Finite Group Databasebeta

← #28336892

Subgroups — #28336892

Order 13696.

360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#13133802C211yesmaximal, characteristic
2C2×#16812088C211yesmaximal, characteristic
3C2×#16812032C212yesmaximal, complemented
4#20292389C212yesmaximal, complemented
5#380×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×#13133802—24no—
8C2×#13133802—22no—
9C22×C107⋊C8C2212yes—
10#16812088—22no—
11#16812088C414yescomplemented
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4—21no—
14C2×C22⋊Dic107—21no—
15#16812032—24no—
16C23×C4×C107C2211yescharacteristic
17C2×C4⋊C4×C107C2212yes—
18C2×C22⋊C4×C107—21no—
19#95×C107—21no—
20C2×Dic214—22no—
21C2×Dic214—22no—
22C2×Dic214—21no—
23C2×Dic214C2×C412yes—
24C22×Dic107—21no—
25C22×Dic107—21no—
26#13133802—28no—
27C2×C107⋊C8—22no—
28C2×C107⋊C8—22no—
29C2×C107⋊C8D414yes—
30C4⋊Dic107—21no—
31C4⋊Dic107C2×C412yes—
32Dic107⋊C4—24no—
33C22⋊Dic107—24no—
34C24×C107D411yescharacteristic
35C22×C4×C107—22no—
36C22×C4×C107—22no—
37C22×C4×C107—22no—
38C22×C4×C107C2×C412yes—
39C22×C4×C107D412yes—
40C22×C4×C107D411yescharacteristic
41C22×C4×C107C2311yescharacteristic
42C4⋊C4×C107—22no—
43C4⋊C4×C107—22no—
44C22⋊C4×C107—42no—
45C23×C107—22no—
46C23×C107—21no—
47C23×C107—22no—
48C23×C107—21no—
49C23×C107C22⋊C412yes—
50C23×C107C2×D411yescharacteristic
51C2×C4×C107—22no—
52C2×C4×C107—24no—
53C2×C4×C107—22no—
54C2×C4×C107—22no—
55C2×C4×C107—42no—
56C2×C4×C107—42no—
57C2×C4×C107—42no—
58C2×C4×C107C22⋊C412yes—
59C2×C4×C107C22×C411yescharacteristic
60C2×C4×C107C2×D412yes—
61C2×C4×C107C2×D411yescharacteristic
62C2×C4×C107#4112yes—
63C2×Dic107—22no—
64C2×Dic107—22no—
65C2×Dic107—42no—
66C2×Dic107—41no—
67C2×Dic107—41no—
68Dic214—24no—
69Dic214—41no—
70Dic214—41no—
71C107⋊C8—24no—
72C22×C107—22no—
73C22×C107—22no—
74C22×C107—22no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107C2×C22⋊C411yescharacteristic
78C22×C107#12112yes—
79C22×C107AGL(1,ℤ/8)11yescharacteristic
80C22×C107#13712yes—
81C22×C107#13711yescharacteristic
82C4×C107—22no—
83C4×C107—82no—
84C4×C107C4×D412yes—
85C4×C107#12011yescharacteristic
86C4×C107#12311yescharacteristic
87Dic107—42no—
88Dic107—42no—
89C2×C107—42no—
90C2×C107#38611yescharacteristic
91C2×C107#41711yescharacteristic
92C2×C107#41911yescharacteristic
93C2×C107#46812yes—
94C2×C107#47012yes—
95#1894—1071nomaximal, Sylow
96C107#189411yescharacteristic, complemented, Sylow
97C22×M16—1071no—
98C2×C22⋊Q8—1071no—
99C2×#103—1072no—
100#380D10711yescharacteristic
101#340—1072no—
102C22×Q8—1071no—
103C23×C4C2×D10711yescharacteristic
104C2×M16—2144no—
105C2×M16—2142no—
106C22×C8—1072no—
107C22⋊Q8—1074no—
108C22⋊Q8—2142no—
109C2×C4⋊C4—1071no—
110C2×C4⋊C4—2141no—
111C2×C4⋊C4C2×D10712yes—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—2141no—
114#103—2144no—
115#95—21no—
116C24C22⋊D10711yescharacteristic
117C2×Q8—1072no—
118C2×Q8—2142no—
119C2×Q8—2142no—
120C2×Q8—2141no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—2141no—
125C22×C4—2141no—
126C22×C4C4×D10712yes—
127C22×C4D42812yes—
128C22×C4C22⋊D10711yescharacteristic
129C22×C4C22×D10711yescharacteristic
130M16—2148no—
131C2×C8—1074no—
132C2×C8—2142no—
133C2×C8—2142no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—1072no—
137C4⋊C4—2144no—
138C4⋊C4—2141no—
139C22⋊C4—42no—
140C22⋊C4—2144no—
141C23—22no—
142C23—21no—
143C23—22no—
144C23—21no—
145C23(C2×D107)⋊C412yes—
146C23C2×C22⋊D10711yescharacteristic
147Q8—4281no—
148Q8—4281no—
149Q8—2144no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—4282no—
158C2×C4—4281no—
159C2×C4—4281no—
160C2×C4—2142no—
161C2×C4—2142no—
162C2×C4(C2×D107)⋊C412yes—
163C2×C4C2×C4×D10711yescharacteristic
164C2×C4D4×D10712yes—
165C2×C4C2×D42811yescharacteristic
166C2×C4#1313381712yes—
167C8—2144no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×(C2×D107)⋊C411yescharacteristic
174C22#1681208112yes—
175C22#1681213511yescharacteristic
176C22#1681213912yes—
177C22#1681214011yescharacteristic
178C4—22no—
179C4—82no—
180C4—4282no—
181C4—4282no—
182C4(C22⋊D107)⋊C412yes—
183C4#1681207911yescharacteristic
184C4#1681209311yescharacteristic
185C2—42no—
186C2#2029250411yescharacteristic
187C2#2029262611yescharacteristic
188C2#2029263511yescharacteristic
189C2#2029287112yes—
190C2#2029288312yes—
subgroupG/H
centerC23C2×C22⋊D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#380×C107C2
socleC23×C107C2×D4
prime radicalC24×C107D4
cosocleC22×C4×C107C23
hypercenter#380D107
hypocenterC107#1894
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16812032C2semidirect
4. #20292389C2semidirect
11. #16812088C4semidirect
96. C107#1894semidirect

Maximal subgroups

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

subgroupindex
C22×#131338022
C2×#168120882
C2×#168120322
#202923892
#380×C1072
#1894107