Finite Group Databasebeta

← #28348375

Subgroups — #28348375

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#20293351C211yesmaximal, characteristic, complemented
2#20293281C211yesmaximal, characteristic
3#20292979C211yesmaximal, characteristic, complemented
4#20292970C212yesmaximal, complemented
5#20292869C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292790C212yesmaximal, complemented
8C4×#13133806C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11#474×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×Dic428C2212yes—
14C2×#13133806C2211yescharacteristic
15C2×C8×D107C2212yes—
16#16812109C2211yescharacteristic
17#16812104—22no—
18#16812098—21no—
19#16812093—22no—
20#16812089C2212yes—
21#16812089C2212yes—
22#16812083—21no—
23Q8×Dic107—21no—
24C4×Dic214C2211yescharacteristic
25C4×D428C2211yescharacteristic
26#16812067C2212yes—
27C2×C4⋊Dic107—21no—
28#16812040C2211yescharacteristic
29#16812034C2212yes—
30#16812032C2211yescharacteristic
31#16812032C2212yes—
32#16812032C2212yes—
33#16812030C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#107×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133817—24no—
42#13133817—42no—
43C2×Dic214C2312yes—
44C2×Dic214C2311yescharacteristic
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C22×Dic107—22no—
48C2×C4×D107C2312yes—
49Dic428—42no—
50#13133806—22no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—42no—
54C4⋊Dic107—21no—
55C4⋊Dic107—21no—
56C4⋊Dic107—22no—
57C4⋊Dic107—22no—
58C4⋊Dic107D411yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—21no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4D412yes—
65C4×Dic107—21no—
66C4×Dic107—22no—
67C2×Q8×C107—21no—
68C2×D4×C107C2311yescharacteristic
69C22×C4×C107—21no—
70C2×C8×C107C2312yes—
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2312yes—
73C4⋊C4×C107C2311yescharacteristic
74C22⋊C4×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C22×D107C2×D412yes—
78Q8×C107—42no—
79D4×C107—21no—
80D4×C107#4112yes—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—44no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—44no—
92C2×Dic107C2×D412yes—
93D428—22no—
94C4×D107—22no—
95C4×D107—22no—
96Dic214—21no—
97Dic214—22no—
98Dic214—21no—
99Dic214—22no—
100C8×C107—22no—
101C107⋊C8—21no—
102C107⋊C8—21no—
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—42no—
113C4×C107#13512yes—
114C4×C107C2×#4111yescharacteristic
115C4×C107H(5,𝔽2)11yescharacteristic
116Dic107—22no—
117Dic107—41no—
118Dic107—42no—
119Dic107—41no—
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#474D10711yescharacteristic, complemented
132#467—1072no—
133#462—1071no—
134#452—1072no—
135C4×SD16—1071no—
136C8×D4—1071no—
137#416—1072no—
138C2×#41—1072no—
139C2×Q16—1072no—
140C2×SD16—1071no—
141C22×C8—1072no—
142C4⋊Q8—1071no—
143C4⋊Q8C2×D10711yescharacteristic
144#126—2142no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1072no—
148C22⋊Q8—1072no—
149#121—2141no—
150C4×Q8—1071no—
151C4×Q8—2141no—
152C4×D4—1071no—
153C4×D4C2×D10711yescharacteristic
154#117—1072no—
155C2×C4⋊C4—2141no—
156#107C2×D10712yes—
157#106—1071no—
158C4⋊C8C2×D10711yescharacteristic
159#103—1072no—
160#103—1072no—
161#103—1071no—
162#103—1072no—
163#102—1071no—
164#102C2×D10712yes—
165C22⋊C8—1072no—
166C4×C8—1071no—
167#41—4282no—
168#41—2144no—
169C2×Q8—21no—
170C2×Q8—1072no—
171C2×Q8—1071no—
172C2×D4—1071no—
173C2×D4—2142no—
174C2×D4C22×D10711yescharacteristic
175C22×C4—21no—
176C22×C4—1072no—
177C22×C4—2142no—
178Q16—4282no—
179SD16—2142no—
180C2×C8—1071no—
181C2×C8—1071no—
182C2×C8—4282no—
183C2×C8C22×D10712yes—
184C4⋊C4—21no—
185C4⋊C4—1071no—
186C4⋊C4—1071no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4C22×D10712yes—
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—1072no—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C42—2142no—
200C42—2141no—
201C42C22×D10711yescharacteristic
202C23—21no—
203C23—1072no—
204Q8—42no—
205Q8—2141no—
206Q8—2142no—
207Q8—2141no—
208Q8—2142no—
209D4—21no—
210D4—4284no—
211D4—2142no—
212D4#1313381912yes—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—1072no—
217C2×C4—4284no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—22no—
227C8—2141no—
228C8—2141no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—41no—
238C4—42no—
239C4—4281no—
240C4—4282no—
241C4—4281no—
242C4—2142no—
243C4#1681212812yes—
244C4C2×#1313381911yescharacteristic
245C4#1681215511yescharacteristic
246C2—42no—
247C2—4282no—
248C2#2029326811yescharacteristic
249C2C2×#1681212811yescharacteristic
250C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
prime radical#20293281C2
hypercenter#474D107
hypocenterC107#3302
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20293351C2semidirect
3. #20292979C2semidirect
4. #20292970C2semidirect
5. #20292869C2semidirect
6. #20292838C2semidirect
7. #20292790C2semidirect
8. C4×#13133806C2semidirect
9. #20292692C2semidirect
10. #20292620C2semidirect
11. #474×C107C2semidirect
126. C107#3302semidirect
131. #474D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202932812
#202929792
#202929702
#202928692
#202928382
#202927902
C4×#131338062
#202926922
#202926202
#474×C1072
#3302107