Finite Group Databasebeta

← #28349127

Subgroups — #28349127

Order 13696.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic214C211yesmaximal, characteristic, complemented
2#20293297C211yesmaximal, characteristic
3#20292973C211yesmaximal, characteristic, complemented
4#20292972C212yesmaximal, complemented
5#20292918C211yesmaximal, characteristic, complemented
6#20292915C212yesmaximal, complemented
7#20292825C212yesmaximal, complemented
8#20292751C211yesmaximal, characteristic, complemented
9SD16×Dic107C212yesmaximal, complemented
10C4×#13133805C211yesmaximal, characteristic, complemented
11#460×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C2×#13133805C2211yescharacteristic
14#16812109—21no—
15#16812109C2211yescharacteristic
16#16812108—22no—
17#16812108C2212yes—
18#16812105C2212yes—
19#16812081—22no—
20#16812080—21no—
21Q8×Dic107C2212yes—
22C4×Dic214—21no—
23C4×Dic214C2211yescharacteristic
24C4×D428C2211yescharacteristic
25D4×Dic107C2212yes—
26#16812041C2212yes—
27#16812040C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812033C2211yescharacteristic
30#16812031C2212yes—
31#16812030C2212yes—
32#16812030C2212yes—
33#16812029C2211yescharacteristic
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×SD16×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133817—44no—
42C2×Dic214—21no—
43C2×Dic214—22no—
44C2×Dic214D411yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C22×Dic107—22no—
49C2×C4×D107—21no—
50#13133805—22no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C4⋊Dic107—21no—
54C4⋊Dic107—22no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—22no—
59Dic107⋊C4D412yes—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—21no—
62C4×Dic107—22no—
63C4×Dic107C2312yes—
64C2×Q8×C107C2311yescharacteristic
65C2×D4×C107—21no—
66C2×D4×C107—21no—
67C2×D4×C107C2312yes—
68SD16×C107—24no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2311yescharacteristic
72C42×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—22no—
75C22×D107—21no—
76Q8×C107—21no—
77Q8×C107#4112yescomplemented
78D4×C107—22no—
79D4×C107—22no—
80D4×C107—42no—
81D4×C107—81no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—44no—
88C22⋊D107—82no—
89C2×Dic107—22no—
90C2×Dic107—21no—
91C2×Dic107—44no—
92C2×Dic107C2×D412yes—
93C2×Dic107C2×D412yes—
94D428—21no—
95D428—21no—
96C4×D107—42no—
97Dic214—82no—
98Dic214—81no—
99Dic214SD1614yescomplemented
100C8×C107—22no—
101C107⋊C8—21no—
102C107⋊C8—21no—
103C22×C107—42no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107C22×D411yescharacteristic
107C2×D107—41no—
108C2×D107—41no—
109C2×D107—41no—
110C4×C107—22no—
111C4×C107—21no—
112C4×C107—41no—
113C4×C107C2×SD1612yes—
114C4×C107C2×#4111yescharacteristic
115C4×C1072−1+411yescharacteristic
116Dic107—22no—
117Dic107—41no—
118Dic107—42no—
119Dic107C2×SD1614yes—
120C2×C107—82no—
121C2×C107#54811yescharacteristic
122C2×C107C22×SD1611yescharacteristic
123C2×C107#57611yescharacteristic
124D107—81no—
125#3362—1071nomaximal, Sylow
126C107#336211yescharacteristic, complemented, Sylow
127Q82—1071no—
128#550—1071no—
129#486—1071no—
130#486—1072no—
131#476—1071no—
132#476—1072no—
133#460—1072no—
134#460D10711yescharacteristic, complemented
135C8×Q8—1071no—
136C4×SD16—1072no—
137C4×SD16—1071no—
138C2×#41—2141no—
139C2×SD16—1071no—
140C2×SD16C2×D10712yes—
141C4⋊Q8—1072no—
142C4⋊Q8—1071no—
143C4⋊Q8—2141no—
144C4⋊Q8—2142no—
145#127—1072no—
146#127C2×D10711yescharacteristic
147#121—2141no—
148#121—2142no—
149C4×Q8—1071no—
150C4×Q8—1072no—
151C4×Q8—2141no—
152C4×Q8C2×D10711yescharacteristic
153C4×D4—1072no—
154C4×D4—1071no—
155#106—1072no—
156#106—1071no—
157C4⋊C8—1072no—
158C4⋊C8C2×D10711yescharacteristic
159#103—1071no—
160#102—1072no—
161#102—1072no—
162#102—1072no—
163#102—1071no—
164#102C2×D10712yes—
165C4×C8—1072no—
166C4×C8—1071no—
167#41—4284no—
168C2×Q8—1071no—
169C2×Q8—2142no—
170C2×Q8—2141no—
171C2×Q8C22×D10711yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1071no—
175C2×D4—2142no—
176C2×D4—2142no—
177C2×D4C22×D10712yes—
178C22×C4—2142no—
179C22×C4—2141no—
180SD16—24no—
181SD16—2142no—
182C2×C8—1071no—
183C2×C8—1071no—
184C2×C8C22×D10712yes—
185C4⋊C4—21no—
186C4⋊C4—1072no—
187C4⋊C4—1071no—
188C4⋊C4—1072no—
189C4⋊C4—1071no—
190C4⋊C4—2142no—
191C4⋊C4—2142no—
192C4⋊C4—2141no—
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—2142no—
195C22⋊C4—2141no—
196C42—21no—
197C42—1072no—
198C42—2142no—
199C42C22×D10711yescharacteristic
200C23—22no—
201C23—2141no—
202Q8—21no—
203Q8—8561no—
204Q8—8562no—
205Q8—1074no—
206Q8#1313381712yescomplemented
207D4—22no—
208D4—22no—
209D4—42no—
210D4—81no—
211D4—8562no—
212D4—4284no—
213D4—2141no—
214D4—2141no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—1072no—
218C2×C4—1072no—
219C2×C4—4284no—
220C2×C4—4282no—
221C2×C4—2142no—
222C2×C4—2141no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—22no—
227C8—2141no—
228C8—2141no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—4281no—
233C22—4281no—
234C22—4281no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—22no—
238C4—41no—
239C4—1074no—
240C4—4281no—
241C4—4282no—
242C4—2142no—
243C4SD16×D10712yes—
244C4C2×#1313381711yescharacteristic
245C4#1681215611yescharacteristic
246C2—82no—
247C2—8561no—
248C2#2029328311yescharacteristic
249C2C2×SD16×D10711yescharacteristic
250C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293297C2
hypercenter#460D107
hypocenterC107#3362
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×Dic214C2semidirect
3. #20292973C2semidirect
4. #20292972C2semidirect
5. #20292918C2semidirect
6. #20292915C2semidirect
7. #20292825C2semidirect
8. #20292751C2semidirect
9. SD16×Dic107C2semidirect
10. C4×#13133805C2semidirect
11. #460×C107C2semidirect
77. Q8×C107#41semidirect
99. Dic214SD16semidirect
126. C107#3362semidirect
134. #460D107semidirect
206. Q8#13133817semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2142
#202932972
#202929732
#202929722
#202929182
#202929152
#202928252
#202927512
SD16×Dic1072
C4×#131338052
#460×C1072
#3362107