Finite Group Databasebeta

← #28336895

Subgroups — #28336895

Order 13696.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16812033C212yesmaximal, complemented
3#20292487C211yesmaximal, characteristic
4#20292389C212yesmaximal, complemented
5C2×C22⋊Q8×C107C211yesmaximal, characteristic
6C2×#13133802—24no—
7C2×#13133802—22no—
8C22×C107⋊C8C2212yes—
9C2×C4⋊Dic107C2212yes—
10C2×C22⋊Dic107—21no—
11#16812033—24no—
12C107⋊#95—21no—
13C22×Q8×C107C2211yescharacteristic
14C23×C4×C107C2211yescharacteristic
15C22⋊Q8×C107—22no—
16C22⋊Q8×C107C414yescomplemented
17C2×C4⋊C4×C107—21no—
18C2×C4⋊C4×C107C2211yescharacteristic
19C2×C22⋊C4×C107—21no—
20C22×Dic107—22no—
21#13133802—28no—
22C2×C107⋊C8—22no—
23C2×C107⋊C8—22no—
24C2×C107⋊C8D414yes—
25C4⋊Dic107—22no—
26C4⋊Dic107—22no—
27C22⋊Dic107—42no—
28C24×C107D411yescharacteristic
29C2×Q8×C107—21no—
30C2×Q8×C107—22no—
31C2×Q8×C107—22no—
32C2×Q8×C107C2×C412yes—
33C22×C4×C107—22no—
34C22×C4×C107—22no—
35C22×C4×C107—21no—
36C22×C4×C107—21no—
37C22×C4×C107C2×C412yes—
38C22×C4×C107D411yescharacteristic
39C22×C4×C107D412yes—
40C22×C4×C107C2311yescharacteristic
41C4⋊C4×C107—24no—
42C4⋊C4×C107—21no—
43C4⋊C4×C107C2×C412yes—
44C22⋊C4×C107—24no—
45C23×C107—22no—
46C23×C107—21no—
47C23×C107—21no—
48C23×C107—22no—
49C23×C107C22⋊C412yes—
50C23×C107C2×D411yescharacteristic
51Q8×C107—24no—
52Q8×C107—41no—
53Q8×C107—41no—
54C2×C4×C107—22no—
55C2×C4×C107—24no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—22no—
60C2×C4×C107—42no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107C22⋊C412yes—
64C2×C4×C107C22×C411yescharacteristic
65C2×C4×C107C2×D412yes—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107#4112yes—
68C2×Dic107—42no—
69C2×Dic107—42no—
70C2×Dic107—42no—
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#13711yescharacteristic
81C22×C107#13712yes—
82C4×C107—22no—
83C4×C107—42no—
84C4×C107—42no—
85C4×C107C4×D412yes—
86C4×C107#12011yescharacteristic
87C4×C107#12311yescharacteristic
88Dic107—82no—
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⋊Q8D10711yescharacteristic
99C2×#103—1072no—
100#380—1071no—
101#340—1072no—
102C22×Q8C2×D10711yescharacteristic
103C23×C4C2×D10711yescharacteristic
104C2×M16—2144no—
105C2×M16—2142no—
106C22×C8—1072no—
107C22⋊Q8—22no—
108C22⋊Q8Dic10714yescomplemented
109C2×C4⋊C4—21no—
110C2×C4⋊C4—1072no—
111C2×C4⋊C4C2×D10711yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—2141no—
114#103—2144no—
115#95—2141no—
116C24C22⋊D10711yescharacteristic
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8C2×Dic10712yes—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—2142no—
126C22×C4C2×Dic10712yes—
127C22×C4C22⋊D10711yescharacteristic
128C22×C4C22⋊D10712yes—
129C22×C4C22×D10711yescharacteristic
130M16—2148no—
131C2×C8—1074no—
132C2×C8—2142no—
133C2×C8—2142no—
134C4⋊C4—24no—
135C4⋊C4—21no—
136C4⋊C4—2142no—
137C4⋊C4—2142no—
138C4⋊C4C2×Dic10712yes—
139C22⋊C4—24no—
140C22⋊C4—4282no—
141C23—22no—
142C23—21no—
143C23—21no—
144C23—22no—
145C23C22⋊Dic10712yes—
146C23C2×C22⋊D10711yescharacteristic
147Q8—24no—
148Q8—41no—
149Q8—41no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—4282no—
160C2×C4—4282no—
161C2×C4—4282no—
162C2×C4C22⋊Dic10712yes—
163C2×C4C22×Dic10711yescharacteristic
164C2×C4D4×D10712yes—
165C2×C4C2×C22⋊D10711yescharacteristic
166C2×C4#1313381712yes—
167C8—2144no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×C22⋊Dic10711yescharacteristic
174C22#1681208112yes—
175C22#1681213411yescharacteristic
176C22#1681213911yescharacteristic
177C22#1681214012yes—
178C4—22no—
179C4—42no—
180C4—42no—
181C4—8562no—
182C4D4×Dic10712yes—
183C4#1681207711yescharacteristic
184C4#1681209011yescharacteristic
185C2—42no—
186C2#2029250711yescharacteristic
187C2#2029262811yescharacteristic
188C2#2029263611yescharacteristic
189C2#2029287512yes—
190C2#2029288312yes—
subgroupG/H
centerC23C2×C22⋊D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×C22⋊Q8×C107C2
socleC23×C107C2×D4
prime radicalC24×C107D4
cosocleC22×C4×C107C23
hypercenterC2×C22⋊Q8D107
hypocenterC107#1894
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812033C2semidirect
4. #20292389C2semidirect
16. C22⋊Q8×C107C4semidirect
96. C107#1894semidirect
108. C22⋊Q8Dic107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338022
C2×#168120332
#202924872
#202923892
C2×C22⋊Q8×C1072
#1894107