Finite Group Databasebeta

← #28349518

Subgroups — #28349518

Order 13696.

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

normal subgroups only

This group has 362 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#20293362C211yesmaximal, characteristic
2#20293352C211yesmaximal, characteristic, complemented
3#20293020C212yesmaximal, complemented
4#20293001C211yesmaximal, characteristic
5#20292851C212yesmaximal, complemented
6#20292831C212yesmaximal, complemented
7#20292828C211yesmaximal, characteristic
8#20292754C211yesmaximal, characteristic, complemented
9#20292730C212yesmaximal, complemented
10C4×#13133808C211yesmaximal, characteristic, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2211yescharacteristic
13C2×#13133807C2212yes—
14C2×#13133805C2212yes—
15C2×#13133805C2212yes—
16#16812107C2212yes—
17#16812106C2211yescharacteristic
18#16812100—21no—
19#16812099—22no—
20#16812098C2212yes—
21#16812097—21no—
22#16812094—22no—
23Q8×Dic107C2212yes—
24C4×Dic214C2211yescharacteristic
25C4×Dic214C2211yescharacteristic
26#16812074—21no—
27C4×D428—21no—
28#16812071C2212yes—
29#16812042C2211yescharacteristic
30Dic107⋊C8C2212yes—
31#16812033C2211yescharacteristic
32#16812032C2211yescharacteristic
33#16812031C2212yes—
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×Q16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214D411yescharacteristic
42C2×Dic214C2311yescharacteristic
43C2×D428—21no—
44C2×D428—21no—
45C2×D428C2312yes—
46C2×C4×D107—22no—
47#13133808—22no—
48#13133807—42no—
49#13133805—24no—
50#13133805—42no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C4⋊Dic107—22no—
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—21no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59Dic107⋊C4D412yes—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64C4×Dic107—22no—
65C4×Dic107C2312yes—
66C2×Q8×C107C2311yescharacteristic
67C2×Q8×C107C2312yes—
68Q16×C107—42no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107—21no—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2311yescharacteristic
74C42×C107—21no—
75C42×C107C2311yescharacteristic
76C22×D107—22no—
77Q8×C107—22no—
78Q8×C107—21no—
79Q8×C107—21no—
80Q8×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91D428—22no—
92D428—22no—
93D428—42no—
94D428—81no—
95C4×D107—82no—
96Dic214—22no—
97Dic214—21no—
98Dic214—21no—
99C8×C107—22no—
100C107⋊C8—21no—
101C107⋊C8#4112yes—
102C22×C107C22×D411yescharacteristic
103C2×D107—42no—
104C2×D107—42no—
105C2×D107—42no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—42no—
109C4×C107—41no—
110C4×C107#13712yes—
111C4×C107C2×#4111yescharacteristic
112C4×C107H(5,𝔽2)11yescharacteristic
113Dic107—22no—
114Dic107—22no—
115Dic107—41no—
116Dic107—41no—
117C2×C107#55011yescharacteristic
118C2×C107C2×#13711yescharacteristic
119C2×C107#57711yescharacteristic
120D107—82no—
121#3400—1071nomaximal, Sylow
122C107#340011yescharacteristic, complemented, Sylow
123#559—1071no—
124#557—1071no—
125#497—1072no—
126#492—1071no—
127#464—1072no—
128#462D10711yescharacteristic, complemented
129#461—1072no—
130#460—1071no—
131#446—1071no—
132#440—1072no—
133C4×Q16—1071no—
134C2×Q16—1071no—
135C2×Q16C2×D10712yes—
136C2×SD16—1072no—
137C2×SD16—1072no—
138C2×SD16—1072no—
139C4⋊Q8—1072no—
140C4⋊Q8C2×D10711yescharacteristic
141#127—1071no—
142#125—2142no—
143#125—2141no—
144#124—1072no—
145#124—2141no—
146#123—2142no—
147C4×Q8—1071no—
148C4×Q8—1072no—
149C4×Q8—1071no—
150C4×Q8—2141no—
151C4×Q8C2×D10711yescharacteristic
152C4×D4—1072no—
153C4×D4—2141no—
154#107—1071no—
155C4⋊C8—1072no—
156C4⋊C8C2×D10711yescharacteristic
157#103—1071no—
158#103—1071no—
159#103C2×D10712yes—
160#102—1072no—
161#97—1072no—
162C4×C8—1071no—
163C2×Q8—1071no—
164C2×Q8—1071no—
165C2×Q8C22×D10711yescharacteristic
166C2×Q8C22×D10712yes—
167C2×D4—1072no—
168C2×D4—2141no—
169C2×D4—2141no—
170C22×C4—2142no—
171Q16—42no—
172Q16—2142no—
173SD16—4282no—
174SD16—4282no—
175SD16—2144no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8C22×D10712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1071no—
183C4⋊C4—1072no—
184C4⋊C4—1071no—
185C4⋊C4—2142no—
186C4⋊C4—2142no—
187C4⋊C4—2142no—
188C4⋊C4—2141no—
189C4⋊C4C22×D10711yescharacteristic
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C42—21no—
194C42—1072no—
195C42—2142no—
196C42—2141no—
197C42C22×D10711yescharacteristic
198C23—2142no—
199Q8—22no—
200Q8—21no—
201Q8—22no—
202Q8—21no—
203Q8—2141no—
204Q8—2141no—
205Q8—2142no—
206D4—8561no—
207D4—4282no—
208D4—2142no—
209D4—2142no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—8562no—
214C2×C4—1072no—
215C2×C4—1072no—
216C2×C4—2141no—
217C2×C4—2141no—
218C2×C4D4×D10711yescharacteristic
219C2×C4D4×D10711yescharacteristic
220C2×C4C23×D10711yescharacteristic
221C8—22no—
222C8—1072no—
223C8—2141no—
224C22—4282no—
225C22—4282no—
226C22—4282no—
227C22C2×D4×D10711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—42no—
232C4—4281no—
233C4—4281no—
234C4—2142no—
235C4—2142no—
236C4#1681214112yes—
237C4#41×D10711yescharacteristic
238C4#1681215411yescharacteristic
239C2—8562no—
240C2#2029329511yescharacteristic
241C2C2×#1681214111yescharacteristic
242C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radical#16812106C22
hypercenter#462D107
hypocenterC107#3400
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #20293352C2semidirect
3. #20293020C2semidirect
5. #20292851C2semidirect
6. #20292831C2semidirect
8. #20292754C2semidirect
9. #20292730C2semidirect
10. C4×#13133808C2semidirect
11. #462×C107C2semidirect
122. C107#3400semidirect
128. #462D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933522
#202930202
#202930012
#202928512
#202928312
#202928282
#202927542
#202927302
C4×#131338082
#462×C1072
#3400107