Finite Group Databasebeta

← #28348328

Subgroups — #28348328

Order 13696.

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

normal subgroups only

This group has 420 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×D428C211yesmaximal, characteristic, complemented
2#20293282C211yesmaximal, characteristic
3#20292978C211yesmaximal, characteristic, complemented
4#20292959C212yesmaximal, complemented
5#20292912C211yesmaximal, characteristic, complemented
6#20292855C212yesmaximal, complemented
7#20292832C211yesmaximal, characteristic, complemented
8#20292764C212yesmaximal, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#103×D107C212yesmaximal, complemented
11C4×SD16×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×Q8×D107C2212yes—
14C2×#13133806C2212yes—
15C2×C8×D107C2212yes—
16#16812109—21no—
17#16812109C2211yescharacteristic
18#16812109C2211yescharacteristic
19#16812091—22no—
20#16812089—22no—
21#16812089C2212yes—
22#16812088—21no—
23#16812080C2212yes—
24C4×D428—21no—
25C4×D428C2211yescharacteristic
26C4⋊C4×D107C2212yes—
27#16812040C2212yes—
28C4⋊(C107⋊C8)C2211yescharacteristic
29#16812034C2212yes—
30#16812034C2212yes—
31#16812033C2212yes—
32#16812032C2212yes—
33(C2×D107)⋊C8C2212yes—
34C2×SD16×C107C2211yescharacteristic
35C4×Q8×C107C2211yescharacteristic
36C4×D4×C107C2211yescharacteristic
37#106×C107C2211yescharacteristic
38#103×C107C2211yescharacteristic
39#102×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic
41#13133818—44no—
42C2×Dic214—21no—
43C2×Dic214—22no—
44C2×Dic214C2312yes—
45Q8×D107—24no—
46Q8×D107—42no—
47C2×D428D411yescharacteristic
48C2×C22⋊D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107C2312yes—
52#13133806—42no—
53C2×C107⋊C8C2312yes—
54C8×D107—42no—
55C4⋊Dic107—21no—
56C4⋊Dic107—21no—
57C4⋊Dic107—21no—
58C4⋊Dic107—22no—
59C4⋊Dic107D411yescharacteristic
60C4⋊Dic107C2312yes—
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4D412yes—
65C2×Q8×C107C2311yescharacteristic
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68SD16×C107—22no—
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2311yescharacteristic
73C4⋊C4×C107C2311yescharacteristic
74C22⋊C4×C107—21no—
75C42×C107—21no—
76C42×C107C2311yescharacteristic
77C23×C107—21no—
78C22×D107C2×D412yes—
79Q8×C107—21no—
80Q8×C107#4112yescomplemented
81D4×C107—21no—
82D4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—42no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107C2×D412yes—
94D428SD1614yescomplemented
95C4×D107—22no—
96C4×D107—22no—
97C4×D107—44no—
98C4×D107—44no—
99Dic214—22no—
100Dic214—22no—
101Dic214—44no—
102Dic214—42no—
103C8×C107—21no—
104C8×C107—21no—
105C107⋊C8—22no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107—41no—
109C22×C107C22×D411yescharacteristic
110C2×D107—22no—
111C2×D107—22no—
112C2×D107C2×SD1614yes—
113C4×C107—22no—
114C4×C107—21no—
115C4×C107—41no—
116C4×C107—41no—
117C4×C107C2×SD1612yes—
118C4×C107C2×#4111yescharacteristic
119C4×C107H(5,𝔽2)11yescharacteristic
120Dic107—22no—
121Dic107—42no—
122Dic107—42no—
123C2×C107—81no—
124C2×C107#54611yescharacteristic
125C2×C107C22×SD1611yescharacteristic
126C2×C107#57811yescharacteristic
127D107—24no—
128#3298—1071nomaximal, Sylow
129C107#329811yescharacteristic, complemented, Sylow
130D4×Q8—1071no—
131#547—1071no—
132#487—1071no—
133#483—1072no—
134#475—1071no—
135#465—1072no—
136#461—1071no—
137#448—1072no—
138C4×SD16D10711yescharacteristic, complemented
139C8×D4—1071no—
140C2×#103—1072no—
141C2×#41—2141no—
142C22×Q8—1072no—
143C2×SD16—1072no—
144C2×SD16C2×D10711yescharacteristic
145C22×C8—1072no—
146C4⋊Q8—1071no—
147C4⋊Q8—1071no—
148C4⋊Q8—2141no—
149#123—2142no—
150C22⋊Q8—1072no—
151C22⋊Q8—2142no—
152C22⋊Q8—2141no—
153#121—1072no—
154C4×Q8C2×D10711yescharacteristic
155C4×D4—1071no—
156C4×D4—2141no—
157C4×D4C2×D10711yescharacteristic
158C2×C4⋊C4—1072no—
159#106—1072no—
160#106C2×D10711yescharacteristic
161C4⋊C8—1071no—
162#103—1072no—
163#103—1072no—
164#103—1072no—
165#103—1072no—
166#103C2×D10711yescharacteristic
167#102C2×D10711yescharacteristic
168C22⋊C8—1072no—
169C4×C8C2×D10711yescharacteristic
170#41—4284no—
171C2×Q8—1072no—
172C2×Q8—4282no—
173C2×Q8—2142no—
174C2×Q8—2144no—
175C2×Q8—2141no—
176C2×Q8C22×D10711yescharacteristic
177C2×D4—1071no—
178C2×D4—2142no—
179C2×D4C22×D10711yescharacteristic
180C22×C4—21no—
181C22×C4—1072no—
182C22×C4—2142no—
183C22×C4—2142no—
184SD16—22no—
185SD16—4282no—
186C2×C8—1072no—
187C2×C8—4282no—
188C2×C8C22×D10711yescharacteristic
189C2×C8C22×D10711yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—1071no—
192C4⋊C4—1072no—
193C4⋊C4—2142no—
194C4⋊C4—2142no—
195C4⋊C4—2141no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4C22×D10711yescharacteristic
199C4⋊C4C22×D10711yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—1072no—
202C22⋊C4—2142no—
203C22⋊C4—2142no—
204C42—21no—
205C42C22×D10711yescharacteristic
206C23—21no—
207C23—1072no—
208Q8—21no—
209Q8—4284no—
210Q8—4282no—
211Q8—2142no—
212Q8—2142no—
213Q8#1313381812yescomplemented
214D4—21no—
215D4—21no—
216D4—1074no—
217D4—4284no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—1072no—
223C2×C4—4284no—
224C2×C4—4284no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4D4×D10711yescharacteristic
230C2×C4D4×D10711yescharacteristic
231C2×C4C23×D10711yescharacteristic
232C8—21no—
233C8—21no—
234C8—2142no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1074no—
239C22—2142no—
240C22—2142no—
241C22C2×D4×D10711yescharacteristic
242C4—21no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—4282no—
247C4—4282no—
248C4—2142no—
249C4SD16×D10712yes—
250C4C2×#1313381811yescharacteristic
251C4#1681215511yescharacteristic
252C2—81no—
253C2—2144no—
254C2#2029327311yescharacteristic
255C2C2×SD16×D10711yescharacteristic
256C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293282C2
hypercenterC4×SD16D107
hypocenterC107#3298
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×D428C2semidirect
3. #20292978C2semidirect
4. #20292959C2semidirect
5. #20292912C2semidirect
6. #20292855C2semidirect
7. #20292832C2semidirect
8. #20292764C2semidirect
9. C8×D428C2semidirect
10. #103×D107C2semidirect
11. C4×SD16×C107C2semidirect
80. Q8×C107#41semidirect
94. D428SD16semidirect
129. C107#3298semidirect
138. C4×SD16D107semidirect
213. Q8#13133818semidirect

Maximal subgroups

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

subgroupindex
Q8×D4282
#202932822
#202929782
#202929592
#202929122
#202928552
#202928322
#202927642
C8×D4282
#103×D1072
C4×SD16×C1072
#3298107