Finite Group Databasebeta

← #28349156

Subgroups — #28349156

Order 13696.

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

normal subgroups only

This group has 348 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#20292983C212yesmaximal, complemented
4#20292975C211yesmaximal, characteristic
5#20292926C212yesmaximal, complemented
6#20292852C212yesmaximal, complemented
7#20292828C211yesmaximal, characteristic
8#20292751C211yesmaximal, characteristic, complemented
9C4×#13133808C211yesmaximal, characteristic, complemented
10#20292714C212yesmaximal, complemented
11#475×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2211yescharacteristic
13C2×#13133807C2212yes—
14#16812107—21no—
15#16812106C2211yescharacteristic
16#16812100C2212yes—
17#16812099—22no—
18#16812098C2212yes—
19#16812097—21no—
20#16812094—22no—
21Q8×Dic107—21no—
22C4×Dic214C2211yescharacteristic
23C4×Dic214C2211yescharacteristic
24#16812074C2212yes—
25C4×D428—21no—
26#16812071C2212yes—
27#16812042C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812033C2211yescharacteristic
30#16812032C2211yescharacteristic
31#16812031C2212yes—
32#16812029C2212yes—
33#16812029C2212yes—
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#106×C107C2212yes—
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—24no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4D412yes—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61C4×Dic107—21no—
62C4×Dic107—22no—
63C4×Dic107C2312yes—
64C2×Q8×C107—21no—
65C2×Q8×C107C2311yescharacteristic
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2312yes—
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C22×D107—22no—
74Q8×C107—21no—
75Q8×C107—21no—
76Q8×C107—42no—
77C2×C4×C107—21no—
78C2×C4×C107—21no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107C2×D412yes—
86C2×Dic107C2×D412yes—
87D428—22no—
88D428—22no—
89D428—42no—
90D428—81no—
91C4×D107—82no—
92Dic214—22no—
93Dic214—21no—
94Dic214#4112yes—
95C8×C107—22no—
96C107⋊C8—21no—
97C107⋊C8—21no—
98C22×C107C22×D411yescharacteristic
99C2×D107—42no—
100C2×D107—42no—
101C2×D107—42no—
102C4×C107—21no—
103C4×C107—41no—
104C4×C107—42no—
105C4×C107—41no—
106C4×C107#13512yes—
107C4×C107C2×#4111yescharacteristic
108C4×C1072−1+411yescharacteristic
109Dic107—22no—
110Dic107—22no—
111Dic107—22no—
112Dic107—41no—
113C2×C107#54811yescharacteristic
114C2×C107C2×#13511yescharacteristic
115C2×C107#57711yescharacteristic
116D107—82no—
117#3365—1071nomaximal, Sylow
118C107#336511yescharacteristic, complemented, Sylow
119#559—1071no—
120#557—1071no—
121#488—1072no—
122#486—1071no—
123#478—1072no—
124#475D10711yescharacteristic, complemented
125#464—1072no—
126#460—1071no—
127C8×Q8—1071no—
128C4×Q16—1071no—
129C4×SD16—1072no—
130C2×Q16—1071no—
131C2×SD16—1072no—
132C4⋊Q8—2141no—
133C4⋊Q8C2×D10711yescharacteristic
134#127—1071no—
135#125—1072no—
136#125—2142no—
137#124—1072no—
138#124—2141no—
139#123—2142no—
140C4×Q8—1071no—
141C4×Q8—1072no—
142C4×Q8—1071no—
143C4×Q8—2141no—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1072no—
146C4×D4—2141no—
147#107—1071no—
148#106C2×D10712yes—
149C4⋊C8—1072no—
150C4⋊C8C2×D10711yescharacteristic
151#103—1071no—
152#103—1071no—
153#103C2×D10712yes—
154#102—1072no—
155#102—1072no—
156#102—1072no—
157C4×C8—1072no—
158C4×C8—1071no—
159C2×Q8—21no—
160C2×Q8—1071no—
161C2×Q8—1071no—
162C2×Q8C22×D10711yescharacteristic
163C2×D4—1072no—
164C2×D4—2141no—
165C2×D4—2141no—
166C22×C4—2142no—
167Q16—2142no—
168SD16—2144no—
169C2×C8—1071no—
170C2×C8—1071no—
171C2×C8C22×D10712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1071no—
175C4⋊C4—1072no—
176C4⋊C4—1071no—
177C4⋊C4—2142no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2141no—
181C4⋊C4C22×D10712yes—
182C4⋊C4C22×D10711yescharacteristic
183C22⋊C4—2142no—
184C22⋊C4—2142no—
185C22⋊C4—2142no—
186C42—21no—
187C42—1072no—
188C42—2142no—
189C42—2141no—
190C42C22×D10711yescharacteristic
191C23—2142no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—1072no—
196Q8—2142no—
197Q8—2141no—
198D4—8561no—
199D4—4282no—
200D4—2142no—
201D4—2142no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—8562no—
206C2×C4—1072no—
207C2×C4—1072no—
208C2×C4—2141no—
209C2×C4—2141no—
210C2×C4D4×D10711yescharacteristic
211C2×C4D4×D10711yescharacteristic
212C2×C4C23×D10711yescharacteristic
213C8—22no—
214C8—2141no—
215C8—2141no—
216C22—4282no—
217C22—4282no—
218C22—4282no—
219C22C2×D4×D10711yescharacteristic
220C4—21no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—4281no—
225C4—2142no—
226C4—2142no—
227C4—2142no—
228C4#1681213012yes—
229C4#41×D10711yescharacteristic
230C4#1681215711yescharacteristic
231C2—8562no—
232C2#2029328711yescharacteristic
233C2C2×#1681213011yescharacteristic
234C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radical#16812106C22
hypercenter#475D107
hypocenterC107#3365
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20293352C2semidirect
3. #20292983C2semidirect
5. #20292926C2semidirect
6. #20292852C2semidirect
8. #20292751C2semidirect
9. C4×#13133808C2semidirect
10. #20292714C2semidirect
11. #475×C107C2semidirect
118. C107#3365semidirect
124. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933522
#202929832
#202929752
#202929262
#202928522
#202928282
#202927512
C4×#131338082
#202927142
#475×C1072
#3365107