Finite Group Databasebeta

← #28345695

Subgroups — #28345695

Order 13696.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16812140C212yesmaximal, complemented
2#20293138C211yesmaximal, characteristic
3#20292783C214yesmaximal, complemented
4#20292766C214yesmaximal, complemented
5#20292636C212yesmaximal, complemented
6#20292591C211yesmaximal, characteristic, complemented
7C23×Q8×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—28no—
10C2×#13133808C2214yescomplemented
11C2×#13133805C2214yescomplemented
12C2×#13133802C2212yes—
13#16812092—21no—
14#16812088C2212yes—
15#16812083C2212yes—
16#16812078—21no—
17C4×C22⋊D107—22no—
18#16812068C2211yescharacteristic
19#16812033C2218yescomplemented
20C22⋊(C107⋊C8)C2214yes—
21C22×Q8×C107—28no—
22C22×Q8×C107—22no—
23C22×Q8×C107—21no—
24C22×Q8×C107C2214yes—
25C22×Q8×C107C2212yes—
26C23×C4×C107—21no—
27C23×C4×C107C2211yescharacteristic
28#13133818—24no—
29C2×Dic214C2311yescharacteristic
30C2×D428C2311yescharacteristic
31C2×C22⋊D107—22no—
32C2×C22⋊D107—21no—
33C2×C4×D107—21no—
34#13133808—28no—
35#13133805—28no—
36#13133802—24no—
37C2×C107⋊C8C2314yes—
38C4⋊Dic107C2312yes—
39Dic107⋊C4—22no—
40C22⋊Dic107—21no—
41C22⋊Dic107—22no—
42(C2×D107)⋊C4—22no—
43C4×Dic107—21no—
44C24×C107D411yescharacteristic
45C2×Q8×C107—216no—
46C2×Q8×C107—28no—
47C2×Q8×C107—216no—
48C2×Q8×C107—24no—
49C2×Q8×C107—24no—
50C2×Q8×C107—22no—
51C2×Q8×C107D418yescomplemented
52C2×Q8×C107C2314yes—
53C22×C4×C107—28no—
54C22×C4×C107—24no—
55C22×C4×C107—22no—
56C22×C4×C107—24no—
57C22×C4×C107D412yes—
58C22×C4×C107D411yescharacteristic
59C22×C4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61C23×C107—22no—
62C23×C107—24no—
63C23×C107C2×D411yescharacteristic
64C23×C107C2×D412yes—
65C22×D107—21no—
66Q8×C107—216no—
67Q8×C107—28no—
68Q8×C107—24no—
69Q8×C107C2×D418yes—
70C2×C4×C107—216no—
71C2×C4×C107—28no—
72C2×C4×C107—22no—
73C2×C4×C107—24no—
74C2×C4×C107—24no—
75C2×C4×C107—24no—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D414yes—
79C2×C4×C107C2411yescharacteristic
80C22⋊D107—42no—
81C22⋊D107—44no—
82C2×Dic107—22no—
83C2×Dic107—21no—
84D428—22no—
85C4×D107—42no—
86Dic214—22no—
87C107⋊C8—24no—
88C22×C107—28no—
89C22×C107—22no—
90C22×C107—22no—
91C22×C107—22no—
92C22×C107#12012yes—
93C22×C107#13714yescomplemented
94C22×C107C22×D411yescharacteristic
95C2×D107—41no—
96C2×D107—42no—
97C4×C107—28no—
98C4×C107—22no—
99C4×C107#12012yes—
100C4×C107C22×D412yes—
101Dic107—41no—
102Dic107—42no—
103C2×C107—22no—
104C2×C107—24no—
105C2×C107C2×#12011yescharacteristic
106C2×C107C2×#13712yes—
107D107—81no—
108#3004—1071nomaximal, Sylow
109C107#300411yescharacteristic, complemented, Sylow
110C23×Q8D10711yescharacteristic, complemented
111C2×#137—1072no—
112#525—1071no—
113#451—1074no—
114#448—1074no—
115#419—1072no—
116#407—1071no—
117C2×#41—1071no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D10714yes—
122C22×Q8C2×D10712yes—
123C23×C4—21no—
124C23×C4C2×D10711yescharacteristic
125#137—2148no—
126C2×Q16—1074no—
127C2×SD16—1074no—
128C2×M16—1072no—
129#123—2141no—
130C22⋊Q8—1072no—
131#121—1072no—
132#120—2141no—
133C4×D4—2142no—
134#117—1071no—
135#103—1078no—
136C22⋊C8—1074no—
137C24C22⋊D10711yescharacteristic
138#41—2144no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—1071no—
146C2×Q8C22⋊D10718yescomplemented
147C2×Q8C22×D10714yes—
148C2×D4—1071no—
149C2×D4—2142no—
150C2×D4—2141no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—2141no—
156C22×C4C22⋊D10712yes—
157C22×C4C22⋊D10711yescharacteristic
158C22×C4C22×D10712yes—
159C22×C4C22×D10711yescharacteristic
160Q16—2148no—
161SD16—2148no—
162M16—2144no—
163C2×C8—1074no—
164C4⋊C4—1072no—
165C4⋊C4—2142no—
166C22⋊C4—2141no—
167C22⋊C4—2142no—
168C22⋊C4—2142no—
169C42—2141no—
170C23—22no—
171C23—24no—
172C23—2141no—
173C23C2×C22⋊D10711yescharacteristic
174C23C2×C22⋊D10712yes—
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—2142no—
179Q8C2×C22⋊D10718yes—
180D4—4282no—
181D4—4284no—
182D4—2142no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—4282no—
190C2×C4—2142no—
191C2×C4—2141no—
192C2×C4C2×C22⋊D10712yes—
193C2×C4C2×C22⋊D10711yescharacteristic
194C2×C4C2×C22⋊D10714yes—
195C2×C4C23×D10711yescharacteristic
196C8—2144no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—4282no—
202C22—4281no—
203C22#1681207812yes—
204C22#1681214014yescomplemented
205C22C22×C22⋊D10711yescharacteristic
206C4—28no—
207C4—22no—
208C4—4282no—
209C4—4281no—
210C4#1681207812yes—
211C4C22×C22⋊D10712yes—
212C2—22no—
213C2—24no—
214C2—8561no—
215C2C2×#1681207811yescharacteristic
216C2C2×#1681214012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC23×Q8×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenterC23×Q8D107
hypocenterC107#3004
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
3. #20292783C2semidirect
4. #20292766C2semidirect
5. #20292636C2semidirect
6. #20292591C2semidirect
7. C23×Q8×C107C2semidirect
10. C2×#13133808C22semidirect
11. C2×#13133805C22semidirect
19. #16812033C22semidirect
51. C2×Q8×C107D4semidirect
93. C22×C107#137semidirect
109. C107#3004semidirect
110. C23×Q8D107semidirect
146. C2×Q8C22⋊D107semidirect
204. C22#16812140semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202931382
#202927832
#202927662
#202926362
#202925912
C23×Q8×C1072
#3004107