Finite Group Databasebeta

← #28346268

Subgroups — #28346268

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
1#20293184C211yesmaximal, characteristic
2C2×C4×Dic214C211yesmaximal, characteristic
3#20292935C214yesmaximal, complemented
4#20292923C212yesmaximal, complemented
5#20292913C212yesmaximal, complemented
6#20292669C212yesmaximal
7#20292637C212yesmaximal
8#423×C107C211yesmaximal, characteristic
9C22×Dic214C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812109C2211yescharacteristic
12#16812101C2212yes—
13#16812088—21no—
14#16812088—21no—
15C4×Dic214—21no—
16C4×Dic214—21no—
17C4×Dic214—21no—
18C4×Dic214—21no—
19C4×Dic214C2212yes—
20C4×Dic214C2212yes—
21#16812068C2212yes—
22C2×C4⋊Dic107C2211yescharacteristic
23C2×Dic107⋊C4—21no—
24C2×C4×Dic107—21no—
25#16812043C2214yes—
26#16812041C2214yes—
27#16812034C2214yes—
28#16812034C2214yes—
29C2×M16×C107C2212yes—
30C2×C42×C107C2211yescharacteristic
31C4⋊C8×C107C2212yes—
32C4⋊C8×C107C2212yes—
33C2×Dic214—21no—
34C2×Dic214—22no—
35C2×Dic214—21no—
36C2×Dic214—21no—
37C2×Dic214Q814yescomplemented
38C2×Dic214C2312yes—
39C22×Dic107—21no—
40C22×Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107—21no—
43C4⋊Dic107—21no—
44C4⋊Dic107C2312yes—
45C4⋊Dic107C2312yes—
46C4⋊Dic107C2312yes—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49Dic107⋊C4—22no—
50Dic107⋊C4—22no—
51C22⋊Dic107—22no—
52C4×Dic107—22no—
53C4×Dic107—22no—
54C4×Dic107—22no—
55C22×C4×C107D411yescharacteristic
56C22×C4×C107D411yescharacteristic
57C22×C4×C107C2311yescharacteristic
58M16×C107—24no—
59C2×C8×C107C2314yes—
60C42×C107D411yescharacteristic
61C42×C107D411yescharacteristic
62C42×C107C2311yescharacteristic
63C42×C107C2311yescharacteristic
64C23×C107C2×D411yescharacteristic
65C2×C4×C107—22no—
66C2×C4×C107—41no—
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×Q812yes—
73C2×C4×C107#4112yes—
74C2×C4×C107C2411yescharacteristic
75C2×Dic107—24no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—42no—
81Dic214—24no—
82Dic214—22no—
83Dic214—42no—
84Dic214—42no—
85Dic214C2×Q814yes—
86C8×C107—24no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107C22⋊Q812yes—
90C22×C107C22×D411yescharacteristic
91C4×C107—21no—
92C4×C107—21no—
93C4×C107—21no—
94C4×C107C22⋊Q812yes—
95C4×C107#13712yes—
96C4×C107C22×Q811yescharacteristic
97C4×C107C2×#4111yescharacteristic
98Dic107—24no—
99Dic107—42no—
100Dic107—42no—
101Dic107—42no—
102C2×C107—22no—
103C2×C107C2×C22⋊Q811yescharacteristic
104C2×C107C2×#13711yescharacteristic
105C2×C107#57511yescharacteristic
106#3078—1071nomaximal, Sylow
107C107#307811yescharacteristic, complemented, Sylow
108#533—1071no—
109C2×C4×Q8—1071no—
110#479—1074no—
111#477—1072no—
112#475—1072no—
113#428—1072no—
114#423D10711yescharacteristic
115#419—1072no—
116C22×Q8—1071no—
117C2×M16C2×D10712yes—
118C4⋊Q8—1071no—
119C4⋊Q8—1071no—
120#125—1072no—
121C22⋊Q8—2141no—
122C22⋊Q8—2141no—
123C4×Q8—1072no—
124C4×Q8—1072no—
125C4×Q8—2141no—
126C4×Q8—2141no—
127C4×Q8—2141no—
128C4×Q8—2141no—
129#117—1072no—
130C2×C4⋊C4—1071no—
131C2×C4⋊C4—2141no—
132C2×C42—2141no—
133C2×C42C2×D10711yescharacteristic
134#107—1074no—
135#106—1074no—
136C4⋊C8C2×D10712yes—
137C4⋊C8C2×D10712yes—
138#103—1074no—
139#103—1074no—
140C2×Q8—1074no—
141C2×Q8—1072no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8—2142no—
146C22×C4—2141no—
147C22×C4—2141no—
148C22×C4D42811yescharacteristic
149C22×C4D42811yescharacteristic
150C22×C4C22×D10711yescharacteristic
151M16—24no—
152C2×C8C22×D10714yes—
153C4⋊C4—1072no—
154C4⋊C4—1072no—
155C4⋊C4—1072no—
156C4⋊C4—2141no—
157C4⋊C4—2142no—
158C4⋊C4—2142no—
159C4⋊C4—2142no—
160C4⋊C4—2142no—
161C4⋊C4—2141no—
162C4⋊C4—2141no—
163C22⋊C4—2142no—
164C42—2142no—
165C42—2142no—
166C42—2142no—
167C42D42811yescharacteristic
168C42D42811yescharacteristic
169C42C22×D10711yescharacteristic
170C42C22×D10711yescharacteristic
171C23C2×D42811yescharacteristic
172Q8—1074no—
173Q8—4282no—
174Q8—4282no—
175Q8—2144no—
176Q8—2142no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—4282no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4—2142no—
183C2×C4—2144no—
184C2×C4—2142no—
185C2×C4C2×D42811yescharacteristic
186C2×C4C2×D42811yescharacteristic
187C2×C4C2×D42811yescharacteristic
188C2×C4C2×D42811yescharacteristic
189C2×C4C2×D42811yescharacteristic
190C2×C4Q8×D10712yes—
191C2×C4#1313381712yes—
192C2×C4C23×D10711yescharacteristic
193C8—24no—
194C22—21no—
195C22—21no—
196C22#1681208912yes—
197C22C22×D42811yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—4282no—
202C4—4282no—
203C4—4282no—
204C4—2144no—
205C4#1681208912yes—
206C4#1681214212yes—
207C4C2×Q8×D10711yescharacteristic
208C4C2×#1313381711yescharacteristic
209C2—22no—
210C2C2×#1681208911yescharacteristic
211C2C2×#1681214211yescharacteristic
212C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#423×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#423D107
hypocenterC107#3078
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
3. #20292935C2semidirect
4. #20292923C2semidirect
5. #20292913C2semidirect
37. 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
#202931842
C2×C4×Dic2142
#202929352
#202929232
#202929132
#202926692
#202926372
#423×C1072
#3078107