Finite Group Databasebeta

← #46838362

Subgroups — #46838362

Order 34688.

246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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#31774633C211yesmaximal, characteristic, complemented
2#31774088C212yesmaximal, complemented
3#31774064C211yesmaximal, characteristic
4#31773686C212yesmaximal, complemented
5#353×C271C211yesmaximal, characteristic, complemented
6#22576205C2211yescharacteristic
7#22576186—21no—
8#22576147—22no—
9#22576144—22no—
10#22576141—21no—
11#22576141C2212yes—
12C4×Dic542—21no—
13D4×Dic271—22no—
14#22576088—21no—
15#22576077C2211yescharacteristic
16#22576074—22no—
17#127×C271C2211yescharacteristic
18#99×C271C2212yes—
19#16864654—21no—
20#16864653—41no—
21C2×Dic542—21no—
22C2×C22⋊D271—22no—
23D4×D271—41no—
24C22×Dic271—21no—
25C22×Dic271D412yes—
26#16864642—41no—
27#16864639—41no—
28#16864638—21no—
29C4⋊Dic271—21no—
30C4⋊Dic271—42no—
31Dic271⋊C4—22no—
32C22⋊Dic271—21no—
33C22⋊Dic271—22no—
34C22⋊Dic271—22no—
35C22⋊Dic271D412yes—
36(C2×D271)⋊C4—42no—
37C4×Dic271—22no—
38C2×D4×C271—22no—
39C2×D4×C271D411yescharacteristic
40C2×D4×C271C2311yescharacteristic
41C22⋊C4×C271—22no—
42C42×C271D411yescharacteristic
43C22×D271—22no—
44Dic542—21no—
45Dic542—42no—
46D1084—21no—
47C22⋊D271—42no—
48C22⋊D271—41no—
49C2×Dic271—22no—
50C2×Dic271—22no—
51C2×Dic271—22no—
52C2×Dic271—42no—
53C2×Dic271—81no—
54C4×D271—41no—
55C271⋊C8—41no—
56C23×C271—21no—
57C23×C271C2×D412yes—
58D4×C271—41no—
59D4×C271—41no—
60D4×C271—44no—
61C2×C4×C271—21no—
62C2×C4×C271—42no—
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—82no—
74Dic271—42no—
75Dic271—41no—
76Dic271—42no—
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#559—2711no—
85#458—2712no—
86#453—2711no—
87#353D27111yescharacteristic, complemented
88#352—2712no—
89H(5,𝔽2)—2711no—
90AGL(1,ℤ/8)—5421no—
91#127C2×D27111yescharacteristic
92#124—5422no—
93#123—2712no—
94#123—5422no—
95#123—5421no—
96C4×Q8—5421no—
97C4×D4—5422no—
98#104—5421no—
99#100—2711no—
100#99—5422no—
101#99C2×D27112yes—
102#41—5421no—
103#41—10841no—
104C2×Q8—5421no—
105C2×D4—22no—
106C2×D4—5422no—
107C2×D4—10841no—
108C2×D4D108411yescharacteristic
109C2×D4C22×D27111yescharacteristic
110C22×C4—2712no—
111C22×C4—5421no—
112SD16—10841no—
113D8—10841no—
114M16—5421no—
115C4⋊C4—5421no—
116C4⋊C4—5422no—
117C4⋊C4—10842no—
118C22⋊C4—22no—
119C22⋊C4—2712no—
120C22⋊C4—5422no—
121C22⋊C4—5421no—
122C22⋊C4—5422no—
123C22⋊C4—10842no—
124C42—5422no—
125C42D108411yescharacteristic
126C23—21no—
127C23—5422no—
128C23D4×D27112yes—
129Q8—5421no—
130Q8—10842no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—5421no—
135D4—10842no—
136D4—10841no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—5422no—
140C2×C4—5422no—
141C2×C4—5422no—
142C2×C4—10841no—
143C2×C4—10842no—
144C2×C4—21681no—
145C2×C4C2×D108411yescharacteristic
146C8—10841no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—10841no—
152C22—21682no—
153C22#2257613011yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—10842no—
158C4—10841no—
159C4—10842no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—21681no—
164C2#3177408511yescharacteristic
subgroupG/H
centerC2#31774085
commutator, cosocleC2×D4×C271C23
FrattiniC2×D4C22×D271
Fitting#353×C271C2
socleC2×C271B(4,𝔽2)
prime radical#31774064C2
hypercenter#353D271
hypocenterC271#2198
LCSG ▷ C2×D4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #31774633C2semidirect
2. #31774088C2semidirect
4. #31773686C2semidirect
5. #353×C271C2semidirect
82. C271#2198semidirect
87. #353D271semidirect

Maximal subgroups

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

subgroupindex
#317746332
#317740882
#317740642
#317736862
#353×C2712
#2198271