Finite Group Databasebeta

← #28337687

Subgroups — #28337687

Order 13696.

290 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#16812036C212yesmaximal, complemented
2#20292628C212yesmaximal, complemented
3#20292492C211yesmaximal, characteristic, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#537×C107C211yesmaximal, characteristic
6C2×#13133802C2212yes—
7C2×C4⋊Dic107C2211yescharacteristic
8C2×C4×Dic107C2211yescharacteristic
9#16812036—24no—
10#16812033—22no—
11#16812030—22no—
12C107⋊#95—22no—
13C2×#41×C107C2212yes—
14C22⋊Q8×C107—22no—
15#121×C107—22no—
16#121×C107—22no—
17C2×C4⋊C4×C107C2211yescharacteristic
18C22×Dic107—21no—
19C22×Dic107—21no—
20#13133802—22no—
21#13133802—22no—
22C2×C107⋊C8—22no—
23C4⋊Dic107—22no—
24C4×Dic107—21no—
25C4×Dic107D412yescomplemented
26#41×C107—22no—
27#41×C107—22no—
28#41×C107—22no—
29#41×C107—22no—
30C2×Q8×C107D412yes—
31C2×D4×C107—22no—
32C2×D4×C107—22no—
33C2×D4×C107D412yes—
34C22×C4×C107—22no—
35C22×C4×C107D412yes—
36C22×C4×C107C2311yescharacteristic
37C4⋊C4×C107—22no—
38C4⋊C4×C107—22no—
39C22⋊C4×C107—24no—
40C23×C107—22no—
41C23×C107C2×D411yescharacteristic
42Q8×C107—22no—
43Q8×C107—22no—
44D4×C107—22no—
45D4×C107—22no—
46D4×C107—42no—
47D4×C107—42no—
48D4×C107—82no—
49C2×C4×C107—22no—
50C2×C4×C107—22no—
51C2×C4×C107—42no—
52C2×C4×C107—42no—
53C2×C4×C107—42no—
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D412yes—
56C2×C4×C107#4112yes—
57C2×C4×C107#4111yescharacteristic
58C2×Dic107—22no—
59C2×Dic107—41no—
60C2×Dic107—41no—
61C2×Dic107—42no—
62C2×Dic107—41no—
63C107⋊C8—42no—
64C22×C107—21no—
65C22×C107—21no—
66C22×C107—42no—
67C22×C107—42no—
68C22×C107—42no—
69C22×C107#12011yescharacteristic
70C22×C107#12111yescharacteristic
71C22×C107#12411yescharacteristic
72C4×C107—42no—
73C4×C107—42no—
74C4×C107#12112yes—
75C4×C107#12312yes—
76Dic107—42no—
77Dic107—81no—
78C2×C107—21no—
79C2×C107—21no—
80C2×C107—82no—
81C2×C107#39411yescharacteristic
82C2×C107#45411yescharacteristic
83C2×C107#45511yescharacteristic
84#2040—1071nomaximal, Sylow
85C107#204011yescharacteristic, complemented, Sylow
86#537D10711yescharacteristic
87C2×#104—1072no—
88#417—1072no—
89#382—1071no—
90#343—1071no—
91C2×#41C2×D10712yes—
92C2×M16—1072no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
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—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8C22⋊D10712yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4C22⋊D10712yes—
111C22×C4—22no—
112C22×C4—2141no—
113C22×C4—2141no—
114C22×C4C22⋊D10712yes—
115C22×C4C22×D10711yescharacteristic
116M16—2142no—
117M16—2142no—
118C2×C8—2142no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—2142no—
122C22⋊C4—24no—
123C42—1072no—
124C42—2141no—
125C23—22no—
126C23C2×C22⋊D10711yescharacteristic
127Q8—22no—
128Q8—22no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—4282no—
140C2×C4—4281no—
141C2×C4—4281no—
142C2×C4—4281no—
143C2×C4—2142no—
144C2×C4D4×D10711yescharacteristic
145C2×C4C2×C22⋊D10712yes—
146C2×C4#1313381712yes—
147C2×C4#1313381711yescharacteristic
148C8—4282no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1681207811yescharacteristic
155C22#1681208011yescharacteristic
156C22#1681209611yescharacteristic
157C4—42no—
158C4—42no—
159C4—8561no—
160C4—4282no—
161C4#1681208112yes—
162C4#1681209012yes—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2029254111yescharacteristic
167C2#2029280011yescharacteristic
168C2#2029280511yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting, prime radical#537×C107C2
socleC22×C107#120
hypercenter#537D107
hypocenterC107#2040
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16812036C2semidirect
2. #20292628C2semidirect
3. #20292492C2semidirect
4. #20292396C2semidirect
25. C4×Dic107D4semidirect
85. C107#2040semidirect

Maximal subgroups

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

subgroupindex
C2×#168120362
#202926282
#202924922
#202923962
#537×C1072
#2040107