Finite Group Databasebeta

← #28349161

Subgroups — #28349161

Order 13696.

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

normal subgroups only

This group has 348 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#20293363C211yesmaximal, characteristic
2#20293356C211yesmaximal, characteristic, complemented
3#20292980C212yesmaximal, complemented
4#20292973C211yesmaximal, characteristic
5#20292929C212yesmaximal, complemented
6#20292913C211yesmaximal, characteristic, complemented
7#20292847C212yesmaximal, complemented
8#20292751C211yesmaximal, characteristic, complemented
9C4×#13133808C211yesmaximal, characteristic, complemented
10SD16×Dic107C212yesmaximal, complemented
11#460×C107C211yesmaximal, characteristic
12C2×#13133808C2211yescharacteristic
13#16812109—21no—
14#16812109C2211yescharacteristic
15#16812100—22no—
16#16812100C2212yes—
17#16812096—21no—
18#16812096C2212yes—
19#16812090—22no—
20Q8×Dic107C2212yes—
21C4×Dic214—21no—
22C4×Dic214C2211yescharacteristic
23C4×Dic214C2211yescharacteristic
24D4×Dic107—21no—
25D4×Dic107C2212yes—
26#16812042C2211yescharacteristic
27#16812041C2212yes—
28Dic107⋊C8C2212yes—
29#16812034C2212yes—
30#16812033C2211yescharacteristic
31#16812032C2211yescharacteristic
32#16812030C2212yes—
33#16812030C2212yes—
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×SD16×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214D411yescharacteristic
43C2×Dic214C2311yescharacteristic
44C22×Dic107—22no—
45#13133808—22no—
46C2×C107⋊C8C2311yescharacteristic
47C2×C107⋊C8C2311yescharacteristic
48C4⋊Dic107—21no—
49C4⋊Dic107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2312yes—
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56Dic107⋊C4D412yes—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59C22⋊Dic107—22no—
60C4×Dic107—21no—
61C4×Dic107—22no—
62C4×Dic107C2312yes—
63C2×Q8×C107C2311yescharacteristic
64C2×D4×C107—21no—
65C2×D4×C107—21no—
66C2×D4×C107C2312yes—
67SD16×C107—24no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C23×C107—22no—
74Q8×C107—21no—
75Q8×C107#4112yes—
76D4×C107—22no—
77D4×C107—22no—
78D4×C107—42no—
79D4×C107—81no—
80C2×C4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—22no—
88C2×Dic107—82no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91Dic214—22no—
92Dic214—21no—
93Dic214—21no—
94Dic214—42no—
95C8×C107—22no—
96C107⋊C8—21no—
97C107⋊C8—21no—
98C22×C107—42no—
99C22×C107—42no—
100C22×C107—42no—
101C22×C107C22×D411yescharacteristic
102C4×C107—22no—
103C4×C107—21no—
104C4×C107—41no—
105C4×C107#13512yes—
106C4×C107C2×#4111yescharacteristic
107C4×C1072−1+411yescharacteristic
108Dic107—22no—
109Dic107—22no—
110Dic107—41no—
111Dic107—42no—
112Dic107—41no—
113C2×C107—82no—
114C2×C107#54811yescharacteristic
115C2×C107C2×#13511yescharacteristic
116C2×C107#57711yescharacteristic
117#3365—1071nomaximal, Sylow
118C107#336511yescharacteristic, complemented, Sylow
119#559—1071no—
120#557—1071no—
121#488—1072no—
122#486—1071no—
123#478—1072no—
124#475—1071no—
125#464—1072no—
126#460D10711yescharacteristic
127C8×Q8—1071no—
128C4×Q16—1071no—
129C4×SD16—1072no—
130C2×Q16—1071no—
131C2×SD16C2×D10712yes—
132C4⋊Q8—1071no—
133C4⋊Q8—2141no—
134#127C2×D10711yescharacteristic
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#106—1072no—
149C4⋊C8—1072no—
150C4⋊C8C2×D10711yescharacteristic
151#103—1071no—
152#103—1072no—
153#103—1071no—
154#102—1072no—
155#102—1072no—
156#102C2×D10712yes—
157C4×C8—1072no—
158C4×C8—1071no—
159C2×Q8—1071no—
160C2×Q8—1071no—
161C2×Q8—2141no—
162C2×Q8C22×D10711yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C22×D10712yes—
166C22×C4—2142no—
167Q16—2142no—
168SD16—24no—
169C2×C8—1071no—
170C2×C8—1071no—
171C2×C8C22×D10712yes—
172C4⋊C4—21no—
173C4⋊C4—1071no—
174C4⋊C4—1072no—
175C4⋊C4—1072no—
176C4⋊C4—1071no—
177C4⋊C4—2142no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2141no—
181C4⋊C4—2141no—
182C4⋊C4C22×D10711yescharacteristic
183C22⋊C4—2142no—
184C22⋊C4—2142no—
185C22⋊C4—2142no—
186C42—21no—
187C42—1072no—
188C42—2142no—
189C42—2141no—
190C42C22×D10711yescharacteristic
191C23—22no—
192Q8—21no—
193Q8—4282no—
194Q8—2141no—
195Q8—2141no—
196Q8—2142no—
197Q8#1313381712yes—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—8562no—
205C2×C4—1072no—
206C2×C4—1072no—
207C2×C4—2141no—
208C2×C4—2141no—
209C2×C4—2142no—
210C2×C4D4×D10711yescharacteristic
211C2×C4D4×D10711yescharacteristic
212C2×C4C23×D10711yescharacteristic
213C8—22no—
214C8—2141no—
215C8—2141no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×D4×D10711yescharacteristic
220C4—21no—
221C4—22no—
222C4—41no—
223C4—4282no—
224C4—4281no—
225C4—4281no—
226C4—2142no—
227C4—2142no—
228C4#1681213012yes—
229C4C2×#1313381711yescharacteristic
230C4#1681215611yescharacteristic
231C2—82no—
232C2#2029328311yescharacteristic
233C2C2×#1681213011yescharacteristic
234C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#127×C107C22
hypercenter#460D107
hypocenterC107#3365
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #20293356C2semidirect
3. #20292980C2semidirect
5. #20292929C2semidirect
6. #20292913C2semidirect
7. #20292847C2semidirect
8. #20292751C2semidirect
9. C4×#13133808C2semidirect
10. SD16×Dic107C2semidirect
118. C107#3365semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933562
#202929802
#202929732
#202929292
#202929132
#202928472
#202927512
C4×#131338082
SD16×Dic1072
#460×C1072
#3365107