Finite Group Databasebeta

← #28347075

Subgroups — #28347075

Order 13696.

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

normal subgroups only

This group has 362 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#20293246C212yesmaximal, complemented
2#20293025C211yesmaximal, characteristic, complemented
3#20292986C212yesmaximal, complemented
4#20292882C211yesmaximal, characteristic, complemented
5#20292877C211yesmaximal, characteristic
6#20292853C212yesmaximal, complemented
7#20292739C211yesmaximal, characteristic, complemented
8#20292734C211yesmaximal, characteristic, complemented
9SD16×Dic107C212yesmaximal, complemented
10#20292581C211yesmaximal, characteristic, complemented
11#495×C107C211yesmaximal, characteristic, complemented
12C2×#13133801C2211yescharacteristic
13C2×C8×D107C2211yescharacteristic
14#16812109C2212yes—
15#16812102—22no—
16#16812095—22no—
17#16812093—22no—
18#16812086—22no—
19#16812085C2212yes—
20#16812080C2212yes—
21Q8×Dic107C2212yes—
22D4×Dic107C2212yes—
23#16812066C2211yescharacteristic
24#16812041C2211yescharacteristic
25#16812041C2211yescharacteristic
26#16812041C2212yes—
27#16812033C2212yes—
28#16812033C2212yes—
29#16812030C2212yes—
30#16812030C2212yes—
31#16812020C2211yescharacteristic
32#16812019C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34C2×Q16×C107C2211yescharacteristic
35C2×SD16×C107C2212yes—
36C2×D8×C107C2211yescharacteristic
37#124×C107C2212yes—
38C4×C8×C107C2211yescharacteristic
39C2×Dic214—22no—
40C2×C22⋊D107—22no—
41C22×Dic107—22no—
42C2×C4×D107C2311yescharacteristic
43#13133801—22no—
44C2×C107⋊C8C2311yescharacteristic
45C2×C107⋊C8C2311yescharacteristic
46C8×D107—22no—
47C4⋊Dic107—22no—
48C4⋊Dic107—22no—
49C4⋊Dic107C2312yes—
50C4⋊Dic107C2312yes—
51Dic107⋊C4—22no—
52Dic107⋊C4D412yes—
53C22⋊Dic107—22no—
54C22⋊Dic107—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4D412yes—
57C4×Dic107—22no—
58C4×Dic107C2311yescharacteristic
59C2×Q8×C107C2312yes—
60C2×D4×C107C2312yes—
61Q16×C107—22no—
62SD16×C107—24no—
63D8×C107—22no—
64C2×C8×C107C2311yescharacteristic
65C2×C8×C107C2311yescharacteristic
66C22⋊C4×C107—22no—
67C22⋊C4×C107—22no—
68C42×C107C2311yescharacteristic
69C23×C107—22no—
70C22×D107C2×D411yescharacteristic
71Q8×C107—22no—
72Q8×C107—22no—
73D4×C107—22no—
74D4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—82no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—82no—
82C2×Dic107C2×D412yes—
83C2×Dic107C2×D411yescharacteristic
84C4×D107—21no—
85C4×D107—21no—
86Dic214—82no—
87C8×C107#4112yes—
88C8×C107#4112yes—
89C107⋊C8—21no—
90C107⋊C8—21no—
91C22×C107—42no—
92C22×C107—42no—
93C22×C107—42no—
94C22×C107C22×D411yescharacteristic
95C2×D107—21no—
96C2×D107—22no—
97C4×C107—22no—
98C4×C107—42no—
99C4×C107C2×#4111yescharacteristic
100C4×C107C2×#4111yescharacteristic
101Dic107—21no—
102Dic107—22no—
103Dic107—42no—
104Dic107—42no—
105C2×C107—82no—
106C2×C107#53211yescharacteristic
107C2×C107#57712yes—
108D107—41no—
109#3173—1071nomaximal, Sylow
110C107#317311yescharacteristic, complemented, Sylow
111#542—1072no—
112#498—1071no—
113#495D10711yescharacteristic, complemented
114#489—1072no—
115#470—1071no—
116#469—1071no—
117#465—1072no—
118#442—1071no—
119#441—1071no—
120C4×SD16—1072no—
121#405—1071no—
122C2×Q16C2×D10711yescharacteristic
123C2×SD16C2×D10712yes—
124C2×D8C2×D10711yescharacteristic
125C2×M16—1071no—
126C22×C8—1071no—
127C4⋊Q8—1072no—
128#126—2142no—
129#124—2142no—
130#124C2×D10712yes—
131#123—2142no—
132C22⋊Q8—1072no—
133C22⋊Q8—2142no—
134#121—1072no—
135C4×Q8—1072no—
136C4×D4—1072no—
137#117—1071no—
138#106—1072no—
139#106—1071no—
140#106—1071no—
141#103—1072no—
142#103—1072no—
143#102—1072no—
144#102—1072no—
145#97—1071no—
146#97—1071no—
147C4×C8—1071no—
148C4×C8C2×D10711yescharacteristic
149C2×Q8—2142no—
150C2×Q8C22×D10712yes—
151C2×D4—2142no—
152C2×D4C22×D10712yes—
153C22×C4—1071no—
154C22×C4—2142no—
155Q16—22no—
156SD16—24no—
157D8—22no—
158M16—2142no—
159C2×C8—1071no—
160C2×C8—1071no—
161C2×C8—2142no—
162C2×C8C22×D10711yescharacteristic
163C2×C8C22×D10711yescharacteristic
164C4⋊C4—1072no—
165C4⋊C4—1072no—
166C4⋊C4—1072no—
167C4⋊C4—2142no—
168C4⋊C4—2142no—
169C4⋊C4—2142no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—1072no—
173C22⋊C4—2142no—
174C22⋊C4—2142no—
175C22⋊C4—2142no—
176C42—1071no—
177C42—2142no—
178C42C22×D10711yescharacteristic
179C23—22no—
180C23—1071no—
181Q8—22no—
182Q8—22no—
183Q8—8562no—
184D4—22no—
185D4—22no—
186D4—8562no—
187C2×C4—22no—
188C2×C4—8562no—
189C2×C4—1072no—
190C2×C4—1071no—
191C2×C4—2142no—
192C2×C4—2142no—
193C2×C4—2141no—
194C2×C4—2141no—
195C2×C4D4×D10712yes—
196C2×C4C23×D10711yescharacteristic
197C8—2141no—
198C8—2141no—
199C8#1313381712yes—
200C8#1313381712yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—2141no—
205C22—2142no—
206C22C2×D4×D10711yescharacteristic
207C4—22no—
208C4—42no—
209C4—4282no—
210C4—4282no—
211C4—2141no—
212C4—2142no—
213C4C2×#1313381711yescharacteristic
214C4C2×#1313381711yescharacteristic
215C2—82no—
216C2—4281no—
217C2#2029317511yescharacteristic
218C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
prime radical#20292877C2
hypercenter#495D107
hypocenterC107#3173
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20293246C2semidirect
2. #20293025C2semidirect
3. #20292986C2semidirect
4. #20292882C2semidirect
6. #20292853C2semidirect
7. #20292739C2semidirect
8. #20292734C2semidirect
9. SD16×Dic107C2semidirect
10. #20292581C2semidirect
11. #495×C107C2semidirect
110. C107#3173semidirect
113. #495D107semidirect

Maximal subgroups

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

subgroupindex
#202932462
#202930252
#202929862
#202928822
#202928772
#202928532
#202927392
#202927342
SD16×Dic1072
#202925812
#495×C1072
#3173107