Finite Group Databasebeta

← #28336929

Subgroups — #28336929

Order 13696.

338 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16812088C211yesmaximal, characteristic
2C2×#16812027C212yesmaximal, complemented
3#20292591C211yesmaximal, characteristic
4#20292395C212yesmaximal, complemented
5C22×M16×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×#13133802C2212yes—
8#16812088—21no—
9#16812088—21no—
10#16812088—21no—
11#16812088—21no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C412yes—
14C2×C22⋊Dic107C412yes—
15#16812027—24no—
16C22⋊(C107⋊C8)—21no—
17C22⋊(C107⋊C8)—21no—
18C23×C4×C107C2211yescharacteristic
19C2×M16×C107—21no—
20C2×M16×C107—24no—
21C2×M16×C107C2212yes—
22C22×C8×C107—21no—
23C2×Dic214—21no—
24C2×Dic214—21no—
25C2×Dic214—22no—
26C2×Dic214—22no—
27C22×Dic107C2×C412yes—
28C22×Dic107C2×C412yes—
29#13133802—22no—
30#13133802—22no—
31C2×C107⋊C8—22no—
32C4⋊Dic107—21no—
33C4⋊Dic107—21no—
34Dic107⋊C4—22no—
35Dic107⋊C4—22no—
36C22⋊Dic107—22no—
37C22⋊Dic107—22no—
38C24×C107C2×C411yescharacteristic
39C22×C4×C107—22no—
40C22×C4×C107—22no—
41C22×C4×C107C2×C411yescharacteristic
42C22×C4×C107D411yescharacteristic
43C22×C4×C107D411yescharacteristic
44C22×C4×C107D411yescharacteristic
45C22×C4×C107D411yescharacteristic
46C22×C4×C107C2311yescharacteristic
47M16×C107—24no—
48M16×C107—22no—
49M16×C107D414yes—
50C2×C8×C107—24no—
51C2×C8×C107—22no—
52C23×C107—22no—
53C23×C107—22no—
54C23×C107—41no—
55C23×C107C22⋊C411yescharacteristic
56C23×C107C22⋊C411yescharacteristic
57C23×C107C22×C411yescharacteristic
58C2×C4×C107—22no—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—21no—
64C2×C4×C107—21no—
65C2×C4×C107C22⋊C411yescharacteristic
66C2×C4×C107C22⋊C411yescharacteristic
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107#4112yes—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—24no—
74C2×Dic107—42no—
75Dic214—41no—
76Dic214—42no—
77Dic214—41no—
78C8×C107—24no—
79C107⋊C8—42no—
80C22×C107—21no—
81C22×C107—21no—
82C22×C107—21no—
83C22×C107—21no—
84C22×C107—21no—
85C22×C107—21no—
86C22×C107—21no—
87C22×C107—44no—
88C22×C107#10112yescomplemented
89C22×C107C2×C22⋊C411yescharacteristic
90C22×C107C4×D412yes—
91C4×C107—21no—
92C4×C107—21no—
93C4×C107#12011yescharacteristic
94C4×C107#12112yes—
95C4×C107#12311yescharacteristic
96Dic107—42no—
97Dic107—42no—
98C2×C107—22no—
99C2×C107—22no—
100C2×C107—41no—
101C2×C107#38611yescharacteristic
102C2×C107C2×#10111yescharacteristic
103C2×C107#41311yescharacteristic
104#1901—1071nomaximal, Sylow
105C107#190111yescharacteristic, complemented, Sylow
106C22×M16D10711yescharacteristic
107C2×C22⋊Q8—1071no—
108C2×#101—1072no—
109#407—1071no—
110#343—1072no—
111C22×Q8—1071no—
112C23×C4C2×D10711yescharacteristic
113C2×M16—21no—
114C2×M16—24no—
115C2×M16—1072no—
116C2×M16C2×D10712yes—
117C22×C8—21no—
118C22⋊Q8—2141no—
119C22⋊Q8—2141no—
120C22⋊Q8—2141no—
121C22⋊Q8—2141no—
122C2×C4⋊C4—1072no—
123C2×C4⋊C4—1071no—
124C2×C22⋊C4—1072no—
125#101—2144no—
126C22⋊C8—2141no—
127C22⋊C8—2141no—
128C24C4×D10711yescharacteristic
129C2×Q8—2142no—
130C2×Q8—2141no—
131C2×Q8—2141no—
132C2×Q8—2142no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—1072no—
136C22×C4—1072no—
137C22×C4C4×D10711yescharacteristic
138C22×C4D42811yescharacteristic
139C22×C4D42811yescharacteristic
140C22×C4C22⋊D10711yescharacteristic
141C22×C4C22⋊D10711yescharacteristic
142C22×C4C22×D10711yescharacteristic
143M16—24no—
144M16—22no—
145M16—2142no—
146M16—2142no—
147M16C22⋊D10714yes—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—2142no—
151C4⋊C4—2142no—
152C4⋊C4—2142no—
153C4⋊C4—2141no—
154C4⋊C4—2141no—
155C22⋊C4—2142no—
156C22⋊C4—2142no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23(C2×D107)⋊C411yescharacteristic
161C23(C2×D107)⋊C411yescharacteristic
162C23C2×C4×D10711yescharacteristic
163Q8—4281no—
164Q8—4282no—
165Q8—4281no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—4282no—
174C2×C4—2142no—
175C2×C4—2142no—
176C2×C4—2144no—
177C2×C4(C2×D107)⋊C411yescharacteristic
178C2×C4(C2×D107)⋊C411yescharacteristic
179C2×C4C2×C22⋊D10712yes—
180C2×C4C2×C22⋊D10711yescharacteristic
181C2×C4C2×D42811yescharacteristic
182C2×C4#1313381812yes—
183C8—24no—
184C8—4282no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1681202712yescomplemented
194C22C2×(C2×D107)⋊C411yescharacteristic
195C22C4×C22⋊D10712yes—
196C4—21no—
197C4—21no—
198C4—4282no—
199C4—4282no—
200C4#1681207811yescharacteristic
201C4#1681208312yes—
202C4#1681209211yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2029250611yescharacteristic
207C2C2×#1681202711yescharacteristic
208C2#2029260911yescharacteristic
subgroupG/H
centerC22C2×(C2×D107)⋊C4
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
FittingC22×M16×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC24×C107C2×C4
cosocleC22×C4×C107C23
hypercenterC22×M16D107
hypocenterC107#1901
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16812027C2semidirect
4. #20292395C2semidirect
88. C22×C107#101semidirect
105. C107#1901semidirect
193. C22#16812027semidirect

Maximal subgroups

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

subgroupindex
C2×#168120882
C2×#168120272
#202925912
#202923952
C22×M16×C1072
#1901107