Finite Group Databasebeta

← #28348392

Subgroups — #28348392

Order 13696.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293280C211yesmaximal, characteristic, complemented
3#20292977C211yesmaximal, characteristic, complemented
4#20292966C212yesmaximal, complemented
5#20292918C211yesmaximal, characteristic, complemented
6#20292853C212yesmaximal, complemented
7#20292771C212yesmaximal, complemented
8C4×#13133803C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#20292622C212yesmaximal, complemented
11#461×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133803C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812095—22no—
17#16812094—22no—
18#16812085C2212yes—
19#16812084—21no—
20#16812083—21no—
21#16812080C2212yes—
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812068—21no—
25#16812067C2212yes—
26C4⋊C4×D107—21no—
27#16812042C2211yescharacteristic
28#16812041C2212yes—
29#16812033C2212yes—
30#16812033C2212yes—
31#16812031C2212yes—
32#16812030C2211yescharacteristic
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×SD16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133819—24no—
42#13133819—42no—
43C2×Dic214—21no—
44C2×D428—22no—
45C2×D428D411yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—22no—
48C2×C4×D107—21no—
49C2×C4×D107—22no—
50C2×C4×D107C2312yes—
51#13133803—22no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—42no—
55C4⋊Dic107—21no—
56C4⋊Dic107D411yescharacteristic
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—22no—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4D412yes—
65C4×Dic107—22no—
66C2×Q8×C107C2312yes—
67C2×D4×C107C2311yescharacteristic
68C22×C4×C107—21no—
69SD16×C107—42no—
70C2×C8×C107C2312yes—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2311yescharacteristic
74C22⋊C4×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C22×D107—21no—
78C22×D107C2×D412yes—
79Q8×C107—22no—
80Q8×C107—22no—
81D4×C107—21no—
82D4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2411yescharacteristic
89C22⋊D107—82no—
90C2×Dic107—22no—
91C2×Dic107—21no—
92C2×Dic107C2×D412yes—
93D428—22no—
94D428—21no—
95D428—44no—
96D428#4112yes—
97C4×D107—22no—
98C4×D107—22no—
99C4×D107—44no—
100C4×D107—42no—
101Dic214—81no—
102C8×C107—22no—
103C107⋊C8—21no—
104C107⋊C8—21no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107C22×D411yescharacteristic
109C2×D107—22no—
110C2×D107—22no—
111C2×D107—22no—
112C2×D107—22no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—21no—
116C4×C107—42no—
117C4×C107—41no—
118C4×C107#13512yes—
119C4×C107C2×#4111yescharacteristic
120C4×C107H(5,𝔽2)11yescharacteristic
121Dic107—22no—
122Dic107—41no—
123Dic107—42no—
124C2×C107—81no—
125C2×C107#54611yescharacteristic
126C2×C107C2×#13511yescharacteristic
127C2×C107#57711yescharacteristic
128D107—42no—
129D107—42no—
130#3303—1071nomaximal, Sylow
131C107#330311yescharacteristic, complemented, Sylow
132#555—1071no—
133#547—1071no—
134#487—1071no—
135#485—1072no—
136#476—1071no—
137#465—1072no—
138#461D10711yescharacteristic, complemented
139#449—1072no—
140C4×D8—1071no—
141C8×D4—1071no—
142#416—1072no—
143C2×#41—1072no—
144C2×SD16C2×D10712yes—
145C2×D8—1071no—
146C22×C8—1072no—
147C4⋊Q8—1071no—
148C4⋊Q8C2×D10711yescharacteristic
149#124—2142no—
150#123—2142no—
151C22⋊Q8—1072no—
152#121—1072no—
153#121—2141no—
154#121—2141no—
155C4×D4—1071no—
156C4×D4—1071no—
157C4×D4C2×D10711yescharacteristic
158#117—1072no—
159#117—2141no—
160C2×C4⋊C4—2141no—
161#107—1071no—
162#106—1072no—
163C4⋊C8C2×D10711yescharacteristic
164#103—1072no—
165#103—1072no—
166#103C2×D10712yes—
167#102—1071no—
168#102—1072no—
169#102—1071no—
170C22⋊C8—1072no—
171C4×C8—1071no—
172#41—4282no—
173#41—2144no—
174C2×Q8—2141no—
175C2×Q8C22×D10712yes—
176C2×D4—1071no—
177C2×D4—1071no—
178C2×D4—2142no—
179C2×D4—2142no—
180C2×D4C22×D10711yescharacteristic
181C22×C4—21no—
182C22×C4—1072no—
183C22×C4—2142no—
184C22×C4—2141no—
185SD16—42no—
186D8—2142no—
187C2×C8—1071no—
188C2×C8—1071no—
189C2×C8—4282no—
190C2×C8C22×D10712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—1071no—
194C4⋊C4—1072no—
195C4⋊C4—1071no—
196C4⋊C4—2142no—
197C4⋊C4—2141no—
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—1072no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C22⋊C4—2142no—
204C22⋊C4—2141no—
205C42—2142no—
206C42C22×D10711yescharacteristic
207C23—21no—
208C23—1072no—
209C23—2141no—
210Q8—22no—
211Q8—22no—
212Q8—8561no—
213D4—21no—
214D4—21no—
215D4—8562no—
216D4—1072no—
217D4—4284no—
218D4—2142no—
219D4—2141no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—1072no—
224C2×C4—4284no—
225C2×C4—4282no—
226C2×C4—2142no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4D4×D10711yescharacteristic
231C2×C4D4×D10711yescharacteristic
232C2×C4C23×D10711yescharacteristic
233C8—22no—
234C8—2141no—
235C8—2141no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—4281no—
240C22—4281no—
241C22—2142no—
242C22—2142no—
243C22—2142no—
244C22—2142no—
245C22C2×D4×D10711yescharacteristic
246C4—21no—
247C4—41no—
248C4—42no—
249C4—4281no—
250C4—4282no—
251C4—2142no—
252C4#1681213012yes—
253C4#41×D10711yescharacteristic
254C4#1681215411yescharacteristic
255C2—81no—
256C2—4282no—
257C2—4282no—
258C2#2029327411yescharacteristic
259C2C2×#1681213011yescharacteristic
260C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
hypercenter#461D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293280C2semidirect
3. #20292977C2semidirect
4. #20292966C2semidirect
5. #20292918C2semidirect
6. #20292853C2semidirect
7. #20292771C2semidirect
8. C4×#13133803C2semidirect
9. #20292692C2semidirect
10. #20292622C2semidirect
11. #461×C107C2semidirect
131. C107#3303semidirect
138. #461D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932802
#202929772
#202929662
#202929182
#202928532
#202927712
C4×#131338032
#202926922
#202926222
#461×C1072
#3303107