Finite Group Databasebeta

← #28337691

Subgroups — #28337691

Order 13696.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16812035C212yesmaximal, complemented
3#20292629C212yesmaximal, complemented
4#20292492C211yesmaximal, characteristic, complemented
5#343×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2212yes—
7#16812089—22no—
8#16812084—22no—
9#16812083—22no—
10C2×C4⋊Dic107C2211yescharacteristic
11C2×C4×Dic107C2211yescharacteristic
12#16812035—24no—
13#16812034—22no—
14#16812031—22no—
15C107⋊#95—22no—
16C2×M16×C107C2212yes—
17C2×C4⋊C4×C107C2211yescharacteristic
18#13133818—22no—
19#13133818—22no—
20#13133818—22no—
21#13133818—22no—
22C2×Dic214D412yes—
23C2×D428—22no—
24C2×D428D412yes—
25C2×C22⋊D107—22no—
26C22×Dic107—21no—
27C22×Dic107—21no—
28C2×C4×D107—22no—
29C4⋊Dic107—22no—
30C4⋊Dic107—22no—
31(C2×D107)⋊C4—24no—
32C4×Dic107—21no—
33C4×Dic107D412yescomplemented
34C22×C4×C107D412yes—
35C22×C4×C107C2311yescharacteristic
36M16×C107—22no—
37M16×C107—22no—
38C2×C8×C107—22no—
39C4⋊C4×C107—22no—
40C23×C107C2×D411yescharacteristic
41C22×D107—22no—
42C2×C4×C107—22no—
43C2×C4×C107—42no—
44C2×C4×C107C2×D411yescharacteristic
45C2×C4×C107C2×D412yes—
46C2×C4×C107#4112yes—
47C2×C4×C107#4111yescharacteristic
48C22⋊D107—42no—
49C22⋊D107—42no—
50C2×Dic107—22no—
51C2×Dic107—22no—
52C2×Dic107—41no—
53C2×Dic107—41no—
54C2×Dic107—42no—
55C2×Dic107—41no—
56D428—22no—
57D428—22no—
58D428—82no—
59C4×D107—42no—
60C4×D107—42no—
61Dic214—22no—
62Dic214—22no—
63C8×C107—42no—
64C22×C107—21no—
65C22×C107—21no—
66C22×C107#12011yescharacteristic
67C22×C107#12111yescharacteristic
68C22×C107#12411yescharacteristic
69C2×D107—42no—
70C2×D107—42no—
71C2×D107—42no—
72C4×C107—42no—
73C4×C107#12112yes—
74C4×C107#12312yes—
75Dic107—42no—
76Dic107—42no—
77Dic107—81no—
78C2×C107—21no—
79C2×C107—21no—
80C2×C107#39411yescharacteristic
81C2×C107#45411yescharacteristic
82C2×C107#45511yescharacteristic
83D107—82no—
84#2040—1071nomaximal, Sylow
85C107#204011yescharacteristic, complemented, Sylow
86#537—1071no—
87C2×#104—1072no—
88#417—1072no—
89#382—1071no—
90#343D10711yescharacteristic, complemented
91C2×#41—1072no—
92C2×M16C2×D10712yes—
93C22⋊Q8—2142no—
94#121—2142no—
95#121—2142no—
96C2×C4⋊C4—1071no—
97C2×C4⋊C4C2×D10711yescharacteristic
98C2×C42—1071no—
99#104—2144no—
100#103—2142no—
101#102—2142no—
102#95—2142no—
103#41—2142no—
104#41—2142no—
105#41—2142no—
106#41—2142no—
107C2×Q8—1072no—
108C2×D4—1072no—
109C2×D4—2142no—
110C2×D4—2142no—
111C22×C4—2141no—
112C22×C4—2141no—
113C22×C4—2142no—
114C22×C4D42812yes—
115C22×C4C22×D10711yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—2142no—
121C4⋊C4—2142no—
122C22⋊C4—2144no—
123C42—1072no—
124C42—2141no—
125C23—2142no—
126C23C2×D42811yescharacteristic
127Q8—2142no—
128Q8—2142no—
129D4—8562no—
130D4—4282no—
131D4—4282no—
132D4—2142no—
133D4—2142no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—4282no—
137C2×C4—4281no—
138C2×C4—4281no—
139C2×C4—4281no—
140C2×C4—4282no—
141C2×C4—4282no—
142C2×C4—2142no—
143C2×C4—2142no—
144C2×C4D4×D10711yescharacteristic
145C2×C4D4×D10712yes—
146C2×C4#1313381812yes—
147C2×C4#1313381911yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—4282no—
152C22—4282no—
153C22—4282no—
154C22#1681207911yescharacteristic
155C22#1681208411yescharacteristic
156C22#1681209711yescharacteristic
157C4—42no—
158C4—8561no—
159C4—4282no—
160C4—4282no—
161C4#1681208212yes—
162C4#1681209412yes—
163C2—21no—
164C2—21no—
165C2—8562no—
166C2#2029254511yescharacteristic
167C2#2029280211yescharacteristic
168C2#2029280711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#343×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#343D107
hypocenterC107#2040
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#16812035C2semidirect
3. #20292629C2semidirect
4. #20292492C2semidirect
5. #343×C107C2semidirect
33. C4×Dic107D4semidirect
85. C107#2040semidirect
90. #343D107semidirect

Maximal subgroups

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

subgroupindex
#202932082
C2×#168120352
#202926292
#202924922
#343×C1072
#2040107