Finite Group Databasebeta

← #28339741

Subgroups — #28339741

Order 13696.

840 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16812071C213yesmaximal, complemented
2C2×(C22⋊D107)⋊C4C213yesmaximal, complemented
3#20292562C213yesmaximal, complemented
4#20292537C211yesmaximal, characteristic, complemented
5#20292536C213yesmaximal, complemented
6#20292472C211yesmaximal, characteristic, complemented
7#399×C107C211yesmaximal, characteristic, complemented
8C22×D428C2213yes—
9C22×C22⋊D107C2213yes—
10C23×Dic107C2211yescharacteristic
11C22×C4×D107C2213yes—
12#16812071—212no—
13(C22⋊D107)⋊C4—212no—
14C2×Dic107⋊C4C2213yes—
15C2×(C2×D107)⋊C4C2216yes—
16C2×(C2×D107)⋊C4C2213yes—
17C2×C4×Dic107C2213yes—
18C107⋊#95C2213yes—
19C2×C4⋊C4×C107C2213yes—
20C2×C22⋊C4×C107C2213yes—
21#95×C107C2211yescharacteristic
22C23×D107C2313yes—
23C2×D428—212no—
24C2×D428—26no—
25C2×C22⋊D107—26no—
26C2×C22⋊D107—26no—
27C2×C22⋊D107—26no—
28C22×Dic107—26no—
29C22×Dic107C2311yescharacteristic
30C22×Dic107C2313yes—
31C22×Dic107C2311yescharacteristic
32C2×C4×D107—26no—
33C2×C4×D107—212no—
34Dic107⋊C4—26no—
35(C2×D107)⋊C4—26no—
36(C2×D107)⋊C4—212no—
37C4×Dic107D4112yescomplemented
38C24×C107C2311yescharacteristic
39C22×C4×C107C2313yes—
40C22×C4×C107C2313yes—
41C4⋊C4×C107—26no—
42C22⋊C4×C107—26no—
43C23×C107—23no—
44C23×C107—23no—
45C23×C107#4112yes—
46C23×C107C2411yescharacteristic
47C22×D107—23no—
48C22×D107—26no—
49C22×D107—23no—
50C22×D107—26no—
51C22×D107#4116yes—
52C2×C4×C107—26no—
53C2×C4×C107—23no—
54C2×C4×C107—26no—
55C2×C4×C107C2×D416yes—
56C22⋊D107—224no—
57C2×Dic107—23no—
58C2×Dic107—23no—
59C2×Dic107—23no—
60C2×Dic107—44no—
61C2×Dic107C2×D416yes—
62C2×Dic107C2×D416yes—
63D428—224no—
64C4×D107—412no—
65C22×C107—26no—
66C22×C107—43no—
67C22×C107—41no—
68C22×C107C22×D413yes—
69C22×C107C2×#4113yes—
70C22×C107C2×#4111yescharacteristic
71C2×D107—212no—
72C2×D107—26no—
73C2×D107—43no—
74C2×D107—46no—
75C2×D107—43no—
76C4×C107—26no—
77C4×C107—43no—
78Dic107—26no—
79Dic107—41no—
80Dic107#12714yes—
81C2×C107—42no—
82C2×C107#52513yes—
83C2×C107#52513yes—
84C2×C107C2×#12711yescharacteristic
85D107—46no—
86#2397—1071nomaximal, Sylow
87C107#239711yescharacteristic, complemented, Sylow
88C2×C4×D4—1073no—
89C2×C4×D4—1073no—
90#399—1073no—
91#399D10711yescharacteristic, complemented
92#392—1073no—
93#392—1071no—
94#376—1071no—
95C22×D4—1073no—
96C22×D4—1073no—
97C23×C4—1071no—
98C23×C4—1073no—
99C4×D4—21412no—
100C4×D4—21412no—
101C2×C4⋊C4—1073no—
102C2×C4⋊C4C2×D10713yes—
103C2×C22⋊C4—1076no—
104C2×C22⋊C4—1073no—
105C2×C22⋊C4C2×D10713yes—
106C2×C42—1073no—
107#95—1073no—
108#95C2×D10711yescharacteristic
109C24—1073no—
110C24C22×D10711yescharacteristic
111C2×D4—21412no—
112C2×D4—2146no—
113C2×D4—2146no—
114C2×D4—2146no—
115C2×D4—2146no—
116C22×C4—1071no—
117C22×C4—1073no—
118C22×C4—1071no—
119C22×C4—2146no—
120C22×C4—21412no—
121C22×C4—2146no—
122C22×C4C22×D10713yes—
123C22×C4C22×D10713yes—
124C4⋊C4—26no—
125C4⋊C4—2146no—
126C22⋊C4—26no—
127C22⋊C4—21412no—
128C22⋊C4—2146no—
129C42—10712no—
130C23—23no—
131C23—23no—
132C23—1076no—
133C23—2143no—
134C23—2146no—
135C23—2146no—
136C23—2143no—
137C23#1313381912yes—
138C23C23×D10711yescharacteristic
139D4—21424no—
140D4—21424no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—26no—
144C2×C4—1076no—
145C2×C4—1076no—
146C2×C4—42812no—
147C2×C4—4284no—
148C2×C4—2143no—
149C2×C4—2143no—
150C2×C4—2143no—
151C2×C4D4×D10716yes—
152C22—26no—
153C22—43no—
154C22—41no—
155C22—4283no—
156C22—4286no—
157C22—4283no—
158C22—21412no—
159C22—2146no—
160C22C2×D4×D10713yes—
161C22#41×D10713yes—
162C22C2×#1313381911yescharacteristic
163C4—26no—
164C4—43no—
165C4—1074no—
166C4—4281no—
167C4—2146no—
168C2—42no—
169C2—4286no—
170C2#2029313513yes—
171C2#2029314013yes—
172C2#127×D10711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
hypercenter#399D107
hypocenterC107#2397
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812071C2semidirect
2. C2×(C22⋊D107)⋊C4C2semidirect
3. #20292562C2semidirect
4. #20292537C2semidirect
5. #20292536C2semidirect
6. #20292472C2semidirect
7. #399×C107C2semidirect
37. C4×Dic107D4semidirect
87. C107#2397semidirect
91. #399D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120712
C2×(C22⋊D107)⋊C42
#202925622
#202925372
#202925362
#202924722
#399×C1072
#2397107