Finite Group Databasebeta

← #46835306

Subgroups — #46835306

Order 34688.

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

normal subgroups only

This group has 266 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#31774528C211yesmaximal, characteristic, complemented
2#31774050C212yesmaximal
3#31773814C211yesmaximal, characteristic
4#31773635C212yesmaximal, complemented
5#324×C271C211yesmaximal, characteristic, complemented
6C22×D1084C2211yescharacteristic
7C2×#16864643—22no—
8#22576159—22no—
9#22576151—22no—
10#22576139—21no—
11#22576139C2212yes—
12#22576119—21no—
13C2×(C2×D271)⋊C4—21no—
14C2×(C2×D271)⋊C4—21no—
15#22576082—22no—
16C271⋊#95C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19C23×D271—21no—
20C2×Dic542—21no—
21C2×Dic542D412yes—
22C2×D1084—22no—
23C2×D1084—42no—
24C22×Dic271—21no—
25#16864643—44no—
26C4⋊Dic271—21no—
27C4⋊Dic271—21no—
28C4⋊Dic271—21no—
29C4⋊Dic271D412yes—
30Dic271⋊C4—22no—
31Dic271⋊C4—22no—
32C22⋊Dic271—22no—
33(C2×D271)⋊C4—42no—
34(C2×D271)⋊C4—42no—
35C4×Dic271—22no—
36C22×C4×C271D411yescharacteristic
37C22×C4×C271D411yescharacteristic
38C22×C4×C271C2311yescharacteristic
39C2×C8×C271—22no—
40C4⋊C4×C271—22no—
41C22×D271—22no—
42C22×D271—41no—
43C22×D271—41no—
44C22×D271—41no—
45C22×D271—41no—
46C22×D271—41no—
47C22×D271—41no—
48Dic542—42no—
49Dic542—81no—
50D1084—24no—
51D1084—42no—
52C2×Dic271—22no—
53C2×Dic271—22no—
54C2×Dic271—41no—
55C2×Dic271—41no—
56C2×Dic271—41no—
57C23×C271C2×D411yescharacteristic
58C2×C4×C271—21no—
59C2×C4×C271—21no—
60C2×C4×C271—21no—
61C2×C4×C271—41no—
62C2×C4×C271—41no—
63C2×C4×C271SD1612yescomplemented
64C2×C4×C271C2×D412yes—
65C8×C271—42no—
66C22×C271—21no—
67C22×C271—21no—
68C22×C271#12011yescharacteristic
69C22×C271C2×SD1611yescharacteristic
70C22×C271#13711yescharacteristic
71C2×D271—42no—
72C2×D271—42no—
73C2×D271—42no—
74C2×D271—81no—
75C2×D271—81no—
76C2×D271—81no—
77C2×D271—81no—
78C4×C271—22no—
79C4×C271—22no—
80C4×C271—41no—
81Dic271—42no—
82Dic271—42no—
83Dic271—81no—
84C2×C271—21no—
85C2×C271—21no—
86C2×C271#44811yescharacteristic
87C2×C271#45411yescharacteristic
88C2×C271B(4,𝔽2)11yescharacteristic
89D271—82no—
90C271#159911yescharacteristic, complemented, Sylow
91#1599—2711nomaximal, Sylow
92#544—2711no—
93#450—2712no—
94#394—2711no—
95#328—2712no—
96#324D27111yescharacteristic, complemented
97C22×D4—2711no—
98C2×SD16—5422no—
99C4⋊Q8—5422no—
100#125—5422no—
101C22⋊Q8—2712no—
102C22⋊Q8—5421no—
103#117—5421no—
104C2×C4⋊C4C2×D27111yescharacteristic
105C2×C22⋊C4—5421no—
106C2×C22⋊C4—5421no—
107#102—5422no—
108C22⋊C8C2×D27112yes—
109#95—2711no—
110C24—5421no—
111C2×Q8—2712no—
112C2×Q8—5421no—
113C2×D4—5422no—
114C2×D4—10842no—
115C22×C4—5421no—
116C22×C4D108411yescharacteristic
117C22×C4D108411yescharacteristic
118C22×C4C22×D27111yescharacteristic
119SD16—10844no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—2712no—
123C4⋊C4—5421no—
124C4⋊C4—5421no—
125C4⋊C4—5422no—
126C4⋊C4—5421no—
127C4⋊C4—5422no—
128C22⋊C4—5422no—
129C22⋊C4—10842no—
130C22⋊C4—10842no—
131C42—5422no—
132C23—5422no—
133C23—10841no—
134C23—10841no—
135C23—10841no—
136C23—10841no—
137C23—10841no—
138C23—10841no—
139C23C2×D108411yescharacteristic
140Q8—10842no—
141Q8—21681no—
142D4—5424no—
143D4—10842no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—5422no—
150C2×C4—5422no—
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—21681no—
163C22—21681no—
164C22—21681no—
165C22—21681no—
166C22#2257613011yescharacteristic
167C22C2×#1686464311yescharacteristic
168C22#2257619311yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—10842no—
173C4—10842no—
174C4—21681no—
175C2—21no—
176C2—21no—
177C2—21682no—
178C2#3177403811yescharacteristic
179C2#3177407211yescharacteristic
180C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radicalC22×D1084C22
hypercenter#324D271
hypocenterC271#1599
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
1. #31774528C2semidirect
4. #31773635C2semidirect
5. #324×C271C2semidirect
63. C2×C4×C271SD16semidirect
90. C271#1599semidirect
96. #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
#317745282
#317740502
#317738142
#317736352
#324×C2712
#1599271