Finite Group Databasebeta

← #28338408

Subgroups — #28338408

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#20293065C211yesmaximal, characteristic, complemented
3#20293062C211yesmaximal, characteristic, complemented
4#20292881C211yesmaximal, characteristic, complemented
5#20292436C211yesmaximal, characteristic, complemented
6#20292408C211yesmaximal, characteristic, complemented
7#360×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×D856C2211yescharacteristic, complemented
16C2×#13133801C2211yescharacteristic
17#16812084—21no—
18#16812054—21no—
19#16812052—21no—
20D1712—21no—
21#16812050—21no—
22#16812049C2211yescharacteristic, complemented
23#16812029—21no—
24#135×C107C2211yescharacteristic
25M32×C107C2211yescharacteristic, complemented
26#106×C107C2211yescharacteristic, complemented
27#13133819—21no—
28#13133818—21no—
29#13133818—41no—
30#13133817—21no—
31#13133817—41no—
32Q8×D107—21no—
33C2×D428—21no—
34C2×D428—41no—
35D4×D107—21no—
36D4×D107—41no—
37C2×C4×D107—41no—
38C2×C4×D107D411yescharacteristic
39Dic428D411yescharacteristic, complemented
40#13133808—21no—
41#13133807—21no—
42#13133806—21no—
43#13133805—21no—
44D856—21no—
45D856—21no—
46D856D411yescharacteristic, complemented
47#13133803—21no—
48#13133801—21no—
49#13133801—21no—
50C2×C107⋊C8D411yescharacteristic
51(C2×D107)⋊C4—41no—
52C107⋊C16—21no—
53#41×C107—21no—
54Q16×C107D411yescharacteristic, complemented
55SD16×C107—21no—
56D8×C107D411yescharacteristic, complemented
57C2×C8×C107C2311yescharacteristic
58C4⋊C4×C107—21no—
59C16×C107—21no—
60C22×D107—41no—
61C22×D107—41no—
62C22×D107D811yescharacteristic
63Q8×C107—21no—
64D4×C107—21no—
65D4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107—41no—
68C2×C4×C107C2×D411yescharacteristic
69C22⋊D107—41no—
70C22⋊D107—41no—
71C22⋊D107—41no—
72C2×Dic107—41no—
73C2×Dic107D811yescharacteristic
74D428—21no—
75D428—41no—
76D428—41no—
77D428—41no—
78D428—82no—
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×C107C2×D811yescharacteristic
91C2×D107—21no—
92C2×D107—41no—
93C2×D107—41no—
94C2×D107—82no—
95C2×D107—81no—
96C2×D107—81no—
97C2×D107—81no—
98C4×C107—41no—
99C4×C107—81no—
100C4×C107#12011yescharacteristic
101C4×C107AGL(1,ℤ/8)11yescharacteristic
102Dic107—21no—
103Dic107—41no—
104C2×C107—21no—
105C2×C107—81no—
106C2×C107#44711yescharacteristic
107D107—41no—
108D107—81no—
109D107—161no—
110#2190—1071nomaximal, Sylow
111C107#219011yescharacteristic, complemented, Sylow
112#575—1071no—
113#509—1071no—
114#509—1071no—
115#469—1071no—
116#360—1071no—
117#360D10711yescharacteristic, complemented
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×D8—1071no—
128C2×M16—1071no—
129#121—2141no—
130SD32—2141no—
131SD32—2141no—
132D16—2141no—
133D16—2141no—
134M32—1071no—
135M32C2×D10711yescharacteristic, complemented
136#106C2×D10711yescharacteristic, complemented
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—4281no—
146C2×D4—4281no—
147C2×D4—2141no—
148C2×D4—2141no—
149C22×C4—1071no—
150C22×C4—4281no—
151Q16—1071no—
152Q16—2141no—
153Q16D42811yescharacteristic, complemented
154SD16—21no—
155SD16—2141no—
156SD16—2141no—
157SD16—2141no—
158D8—1071no—
159D8—2141no—
160D8—2141no—
161D8—2141no—
162D8D42811yescharacteristic, complemented
163M16—2141no—
164M16—2141no—
165C2×C8—1071no—
166C2×C8C22×D10711yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—4281no—
169C16—21no—
170C16—2141no—
171C23—1071no—
172C23—4281no—
173C23—4281no—
174Q8—21no—
175Q8—4281no—
176Q8—2141no—
177D4—21no—
178D4—41no—
179D4—8562no—
180D4—4281no—
181D4—4281no—
182D4—4281no—
183D4—4281no—
184D4—4281no—
185D4—4281no—
186D4—2141no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1071no—
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×D42811yescharacteristic
200C22—41no—
201C22—8561no—
202C22—8561no—
203C22—8562no—
204C22—8561no—
205C22—4281no—
206C22—4281no—
207C22—2141no—
208C22D8×D10711yescharacteristic
209C4—41no—
210C4—81no—
211C4—4281no—
212C4—2141no—
213C4#1681207911yescharacteristic
214C4#1681213611yescharacteristic
215C2—21no—
216C2—81no—
217C2—8561no—
218C2—4281no—
219C2—17121no—
220C2#2029276211yescharacteristic
subgroupG/H
centerC2#20292762
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#360×C107C2
socleC2×C107#447
hypercenter#360D107
hypocenterC107#2190
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #20293424C2semidirect
2. #20293065C2semidirect
3. #20293062C2semidirect
4. #20292881C2semidirect
5. #20292436C2semidirect
6. #20292408C2semidirect
7. #360×C107C2semidirect
15. C2×D856C22semidirect
22. #16812049C22semidirect
25. M32×C107C22semidirect
26. #106×C107C22semidirect
39. Dic428D4semidirect
46. D856D4semidirect
54. Q16×C107D4semidirect
56. D8×C107D4semidirect
111. C107#2190semidirect
117. #360D107semidirect
135. M32C2×D107semidirect
136. #106C2×D107semidirect
153. Q16D428semidirect
162. D8D428semidirect

Maximal subgroups

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

subgroupindex
#202934242
#202930652
#202930622
#202928812
#202924362
#202924082
#360×C1072
#2190107