Finite Group Databasebeta

← #28336955

Subgroups — #28336955

Order 13696.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16812140C211yesmaximal, characteristic
2#20292645C211yesmaximal, characteristic
3#20292636C211yesmaximal, characteristic, complemented
4#20292600C211yesmaximal, characteristic
5#20292397C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7#519×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—22no—
10#16812140—22no—
11C2×#13133808C412yescomplemented
12C2×#13133805C412yescomplemented
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812068C2211yescharacteristic
16#16812037—22no—
17#16812033—22no—
18#16812023—22no—
19C22×Q8×C107C2211yescharacteristic
20C4×Q8×C107—22no—
21C4×Q8×C107—22no—
22#117×C107—22no—
23#117×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#13133818—21no—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29C2×Dic214C2×C411yescharacteristic
30C2×D428C2×C411yescharacteristic
31C2×C22⋊D107—21no—
32C2×C4×D107—21no—
33#13133808—24no—
34#13133805—24no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38#13133802—21no—
39C2×C107⋊C8—21no—
40C2×C107⋊C8C2×C412yes—
41C4⋊Dic107—21no—
42C22⋊Dic107—21no—
43C4×Dic107—21no—
44C2×Q8×C107—21no—
45C2×Q8×C107—21no—
46C2×Q8×C107—21no—
47C2×Q8×C107C2×C412yes—
48C2×Q8×C107D412yes—
49C2×Q8×C107D412yes—
50C22×C4×C107—21no—
51C22×C4×C107C2311yescharacteristic
52C4⋊C4×C107—24no—
53C4⋊C4×C107—21no—
54C4⋊C4×C107—21no—
55C22⋊C4×C107—21no—
56C22⋊C4×C107D412yes—
57C42×C107—24no—
58C42×C107—21no—
59C42×C107D412yes—
60C23×C107C2×D411yescharacteristic
61C22×D107—21no—
62Q8×C107—22no—
63Q8×C107—22no—
64Q8×C107—22no—
65Q8×C107C22⋊C414yes—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107C22×C411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107#4111yescharacteristic
76C2×C4×C107#4111yescharacteristic
77C22⋊D107—41no—
78C22⋊D107—41no—
79C2×Dic107—21no—
80C2×Dic107—41no—
81D428—21no—
82D428—21no—
83C4×D107—41no—
84C4×D107—41no—
85Dic214—21no—
86Dic214—21no—
87C107⋊C8—22no—
88C107⋊C8—41no—
89C22×C107—41no—
90C22×C107C4×D411yescharacteristic
91C22×C107#12011yescharacteristic
92C22×C107#12111yescharacteristic
93C2×D107—41no—
94C2×D107—81no—
95C4×C107—24no—
96C4×C107—42no—
97C4×C107—42no—
98C4×C107C2×C22⋊C411yescharacteristic
99C4×C107C4×D411yescharacteristic
100C4×C107#12111yescharacteristic
101C4×C107#12311yescharacteristic
102Dic107—41no—
103Dic107—81no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107#38611yescharacteristic
108D107—81no—
109#1905—1071nomaximal, Sylow
110C107#190511yescharacteristic, complemented, Sylow
111C2×#137—1071no—
112#519D10711yescharacteristic, complemented
113#421—1071no—
114#419—1071no—
115#410—1071no—
116#344—1071no—
117H(3,ℤ/4)—1071no—
118C2×#41—1071no—
119C22×Q8C2×D10711yescharacteristic
120#137—2142no—
121#137—2142no—
122C2×Q16—1072no—
123C2×SD16—1072no—
124C2×M16—1071no—
125C2×M16—1071no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—22no—
129#117—1071no—
130#117C2×D10711yescharacteristic
131#117C2×D10711yescharacteristic
132#104—2142no—
133#103—2142no—
134#99—2142no—
135#41—2141no—
136#41—2141no—
137#41—2141no—
138#41—2141no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—1071no—
143C2×Q8C4×D10712yes—
144C2×Q8D42812yes—
145C2×Q8C22⋊D10712yes—
146C2×D4—1071no—
147C2×D4—2141no—
148C22×C4—21no—
149C22×C4—2141no—
150C22×C4C22×D10711yescharacteristic
151Q16—2144no—
152SD16—2144no—
153M16—2141no—
154M16—2141no—
155M16—2141no—
156M16—2141no—
157C2×C8—1072no—
158C2×C8—2141no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—2141no—
163C22⋊C4—21no—
164C22⋊C4—2141no—
165C22⋊C4C22⋊D10712yes—
166C42—24no—
167C42—21no—
168C42—2141no—
169C42C22⋊D10712yes—
170C23—2141no—
171C23C2×C22⋊D10711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8—2141no—
176Q8—2141no—
177Q8(C2×D107)⋊C414yes—
178D4—4281no—
179D4—4281no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—4281no—
188C2×C4—4281no—
189C2×C4—4281no—
190C2×C4—2141no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4C2×C22⋊D10711yescharacteristic
193C2×C4C2×C22⋊D10711yescharacteristic
194C2×C4C2×D42811yescharacteristic
195C2×C4#1313381811yescharacteristic
196C2×C4#1313381811yescharacteristic
197C8—4281no—
198C8—2142no—
199C22—41no—
200C22—8561no—
201C22—4281no—
202C22C4×C22⋊D10711yescharacteristic
203C22#1681207811yescharacteristic
204C22#1681208311yescharacteristic
205C4—24no—
206C4—42no—
207C4—42no—
208C4—8561no—
209C4—4281no—
210C4C2×(C2×D107)⋊C411yescharacteristic
211C4C4×C22⋊D10711yescharacteristic
212C4#1681208311yescharacteristic
213C4#1681209211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—8561no—
218C2#2029250611yescharacteristic
subgroupG/H
centerC2#20292506
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#519×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenter#519D107
hypocenterC107#1905
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #20292636C2semidirect
5. #20292397C2semidirect
6. #20292383C2semidirect
7. #519×C107C2semidirect
11. C2×#13133808C4semidirect
12. C2×#13133805C4semidirect
110. C107#1905semidirect
112. #519D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202926452
#202926362
#202926002
#202923972
#202923832
#519×C1072
#1905107