Finite Group Databasebeta

← #28335566

Subgroups — #28335566

Order 13696.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D428C211yesmaximal, characteristic, complemented
2#20292974C211yesmaximal, characteristic, complemented
3#20292827C211yesmaximal, characteristic, complemented
4#20292823C211yesmaximal, characteristic
5#20292372C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7Q8⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107—21no—
9C2×#13133805—21no—
10C2×D856—21no—
11#16812109—21no—
12#16812109C2211yescharacteristic
13#16812106C2211yescharacteristic
14#16812089—22no—
15#16812089—21no—
16C4×D428—21no—
17C4×D428C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812031—22no—
20#16812031—21no—
21#16812029—21no—
22C4×Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—22no—
26C2×Dic214—21no—
27Q8×D107—42no—
28Q8×D107—42no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428D411yescharacteristic
33C2×C4×D107—22no—
34C2×C4×D107—21no—
35#13133805—42no—
36D856—42no—
37C2×C107⋊C8—21no—
38C4⋊Dic107—22no—
39C4⋊Dic107—21no—
40C4⋊Dic107D411yescharacteristic
41C4⋊Dic107D412yes—
42(C2×D107)⋊C4—22no—
43(C2×D107)⋊C4—21no—
44C2×Q8×C107D411yescharacteristic
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C4⋊C4×C107—21no—
48C4⋊C4×C107D411yescharacteristic
49C42×C107—21no—
50C42×C107C2311yescharacteristic
51C22×D107—21no—
52C22×D107—41no—
53Q8×C107—21no—
54Q8×C107D812yescomplemented
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C2×Dic107—22no—
61C2×Dic107—21no—
62D428—21no—
63D428—42no—
64D428—41no—
65D428—41no—
66D428—41no—
67D428—41no—
68D428SD1612yescomplemented
69C4×D107—44no—
70C4×D107—41no—
71C4×D107—41no—
72Dic214—44no—
73Dic214—41no—
74Dic214—41no—
75C8×C107—41no—
76C8×C107—41no—
77C107⋊C8—41no—
78C22×C107#12011yescharacteristic
79C2×D107—22no—
80C2×D107—41no—
81C2×D107—41no—
82C2×D107—81no—
83C2×D107—81no—
84C2×D107—81no—
85C4×C107—22no—
86C4×C107—21no—
87C4×C107—41no—
88C4×C107C2×D811yescharacteristic
89C4×C107C2×SD1611yescharacteristic
90C4×C107AGL(1,ℤ/8)11yescharacteristic
91C4×C107#13711yescharacteristic
92Dic107—41no—
93Dic107—42no—
94C2×C107#44711yescharacteristic
95C2×C107#44811yescharacteristic
96C2×C107#45411yescharacteristic
97D107—42no—
98D107—161no—
99#1627—1071nomaximal, Sylow
100C107#162711yescharacteristic, complemented, Sylow
101D4×Q8—1071no—
102#486—1071no—
103#460—1071no—
104#459—1071no—
105#332—1071no—
106Q8⋊C8D10711yescharacteristic, complemented
107D4⋊C8—1071no—
108C22×Q8—2141no—
109C2×SD16—2141no—
110C2×D8—2141no—
111C4⋊Q8—1071no—
112C4⋊Q8—2141no—
113#127—1071no—
114C22⋊Q8—2142no—
115C22⋊Q8—2141no—
116C4×Q8C2×D10711yescharacteristic
117C4×D4—1071no—
118C4×D4—2141no—
119C4⋊C8—1071no—
120C4⋊C8C2×D10711yescharacteristic
121#102—2141no—
122#102—2142no—
123#102—2141no—
124C4×C8C2×D10711yescharacteristic
125C2×Q8—4282no—
126C2×Q8—4282no—
127C2×Q8—2142no—
128C2×Q8—2141no—
129C2×Q8D42811yescharacteristic
130C2×D4—1071no—
131C2×D4—2141no—
132C2×D4—2141no—
133C2×D4—2141no—
134C22×C4—2142no—
135C22×C4—2141no—
136SD16—4282no—
137D8—4282no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2141no—
141C4⋊C4—21no—
142C4⋊C4—1071no—
143C4⋊C4—1072no—
144C4⋊C4—2142no—
145C4⋊C4—2141no—
146C4⋊C4D42811yescharacteristic
147C22⋊C4—2142no—
148C22⋊C4—2141no—
149C42—21no—
150C42C22×D10711yescharacteristic
151C23—4281no—
152C23—2141no—
153Q8—21no—
154Q8—4284no—
155Q8—4281no—
156Q8—4281no—
157Q8D85612yescomplemented
158D4—1072no—
159D4—4282no—
160D4—4281no—
161D4—4281no—
162D4—4281no—
163D4—4281no—
164D4—2141no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—4284no—
168C2×C4—4281no—
169C2×C4—4281no—
170C2×C4—2142no—
171C2×C4—2141no—
172C2×C4D4×D10711yescharacteristic
173C2×C4D4×D10711yescharacteristic
174C2×C4C2×D42811yescharacteristic
175C8—41no—
176C8—41no—
177C8—4281no—
178C22—8561no—
179C22—8561no—
180C22—8561no—
181C22—4281no—
182C22—4281no—
183C22—2142no—
184C22#1681207911yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—4282no—
189C4—4281no—
190C4C2×D85611yescharacteristic
191C4SD16×D10711yescharacteristic
192C4#1681213711yescharacteristic
193C4#1681214111yescharacteristic
194C2—4282no—
195C2—17121no—
196C2#2029276311yescharacteristic
197C2#2029276711yescharacteristic
198C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingQ8⋊C8×C107C2
socleC22×C107#120
prime radical#20292823C2
hypercenterQ8⋊C8D107
hypocenterC107#1627
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. Q8×D428C2semidirect
2. #20292974C2semidirect
3. #20292827C2semidirect
5. #20292372C2semidirect
6. #20292356C2semidirect
7. Q8⋊C8×C107C2semidirect
54. Q8×C107D8semidirect
68. D428SD16semidirect
100. C107#1627semidirect
106. Q8⋊C8D107semidirect
157. Q8D856semidirect

Maximal subgroups

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

subgroupindex
Q8×D4282
#202929742
#202928272
#202928232
#202923722
#202923562
Q8⋊C8×C1072
#1627107