Finite Group Databasebeta

← #46838360

Subgroups — #46838360

Order 34688.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#31774086C212yesmaximal, complemented
2#31774064C211yesmaximal, characteristic
3#31773690C211yesmaximal, characteristic, complemented
4#31773687C212yesmaximal, complemented
5#559×C271C211yesmaximal, characteristic, complemented
6#22576205C2211yescharacteristic
7#22576186—21no—
8#22576143—22no—
9#22576088—21no—
10#22576077C2211yescharacteristic
11#22576075C2212yes—
12#22576074—22no—
13#127×C271C2211yescharacteristic
14#124×C271—22no—
15#123×C271—21no—
16#123×C271C2212yes—
17C4×Q8×C271—21no—
18C4×D4×C271—22no—
19#16864654—21no—
20#16864653—41no—
21C2×C22⋊D271—22no—
22D4×D271—41no—
23#16864642—41no—
24#16864639—41no—
25#16864638—21no—
26Dic271⋊C4—42no—
27C22⋊Dic271—22no—
28(C2×D271)⋊C4—42no—
29C2×Q8×C271—21no—
30C2×D4×C271—22no—
31C2×D4×C271D411yescharacteristic
32C2×D4×C271C2311yescharacteristic
33C22×C4×C271—21no—
34C22×C4×C271D412yes—
35C4⋊C4×C271—21no—
36C4⋊C4×C271—22no—
37C22⋊C4×C271—22no—
38C22⋊C4×C271—21no—
39C22⋊C4×C271—22no—
40C22⋊C4×C271D412yes—
41C42×C271—22no—
42C42×C271D411yescharacteristic
43C22×D271—22no—
44Dic542—21no—
45D1084—21no—
46C22⋊D271—42no—
47C22⋊D271—41no—
48C2×Dic271—22no—
49C2×Dic271—42no—
50C4×D271—41no—
51C271⋊C8—41no—
52C23×C271—21no—
53C23×C271C2×D412yes—
54Q8×C271—42no—
55D4×C271—41no—
56D4×C271—41no—
57D4×C271—44no—
58C2×C4×C271—21no—
59C2×C4×C271—22no—
60C2×C4×C271—22no—
61C2×C4×C271—42no—
62C2×C4×C271—81no—
63C2×C4×C271C2×D411yescharacteristic
64C22×C271—22no—
65C22×C271—42no—
66C22×C271—42no—
67C22×C271—41no—
68C22×C271#12011yescharacteristic
69C2×D271—41no—
70C2×D271—82no—
71C4×C271—22no—
72C4×C271—21no—
73C4×C271—42no—
74C4×C271—42no—
75Dic271—41no—
76Dic271—82no—
77C2×C271—21no—
78C2×C271—42no—
79C2×C271—81no—
80C2×C271B(4,𝔽2)11yescharacteristic
81D271—81no—
82C271#219811yescharacteristic, complemented, Sylow
83#2198—2711nomaximal, Sylow
84#559D27111yescharacteristic, complemented
85#458—2712no—
86#453—2711no—
87#353—2711no—
88#352—2712no—
89H(5,𝔽2)—2711no—
90AGL(1,ℤ/8)—5421no—
91#127C2×D27111yescharacteristic
92#124—22no—
93#123—21no—
94#123—5422no—
95#123C2×D27112yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—5421no—
99#100—2711no—
100#99—2712no—
101#99—5422no—
102#41—5421no—
103#41—10841no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—5422no—
107C2×D4—10841no—
108C2×D4C22⋊D27111yescharacteristic
109C2×D4C22×D27111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D27112yes—
112SD16—10841no—
113D8—10841no—
114M16—5421no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—10842no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—5422no—
122C22⋊C4—10842no—
123C22⋊C4C22⋊D27112yes—
124C42—22no—
125C42C22⋊D27111yescharacteristic
126C23—21no—
127C23—5422no—
128C23C2×C22⋊D27112yes—
129Q8—42no—
130Q8—5421no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—5421no—
135D4—10842no—
136D4—10841no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—5422no—
143C2×C4—10841no—
144C2×C4—10842no—
145C2×C4C2×C22⋊D27111yescharacteristic
146C8—10841no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—10841no—
152C22—21682no—
153C22#2257612911yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—10841no—
159C4—21682no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—21681no—
164C2#3177408411yescharacteristic
subgroupG/H
centerC2#31774084
commutator, cosocleC2×D4×C271C23
FrattiniC2×D4C22×D271
Fitting#559×C271C2
socleC2×C271B(4,𝔽2)
prime radical#31774064C2
hypercenter#559D271
hypocenterC271#2198
LCSG ▷ C2×D4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #31774086C2semidirect
3. #31773690C2semidirect
4. #31773687C2semidirect
5. #559×C271C2semidirect
82. C271#2198semidirect
84. #559D271semidirect

Maximal subgroups

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

subgroupindex
#317740862
#317740642
#317736902
#317736872
#559×C2712
#2198271