Finite Group Databasebeta

← #28336962

Subgroups — #28336962

Order 13696.

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

normal subgroups only

This group has 332 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
1#20293425C211yesmaximal, characteristic
2#20292644C211yesmaximal, characteristic, complemented
3C2×#16812037C211yesmaximal, characteristic, complemented
4#20292600C211yesmaximal, characteristic, complemented
5#20292397C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7C4×#41×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—21no—
10#16812139C412yescomplemented
11#16812135—21no—
12#16812134C412yescomplemented
13#16812129—22no—
14C2×#13133802C2211yescharacteristic
15C2×#13133802C2211yescharacteristic
16#16812068C2211yescharacteristic
17#16812037—21no—
18#16812037—21no—
19#16812036—22no—
20#16812023—22no—
21C2×#41×C107C2211yescharacteristic
22C4×Q8×C107—21no—
23C4×D4×C107—21no—
24C4×D4×C107—22no—
25#117×C107—21no—
26#117×C107C2211yescharacteristic
27C2×C42×C107—21no—
28C2×C42×C107C2211yescharacteristic
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33C2×Dic214C2×C411yescharacteristic, complemented
34C2×D428C2×C411yescharacteristic, complemented
35C2×C22⋊D107—21no—
36C2×C4×D107—21no—
37#13133808—22no—
38#13133806—22no—
39#13133805—22no—
40#13133803—22no—
41#13133802—21no—
42#13133802—21no—
43#13133802—21no—
44#13133802C2×C412yescomplemented
45C2×C107⋊C8—21no—
46C2×C107⋊C8—21no—
47C4⋊Dic107—21no—
48C22⋊Dic107—21no—
49C4×Dic107—21no—
50#41×C107—21no—
51#41×C107—21no—
52#41×C107C2×C412yes—
53#41×C107D412yes—
54C2×Q8×C107D411yescharacteristic
55C2×D4×C107—21no—
56C2×D4×C107D411yescharacteristic
57C22×C4×C107—21no—
58C22×C4×C107—22no—
59C22×C4×C107—21no—
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107—21no—
63C22⋊C4×C107—22no—
64C22⋊C4×C107D412yes—
65C42×C107—22no—
66C42×C107—21no—
67C42×C107—21no—
68C42×C107D411yescharacteristic
69C42×C107D411yescharacteristic
70C23×C107—21no—
71C23×C107C2×D411yescharacteristic
72C22×D107—21no—
73Q8×C107—21no—
74Q8×C107C22⋊C412yes—
75D4×C107—22no—
76D4×C107—22no—
77D4×C107—21no—
78D4×C107C22⋊C412yes—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—42no—
86C2×C4×C107—44no—
87C2×C4×C107C22×C411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107#4111yescharacteristic
92C2×C4×C107#4111yescharacteristic
93C22⋊D107—41no—
94C22⋊D107—41no—
95C2×Dic107—21no—
96C2×Dic107—41no—
97D428—21no—
98D428—21no—
99C4×D107—41no—
100C4×D107—41no—
101Dic214—21no—
102Dic214—21no—
103C107⋊C8—22no—
104C107⋊C8—41no—
105C22×C107—22no—
106C22×C107—41no—
107C22×C107—42no—
108C22×C107C4×D411yescharacteristic
109C22×C107#12011yescharacteristic
110C22×C107#12111yescharacteristic
111C2×D107—41no—
112C2×D107—81no—
113C4×C107—22no—
114C4×C107—22no—
115C4×C107—42no—
116C4×C107—41no—
117C4×C107C2×C22⋊C411yescharacteristic
118C4×C107C4×D411yescharacteristic
119C4×C107#12111yescharacteristic
120C4×C107#12311yescharacteristic
121Dic107—41no—
122Dic107—81no—
123C2×C107—21no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107—42no—
127C2×C107#38611yescharacteristic
128D107—81no—
129#1906—1071nomaximal, Sylow
130C107#190611yescharacteristic, complemented, Sylow
131#575—1071no—
132C4×#41D10711yescharacteristic, complemented
133#421—1071no—
134C2×#104—1071no—
135#410—1071no—
136#344—1071no—
137#339—1071no—
138C2×#41—1071no—
139C2×#41C2×D10711yescharacteristic
140#137—1072no—
141#137—2141no—
142AGL(1,ℤ/8)—1072no—
143AGL(1,ℤ/8)—2141no—
144#135—2142no—
145C2×M16—1071no—
146C2×M16—1071no—
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117—1071no—
152#117C2×D10711yescharacteristic
153C2×C42—21no—
154C2×C42C2×D10711yescharacteristic
155#104—2142no—
156#104—2141no—
157#104—2141no—
158#99—2142no—
159#41—21no—
160#41—21no—
161#41—2141no—
162#41—2141no—
163#41—2141no—
164#41—2141no—
165#41C4×D10712yes—
166#41D42812yes—
167C2×Q8—1071no—
168C2×Q8C22⋊D10711yescharacteristic
169C2×D4—21no—
170C2×D4—1071no—
171C2×D4—2141no—
172C2×D4C22⋊D10711yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—2141no—
177C22×C4C22×D10711yescharacteristic
178Q16—2142no—
179SD16—2142no—
180SD16—2142no—
181D8—2142no—
182M16—1072no—
183M16—2141no—
184M16—2141no—
185M16—2141no—
186C2×C8—2141no—
187C2×C8—2141no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—2141no—
191C22⋊C4—22no—
192C22⋊C4—2141no—
193C22⋊C4C22⋊D10712yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—2141no—
198C42C22⋊D10711yescharacteristic
199C42C22⋊D10711yescharacteristic
200C23—21no—
201C23—2141no—
202C23C2×C22⋊D10711yescharacteristic
203Q8—21no—
204Q8—2141no—
205Q8—2141no—
206Q8(C2×D107)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—4281no—
211D4—4281no—
212D4—2141no—
213D4—2141no—
214D4(C2×D107)⋊C412yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—4281no—
224C2×C4—4281no—
225C2×C4—4281no—
226C2×C4—2141no—
227C2×C4C2×C4×D10711yescharacteristic
228C2×C4C2×C22⋊D10711yescharacteristic
229C2×C4C2×C22⋊D10711yescharacteristic
230C2×C4C2×D42811yescharacteristic
231C2×C4#1313381811yescharacteristic
232C2×C4#1313381811yescharacteristic
233C8—4281no—
234C8—2142no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—8561no—
239C22—4281no—
240C22C4×C22⋊D10711yescharacteristic
241C22#1681207811yescharacteristic
242C22#1681208311yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—8561no—
248C4—4281no—
249C4C2×(C2×D107)⋊C411yescharacteristic
250C4C4×C22⋊D10711yescharacteristic
251C4#1681208311yescharacteristic
252C4#1681209211yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—8561no—
258C2#2029250611yescharacteristic
subgroupG/H
centerC4#16812083
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC4×#41×C107C2
socleC2×C107#386
prime radical#20293425C2
cosocleC22×C4×C107C23
hypercenterC4×#41D107
hypocenterC107#1906
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #20292644C2semidirect
3. C2×#16812037C2semidirect
4. #20292600C2semidirect
5. #20292397C2semidirect
6. #20292387C2semidirect
7. C4×#41×C107C2semidirect
10. #16812139C4semidirect
12. #16812134C4semidirect
33. C2×Dic214C2×C4semidirect
34. C2×D428C2×C4semidirect
44. #13133802C2×C4semidirect
130. C107#1906semidirect
132. C4×#41D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202926442
C2×#168120372
#202926002
#202923972
#202923872
C4×#41×C1072
#1906107