Finite Group Databasebeta

← #46835400

Subgroups — #46835400

Order 34688.

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

normal subgroups only

This group has 246 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#31774215C212yesmaximal, complemented
3#31773630C212yesmaximal, complemented
4#31773623C211yesmaximal, characteristic, complemented
5#400×C271C211yesmaximal, characteristic, complemented
6C2×#16864654—21no—
7#22576140—22no—
8#22576135—22no—
9#22576134—21no—
10#22576134C2212yes—
11#22576091—22no—
12#22576080—22no—
13C22⋊(C271⋊C8)C2212yes—
14C2×C4⋊C4×C271C2211yescharacteristic
15C2×C4⋊C4×C271C2211yescharacteristic
16#95×C271—21no—
17#95×C271—21no—
18#95×C271C2211yescharacteristic
19#16864654—42no—
20#16864654—42no—
21C2×Dic542—21no—
22C2×D1084—21no—
23C2×D1084—21no—
24C2×D1084D412yes—
25C2×C22⋊D271—22no—
26C2×C4×D271—22no—
27C2×C271⋊C8—22no—
28C4⋊Dic271—21no—
29C4⋊Dic271D412yes—
30(C2×D271)⋊C4—22no—
31(C2×D271)⋊C4—22no—
32C22×C4×C271—21no—
33C22×C4×C271—21no—
34C22×C4×C271D411yescharacteristic
35C22×C4×C271D411yescharacteristic
36C22×C4×C271C2311yescharacteristic
37C4⋊C4×C271—22no—
38C4⋊C4×C271—22no—
39C22×D271—22no—
40Dic542—42no—
41D1084—42no—
42D1084—42no—
43D1084—81no—
44C22⋊D271—42no—
45C22⋊D271—42no—
46C2×Dic271—22no—
47C4×D271—44no—
48C271⋊C8—42no—
49C23×C271C2×D411yescharacteristic
50C2×C4×C271—21no—
51C2×C4×C271—21no—
52C2×C4×C271—21no—
53C2×C4×C271—41no—
54C2×C4×C271—41no—
55C2×C4×C271—41no—
56C2×C4×C271—41no—
57C2×C4×C271—42no—
58C2×C4×C271—41no—
59C2×C4×C271—41no—
60C2×C4×C271SD1612yes—
61C2×C4×C271C2×D412yes—
62C22×C271—21no—
63C22×C271—21no—
64C22×C271#12011yescharacteristic
65C22×C271C2×SD1611yescharacteristic
66C22×C271AGL(1,ℤ/8)11yescharacteristic
67C2×D271—42no—
68C2×D271—42no—
69C2×D271—42no—
70C4×C271—22no—
71C4×C271—22no—
72C4×C271—41no—
73C4×C271—81no—
74C4×C271—81no—
75Dic271—42no—
76C2×C271—21no—
77C2×C271—21no—
78C2×C271#45011yescharacteristic
79C2×C271#45411yescharacteristic
80C2×C271#45811yescharacteristic
81D271—82no—
82C271#161611yescharacteristic, complemented, Sylow
83#1616—2711nomaximal, Sylow
84#537—2711no—
85#481—2712no—
86#400D27111yescharacteristic, complemented
87#327—2712no—
88#324—2711no—
89C2×#41—5421no—
90C22⋊Q8—5422no—
91#121—2712no—
92#121—5422no—
93#121—5421no—
94C2×C4⋊C4C2×D27111yescharacteristic
95C2×C4⋊C4C2×D27111yescharacteristic
96#106—5422no—
97#102—5422no—
98C22⋊C8—2712no—
99#95—21no—
100#95—21no—
101#95C2×D27111yescharacteristic
102#41—10842no—
103#41—10842no—
104C2×Q8—5421no—
105C2×D4—2712no—
106C2×D4—5421no—
107C2×D4—5421no—
108C2×D4—5422no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—5422no—
112C22×C4C22⋊D27111yescharacteristic
113C22×C4C22⋊D27111yescharacteristic
114C22×C4C22×D27111yescharacteristic
115C2×C8—5422no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—2712no—
119C4⋊C4—5421no—
120C22⋊C4—5422no—
121C22⋊C4—5422no—
122C23—5422no—
123C23C2×C22⋊D27111yescharacteristic
124Q8—10842no—
125D4—10842no—
126D4—10842no—
127D4—10842no—
128D4—10842no—
129D4—21681no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—5422no—
141C2×C4—10844no—
142C2×C4#1686464112yes—
143C2×C4D4×D27112yes—
144C8—10842no—
145C22—21no—
146C22—21no—
147C22—10842no—
148C22—10842no—
149C22—10842no—
150C22#2257612811yescharacteristic
151C22C2×#1686464111yescharacteristic
152C22#2257618511yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—10842no—
159C2—21no—
160C2—21no—
161C2—21682no—
162C2#3177404711yescharacteristic
163C2#3177406911yescharacteristic
164C2#3177408611yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#400×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#400D271
hypocenterC271#1616
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #31774215C2semidirect
3. #31773630C2semidirect
4. #31773623C2semidirect
5. #400×C271C2semidirect
82. C271#1616semidirect
86. #400D271semidirect

Maximal subgroups

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

subgroupindex
#317744782
#317742152
#317736302
#317736232
#400×C2712
#1616271