13#16812106C2211yescharacteristic
14#16812105—21no—
15#16812083—21no—
16#16812082—22no—
17C4×Dic214C2211yescharacteristic
18#16812071—21no—
19C4⋊(C107⋊C8)C2211yescharacteristic
20#16812031—21no—
21#16812029—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25#13133818—42no—
26#13133818—42no—
27C2×Dic214D411yescharacteristic
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428—21no—
33C2×D428D412yes—
34C2×C22⋊D107—22no—
35C2×C22⋊D107—22no—
36C2×C4×D107—22no—
37#13133807—42no—
38D856—44no—
39#13133803—42no—
40C2×C107⋊C8—21no—
41C4⋊Dic107D411yescharacteristic
42Dic107⋊C4—21no—
43(C2×D107)⋊C4—22no—
44C4×Dic107—21no—
45C2×D4×C107D411yescharacteristic
46C22×C4×C107—21no—
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C4⋊C4×C107D411yescharacteristic
50C22⋊C4×C107—21no—
51C42×C107C2311yescharacteristic
52C23×C107—21no—
53C22×D107—22no—
54C22×D107—41no—
55D4×C107—21no—
56D4×C107—21no—
57C2×C4×C107—21no—
58C2×C4×C107—41no—
59C2×C4×C107—41no—
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C2×C4×C107C2×D411yescharacteristic
63C22⋊D107—44no—
64C22⋊D107—82no—
65C2×Dic107—21no—
66C2×Dic107—21no—
67D428—42no—
68D428—42no—
69D428—41no—
70D428—41no—
71D428—41no—
72D428—41no—
73D428—41no—
74D428—41no—
75D428—41no—
76D428—41no—
77C4×D107—44no—
78Dic214—21no—
79Dic214D812yescomplemented
80C8×C107—22no—
81C8×C107—41no—
82C107⋊C8—41no—
83C22×C107—41no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107#12011yescharacteristic
87C2×D107—42no—
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—81no—
91C2×D107—81no—
92C2×D107—81no—
93C4×C107—21no—
94C4×C107—41no—
95C4×C107C2×D811yescharacteristic
96C4×C107#13511yescharacteristic
97C4×C107AGL(1,ℤ/8)11yescharacteristic
98C4×C107AGL(1,ℤ/8)11yescharacteristic
99Dic107—22no—
100Dic107—41no—
101C2×C107—81no—
102C2×C107#44711yescharacteristic
103C2×C107#44911yescharacteristic
104C2×C107#45311yescharacteristic
105D107—82no—
106D107—161no—
107#1621—1071nomaximal, Sylow
108C107#162111yescharacteristic, complemented, Sylow
109#550—1071no—
110#493—1071no—
111#460—1071no—
112#459—1071no—
113#331—1071no—
114Q8⋊C8—1071no—
115D4⋊C8D10711yescharacteristic, complemented
116C2×#41—2141no—
117C2×SD16—2141no—
118C2×D8—2142no—
119C2×D8—2141no—
120#127—1071no—
121#127—1071no—
122#127—2141no—
123#121—2142no—
124#121—2141no—
125C4×Q8—1071no—
126C4×D4—2141no—
127C4×D4C2×D10711yescharacteristic
128C4⋊C8—1071no—
129C4⋊C8C2×D10711yescharacteristic
130#102—2141no—
131#102—2141no—
132C4×C8C2×D10711yescharacteristic
133#41—4282no—
134#41—4282no—
135C2×Q8—1071no—
136C2×D4—1072no—
137C2×D4—2142no—
138C2×D4—2142no—
139C2×D4—2141no—
140C2×D4—2141no—
141C2×D4—2141no—
142C2×D4—2141no—
143C2×D4—2141no—
144C2×D4D42811yescharacteristic
145C22×C4—21no—
146C22×C4—2142no—
147SD16—4282no—
148D8—4284no—
149D8—4282no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2141no—
153C4⋊C4—1071no—
154C4⋊C4—2141no—
155C4⋊C4D42811yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—2142no—
158C42—2141no—
159C42C22×D10711yescharacteristic
160C23—21no—
161C23—4281no—
162C23—2142no—
163Q8—1072no—
164Q8—2141no—
165D4—21no—
166D4—21no—
167D4—8562no—
168D4—4282no—
169D4—4282no—
170D4—4284no—
171D4—4281no—
172D4—4281no—
173D4—4281no—
174D4—4281no—
175D4—4281no—
176D4—4281no—
177D4—4281no—
178D4—4281no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—4284no—
183C2×C4—2141no—
184C2×C4—2141no—
185C2×C4D4×D10711yescharacteristic
186C2×C4D4×D10711yescharacteristic
187C2×C4C2×D42811yescharacteristic
188C8—22no—
189C8—41no—
190C8—4281no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—8561no—
195C22—8561no—
196C22—8561no—
197C22—4282no—
198C22—4282no—
199C22—4282no—
200C22#1681207911yescharacteristic
201C4—21no—
202C4—41no—
203C4—4281no—
204C4—2142no—
205C4D8×D10711yescharacteristic
206C4#1681213111yescharacteristic
207C4#1681213611yescharacteristic
208C4#1681213711yescharacteristic
209C2—81no—
210C2—8562no—
211C2—17121no—
212C2#2029276211yescharacteristic
213C2#2029277411yescharacteristic
214C2#2029279611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
hypercenterD4⋊C8D107
hypocenterC107#1621
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20293298C2semidirect
2. #20293004C2semidirect
3. #20292828C2semidirect
4. #20292822C2semidirect
5. #20292370C2semidirect
6. #20292359C2semidirect
7. D4⋊C8×C107C2semidirect
79. Dic214D8semidirect
108. C107#1621semidirect
115. D4⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202932982
#202930042
#202928282
#202928222
#202923702
#202923592
D4⋊C8×C1072
#1621107