Finite Group Databasebeta

← #46835335

Subgroups — #46835335

Order 34688.

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

normal subgroups only

This group has 282 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#31774478C211yesmaximal, characteristic
2#31774038C212yesmaximal, complemented
3#31773817C211yesmaximal, characteristic, complemented
4#31773632C212yesmaximal, complemented
5#324×C271C211yesmaximal, characteristic, complemented
6C2×#16864654—21no—
7C22×Dic542C2211yescharacteristic
8C2×#16864643—22no—
9#22576140—22no—
10#22576135—22no—
11#22576134—21no—
12#22576134C2212yes—
13C2×Dic271⋊C4—21no—
14#22576085—22no—
15C271⋊#95—21no—
16C271⋊#95C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19#16864654—42no—
20#16864654—42no—
21C2×Dic542—21no—
22C2×Dic542—22no—
23C2×Dic542—42no—
24C2×D1084—21no—
25C2×D1084—21no—
26C2×D1084D412yes—
27C2×C22⋊D271—22no—
28C22×Dic271—21no—
29C22×Dic271—21no—
30C2×C4×D271—22no—
31#16864643—44no—
32C4⋊Dic271—21no—
33C4⋊Dic271D412yes—
34Dic271⋊C4—42no—
35(C2×D271)⋊C4—22no—
36(C2×D271)⋊C4—22no—
37C22×C4×C271D411yescharacteristic
38C22×C4×C271D411yescharacteristic
39C22×C4×C271C2311yescharacteristic
40C2×C8×C271—22no—
41C4⋊C4×C271—22no—
42C22×D271—22no—
43Dic542—24no—
44Dic542—42no—
45Dic542—42no—
46D1084—42no—
47D1084—42no—
48D1084—81no—
49C22⋊D271—42no—
50C22⋊D271—42no—
51C2×Dic271—22no—
52C2×Dic271—22no—
53C2×Dic271—22no—
54C2×Dic271—41no—
55C2×Dic271—41no—
56C2×Dic271—41no—
57C2×Dic271—41no—
58C4×D271—44no—
59C23×C271C2×D411yescharacteristic
60C2×C4×C271—21no—
61C2×C4×C271—21no—
62C2×C4×C271—21no—
63C2×C4×C271—41no—
64C2×C4×C271—41no—
65C2×C4×C271SD1612yescomplemented
66C2×C4×C271C2×D412yes—
67C8×C271—42no—
68C22×C271—21no—
69C22×C271—21no—
70C22×C271#12011yescharacteristic
71C22×C271C2×SD1611yescharacteristic
72C22×C271AGL(1,ℤ/8)11yescharacteristic
73C2×D271—42no—
74C2×D271—42no—
75C2×D271—42no—
76C4×C271—22no—
77C4×C271—22no—
78C4×C271—41no—
79Dic271—42no—
80Dic271—42no—
81Dic271—42no—
82C2×C271—21no—
83C2×C271—21no—
84C2×C271#45011yescharacteristic
85C2×C271#45611yescharacteristic
86C2×C271B(4,𝔽2)11yescharacteristic
87D271—82no—
88C271#160411yescharacteristic, complemented, Sylow
89#1604—2711nomaximal, Sylow
90#537—2711no—
91#448—2712no—
92#395—2711no—
93#327—2712no—
94#324D27111yescharacteristic, complemented
95C2×#41—5421no—
96C22×Q8—2711no—
97C2×SD16—5422no—
98C22⋊Q8—5422no—
99#121—2712no—
100#121—5422no—
101#121—5421no—
102C2×C4⋊C4—5421no—
103C2×C4⋊C4C2×D27111yescharacteristic
104#103—5422no—
105C22⋊C8C2×D27112yes—
106#95—2711no—
107#95—5421no—
108#41—10842no—
109#41—10842no—
110C2×Q8—5421no—
111C2×Q8—5422no—
112C2×Q8—10842no—
113C2×D4—2712no—
114C2×D4—5421no—
115C2×D4—5421no—
116C2×D4—5422no—
117C22×C4—5422no—
118C22×C4—5421no—
119C22×C4—5421no—
120C22×C4D108411yescharacteristic
121C22×C4D108411yescharacteristic
122C22×C4C22×D27111yescharacteristic
123SD16—10844no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—2712no—
127C4⋊C4—5421no—
128C4⋊C4—10842no—
129C22⋊C4—5422no—
130C22⋊C4—5422no—
131C23—5422no—
132C23C2×D108411yescharacteristic
133Q8—5424no—
134Q8—10842no—
135Q8—10842no—
136D4—10842no—
137D4—10842no—
138D4—10842no—
139D4—10842no—
140D4—21681no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—5422no—
147C2×C4—5422no—
148C2×C4—5422no—
149C2×C4—10844no—
150C2×C4—10841no—
151C2×C4—10841no—
152C2×C4—10841no—
153C2×C4—10841no—
154C2×C4#1686464312yescomplemented
155C2×C4D4×D27112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—10842no—
160C22—10842no—
161C22—10842no—
162C22#2257613011yescharacteristic
163C22C2×#1686464311yescharacteristic
164C22#2257618811yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—10842no—
169C4—10842no—
170C4—10842no—
171C2—21no—
172C2—21no—
173C2—21682no—
174C2#3177405011yescharacteristic
175C2#3177408211yescharacteristic
176C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#324D271
hypocenterC271#1604
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #31774038C2semidirect
3. #31773817C2semidirect
4. #31773632C2semidirect
5. #324×C271C2semidirect
65. C2×C4×C271SD16semidirect
88. C271#1604semidirect
94. #324D271semidirect
154. C2×C4#16864643semidirect

Maximal subgroups

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

subgroupindex
#317744782
#317740382
#317738172
#317736322
#324×C2712
#1604271