Finite Group Databasebeta

← #28338410

Subgroups — #28338410

Order 13696.

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

normal subgroups only

This group has 216 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
2#20293068C211yesmaximal, characteristic, complemented
3#20292882C211yesmaximal, characteristic, complemented
4#20292438C211yesmaximal, characteristic, complemented
5#20292437C211yesmaximal, characteristic, complemented
6#20292408C211yesmaximal, characteristic, complemented
7#510×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#16812085—21no—
17#16812055—21no—
18#16812052—21no—
19#16812049C2211yescharacteristic, complemented
20#16812041C2211yescharacteristic, complemented
21#16812033—21no—
22#135×C107C2211yescharacteristic
23C2×Q16×C107C2211yescharacteristic, complemented
24Q32×C107—21no—
25SD32×C107—21no—
26M32×C107C2211yescharacteristic, complemented
27#13133819—21no—
28#13133818—21no—
29#13133818—41no—
30#13133817—21no—
31#13133817—41no—
32Q8×D107—21no—
33D4×D107—21no—
34D4×D107—41no—
35C2×C4×D107—41no—
36C2×C4×D107D411yescharacteristic
37Dic428D411yescharacteristic
38#13133808—21no—
39#13133807—21no—
40#13133806—21no—
41#13133805—21no—
42D856D411yescharacteristic
43#13133803—21no—
44#13133801—21no—
45#13133801—21no—
46C2×C107⋊C8D411yescharacteristic
47C4⋊Dic107—21no—
48Dic107⋊C4—41no—
49(C2×D107)⋊C4—41no—
50C107⋊C16—21no—
51#41×C107—21no—
52C2×Q8×C107—21no—
53Q16×C107—21no—
54Q16×C107—21no—
55Q16×C107D411yescharacteristic
56SD16×C107—21no—
57D8×C107D411yescharacteristic
58C2×C8×C107C2311yescharacteristic
59C16×C107—21no—
60C22×D107—41no—
61C22×D107D811yescharacteristic
62Q8×C107—21no—
63Q8×C107—41no—
64Q8×C107—41no—
65D4×C107—21no—
66D4×C107—41no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107C2×D411yescharacteristic
70C22⋊D107—41no—
71C22⋊D107—41no—
72C22⋊D107—41no—
73C2×Dic107—41no—
74C2×Dic107—41no—
75C2×Dic107D811yescharacteristic
76D428—21no—
77D428—41no—
78C4×D107—21no—
79C4×D107—21no—
80C4×D107—41no—
81C4×D107—41no—
82C4×D107—41no—
83Dic214—21no—
84Dic214—41no—
85C8×C107C2×D411yescharacteristic
86C8×C107C2×D411yescharacteristic
87C107⋊C8—21no—
88C22×C107—41no—
89C22×C107C2×D811yescharacteristic
90C2×D107—21no—
91C2×D107—41no—
92C2×D107—41no—
93C2×D107—82no—
94C4×C107—41no—
95C4×C107—81no—
96C4×C107#12011yescharacteristic
97C4×C107AGL(1,ℤ/8)11yescharacteristic
98Dic107—21no—
99Dic107—41no—
100Dic107—81no—
101C2×C107—21no—
102C2×C107—81no—
103C2×C107#44711yescharacteristic
104D107—41no—
105D107—81no—
106#2191—1071nomaximal, Sylow
107C107#219111yescharacteristic, complemented, Sylow
108#575—1071no—
109#510—1071no—
110#510D10711yescharacteristic, complemented
111#470—1071no—
112#360—1071no—
113#360—1071no—
114#349—1071no—
115C2×#41—2141no—
116#137—2141no—
117#137—2141no—
118AGL(1,ℤ/8)—2141no—
119AGL(1,ℤ/8)—2141no—
120#135—1071no—
121#135—2141no—
122#135C2×D10711yescharacteristic
123C2×Q16C2×D10711yescharacteristic, complemented
124C2×M16—1071no—
125C22⋊Q8—2141no—
126Q32—21no—
127Q32—2141no—
128SD32—21no—
129SD32—2141no—
130M32—1071no—
131M32C2×D10711yescharacteristic, complemented
132#106—1071no—
133#103—2141no—
134#41—21no—
135#41—4281no—
136#41—4281no—
137#41—2141no—
138#41—2141no—
139#41—2141no—
140C2×Q8—21no—
141C2×Q8—2141no—
142C2×D4—4281no—
143C2×D4—2141no—
144C22×C4—1071no—
145C22×C4—4281no—
146Q16—21no—
147Q16—21no—
148Q16—1071no—
149Q16—2141no—
150Q16C22⋊D10711yescharacteristic
151SD16—21no—
152SD16—2141no—
153SD16—2141no—
154SD16—2141no—
155D8—1071no—
156D8—2141no—
157D8C22⋊D10711yescharacteristic
158M16—2141no—
159M16—2141no—
160C2×C8—1071no—
161C2×C8C22×D10711yescharacteristic
162C4⋊C4—4281no—
163C4⋊C4—2141no—
164C22⋊C4—4281no—
165C16—21no—
166C16—2141no—
167C23—1071no—
168C23—4281no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—4281no—
173Q8—2141no—
174D4—21no—
175D4—41no—
176D4—4281no—
177D4—4281no—
178D4—4281no—
179D4—4281no—
180D4—2141no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—1071no—
184C2×C4—4281no—
185C2×C4—4281no—
186C2×C4—4281no—
187C2×C4—4281no—
188C2×C4—4281no—
189C2×C4—2141no—
190C2×C4—2141no—
191C2×C4D4×D10711yescharacteristic
192C8—2141no—
193C8D4×D10711yescharacteristic
194C8C2×C22⋊D10711yescharacteristic
195C22—41no—
196C22—8562no—
197C22—4281no—
198C22—4281no—
199C22—2141no—
200C22D8×D10711yescharacteristic
201C4—41no—
202C4—81no—
203C4—8561no—
204C4—4281no—
205C4—2141no—
206C4#1681207711yescharacteristic
207C4#1681213311yescharacteristic
208C2—21no—
209C2—81no—
210C2—8561no—
211C2—4281no—
212C2#2029275911yescharacteristic
subgroupG/H
centerC2#20292759
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#510×C107C2
socleC2×C107#447
prime radical#20293424C2
hypercenter#510D107
hypocenterC107#2191
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #20293068C2semidirect
3. #20292882C2semidirect
4. #20292438C2semidirect
5. #20292437C2semidirect
6. #20292408C2semidirect
7. #510×C107C2semidirect
19. #16812049C22semidirect
20. #16812041C22semidirect
23. C2×Q16×C107C22semidirect
26. M32×C107C22semidirect
107. C107#2191semidirect
110. #510D107semidirect
123. C2×Q16C2×D107semidirect
131. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934242
#202930682
#202928822
#202924382
#202924372
#202924082
#510×C1072
#2191107