Finite Group Databasebeta

← #28346851

Subgroups — #28346851

Order 13696.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293175C211yesmaximal, characteristic
2C4⋊Q8×D107C211yesmaximal, characteristic, complemented
3#20292985C214yesmaximal, complemented
4#20292979C211yesmaximal, characteristic, complemented
5#20292978C211yesmaximal, characteristic, complemented
6#20292973C211yesmaximal, characteristic, complemented
7#20292625C214yesmaximal, complemented
8C4×#13133801C211yesmaximal, characteristic, complemented
9#486×C107C211yesmaximal, characteristic, complemented
10C2×#13133817C2212yes—
11C2×Q8×D107—21no—
12C2×#13133801C2212yes—
13#16812109C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812108—21no—
16#16812108C2212yes—
17#16812107—21no—
18#16812096C2212yes—
19#16812080—21no—
20#16812080—21no—
21D4×Dic107—21no—
22D4×Dic107—21no—
23C4⋊C4×D107—21no—
24C4⋊C4×D107C2212yes—
25C42×D107C2211yescharacteristic
26#16812034C2214yes—
27#16812032C2214yes—
28#16812030C2214yes—
29#16812019C2212yes—
30C4×C107⋊C8C2211yescharacteristic
31C4⋊Q8×C107C2211yescharacteristic
32#127×C107C2211yescharacteristic
33#102×C107C2214yes—
34C4×C8×C107C2211yescharacteristic
35#13133817—28no—
36C2×Dic214—21no—
37C2×Dic214—22no—
38C2×Dic214C2312yes—
39Q8×D107—44no—
40C2×C22⋊D107—22no—
41C22×Dic107—22no—
42C2×C4×D107—22no—
43C2×C4×D107D411yescharacteristic
44C2×C4×D107D411yescharacteristic
45C2×C4×D107C2311yescharacteristic
46#13133801—24no—
47C2×C107⋊C8C2312yes—
48C4⋊Dic107—21no—
49C4⋊Dic107—21no—
50C4⋊Dic107—21no—
51C4⋊Dic107C2312yes—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54C22⋊Dic107—22no—
55C22⋊Dic107—22no—
56C4×Dic107D411yescharacteristic
57C4×Dic107D411yescharacteristic
58C4×Dic107C2311yescharacteristic
59C2×Q8×C107—21no—
60C2×D4×C107—21no—
61C2×D4×C107—21no—
62C2×D4×C107C2312yes—
63C2×C8×C107C2312yes—
64C4⋊C4×C107—21no—
65C4⋊C4×C107C2312yes—
66C42×C107C2311yescharacteristic
67C23×C107—22no—
68C22×D107C2×D411yescharacteristic
69Q8×C107—42no—
70D4×C107—24no—
71D4×C107—42no—
72D4×C107—42no—
73C2×C4×C107—22no—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2411yescharacteristic
77C22⋊D107—44no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—44no—
81C2×Dic107C2×D411yescharacteristic
82C2×Dic107C2×D411yescharacteristic
83C2×Dic107C2×D411yescharacteristic
84C4×D107—22no—
85C4×D107—22no—
86C4×D107—44no—
87C4×D107#4114yes—
88Dic214—24no—
89Dic214—44no—
90Dic214—42no—
91C8×C107—22no—
92C107⋊C8—22no—
93C22×C107—42no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107C22×D411yescharacteristic
97C2×D107—21no—
98C2×D107—21no—
99C2×D107#12412yes—
100C4×C107—42no—
101C4×C107AGL(1,ℤ/8)12yes—
102C4×C107#13712yes—
103C4×C107C2×#4112yes—
104Dic107—21no—
105Dic107—21no—
106Dic107—42no—
107Dic107—42no—
108Dic107#12412yes—
109C2×C107—82no—
110C2×C107C2×#12411yescharacteristic
111C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C107C2×#13711yescharacteristic
113D107—22no—
114#3141—1071nomaximal, Sylow
115C107#314111yescharacteristic, complemented, Sylow
116#532—1071no—
117C2×C4⋊Q8—1071no—
118#489—1074no—
119#487—1071no—
120#487—1071no—
121#486—1071no—
122#486D10711yescharacteristic, complemented
123#417—1074no—
124C4×M16—1071no—
125C2×#41—1072no—
126C22×Q8—2141no—
127C2×M16—1072no—
128C4⋊Q8—1072no—
129C4⋊Q8—1071no—
130C4⋊Q8—1071no—
131C4⋊Q8—2141no—
132C4⋊Q8—2141no—
133C4⋊Q8C2×D10711yescharacteristic
134#127C2×D10711yescharacteristic
135#124—1072no—
136#121—2141no—
137#121—2141no—
138C4×D4—2141no—
139C4×D4—2141no—
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—2141no—
142C2×C42—1071no—
143#103—1074no—
144#103—1074no—
145#102—1074no—
146#102C2×D10714yes—
147#97—1072no—
148C4×C8—1071no—
149C4×C8C2×D10711yescharacteristic
150#41—2148no—
151C2×Q8—21no—
152C2×Q8—1072no—
153C2×Q8—4284no—
154C2×Q8—2142no—
155C2×Q8—2141no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—2142no—
159C2×D4C22×D10712yes—
160C22×C4—1071no—
161C22×C4—1071no—
162C22×C4—1071no—
163C22×C4—2142no—
164C22×C4—2142no—
165M16—2144no—
166C2×C8—1072no—
167C2×C8C22×D10712yes—
168C4⋊C4—21no—
169C4⋊C4—1072no—
170C4⋊C4—2142no—
171C4⋊C4—2142no—
172C4⋊C4—2141no—
173C4⋊C4—2141no—
174C4⋊C4—2141no—
175C4⋊C4C22×D10712yes—
176C22⋊C4—2142no—
177C22⋊C4—2142no—
178C42—1071no—
179C42—1071no—
180C42—1071no—
181C42C22×D10711yescharacteristic
182C23—22no—
183C23—1071no—
184Q8—42no—
185Q8—4284no—
186Q8—4282no—
187Q8—2144no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—4284no—
192C2×C4—22no—
193C2×C4—1071no—
194C2×C4—1074no—
195C2×C4—1071no—
196C2×C4—1071no—
197C2×C4—4284no—
198C2×C4—4284no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4D4×D10711yescharacteristic
204C2×C4D4×D10711yescharacteristic
205C2×C4C23×D10711yescharacteristic
206C8—22no—
207C8—2142no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—1072no—
212C22—2141no—
213C22—2141no—
214C22C2×D4×D10711yescharacteristic
215C4—42no—
216C4—1072no—
217C4—4282no—
218C4—4282no—
219C4—2141no—
220C4—2141no—
221C4#1681213312yes—
222C4#1681213812yes—
223C4#41×D10712yes—
224C2—82no—
225C2—2142no—
226C2#124×D10711yescharacteristic
227C2C2×#1681213311yescharacteristic
228C2C2×#1681213811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#486×C107C2
socleC22×C107C22×D4
prime radical#20293175C2
hypercenter#486D107
hypocenterC107#3141
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D107C2semidirect
3. #20292985C2semidirect
4. #20292979C2semidirect
5. #20292978C2semidirect
6. #20292973C2semidirect
7. #20292625C2semidirect
8. C4×#13133801C2semidirect
9. #486×C107C2semidirect
115. C107#3141semidirect
122. #486D107semidirect

Maximal subgroups

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

subgroupindex
#202931752
C4⋊Q8×D1072
#202929852
#202929792
#202929782
#202929732
#202926252
C4×#131338012
#486×C1072
#3141107