Finite Group Databasebeta

← #46838364

Subgroups — #46838364

Order 34688.

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

normal subgroups only

This group has 270 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#31774639C211yesmaximal, characteristic, complemented
2#31774084C212yesmaximal, complemented
3#31773692C211yesmaximal, characteristic, complemented
4#31773685C212yesmaximal, complemented
5#455×C271C211yesmaximal, characteristic, complemented
6#22576149—21no—
7#22576148—21no—
8#22576146—22no—
9#22576130—22no—
10#22576128—22no—
11#22576128C2212yes—
12#22576075—22no—
13#22576075C2212yes—
14H(5,𝔽2)×C271C2211yescharacteristic
15#137×C271—21no—
16#124×C271C2211yescharacteristic
17#104×C271—21no—
18#100×C271C2211yescharacteristic
19C23×D271D412yescomplemented
20C2×Dic542—21no—
21C2×D1084—21no—
22C2×D1084—21no—
23C2×C22⋊D271—22no—
24C2×C22⋊D271—42no—
25C22⋊Dic271—22no—
26C22⋊Dic271D412yescomplemented
27(C2×D271)⋊C4—22no—
28(C2×D271)⋊C4—22no—
29(C2×D271)⋊C4—22no—
30(C2×D271)⋊C4—42no—
31C4×Dic271—21no—
32#41×C271—21no—
33#41×C271—41no—
34C2×Q8×C271D411yescharacteristic, complemented
35C2×D4×C271—22no—
36C2×D4×C271—41no—
37C2×D4×C271C2311yescharacteristic
38Q16×C271—41no—
39SD16×C271—41no—
40M16×C271—21no—
41C22⋊C4×C271—22no—
42C42×C271D411yescharacteristic, complemented
43C22×D271—22no—
44C22×D271—42no—
45C22×D271—82no—
46Dic542—81no—
47D1084—81no—
48D1084—81no—
49C22⋊D271—82no—
50C22⋊D271—84no—
51C2×Dic271—22no—
52C2×Dic271—42no—
53C23×C271—22no—
54C23×C271C2×D412yes—
55Q8×C271—21no—
56Q8×C271—41no—
57D4×C271—21no—
58D4×C271—41no—
59D4×C271—42no—
60D4×C271—41no—
61C2×C4×C271—22no—
62C2×C4×C271—21no—
63C2×C4×C271—21no—
64C2×C4×C271—41no—
65C2×C4×C271C2×D411yescharacteristic
66C8×C271—41no—
67C22×C271—22no—
68C22×C271—42no—
69C22×C271—41no—
70C22×C271—82no—
71C22×C271#12011yescharacteristic
72C2×D271—42no—
73C2×D271—42no—
74C2×D271—44no—
75C2×D271—42no—
76C2×D271—82no—
77C4×C271—21no—
78C4×C271—41no—
79C4×C271—41no—
80C4×C271—41no—
81Dic271—42no—
82Dic271—82no—
83C2×C271—21no—
84C2×C271—42no—
85C2×C271—81no—
86C2×C271B(4,𝔽2)11yescharacteristic
87D271—82no—
88C271#219911yescharacteristic, complemented, Sylow
89#2199—2711nomaximal, Sylow
90H(3,𝔽4)—2711no—
91B(4,𝔽2)—2712no—
92#455D27111yescharacteristic, complemented
93#354—2711no—
94AΣL(1,𝔽16)—2712no—
95H(5,𝔽2)C2×D27111yescharacteristic
96#137—21no—
97#124—5422no—
98#124—5421no—
99#124—5421no—
100#124C2×D27111yescharacteristic
101#120—2712no—
102#120—5422no—
103#120—5422no—
104#104—21no—
105#100C2×D27111yescharacteristic
106#99—2712no—
107#99—5422no—
108C24—2712no—
109#41—21no—
110#41—41no—
111C2×Q8—5421no—
112C2×Q8C22⋊D27111yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—5421no—
116C2×D4—5421no—
117C2×D4—5422no—
118C2×D4—10842no—
119C2×D4C22×D27111yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—2712no—
125C22⋊C4—5422no—
126C22⋊C4—5422no—
127C22⋊C4—5422no—
128C22⋊C4—5422no—
129C22⋊C4—10842no—
130C42—5421no—
131C42C22⋊D27111yescharacteristic, complemented
132C23—22no—
133C23—5422no—
134C23—10842no—
135C23—21682no—
136C23D4×D27112yes—
137Q8—21no—
138Q8—41no—
139Q8—21681no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—21681no—
145D4—21681no—
146D4—21682no—
147D4—21684no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—5422no—
153C2×C4—10842no—
154C2×C4C2×C22⋊D27111yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—10842no—
161C22—10842no—
162C22—10844no—
163C22—10842no—
164C22—21682no—
165C22#2257612811yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—10842no—
171C4—21682no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—21682no—
176C2#3177408311yescharacteristic
subgroupG/H
centerC2#31774083
commutator, cosocleC2×D4×C271C23
FrattiniC2×D4C22×D271
Fitting#455×C271C2
socleC2×C271B(4,𝔽2)
hypercenter#455D271
hypocenterC271#2199
LCSG ▷ C2×D4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #31774639C2semidirect
2. #31774084C2semidirect
3. #31773692C2semidirect
4. #31773685C2semidirect
5. #455×C271C2semidirect
19. C23×D271D4semidirect
26. C22⋊Dic271D4semidirect
34. C2×Q8×C271D4semidirect
42. C42×C271D4semidirect
88. C271#2199semidirect
92. #455D271semidirect
112. C2×Q8C22⋊D271semidirect
131. C42C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317746392
#317740842
#317736922
#317736852
#455×C2712
#2199271