Finite Group Databasebeta

← #28334459

Subgroups — #28334459

Order 13696.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#20292536C211yesmaximal, characteristic
2#20292394C211yesmaximal, characteristic, complemented
3#323×C107C211yesmaximal, characteristic, complemented
4C22×D428C411yescharacteristic, complemented
5C22×C22⋊D107—21no—
6C2×C22⋊Dic107—21no—
7C2×(C2×D107)⋊C4—21no—
8C107⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C107C2211yescharacteristic
10C22⋊C8×C107—21no—
11C23×D107—21no—
12C2×D428—41no—
13C2×D428—42no—
14C2×C22⋊D107—42no—
15C2×C22⋊D107—42no—
16C2×C22⋊D107—41no—
17C2×C22⋊D107—41no—
18C22×Dic107—21no—
19C22×Dic107—41no—
20C22⋊Dic107—22no—
21C22⋊Dic107—41no—
22(C2×D107)⋊C4—42no—
23C24×C107D411yescharacteristic
24C22×C4×C107C2×C411yescharacteristic
25C22×C4×C107D411yescharacteristic
26C2×C8×C107—41no—
27C22⋊C4×C107—41no—
28C23×C107—21no—
29C23×C107—21no—
30C23×C107—41no—
31C23×C107—41no—
32C23×C107C22⋊C411yescharacteristic
33C23×C107D811yescharacteristic
34C23×C107SD1611yescharacteristic
35C22×D107—22no—
36C22×D107—41no—
37C22×D107—41no—
38C22×D107—41no—
39C22×D107—41no—
40C22×D107—41no—
41C22×D107—41no—
42C2×C4×C107—21no—
43C2×C4×C107—21no—
44C2×C4×C107—41no—
45C2×C4×C107—41no—
46C22⋊D107—44no—
47C22⋊D107—44no—
48C2×Dic107—22no—
49C2×Dic107—41no—
50C2×Dic107—41no—
51C2×Dic107—42no—
52C2×Dic107—81no—
53C2×Dic107—81no—
54D428—42no—
55D428—42no—
56C8×C107—81no—
57C22×C107—21no—
58C22×C107—21no—
59C22×C107—21no—
60C22×C107—21no—
61C22×C107—41no—
62C22×C107—41no—
63C22×C107—41no—
64C22×C107—41no—
65C22×C107—81no—
66C22×C107#9911yescharacteristic
67C22×C107#10211yescharacteristic
68C22×C107#10411yescharacteristic
69C2×D107—42no—
70C2×D107—42no—
71C2×D107—42no—
72C2×D107—81no—
73C2×D107—81no—
74C2×D107—81no—
75C2×D107—81no—
76C4×C107—41no—
77C4×C107—41no—
78Dic107—42no—
79Dic107—82no—
80C2×C107—21no—
81C2×C107—21no—
82C2×C107—41no—
83C2×C107—41no—
84C2×C107#32711yescharacteristic
85C2×C107AΣL(1,𝔽16)11yescharacteristic
86C2×C107#35311yescharacteristic
87D107—82no—
88#1343—1071nomaximal, Sylow
89C107#134311yescharacteristic, complemented, Sylow
90#392—1071no—
91#342—1071no—
92#323D10711yescharacteristic, complemented
93C22×D4—1071no—
94C22×D4—2141no—
95C2×C22⋊C4—2141no—
96C2×C22⋊C4—2141no—
97C2×C22⋊C4C2×D10711yescharacteristic
98C22⋊C8—21no—
99#95—1071no—
100C24—2141no—
101C24D42811yescharacteristic
102C2×D4—4281no—
103C2×D4—4282no—
104C2×D4—4282no—
105C2×D4—4281no—
106C2×D4—4281no—
107C2×D4—4282no—
108C22×C4—4281no—
109C22×C4—2141no—
110C22×C4C4×D10711yescharacteristic
111C22×C4C22⋊D10711yescharacteristic
112C2×C8—41no—
113C22⋊C4—41no—
114C22⋊C4—4282no—
115C22⋊C4—4281no—
116C22⋊C4—2142no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—4281no—
122C23—4281no—
123C23—4281no—
124C23—4281no—
125C23—4281no—
126C23—4281no—
127C23—2142no—
128C23(C2×D107)⋊C411yescharacteristic
129C23D85611yescharacteristic
130C23#1313380711yescharacteristic
131D4—4282no—
132D4—4282no—
133D4—4284no—
134D4—4284no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—8561no—
140C2×C4—8561no—
141C2×C4—4282no—
142C2×C4—4281no—
143C2×C4—4281no—
144C2×C4—2142no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—8561no—
156C22—8561no—
157C22—8561no—
158C22—8561no—
159C22—4282no—
160C22—4282no—
161C22—4282no—
162C22#1681202311yescharacteristic
163C22#1681203111yescharacteristic
164C22#1681203511yescharacteristic
165C4—41no—
166C4—41no—
167C4—8562no—
168C4—4282no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—8562no—
174C2#2029236211yescharacteristic
175C2#2029241511yescharacteristic
176C2#2029241911yescharacteristic
subgroupG/H
centerC22#16812023
commutatorC22×C4×C107C2×C4
FrattiniC2×C22⋊C4C2×D107
Fitting#323×C107C2
socleC22×C107#99
prime radical#20292536C2
cosocleC2×C22⋊C4×C107C22
hypercenter#323D107
hypocenterC107#1343
LCSG ▷ C22×C4×C107 ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292394C2semidirect
3. #323×C107C2semidirect
4. C22×D428C4semidirect
8. C107⋊#95C4semidirect
89. C107#1343semidirect
92. #323D107semidirect

Maximal subgroups

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

subgroupindex
#202925362
#202923942
#323×C1072
#1343107