Finite Group Databasebeta

← #28338405

Subgroups — #28338405

Order 13696.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#20293424C211yesmaximal, characteristic, complemented
2#20293063C211yesmaximal, characteristic, complemented
3#20292877C211yesmaximal, characteristic, complemented
4#20292438C211yesmaximal, characteristic, complemented
5#20292437C211yesmaximal, characteristic, complemented
6#20292408C211yesmaximal, characteristic, complemented
7#509×C107C211yesmaximal, characteristic, complemented
8#41×D107—21no—
9#16812141—21no—
10#16812138—21no—
11#16812136—21no—
12#16812133—21no—
13#16812131C2211yescharacteristic
14#16812129—21no—
15C2×#13133801C2211yescharacteristic
16#16812080—21no—
17#16812053—21no—
18#16812050—21no—
19#16812049C2211yescharacteristic, complemented
20#16812041C2211yescharacteristic, complemented
21#16812030—21no—
22#135×C107C2211yescharacteristic
23C2×D8×C107C2211yescharacteristic, complemented
24SD32×C107—21no—
25D16×C107—21no—
26M32×C107C2211yescharacteristic, complemented
27#13133819—21no—
28#13133818—21no—
29#13133818—41no—
30#13133817—21no—
31#13133817—41no—
32Q8×D107—21no—
33C2×C22⋊D107—41no—
34D4×D107—21no—
35D4×D107—41no—
36C2×C4×D107—41no—
37C2×C4×D107D411yescharacteristic
38Dic428D411yescharacteristic, complemented
39#13133808—21no—
40#13133807—21no—
41#13133806—21no—
42#13133805—21no—
43D856D411yescharacteristic, complemented
44#13133803—21no—
45#13133801—21no—
46#13133801—21no—
47C2×C107⋊C8D411yescharacteristic
48C4⋊Dic107—21no—
49C22⋊Dic107—41no—
50C107⋊C16—21no—
51#41×C107—21no—
52C2×D4×C107—21no—
53Q16×C107D411yescharacteristic, complemented
54SD16×C107—21no—
55D8×C107—21no—
56D8×C107—21no—
57D8×C107D411yescharacteristic, complemented
58C2×C8×C107C2311yescharacteristic
59C16×C107—21no—
60C23×C107—41no—
61C22×D107—41no—
62C22×D107D811yescharacteristic
63Q8×C107—21no—
64D4×C107—21no—
65D4×C107—41no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107C2×D411yescharacteristic
70C22⋊D107—41no—
71C22⋊D107—41no—
72C22⋊D107—41no—
73C22⋊D107—82no—
74C2×Dic107—41no—
75C2×Dic107—41no—
76C2×Dic107D811yescharacteristic
77D428—21no—
78D428—41no—
79C4×D107—21no—
80C4×D107—21no—
81C4×D107—41no—
82C4×D107—41no—
83C4×D107—41no—
84Dic214—21no—
85Dic214—41no—
86C8×C107C2×D411yescharacteristic
87C8×C107C2×D411yescharacteristic
88C107⋊C8—21no—
89C22×C107—41no—
90C22×C107—81no—
91C22×C107—81no—
92C22×C107—81no—
93C22×C107C2×D811yescharacteristic
94C2×D107—21no—
95C2×D107—41no—
96C2×D107—41no—
97C2×D107—82no—
98C4×C107—41no—
99C4×C107#12011yescharacteristic
100C4×C107AGL(1,ℤ/8)11yescharacteristic
101Dic107—21no—
102Dic107—41no—
103Dic107—81no—
104C2×C107—21no—
105C2×C107—81no—
106C2×C107—161no—
107C2×C107#44711yescharacteristic
108D107—41no—
109D107—81no—
110#2190—1071nomaximal, Sylow
111C107#219011yescharacteristic, complemented, Sylow
112#575—1071no—
113#509—1071no—
114#509D10711yescharacteristic, complemented
115#469—1071no—
116#360—1071no—
117#360—1071no—
118#349—1071no—
119C2×#41—2141no—
120#137—2141no—
121#137—2141no—
122AGL(1,ℤ/8)—2141no—
123AGL(1,ℤ/8)—2141no—
124#135—1071no—
125#135—2141no—
126#135C2×D10711yescharacteristic
127C2×D8C2×D10711yescharacteristic, complemented
128C2×M16—1071no—
129#121—2141no—
130SD32—21no—
131SD32—2141no—
132D16—21no—
133D16—2141no—
134M32—1071no—
135M32C2×D10711yescharacteristic, complemented
136#106—1071no—
137#102—2141no—
138#41—21no—
139#41—4281no—
140#41—4281no—
141#41—2141no—
142#41—2141no—
143#41—2141no—
144C2×Q8—2141no—
145C2×D4—21no—
146C2×D4—4281no—
147C2×D4—4281no—
148C2×D4—2141no—
149C22×C4—1071no—
150C22×C4—4281no—
151Q16—1071no—
152Q16—2141no—
153Q16C22⋊D10711yescharacteristic, complemented
154SD16—21no—
155SD16—2141no—
156SD16—2141no—
157SD16—2141no—
158D8—21no—
159D8—21no—
160D8—1071no—
161D8—2141no—
162D8C22⋊D10711yescharacteristic, complemented
163M16—2141no—
164M16—2141no—
165C2×C8—1071no—
166C2×C8C22×D10711yescharacteristic
167C4⋊C4—2141no—
168C22⋊C4—4281no—
169C16—21no—
170C16—2141no—
171C23—41no—
172C23—1071no—
173C23—4281no—
174Q8—21no—
175Q8—4281no—
176Q8—2141no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—8562no—
182D4—4281no—
183D4—4281no—
184D4—4281no—
185D4—4281no—
186D4—2141no—
187C2×C4—41no—
188C2×C4—1071no—
189C2×C4—4281no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4—4281no—
193C2×C4—4281no—
194C2×C4—2141no—
195C2×C4—2141no—
196C2×C4D4×D10711yescharacteristic
197C8—2141no—
198C8D4×D10711yescharacteristic
199C8C2×C22⋊D10711yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—8562no—
205C22—4281no—
206C22—4281no—
207C22—2141no—
208C22D8×D10711yescharacteristic
209C4—41no—
210C4—8561no—
211C4—4281no—
212C4—2141no—
213C4#1681207711yescharacteristic
214C4#1681213311yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—8561no—
219C2—4281no—
220C2#2029275911yescharacteristic
subgroupG/H
centerC2#20292759
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#509×C107C2
socleC2×C107#447
hypercenter#509D107
hypocenterC107#2190
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20293424C2semidirect
2. #20293063C2semidirect
3. #20292877C2semidirect
4. #20292438C2semidirect
5. #20292437C2semidirect
6. #20292408C2semidirect
7. #509×C107C2semidirect
19. #16812049C22semidirect
20. #16812041C22semidirect
23. C2×D8×C107C22semidirect
26. M32×C107C22semidirect
38. Dic428D4semidirect
43. D856D4semidirect
53. Q16×C107D4semidirect
57. D8×C107D4semidirect
111. C107#2190semidirect
114. #509D107semidirect
127. C2×D8C2×D107semidirect
135. M32C2×D107semidirect
153. Q16C22⋊D107semidirect
162. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934242
#202930632
#202928772
#202924382
#202924372
#202924082
#509×C1072
#2190107