Finite Group Databasebeta

← #28336963

Subgroups — #28336963

Order 13696.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293427C211yesmaximal, characteristic
2#20293095C211yesmaximal, characteristic, complemented
3#20292643C211yesmaximal, characteristic, complemented
4C2×#16812035C211yesmaximal, characteristic, complemented
5#20292600C211yesmaximal, characteristic, complemented
6#20292388C211yesmaximal, characteristic, complemented
7#344×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133818C2211yescharacteristic
10#16812142—21no—
11#16812142C412yescomplemented
12#16812137—21no—
13#16812137C412yescomplemented
14#16812131—22no—
15#16812074—21no—
16#16812071—21no—
17(C22⋊D107)⋊C4—22no—
18#16812068C2211yescharacteristic
19#16812066—21no—
20C2×C4×Dic107C2211yescharacteristic
21C42×D107—21no—
22#16812035—22no—
23#16812035—21no—
24#16812035—21no—
25#16812023—22no—
26C2×M16×C107C2211yescharacteristic
27C2×M16×C107C2211yescharacteristic
28#117×C107C2211yescharacteristic
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33#13133818—21no—
34#13133818—21no—
35#13133818C2×C412yes—
36#13133818D412yes—
37C2×Dic214C2×C411yescharacteristic, complemented
38C2×Dic214D411yescharacteristic
39C2×D428C2×C411yescharacteristic, complemented
40C2×D428D411yescharacteristic
41C2×C22⋊D107—21no—
42C2×C22⋊D107—21no—
43C22×Dic107—21no—
44C2×C4×D107—21no—
45C2×C4×D107—22no—
46C2×C4×D107—21no—
47Dic428—22no—
48#13133807—22no—
49#13133807—22no—
50D856—22no—
51C4⋊Dic107—21no—
52Dic107⋊C4—22no—
53C22⋊Dic107—21no—
54(C2×D107)⋊C4—22no—
55C4×Dic107—22no—
56C4×Dic107—21no—
57C4×Dic107—21no—
58C4×Dic107D411yescharacteristic
59C4×Dic107D411yescharacteristic
60C22×C4×C107C2311yescharacteristic
61M16×C107—21no—
62M16×C107—21no—
63M16×C107—21no—
64M16×C107C2×C412yescomplemented
65C2×C8×C107—21no—
66C2×C8×C107—21no—
67C4⋊C4×C107—21no—
68C22⋊C4×C107D412yes—
69C42×C107—21no—
70C23×C107C2×D411yescharacteristic
71C22×D107—21no—
72C22×D107—21no—
73C2×C4×C107—22no—
74C2×C4×C107C22×C411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107#4111yescharacteristic
79C2×C4×C107#4111yescharacteristic
80C22⋊D107—22no—
81C22⋊D107—22no—
82C22⋊D107—41no—
83C22⋊D107—41no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—42no—
89C2×Dic107—41no—
90D428—21no—
91D428—21no—
92D428—21no—
93D428C22⋊C412yes—
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—44no—
97C4×D107—41no—
98C4×D107—41no—
99Dic214—21no—
100Dic214—21no—
101Dic214—21no—
102Dic214C22⋊C412yes—
103C8×C107—22no—
104C8×C107—41no—
105C22×C107—41no—
106C22×C107C4×D411yescharacteristic
107C22×C107#12011yescharacteristic
108C22×C107#12111yescharacteristic
109C2×D107—22no—
110C2×D107—42no—
111C2×D107—41no—
112C2×D107—81no—
113C4×C107—42no—
114C4×C107C2×C22⋊C411yescharacteristic
115C4×C107C4×D411yescharacteristic
116C4×C107#12111yescharacteristic
117C4×C107#12311yescharacteristic
118Dic107—22no—
119Dic107—22no—
120Dic107—41no—
121Dic107—41no—
122Dic107—81no—
123C2×C107—21no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107#38611yescharacteristic
127D107—42no—
128D107—81no—
129#1906—1071nomaximal, Sylow
130C107#190611yescharacteristic, complemented, Sylow
131#575—1071no—
132C4×#41—1071no—
133#421—1071no—
134C2×#104—1071no—
135#410—1071no—
136#344D10711yescharacteristic, complemented
137#339—1071no—
138C2×#41—1071no—
139C2×#41—1071no—
140#137—1072no—
141#137—2141no—
142AGL(1,ℤ/8)—1072no—
143AGL(1,ℤ/8)—2141no—
144#135—2142no—
145C2×M16C2×D10711yescharacteristic
146C2×M16C2×D10711yescharacteristic
147C4×Q8—2141no—
148C4×D4—2141no—
149C4×D4—2142no—
150#117—1071no—
151#117—2141no—
152#117C2×D10711yescharacteristic
153C2×C42—1071no—
154C2×C42—2141no—
155#104—2141no—
156#104—2141no—
157#104—2142no—
158#99—2142no—
159#41—1072no—
160#41—1072no—
161#41—2141no—
162#41—2141no—
163#41—2141no—
164#41—2141no—
165#41—2141no—
166#41—2141no—
167C2×Q8—1071no—
168C2×Q8—1071no—
169C2×D4—1071no—
170C2×D4—1071no—
171C2×D4—2141no—
172C2×D4—2141no—
173C22×C4—2142no—
174C22×C4—2141no—
175C22×C4—2141no—
176C22×C4—2141no—
177C22×C4C22×D10711yescharacteristic
178Q16—2142no—
179SD16—2142no—
180SD16—2142no—
181D8—2142no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D10712yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—2142no—
190C4⋊C4—2141no—
191C22⋊C4—2142no—
192C22⋊C4—2141no—
193C22⋊C4D42812yes—
194C42—21no—
195C42—1071no—
196C42—1071no—
197C42—2142no—
198C42—2141no—
199C42—2141no—
200C23—2141no—
201C23—2141no—
202C23C2×D42811yescharacteristic
203Q8—1072no—
204Q8—2141no—
205Q8—2141no—
206Q8—2141no—
207D4—1072no—
208D4—4281no—
209D4—4281no—
210D4—2142no—
211D4—2142no—
212D4—2141no—
213D4—2141no—
214D4—2141no—
215C2×C4—22no—
216C2×C4—4284no—
217C2×C4—4281no—
218C2×C4—4282no—
219C2×C4—4281no—
220C2×C4—4281no—
221C2×C4—2141no—
222C2×C4—2141no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4C2×C4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4D4×D10711yescharacteristic
230C2×C4D4×D10711yescharacteristic
231C2×C4#1313381811yescharacteristic
232C2×C4#1313381911yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—8561no—
237C22—4282no—
238C22—4281no—
239C22—2142no—
240C22C4×D42811yescharacteristic
241C22#1681207911yescharacteristic
242C22#1681208411yescharacteristic
243C4—42no—
244C4—8561no—
245C4—4281no—
246C4—4281no—
247C4—2142no—
248C4—2142no—
249C4C22⋊C4×D10711yescharacteristic
250C4#1681207111yescharacteristic
251C4#1681208211yescharacteristic
252C4#1681209411yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—8561no—
257C2—4282no—
258C2#2029250511yescharacteristic
subgroupG/H
centerC4#16812082
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#344×C107C2
socleC2×C107#386
prime radical#20293427C2
cosocleC22×C4×C107C23
hypercenter#344D107
hypocenterC107#1906
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #20293095C2semidirect
3. #20292643C2semidirect
4. C2×#16812035C2semidirect
5. #20292600C2semidirect
6. #20292388C2semidirect
7. #344×C107C2semidirect
11. #16812142C4semidirect
13. #16812137C4semidirect
37. C2×Dic214C2×C4semidirect
39. C2×D428C2×C4semidirect
64. M16×C107C2×C4semidirect
130. C107#1906semidirect
136. #344D107semidirect
185. M16C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202934272
#202930952
#202926432
C2×#168120352
#202926002
#202923882
#344×C1072
#1906107