Finite Group Databasebeta

← #28335466

Subgroups — #28335466

Order 13696.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293208C211yesmaximal, characteristic
2#20292941C212yesmaximal, complemented
3#20292557C211yesmaximal, characteristic, complemented
4#20292362C212yesmaximal, complemented
5#324×C107C211yesmaximal, characteristic, complemented
6C2×#13133818—21no—
7#16812089—22no—
8#16812084—22no—
9#16812083—21no—
10#16812083C2212yes—
11C2×C4⋊Dic107C2211yescharacteristic
12C2×Dic107⋊C4—21no—
13#16812043—22no—
14#16812031—22no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19#13133818—42no—
20#13133818—42no—
21C2×Dic214—21no—
22C2×D428—21no—
23C2×D428—21no—
24C2×D428D412yes—
25C2×C22⋊D107—22no—
26C22×Dic107—21no—
27C22×Dic107—21no—
28C2×C4×D107—22no—
29C4⋊Dic107—21no—
30C4⋊Dic107—22no—
31C4⋊Dic107D412yes—
32Dic107⋊C4—42no—
33(C2×D107)⋊C4—22no—
34(C2×D107)⋊C4—22no—
35C22×C4×C107D411yescharacteristic
36C22×C4×C107D411yescharacteristic
37C22×C4×C107C2311yescharacteristic
38C2×C8×C107—22no—
39C4⋊C4×C107—22no—
40C23×C107C2×D411yescharacteristic
41C22×D107—22no—
42C2×C4×C107—21no—
43C2×C4×C107—21no—
44C2×C4×C107—21no—
45C2×C4×C107—41no—
46C2×C4×C107—41no—
47C2×C4×C107D812yes—
48C2×C4×C107C2×D412yes—
49C22⋊D107—42no—
50C22⋊D107—42no—
51C2×Dic107—22no—
52C2×Dic107—22no—
53C2×Dic107—41no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56C2×Dic107—41no—
57C2×Dic107—41no—
58D428—42no—
59D428—42no—
60D428—81no—
61C4×D107—44no—
62Dic214—42no—
63C8×C107—42no—
64C22×C107—21no—
65C22×C107—21no—
66C22×C107#12011yescharacteristic
67C22×C107C2×D811yescharacteristic
68C22×C107AGL(1,ℤ/8)11yescharacteristic
69C2×D107—42no—
70C2×D107—42no—
71C2×D107—42no—
72C4×C107—22no—
73C4×C107—22no—
74C4×C107—41no—
75Dic107—42no—
76Dic107—42no—
77Dic107—81no—
78C2×C107—21no—
79C2×C107—21no—
80C2×C107#44711yescharacteristic
81C2×C107#45611yescharacteristic
82C2×C107#45811yescharacteristic
83D107—82no—
84#1610—1071nomaximal, Sylow
85C107#161011yescharacteristic, complemented, Sylow
86#537—1071no—
87#480—1072no—
88#398—1071no—
89#327—1072no—
90#324D10711yescharacteristic, complemented
91C2×#41—2141no—
92C22⋊Q8—2142no—
93#121—1072no—
94#121—2142no—
95#121—2141no—
96C2×C4⋊C4—1071no—
97C2×C4⋊C4—2141no—
98C2×C4⋊C4C2×D10711yescharacteristic
99#107—2142no—
100#102—2142no—
101C22⋊C8C2×D10712yes—
102#95—1071no—
103#95—2141no—
104#41—4282no—
105#41—4282no—
106C2×Q8—2141no—
107C2×D4—1072no—
108C2×D4—2141no—
109C2×D4—2141no—
110C2×D4—2142no—
111C22×C4—2142no—
112C22×C4—2141no—
113C22×C4—2141no—
114C22×C4D42811yescharacteristic
115C22×C4D42811yescharacteristic
116C22×C4C22×D10711yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—1072no—
120C4⋊C4—4282no—
121C4⋊C4—2141no—
122C4⋊C4—2142no—
123C22⋊C4—2142no—
124C22⋊C4—2142no—
125C23—2142no—
126C23C2×D42811yescharacteristic
127Q8—4282no—
128D4—8561no—
129D4—4282no—
130D4—4282no—
131D4—4282no—
132D4—4282no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—4284no—
139C2×C4—4281no—
140C2×C4—4281no—
141C2×C4—4281no—
142C2×C4—4281no—
143C2×C4—4281no—
144C2×C4—2142no—
145C2×C4—2142no—
146C2×C4D85612yes—
147C2×C4D4×D10712yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—4282no—
152C22—4282no—
153C22—4282no—
154C22#1681207911yescharacteristic
155C22C2×D85611yescharacteristic
156C22#1681213711yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—8561no—
161C4—4282no—
162C4—4282no—
163C2—21no—
164C2—21no—
165C2—8562no—
166C2#2029276311yescharacteristic
167C2#2029281211yescharacteristic
168C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#324×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#324D107
hypocenterC107#1610
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20292941C2semidirect
3. #20292557C2semidirect
4. #20292362C2semidirect
5. #324×C107C2semidirect
85. C107#1610semidirect
90. #324D107semidirect

Maximal subgroups

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

subgroupindex
#202932082
#202929412
#202925572
#202923622
#324×C1072
#1610107