Finite Group Databasebeta

← #28336909

Subgroups — #28336909

Order 13696.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#20293138C211yesmaximal, characteristic
2C2×#16812037C212yesmaximal, complemented
3#20292591C211yesmaximal, characteristic, complemented
4#20292387C212yesmaximal, complemented
5C2×#117×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2211yescharacteristic
7C2×#13133802C2212yes—
8#16812092—21no—
9#16812088C412yescomplemented
10#16812083C412yescomplemented
11#16812078—21no—
12C4×C22⋊D107—22no—
13#16812068C2211yescharacteristic
14#16812037—24no—
15C22⋊(C107⋊C8)—22no—
16C23×C4×C107C2211yescharacteristic
17#117×C107—24no—
18C2×C4⋊C4×C107—21no—
19C2×C22⋊C4×C107—21no—
20C2×C42×C107C2212yes—
21#13133818—22no—
22#13133818—22no—
23C2×Dic214C2×C411yescharacteristic
24C2×D428C2×C411yescharacteristic
25C2×C22⋊D107—22no—
26C2×C22⋊D107—21no—
27C2×C4×D107—21no—
28#13133802—22no—
29#13133802—22no—
30C2×C107⋊C8—22no—
31C4⋊Dic107C2×C412yes—
32Dic107⋊C4—22no—
33C22⋊Dic107—22no—
34C22⋊Dic107—21no—
35(C2×D107)⋊C4—22no—
36C4×Dic107—21no—
37C24×C107D411yescharacteristic
38C22×C4×C107—22no—
39C22×C4×C107—22no—
40C22×C4×C107—22no—
41C22×C4×C107C2×C412yes—
42C22×C4×C107D412yes—
43C22×C4×C107D411yescharacteristic
44C22×C4×C107C2311yescharacteristic
45C4⋊C4×C107—24no—
46C22⋊C4×C107—24no—
47C42×C107—22no—
48C42×C107D414yescomplemented
49C23×C107—21no—
50C23×C107—21no—
51C23×C107—42no—
52C23×C107C22⋊C412yes—
53C23×C107C2×D411yescharacteristic
54C22×D107—21no—
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—24no—
58C2×C4×C107—24no—
59C2×C4×C107—24no—
60C2×C4×C107—44no—
61C2×C4×C107C22⋊C412yes—
62C2×C4×C107C22×C411yescharacteristic
63C2×C4×C107C2×D412yes—
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107#4112yes—
66C22⋊D107—44no—
67C22⋊D107—41no—
68C22⋊D107—41no—
69C2×Dic107—22no—
70C2×Dic107—21no—
71D428—22no—
72C4×D107—42no—
73Dic214—22no—
74C107⋊C8—42no—
75C22×C107—22no—
76C22×C107—22no—
77C22×C107—22no—
78C22×C107—22no—
79C22×C107—42no—
80C22×C107—42no—
81C22×C107C2×C22⋊C411yescharacteristic
82C22×C107#12011yescharacteristic
83C22×C107#12311yescharacteristic
84C2×D107—41no—
85C2×D107—42no—
86C4×C107—22no—
87C4×C107—44no—
88C4×C107C4×D412yes—
89C4×C107#12112yes—
90Dic107—41no—
91Dic107—42no—
92C2×C107—21no—
93C2×C107—21no—
94C2×C107—42no—
95C2×C107#38611yescharacteristic
96C2×C107#42112yes—
97D107—81no—
98#1897—1071nomaximal, Sylow
99C107#189711yescharacteristic, complemented, Sylow
100#525—1071no—
101C2×#117D10711yescharacteristic, complemented
102C2×#104—1072no—
103#407—1071no—
104#339—1072no—
105C2×#41—1071no—
106C23×C4C2×D10711yescharacteristic
107C2×M16—1072no—
108#123—2141no—
109C22⋊Q8—1072no—
110#121—1072no—
111#120—2141no—
112C4×D4—2142no—
113#117—24no—
114#117—1071no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D10712yes—
118#104—2144no—
119C22⋊C8—2142no—
120C24C22⋊D10711yescharacteristic
121#41—2142no—
122#41—2142no—
123C2×Q8—1071no—
124C2×D4—1071no—
125C2×D4—2142no—
126C2×D4—2141no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—2141no—
131C22×C4C4×D10712yes—
132C22×C4D42812yes—
133C22×C4C22⋊D10711yescharacteristic
134C22×C4C22×D10711yescharacteristic
135M16—2142no—
136M16—2142no—
137C2×C8—2142no—
138C4⋊C4—24no—
139C4⋊C4—1072no—
140C4⋊C4—2142no—
141C22⋊C4—24no—
142C22⋊C4—2142no—
143C22⋊C4—2141no—
144C22⋊C4—2142no—
145C42—22no—
146C42—2141no—
147C42C22⋊D10714yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2141no—
152C23(C2×D107)⋊C412yes—
153C23C2×C22⋊D10711yescharacteristic
154Q8—2142no—
155D4—4284no—
156D4—4281no—
157D4—4281no—
158D4—2142no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—4282no—
166C2×C4—2142no—
167C2×C4—2141no—
168C2×C4(C2×D107)⋊C412yes—
169C2×C4C2×C4×D10711yescharacteristic
170C2×C4C2×C22⋊D10712yes—
171C2×C4C2×D42811yescharacteristic
172C2×C4#1313381812yes—
173C8—4282no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—4282no—
181C22—4281no—
182C22C2×(C2×D107)⋊C411yescharacteristic
183C22#1681207811yescharacteristic
184C22#1681209211yescharacteristic
185C4—22no—
186C4—44no—
187C4—4282no—
188C4—4281no—
189C4C4×C22⋊D10712yes—
190C4#1681208312yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—8561no—
195C2#2029250611yescharacteristic
196C2#2029264512yes—
subgroupG/H
centerC2×C4C2×D428
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#117×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293138C2
cosocleC22×C4×C107C23
hypercenterC2×#117D107
hypocenterC107#1897
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16812037C2semidirect
3. #20292591C2semidirect
4. #20292387C2semidirect
5. C2×#117×C107C2semidirect
9. #16812088C4semidirect
10. #16812083C4semidirect
48. C42×C107D4semidirect
99. C107#1897semidirect
101. C2×#117D107semidirect
147. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
C2×#168120372
#202925912
#202923872
C2×#117×C1072
#1897107