Finite Group Databasebeta

← #28345992

Subgroups — #28345992

Order 13696.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812142C211yesmaximal, characteristic
2C22×Dic428C211yesmaximal, characteristic, complemented
3#20293203C211yesmaximal, characteristic
4#20293104C211yesmaximal, characteristic, complemented
5#20292850C214yesmaximal, complemented
6#20292838C214yesmaximal, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic
9#424×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C22×Dic214C2211yescharacteristic
13#16812142—24no—
14C2×Dic428—24no—
15C2×Dic428C2214yes—
16C2×Dic428C2212yes—
17C2×#13133807C2212yes—
18#16812109C2212yes—
19#16812097C2212yes—
20#16812093—22no—
21#16812089—22no—
22C4×Dic214C2214yes—
23#16812074—22no—
24#16812068C2211yescharacteristic
25#16812065—22no—
26#16812034C2212yes—
27#16812034C2214yes—
28#16812031C2212yes—
29C2×M16×C107C2211yescharacteristic
30C22×C8×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32C4⋊C8×C107C2214yes—
33#13133818—22no—
34#13133818—22no—
35C2×Dic214—21no—
36C2×Dic214—22no—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214D414yes—
41C2×Dic214C2312yes—
42C2×Dic214C2311yescharacteristic
43C2×Dic214C2312yes—
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—21no—
46C22×Dic107—21no—
47C22×Dic107—21no—
48C2×C4×D107—21no—
49Dic428—28no—
50Dic428—24no—
51#13133807—24no—
52C4⋊Dic107—24no—
53C4⋊Dic107C2312yes—
54Dic107⋊C4—24no—
55C22⋊Dic107—21no—
56(C2×D107)⋊C4—24no—
57C4×Dic107—24no—
58C4×Dic107—21no—
59C22×C4×C107C2311yescharacteristic
60M16×C107—22no—
61C2×C8×C107—22no—
62C2×C8×C107C2312yes—
63C2×C8×C107C2312yes—
64C4⋊C4×C107D412yes—
65C22⋊C4×C107D412yes—
66C42×C107C2312yes—
67C23×C107C2×D411yescharacteristic
68C22×D107—21no—
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D414yes—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107#4112yes—
73C2×C4×C107C2411yescharacteristic
74C22⋊D107—42no—
75C2×Dic107—24no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—21no—
80C2×Dic107—42no—
81D428—21no—
82D428—21no—
83C4×D107—42no—
84Dic214—24no—
85Dic214—22no—
86Dic214—22no—
87Dic214—21no—
88Dic214—22no—
89Dic214—21no—
90Dic214—42no—
91Dic214C2×D414yes—
92C8×C107—22no—
93C8×C107—22no—
94C22×C107—22no—
95C22×C107#12112yes—
96C22×C107C22×D411yescharacteristic
97C2×D107—41no—
98C2×D107—42no—
99C4×C107—24no—
100C4×C107#12112yes—
101C4×C107C22×D411yescharacteristic
102C4×C107C2×#4111yescharacteristic
103Dic107—24no—
104Dic107—42no—
105Dic107—41no—
106Dic107—42no—
107C2×C107—22no—
108C2×C107C2×#12111yescharacteristic
109C2×C107#57812yes—
110D107—81no—
111#3044—1071nomaximal, Sylow
112C107#304411yescharacteristic, complemented, Sylow
113C2×#137—1071no—
114C22×Q16—1071no—
115#536—1071no—
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424D10711yescharacteristic, complemented
120#419—1071no—
121#416—1071no—
122C2×#41—1071no—
123C22×Q8—1071no—
124C22×Q8—1071no—
125#137—2144no—
126C2×Q16—1074no—
127C2×Q16—1072no—
128C2×Q16—2144no—
129C2×SD16—1072no—
130C2×M16C2×D10711yescharacteristic
131C22×C8C2×D10711yescharacteristic
132C4⋊Q8—1072no—
133#124—1072no—
134#123—2142no—
135C22⋊Q8—2142no—
136C4×Q8—1074no—
137C4×Q8—2142no—
138#117—1071no—
139#117—2142no—
140#117C2×D10711yescharacteristic
141C4⋊C8C2×D10714yes—
142#103—1072no—
143#103—1074no—
144#102—1072no—
145#41—2142no—
146#41—2142no—
147C2×Q8—1074no—
148C2×Q8—1072no—
149C2×Q8—1072no—
150C2×Q8—1071no—
151C2×Q8—2142no—
152C2×Q8—2142no—
153C2×Q8—2141no—
154C2×Q8—2141no—
155C2×Q8—2142no—
156C2×D4—1071no—
157C2×D4—2141no—
158C22×C4—2141no—
159C22×C4—2141no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—2148no—
163Q16—2144no—
164SD16—2144no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D10712yes—
168C2×C8C22×D10712yes—
169C4⋊C4—1072no—
170C4⋊C4—2144no—
171C4⋊C4—2144no—
172C4⋊C4D42812yes—
173C22⋊C4—2141no—
174C22⋊C4—2144no—
175C22⋊C4D42812yes—
176C42—2141no—
177C42—2144no—
178C42C22×D10712yes—
179C23—2141no—
180C23C2×D42811yescharacteristic
181Q8—1074no—
182Q8—4282no—
183Q8—2142no—
184Q8—2141no—
185Q8—2142no—
186Q8—2141no—
187Q8—2144no—
188Q8—2142no—
189D4—4282no—
190D4—2141no—
191D4—2141no—
192C2×C4—4282no—
193C2×C4—4282no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4—2141no—
197C2×C4—2144no—
198C2×C4—2142no—
199C2×C4D4×D10712yes—
200C2×C4C2×D42814yes—
201C2×C4C2×D42811yescharacteristic
202C2×C4#1313381912yes—
203C2×C4C23×D10711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—4281no—
208C22—4282no—
209C22#1681208412yes—
210C22C22×D42811yescharacteristic
211C4—24no—
212C4—4281no—
213C4—4282no—
214C4—4282no—
215C4—2144no—
216C4#1681208412yes—
217C4C2×D4×D10711yescharacteristic
218C4C2×#1313381911yescharacteristic
219C2—22no—
220C2—8561no—
221C2C2×#1681208411yescharacteristic
222C2#2029344412yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#424D107
hypocenterC107#3044
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic428C2semidirect
4. #20293104C2semidirect
5. #20292850C2semidirect
6. #20292838C2semidirect
7. #20292637C2semidirect
9. #424×C107C2semidirect
112. C107#3044semidirect
119. #424D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C22×Dic4282
#202932032
#202931042
#202928502
#202928382
#202926372
#202926242
#424×C1072
#3044107