Finite Group Databasebeta

← #28346119

Subgroups — #28346119

Order 13696.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16812115C211yesmaximal, characteristic, complemented
2#20293205C212yesmaximal, complemented
3#20292882C212yesmaximal, complemented
4#20292877C212yesmaximal, complemented
5#20292853C214yesmaximal, complemented
6C2×#16812041C211yesmaximal, characteristic, complemented
7#20292628C212yesmaximal, complemented
8C2×#135×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2211yescharacteristic
10#16812115—24no—
11C2×#13133802C2211yescharacteristic
12C2×#13133801C2212yes—
13C2×C8×D107C2212yes—
14#16812088—22no—
15#16812085C2214yes—
16#16812083—22no—
17#16812081—24no—
18#16812080C2214yes—
19C2×C4⋊Dic107C2212yes—
20#16812041C2212yes—
21#16812041C2212yes—
22#16812033C2214yes—
23#16812030C2214yes—
24C2×#41×C107C2212yes—
25#135×C107—24no—
26C2×Q16×C107C2211yescharacteristic
27C2×SD16×C107C2212yes—
28C2×D8×C107C2211yescharacteristic
29C22×C8×C107C2211yescharacteristic
30#13133818—22no—
31#13133818—22no—
32C2×Dic214D411yescharacteristic
33C2×D428D411yescharacteristic
34C2×C22⋊D107—24no—
35C2×C22⋊D107D412yes—
36C22×Dic107—22no—
37C2×C4×D107C2312yes—
38#13133802—22no—
39#13133801—24no—
40C2×C107⋊C8C2312yes—
41C8×D107—24no—
42C4⋊Dic107—22no—
43C4⋊Dic107C2314yes—
44Dic107⋊C4—24no—
45C22⋊Dic107—24no—
46(C2×D107)⋊C4—24no—
47#41×C107—24no—
48#41×C107—24no—
49C2×Q8×C107C2312yes—
50C2×D4×C107—22no—
51C2×D4×C107C2312yes—
52C22×C4×C107—22no—
53C22×C4×C107C2311yescharacteristic
54Q16×C107—22no—
55SD16×C107—24no—
56D8×C107—22no—
57C2×C8×C107D414yescomplemented
58C2×C8×C107C2311yescharacteristic
59C2×C8×C107C2311yescharacteristic
60C23×C107—22no—
61C23×C107C2×D411yescharacteristic
62C22×D107C2×D412yes—
63Q8×C107—22no—
64Q8×C107—22no—
65D4×C107—22no—
66D4×C107—22no—
67D4×C107—44no—
68C2×C4×C107—22no—
69C2×C4×C107—44no—
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D412yes—
72C2×C4×C107#4112yes—
73C2×C4×C107C2411yescharacteristic
74C22⋊D107—24no—
75C22⋊D107—84no—
76C2×Dic107—24no—
77C2×Dic107—82no—
78C2×Dic107C2×D412yes—
79D428—22no—
80C4×D107—22no—
81C4×D107—22no—
82Dic214—22no—
83C8×C107C2×D412yes—
84C8×C107C2×D412yes—
85C107⋊C8—22no—
86C22×C107—22no—
87C22×C107—42no—
88C22×C107—42no—
89C22×C107—42no—
90C22×C107#12112yes—
91C22×C107C22×D411yescharacteristic
92C2×D107—22no—
93C2×D107—24no—
94C4×C107—42no—
95C4×C107#12112yes—
96C4×C107C22×D411yescharacteristic
97C4×C107C2×#4111yescharacteristic
98Dic107—22no—
99Dic107—44no—
100C2×C107—22no—
101C2×C107—82no—
102C2×C107C2×#12111yescharacteristic
103C2×C107#57712yes—
104D107—42no—
105#3060—1071nomaximal, Sylow
106C107#306011yescharacteristic, complemented, Sylow
107C2×#135D10711yescharacteristic, complemented
108C2×#131—1071no—
109#537—1072no—
110#470—1072no—
111#469—1072no—
112#465—1074no—
113C2×#106—1071no—
114#417—1072no—
115C2×#41—1071no—
116C2×#41C2×D10712yes—
117#135—24no—
118C2×Q16C2×D10711yescharacteristic
119C2×SD16C2×D10712yes—
120C2×D8C2×D10711yescharacteristic
121#131—2144no—
122C2×M16—1071no—
123C2×M16—1072no—
124C22×C8—1072no—
125C22×C8C2×D10711yescharacteristic
126C22⋊Q8—1074no—
127C22⋊Q8—2142no—
128#121—1074no—
129#121—2142no—
130#121—2144no—
131C2×C4⋊C4—1072no—
132#106—1072no—
133#106—1072no—
134#103—1074no—
135#102—1074no—
136#41—24no—
137#41—24no—
138#41—2142no—
139#41—2142no—
140C2×Q8—1071no—
141C2×Q8C22×D10712yes—
142C2×D4—22no—
143C2×D4—1072no—
144C2×D4—1071no—
145C2×D4—2144no—
146C2×D4C22×D10712yes—
147C22×C4—22no—
148C22×C4—1072no—
149C22×C4—2142no—
150C22×C4C22×D10711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154M16—2142no—
155M16—2144no—
156C2×C8—1072no—
157C2×C8—2144no—
158C2×C8C22⋊D10714yescomplemented
159C2×C8C22×D10711yescharacteristic
160C2×C8C22×D10711yescharacteristic
161C4⋊C4—1074no—
162C4⋊C4—2144no—
163C4⋊C4—2142no—
164C22⋊C4—2144no—
165C22⋊C4—2144no—
166C23—22no—
167C23—1072no—
168C23D4×D10711yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—2142no—
172D4—22no—
173D4—22no—
174D4—44no—
175D4—8564no—
176D4—2144no—
177D4—2142no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—8562no—
181C2×C4—1072no—
182C2×C4—2144no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4D4×D10711yescharacteristic
186C2×C4C2×C22⋊D10712yes—
187C2×C4#1313381712yes—
188C2×C4C23×D10711yescharacteristic
189C8—2142no—
190C8C2×C22⋊D10712yes—
191C8C2×C22⋊D10712yes—
192C22—22no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—2144no—
197C22—2142no—
198C22#1681208012yes—
199C22C2×D4×D10711yescharacteristic
200C4—42no—
201C4—4284no—
202C4—2142no—
203C4#1681208012yes—
204C4C22×C22⋊D10711yescharacteristic
205C4C2×#1313381711yescharacteristic
206C2—22no—
207C2—82no—
208C2—4282no—
209C2C2×#1681208011yescharacteristic
210C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#135×C107C2
socleC22×C107C22×D4
hypercenterC2×#135D107
hypocenterC107#3060
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16812115C2semidirect
2. #20293205C2semidirect
3. #20292882C2semidirect
4. #20292877C2semidirect
5. #20292853C2semidirect
6. C2×#16812041C2semidirect
7. #20292628C2semidirect
8. C2×#135×C107C2semidirect
57. C2×C8×C107D4semidirect
106. C107#3060semidirect
107. C2×#135D107semidirect
158. C2×C8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121152
#202932052
#202928822
#202928772
#202928532
C2×#168120412
#202926282
C2×#135×C1072
#3060107