Finite Group Databasebeta

← #28348380

Subgroups — #28348380

Order 13696.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293350C211yesmaximal, characteristic, complemented
2#20293281C211yesmaximal, characteristic
3#20292978C211yesmaximal, characteristic, complemented
4#20292971C212yesmaximal, complemented
5#20292907C211yesmaximal, characteristic, complemented
6#20292867C212yesmaximal, complemented
7#20292837C211yesmaximal, characteristic, complemented
8#20292786C212yesmaximal, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11C4×SD16×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×#13133808C2212yes—
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812109C2211yescharacteristic
17#16812104—22no—
18#16812097—21no—
19#16812093—22no—
20#16812089C2212yes—
21#16812085C2212yes—
22#16812083—21no—
23C4×Dic214—21no—
24C4×D428C2211yescharacteristic
25#16812067C2212yes—
26C2×C4⋊Dic107—21no—
27#16812042C2212yes—
28C4⋊(C107⋊C8)C2211yescharacteristic
29#16812034C2212yes—
30#16812034C2212yes—
31#16812032C2212yes—
32#16812030C2212yes—
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#13133817—24no—
42#13133817—42no—
43C2×Dic214—21no—
44C2×Dic214C2312yes—
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C22×Dic107—22no—
48C2×C4×D107C2312yes—
49#13133808—42no—
50C2×C107⋊C8C2312yes—
51C8×D107—42no—
52C4⋊Dic107—21no—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107—22no—
56C4⋊Dic107D411yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4D412yes—
63C4×Dic107—22no—
64C2×Q8×C107C2311yescharacteristic
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67SD16×C107—22no—
68C2×C8×C107C2311yescharacteristic
69C2×C8×C107C2311yescharacteristic
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C42×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C22×D107C2×D412yes—
78Q8×C107—21no—
79Q8×C107—21no—
80D4×C107—21no—
81D4×C107#4112yes—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—42no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2411yescharacteristic
89C22⋊D107—44no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—44no—
93C2×Dic107C2×D412yes—
94D428—22no—
95C4×D107—22no—
96C4×D107—22no—
97Dic214—22no—
98Dic214—22no—
99Dic214—42no—
100C8×C107—21no—
101C8×C107—21no—
102C107⋊C8—22no—
103C22×C107—22no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107C22×D411yescharacteristic
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—22no—
110C4×C107—21no—
111C4×C107—41no—
112C4×C107—41no—
113C4×C107—41no—
114C4×C107#13512yes—
115C4×C107C2×#4111yescharacteristic
116C4×C107H(5,𝔽2)11yescharacteristic
117Dic107—22no—
118Dic107—42no—
119Dic107—42no—
120C2×C107—42no—
121C2×C107#54611yescharacteristic
122C2×C107C2×#13511yescharacteristic
123C2×C107#57811yescharacteristic
124D107—42no—
125#3302—1071nomaximal, Sylow
126C107#330211yescharacteristic, complemented, Sylow
127#556—1071no—
128#547—1071no—
129#487—1071no—
130#485—1072no—
131#474—1071no—
132#467—1072no—
133#462—1071no—
134#452—1072no—
135C4×SD16D10711yescharacteristic, complemented
136C8×D4—1071no—
137#416—1072no—
138C2×#41—1072no—
139C2×Q16—1072no—
140C2×SD16C2×D10711yescharacteristic
141C22×C8—1072no—
142C4⋊Q8—1071no—
143C4⋊Q8—1071no—
144#126—2142no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1072no—
148C22⋊Q8—1072no—
149#121—2141no—
150C4×Q8—2141no—
151C4×Q8C2×D10711yescharacteristic
152C4×D4—1071no—
153C4×D4C2×D10711yescharacteristic
154#117—1072no—
155C2×C4⋊C4—2141no—
156#107—1072no—
157#106C2×D10711yescharacteristic
158C4⋊C8—1071no—
159#103—1072no—
160#103—1072no—
161#103—1072no—
162#103C2×D10711yescharacteristic
163#102—1072no—
164#102C2×D10711yescharacteristic
165C22⋊C8—1072no—
166C4×C8C2×D10711yescharacteristic
167#41—4282no—
168#41—2144no—
169C2×Q8—1072no—
170C2×Q8—2141no—
171C2×Q8C22×D10711yescharacteristic
172C2×D4—1071no—
173C2×D4—2142no—
174C2×D4C22×D10711yescharacteristic
175C22×C4—21no—
176C22×C4—1072no—
177C22×C4—2142no—
178Q16—4282no—
179SD16—22no—
180C2×C8—1072no—
181C2×C8—4282no—
182C2×C8C22×D10711yescharacteristic
183C2×C8C22×D10711yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—1071no—
186C4⋊C4—1072no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4C22×D10711yescharacteristic
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—1072no—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C42—21no—
200C42—2142no—
201C42C22×D10711yescharacteristic
202C23—21no—
203C23—1072no—
204Q8—21no—
205Q8—21no—
206Q8—4282no—
207Q8—2142no—
208Q8—2142no—
209D4—21no—
210D4—4284no—
211D4—2142no—
212D4#1313381812yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—1072no—
218C2×C4—4284no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—21no—
227C8—21no—
228C8—2142no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—4282no—
241C4—4282no—
242C4—2142no—
243C4#1681213012yes—
244C4C2×#1313381811yescharacteristic
245C4#1681215511yescharacteristic
246C2—42no—
247C2—4282no—
248C2#2029327311yescharacteristic
249C2C2×#1681213011yescharacteristic
250C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293281C2
hypercenterC4×SD16D107
hypocenterC107#3302
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. #20293350C2semidirect
3. #20292978C2semidirect
4. #20292971C2semidirect
5. #20292907C2semidirect
6. #20292867C2semidirect
7. #20292837C2semidirect
8. #20292786C2semidirect
9. C8×D428C2semidirect
10. #20292620C2semidirect
11. C4×SD16×C107C2semidirect
126. C107#3302semidirect
135. C4×SD16D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202932812
#202929782
#202929712
#202929072
#202928672
#202928372
#202927862
C8×D4282
#202926202
C4×SD16×C1072
#3302107