Finite Group Databasebeta

← #28337685

Subgroups — #28337685

Order 13696.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#20293208C211yesmaximal, characteristic
2C2×#16812037C212yesmaximal, complemented
3#20292626C212yesmaximal, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#384×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2212yes—
7C2×#13133802C2212yes—
8#16812089—22no—
9#16812084—22no—
10#16812083—22no—
11#16812037—24no—
12#16812032—22no—
13#16812029—22no—
14C2×C4⋊C4×C107—22no—
15C2×C4⋊C4×C107C2211yescharacteristic
16C2×C4⋊C4×C107C2211yescharacteristic
17C2×C42×C107C2211yescharacteristic
18#13133818—22no—
19#13133818—22no—
20#13133818—22no—
21#13133818—22no—
22C2×Dic214D412yes—
23C2×D428—22no—
24C2×D428D412yes—
25C2×C22⋊D107—22no—
26C2×C4×D107—22no—
27#13133802—22no—
28#13133802—22no—
29C2×C107⋊C8—22no—
30C4⋊Dic107—22no—
31(C2×D107)⋊C4—24no—
32C22×C4×C107—21no—
33C22×C4×C107—21no—
34C22×C4×C107D412yes—
35C22×C4×C107C2311yescharacteristic
36C4⋊C4×C107—22no—
37C4⋊C4×C107—22no—
38C4⋊C4×C107—44no—
39C42×C107—21no—
40C42×C107D412yescomplemented
41C23×C107C2×D411yescharacteristic
42C22×D107—22no—
43C2×C4×C107—22no—
44C2×C4×C107—22no—
45C2×C4×C107—22no—
46C2×C4×C107—22no—
47C2×C4×C107—42no—
48C2×C4×C107—42no—
49C2×C4×C107—41no—
50C2×C4×C107C2×D412yes—
51C2×C4×C107C2×D411yescharacteristic
52C2×C4×C107#4111yescharacteristic
53C2×C4×C107#4112yes—
54C22⋊D107—42no—
55C22⋊D107—42no—
56C2×Dic107—22no—
57D428—22no—
58D428—22no—
59D428—82no—
60C4×D107—42no—
61C4×D107—42no—
62Dic214—22no—
63Dic214—22no—
64C107⋊C8—42no—
65C22×C107—21no—
66C22×C107—21no—
67C22×C107#12011yescharacteristic
68C22×C107#12111yescharacteristic
69C22×C107#12411yescharacteristic
70C2×D107—42no—
71C2×D107—42no—
72C2×D107—42no—
73C4×C107—24no—
74C4×C107—42no—
75C4×C107—81no—
76C4×C107#12112yes—
77C4×C107#12312yes—
78Dic107—42no—
79C2×C107—21no—
80C2×C107—21no—
81C2×C107#39411yescharacteristic
82C2×C107#45311yescharacteristic
83C2×C107#45611yescharacteristic
84D107—82no—
85#2039—1071nomaximal, Sylow
86C107#203911yescharacteristic, complemented, Sylow
87#537—1071no—
88C2×#104—1072no—
89#417—1072no—
90#384D10711yescharacteristic, complemented
91#343—1071no—
92C2×#41—1072no—
93C2×M16—1072no—
94C22⋊Q8—2142no—
95#121—2142no—
96#121—2142no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D10711yescharacteristic
99C2×C4⋊C4C2×D10711yescharacteristic
100C2×C42C2×D10711yescharacteristic
101#104—2144no—
102#103—2142no—
103#102—2142no—
104#41—2142no—
105#41—2142no—
106#41—2142no—
107#41—2142no—
108C2×Q8—1072no—
109C2×D4—1072no—
110C2×D4—2142no—
111C2×D4—2142no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2142no—
115C22×C4C22⋊D10712yes—
116C22×C4C22×D10711yescharacteristic
117M16—2142no—
118M16—2142no—
119C2×C8—2142no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—2142no—
124C22⋊C4—2144no—
125C42—21no—
126C42D42812yescomplemented
127C23—2142no—
128C23C2×C22⋊D10711yescharacteristic
129Q8—2142no—
130Q8—2142no—
131D4—8562no—
132D4—4282no—
133D4—4282no—
134D4—2142no—
135D4—2142no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—4282no—
144C2×C4—4282no—
145C2×C4—2142no—
146C2×C4D4×D10712yes—
147C2×C4C2×D42811yescharacteristic
148C2×C4#1313381811yescharacteristic
149C2×C4#1313381912yes—
150C8—4282no—
151C22—21no—
152C22—21no—
153C22—4282no—
154C22—4282no—
155C22—4282no—
156C22#1681207711yescharacteristic
157C22#1681208311yescharacteristic
158C22#1681209811yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—4282no—
163C4#1681208412yes—
164C4#1681209412yes—
165C2—21no—
166C2—21no—
167C2—8562no—
168C2#2029254411yescharacteristic
169C2#2029279411yescharacteristic
170C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#384×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#384D107
hypocenterC107#2039
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#16812037C2semidirect
3. #20292626C2semidirect
4. #20292396C2semidirect
5. #384×C107C2semidirect
40. C42×C107D4semidirect
86. C107#2039semidirect
90. #384D107semidirect
126. C42D428semidirect

Maximal subgroups

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

subgroupindex
#202932082
C2×#168120372
#202926262
#202923962
#384×C1072
#2039107