Finite Group Databasebeta

← #28346262

Subgroups — #28346262

Order 13696.

376 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×Dic214C211yesmaximal, characteristic
2#20292931C214yesmaximal, complemented
3#20292919C212yesmaximal, complemented
4#20292909C212yesmaximal, complemented
5#20292670C212yesmaximal
6#20292651C211yesmaximal, characteristic
7#20292635C212yesmaximal
8#533×C107C211yesmaximal, characteristic
9C22×Dic214C2211yescharacteristic
10C2×#13133802C2212yes—
11C4×Dic214—21no—
12C4×Dic214—21no—
13C4×Dic214C2212yes—
14C4×Dic214C2212yes—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×Dic107⋊C4—21no—
17C2×C4×Dic107—21no—
18#16812042C2214yes—
19#16812040C2214yes—
20C4⋊(C107⋊C8)C2212yes—
21C4⋊(C107⋊C8)C2212yes—
22#16812032C2214yes—
23#16812032C2214yes—
24C4⋊Q8×C107C2211yescharacteristic
25C4⋊Q8×C107C2211yescharacteristic
26#125×C107C2212yes—
27C22⋊Q8×C107—21no—
28C22⋊Q8×C107—21no—
29C4×Q8×C107—21no—
30C4×Q8×C107—21no—
31#117×C107C2212yes—
32C2×C42×C107C2211yescharacteristic
33C2×Dic214—21no—
34C2×Dic214—22no—
35C2×Dic214Q814yescomplemented
36C2×Dic214C2312yes—
37C22×Dic107—21no—
38C22×Dic107—21no—
39#13133802—24no—
40C2×C107⋊C8C2314yes—
41C4⋊Dic107—21no—
42C4⋊Dic107C2312yes—
43Dic107⋊C4—22no—
44Dic107⋊C4—22no—
45C4×Dic107—22no—
46C4×Dic107—22no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C22×C4×C107D411yescharacteristic
50C22×C4×C107D411yescharacteristic
51C22×C4×C107C2311yescharacteristic
52C4⋊C4×C107—22no—
53C4⋊C4×C107—22no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C4⋊C4×C107C2312yes—
57C4⋊C4×C107C2312yes—
58C22⋊C4×C107—22no—
59C42×C107—22no—
60C42×C107D411yescharacteristic
61C42×C107D411yescharacteristic
62C42×C107C2311yescharacteristic
63C42×C107C2311yescharacteristic
64C23×C107C2×D411yescharacteristic
65Q8×C107—42no—
66Q8×C107—42no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—41no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×Q812yes—
77C2×C4×C107#4112yes—
78C2×C4×C107C2411yescharacteristic
79C2×Dic107—24no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—42no—
83Dic214—24no—
84Dic214—22no—
85Dic214C2×Q814yes—
86C107⋊C8—24no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107C22⋊Q812yes—
90C22×C107C22×D411yescharacteristic
91C4×C107—21no—
92C4×C107—21no—
93C4×C107—21no—
94C4×C107—42no—
95C4×C107—42no—
96C4×C107C22⋊Q812yes—
97C4×C107#13712yes—
98C4×C107C22×Q811yescharacteristic
99C4×C107C2×#4111yescharacteristic
100Dic107—24no—
101Dic107—42no—
102C2×C107—22no—
103C2×C107C2×C22⋊Q811yescharacteristic
104C2×C107C2×#13711yescharacteristic
105C2×C107#57511yescharacteristic
106#3078—1071nomaximal, Sylow
107C107#307811yescharacteristic, complemented, Sylow
108#533D10711yescharacteristic
109C2×C4×Q8—1071no—
110#479—1074no—
111#477—1072no—
112#475—1072no—
113#428—1072no—
114#423—1071no—
115#419—1072no—
116C22×Q8—1071no—
117C2×M16—1072no—
118C4⋊Q8C2×D10711yescharacteristic
119C4⋊Q8C2×D10711yescharacteristic
120#125C2×D10712yes—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—1072no—
126C4×Q8—1072no—
127C4×Q8—2141no—
128C4×Q8—2141no—
129#117C2×D10712yes—
130C2×C4⋊C4—1071no—
131C2×C4⋊C4—2141no—
132C2×C42—2141no—
133C2×C42C2×D10711yescharacteristic
134#107—1074no—
135#106—1074no—
136C4⋊C8—1072no—
137C4⋊C8—1072no—
138#103—1074no—
139#103—1074no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—1074no—
143C2×Q8—1072no—
144C2×Q8—2141no—
145C2×Q8—2142no—
146C22×C4—2141no—
147C22×C4—2141no—
148C22×C4C22⋊D10711yescharacteristic
149C22×C4C22⋊D10711yescharacteristic
150C22×C4C22×D10711yescharacteristic
151M16—2144no—
152C2×C8—1074no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—1072no—
158C4⋊C4—2141no—
159C4⋊C4—2142no—
160C4⋊C4—2142no—
161C4⋊C4C22×D10712yes—
162C4⋊C4C22×D10712yes—
163C22⋊C4—22no—
164C42—22no—
165C42—2142no—
166C42—2142no—
167C42C22⋊D10711yescharacteristic
168C42C22⋊D10711yescharacteristic
169C42C22×D10711yescharacteristic
170C42C22×D10711yescharacteristic
171C23C2×C22⋊D10711yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—1074no—
175Q8—2144no—
176Q8—2142no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—4282no—
182C2×C4—2142no—
183C2×C4—2144no—
184C2×C4—2142no—
185C2×C4C2×C22⋊D10711yescharacteristic
186C2×C4C2×C22⋊D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C2×C4C2×C22⋊D10711yescharacteristic
189C2×C4C2×C22⋊D10711yescharacteristic
190C2×C4Q8×D10712yes—
191C2×C4#1313381912yes—
192C2×C4C23×D10711yescharacteristic
193C8—2144no—
194C22—21no—
195C22—21no—
196C22#1681208512yes—
197C22C22×C22⋊D10711yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—42no—
203C4—4282no—
204C4—2144no—
205C4#1681208512yes—
206C4#1681213912yes—
207C4C2×Q8×D10711yescharacteristic
208C4C2×#1313381911yescharacteristic
209C2—22no—
210C2C2×#1681208511yescharacteristic
211C2C2×#1681213911yescharacteristic
212C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#533×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#533D107
hypocenterC107#3078
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20292931C2semidirect
3. #20292919C2semidirect
4. #20292909C2semidirect
35. C2×Dic214Q8semidirect
107. C107#3078semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2142
#202929312
#202929192
#202929092
#202926702
#202926512
#202926352
#533×C1072
#3078107