Finite Group Databasebeta

← #28348412

Subgroups — #28348412

Order 13696.

408 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293342C211yesmaximal, characteristic, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20292974C211yesmaximal, characteristic, complemented
4#20292952C212yesmaximal, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292863C212yesmaximal, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292774C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10#20292624C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C2×#13133807C2211yescharacteristic
14C2×D856C2212yes—
15#16812109C2211yescharacteristic
16#16812106C2211yescharacteristic
17#16812105—21no—
18#16812095—22no—
19#16812083C2212yes—
20#16812083C2212yes—
21#16812082—22no—
22#16812080—21no—
23C4×Dic214C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812071—21no—
26#16812068C2212yes—
27#16812067—21no—
28#16812043C2212yes—
29#16812041C2211yescharacteristic
30#16812034C2211yescharacteristic
31#16812034C2212yes—
32#16812031C2212yes—
33#16812031C2212yes—
34#16812031C2212yes—
35#16812031C2211yescharacteristic
36C22×C8×C107C2212yes—
37C4×D4×C107C2211yescharacteristic
38C4⋊C8×C107C2211yescharacteristic
39C22⋊C8×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133818—24no—
42#13133818—42no—
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—21no—
47C2×D428C2311yescharacteristic
48C2×D428C2312yes—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54#13133807—22no—
55D856—42no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2312yes—
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—21no—
61C22⋊Dic107—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—21no—
65C4×Dic107—22no—
66C4×Dic107—21no—
67C2×D4×C107D411yescharacteristic
68C22×C4×C107C2312yes—
69C2×C8×C107—42no—
70C2×C8×C107C2312yes—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107D411yescharacteristic
74C22⋊C4×C107D412yes—
75C42×C107C2311yescharacteristic
76C23×C107C2×D412yes—
77C22×D107—21no—
78C22×D107—22no—
79D4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—44no—
87C22⋊D107—82no—
88C22⋊D107—82no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93D428—21no—
94D428—22no—
95D428—21no—
96D428—22no—
97D428—42no—
98D428—81no—
99C4×D107—44no—
100C4×D107—42no—
101Dic214—21no—
102Dic214—81no—
103Dic214#4112yes—
104C8×C107—21no—
105C8×C107—21no—
106C8×C107—22no—
107C22×C107—22no—
108C22×C107—22no—
109C22×C107—22no—
110C22×C107C22×D411yescharacteristic
111C2×D107—42no—
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—42no—
115C2×D107—42no—
116C2×D107—41no—
117C4×C107—22no—
118C4×C107—21no—
119C4×C107#13512yes—
120C4×C107C2×#4111yescharacteristic
121C4×C107H(5,𝔽2)11yescharacteristic
122Dic107—22no—
123Dic107—41no—
124Dic107—42no—
125Dic107—41no—
126C2×C107—42no—
127C2×C107#54611yescharacteristic
128C2×C107C2×#13511yescharacteristic
129C2×C107#57611yescharacteristic
130D107—82no—
131D107—81no—
132#3305—1071nomaximal, Sylow
133C107#330511yescharacteristic, complemented, Sylow
134#555—1071no—
135#550—1071no—
136#486—1071no—
137#482—1072no—
138#477—1071no—
139#466—1072no—
140#459—1071no—
141#449—1072no—
142C4×SD16—1071no—
143C8×D4D10711yescharacteristic, complemented
144#416—1072no—
145C2×#41—1072no—
146C2×SD16—1071no—
147C2×D8—1072no—
148C22×C8C2×D10712yes—
149C4⋊Q8—1071no—
150#127—1071no—
151#127—2141no—
152#124—2142no—
153#121—1072no—
154#121—1072no—
155#121—2141no—
156#121—2142no—
157C4×Q8—1071no—
158C4×D4—1071no—
159C4×D4—2141no—
160C4×D4C2×D10711yescharacteristic
161#117—1072no—
162#117—2141no—
163#107—1072no—
164#106—1071no—
165C4⋊C8C2×D10711yescharacteristic
166#103—1071no—
167#103—1072no—
168#102—1072no—
169#102—1072no—
170#102—1071no—
171#102—1072no—
172C22⋊C8C2×D10712yes—
173C4×C8C2×D10711yescharacteristic
174#41—4282no—
175#41—2144no—
176C2×Q8—1071no—
177C2×Q8—2141no—
178C2×D4—1072no—
179C2×D4—1071no—
180C2×D4—2142no—
181C2×D4—2142no—
182C2×D4—2142no—
183C2×D4—2141no—
184C2×D4—2141no—
185C2×D4D42811yescharacteristic
186C22×C4—2141no—
187C22×C4—2142no—
188C22×C4C22×D10712yes—
189SD16—2142no—
190D8—4282no—
191C2×C8—42no—
192C2×C8C22×D10711yescharacteristic
193C2×C8C22×D10711yescharacteristic
194C2×C8C22×D10712yes—
195C4⋊C4—1071no—
196C4⋊C4—1072no—
197C4⋊C4—1071no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4D42811yescharacteristic
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C22⋊C4—2141no—
204C22⋊C4—2142no—
205C22⋊C4D42812yes—
206C42—2142no—
207C42—2141no—
208C42C22×D10711yescharacteristic
209C23—2141no—
210C23—2142no—
211C23C2×D42812yes—
212Q8—8561no—
213Q8—1072no—
214Q8—2141no—
215D4—22no—
216D4—8561no—
217D4—8562no—
218D4—8562no—
219D4—4284no—
220D4—4282no—
221D4—2141no—
222D4—2142no—
223D4—2141no—
224D4—2142no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—4284no—
228C2×C4—4282no—
229C2×C4—2141no—
230C2×C4—2141no—
231C2×C4—2142no—
232C2×C4—2141no—
233C2×C4C2×D42812yes—
234C2×C4C2×D42811yescharacteristic
235C2×C4C2×D42811yescharacteristic
236C2×C4C23×D10711yescharacteristic
237C8—21no—
238C8—21no—
239C8—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—4281no—
244C22—4282no—
245C22—4282no—
246C22—4281no—
247C22—4282no—
248C22—4281no—
249C22C22×D42811yescharacteristic
250C4—21no—
251C4—22no—
252C4—4281no—
253C4—4281no—
254C4—4282no—
255C4—2142no—
256C4#1681213112yes—
257C4#41×D10711yescharacteristic
258C4#1681215411yescharacteristic
259C2—42no—
260C2—8561no—
261C2—8562no—
262C2#2029327211yescharacteristic
263C2C2×#1681213111yescharacteristic
264C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107#3305
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293298C2semidirect
3. #20292974C2semidirect
4. #20292952C2semidirect
5. #20292923C2semidirect
6. #20292863C2semidirect
7. #20292823C2semidirect
8. #20292774C2semidirect
9. C4×#13133807C2semidirect
10. #20292624C2semidirect
11. C8×D4×C107C2semidirect
133. C107#3305semidirect
143. C8×D4D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932982
#202929742
#202929522
#202929232
#202928632
#202928232
#202927742
C4×#131338072
#202926242
C8×D4×C1072
#3305107