Finite Group Databasebeta

← #28335696

Subgroups — #28335696

Order 13696.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#20293030C211yesmaximal, characteristic, complemented
2#20292907C211yesmaximal, characteristic, complemented
3#20292832C211yesmaximal, characteristic, complemented
4#20292373C211yesmaximal, characteristic, complemented
5D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
7#547×C107C211yesmaximal, characteristic
8C2×#13133806—21no—
9#16812109C2211yescharacteristic
10#16812042—21no—
11#16812042—22no—
12C4⋊(C107⋊C8)C2211yescharacteristic
13C4⋊(C107⋊C8)C2211yescharacteristic
14#16812032—21no—
15#16812030—21no—
16C4×C107⋊C8C2211yescharacteristic
17C2×#41×C107—21no—
18C4⋊Q8×C107C2211yescharacteristic
19#123×C107—22no—
20C22⋊Q8×C107—21no—
21#121×C107—21no—
22C4×D4×C107C2211yescharacteristic
23C4×D4×C107C2211yescharacteristic
24C2×C4⋊C4×C107—21no—
25C2×Dic214—21no—
26#13133806—42no—
27C2×C107⋊C8—21no—
28C2×C107⋊C8—21no—
29C2×C107⋊C8—21no—
30C4⋊Dic107—21no—
31C4⋊Dic107—21no—
32#41×C107—42no—
33#41×C107—42no—
34C2×Q8×C107—21no—
35C2×D4×C107—22no—
36C2×D4×C107D411yescharacteristic
37C2×D4×C107D411yescharacteristic
38C22×C4×C107—22no—
39C22×C4×C107—21no—
40C22×C4×C107—21no—
41C4⋊C4×C107—22no—
42C4⋊C4×C107—21no—
43C4⋊C4×C107D411yescharacteristic
44C4⋊C4×C107D411yescharacteristic
45C4⋊C4×C107D412yes—
46C22⋊C4×C107—22no—
47C22⋊C4×C107—21no—
48C22⋊C4×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—21no—
51C23×C107—21no—
52Q8×C107—41no—
53Q8×C107—41no—
54D4×C107—21no—
55D4×C107—21no—
56D4×C107—21no—
57D4×C107—44no—
58D4×C107D812yes—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—22no—
62C2×C4×C107—44no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×Dic107—41no—
71Dic214—41no—
72Dic214—41no—
73C107⋊C8—22no—
74C107⋊C8—41no—
75C107⋊C8—41no—
76C22×C107—22no—
77C22×C107—41no—
78C22×C107—41no—
79C22×C107—41no—
80C22×C107—41no—
81C22×C107—41no—
82C22×C107#12011yescharacteristic
83C4×C107—21no—
84C4×C107—41no—
85C4×C107—42no—
86C4×C107—41no—
87C4×C107C2×D811yescharacteristic
88C4×C107#13511yescharacteristic
89C4×C107AGL(1,ℤ/8)11yescharacteristic
90C4×C107#13711yescharacteristic
91Dic107—81no—
92C2×C107—42no—
93C2×C107—81no—
94C2×C107#44711yescharacteristic
95C2×C107#45211yescharacteristic
96C2×C107#45611yescharacteristic
97#1647—1071nomaximal, Sylow
98C107#164711yescharacteristic, complemented, Sylow
99#547D10711yescharacteristic
100#500—1071no—
101#474—1071no—
102#461—1071no—
103#332—1071no—
104D4⋊C8—1071no—
105D4⋊C8—1071no—
106C2×#41—21no—
107C2×SD16—2141no—
108C4⋊Q8—1071no—
109C4⋊Q8C2×D10711yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D10711yescharacteristic
114C4×D4C2×D10711yescharacteristic
115C2×C4⋊C4—21no—
116#107—2141no—
117#107—2142no—
118C4⋊C8—1071no—
119C4⋊C8—1071no—
120#103—2141no—
121#102—2141no—
122C4×C8—1071no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—2141no—
127C2×D4—22no—
128C2×D4C22⋊D10711yescharacteristic
129C2×D4C22⋊D10711yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—4282no—
134C2×C8—2141no—
135C2×C8—2141no—
136C2×C8—2141no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—2141no—
140C4⋊C4—2141no—
141C4⋊C4C22⋊D10711yescharacteristic
142C4⋊C4C22⋊D10711yescharacteristic
143C4⋊C4C22⋊D10712yes—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D10711yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—4281no—
153Q8—4281no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1313380312yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—4281no—
168C2×C4C2×C22⋊D10711yescharacteristic
169C2×C4C2×C22⋊D10711yescharacteristic
170C2×C4C2×C22⋊D10711yescharacteristic
171C8—4281no—
172C8—4281no—
173C8—2142no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1681207811yescharacteristic
181C4—21no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—8561no—
186C4C2×#1313380311yescharacteristic
187C4#1681212911yescharacteristic
188C4#1681213511yescharacteristic
189C4#1681213911yescharacteristic
190C2—42no—
191C2—81no—
192C2#2029276111yescharacteristic
193C2#2029278911yescharacteristic
194C2#2029281011yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting, prime radical#547×C107C2
socleC22×C107#120
hypercenter#547D107
hypocenterC107#1647
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20293030C2semidirect
2. #20292907C2semidirect
3. #20292832C2semidirect
4. #20292373C2semidirect
5. D4⋊(C107⋊C8)C2semidirect
6. D4⋊(C107⋊C8)C2semidirect
98. C107#1647semidirect

Maximal subgroups

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

subgroupindex
#202930302
#202929072
#202928322
#202923732
D4⋊(C107⋊C8)2
#547×C1072
#1647107