Finite Group Databasebeta

← #46838034

Subgroups — #46838034

Order 34688.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#31774460C211yesmaximal, characteristic
2#31773869C212yesmaximal, complemented
3#31773690C211yesmaximal, characteristic, complemented
4#31773689C212yesmaximal, complemented
5#353×C271C211yesmaximal, characteristic, complemented
6C2×#16864653C2211yescharacteristic
7C2×D4×D271C411yescharacteristic, complemented
8#22576156C2211yescharacteristic
9#22576148C411yescharacteristic, complemented
10#22576147C411yescharacteristic, complemented
11#22576132—22no—
12#22576128—22no—
13#22576117C411yescharacteristic, complemented
14#22576116—22no—
15#22576075C2212yescomplemented
16#22576074—22no—
17#127×C271C2211yescharacteristic
18#99×C271C2212yescomplemented
19C23×D271—21no—
20#16864653—42no—
21C2×Dic542C2×C411yescharacteristic
22C2×D1084C2×C411yescharacteristic
23C2×C22⋊D271—21no—
24C2×C22⋊D271—22no—
25C2×C22⋊D271D412yes—
26D4×D271—42no—
27C22×Dic271D412yes—
28C2×C4×D271C2×C411yescharacteristic
29Dic271⋊C4—21no—
30Dic271⋊C4—42no—
31C22⋊Dic271—22no—
32C22⋊Dic271—22no—
33(C2×D271)⋊C4—21no—
34(C2×D271)⋊C4—22no—
35C4×Dic271—42no—
36C4×Dic271C2×C411yescharacteristic
37C2×D4×C271—22no—
38C2×D4×C271C2×C411yescharacteristic
39C2×D4×C271C2311yescharacteristic
40C22⋊C4×C271—22no—
41C42×C271C2×C411yescharacteristic
42C22×D271—21no—
43C22×D271—42no—
44C22×D271—82no—
45C22×D271C22⋊C411yescharacteristic
46Dic542—41no—
47D1084—41no—
48C22⋊D271—42no—
49C22⋊D271—42no—
50C22⋊D271—82no—
51C2×Dic271—22no—
52C2×Dic271—22no—
53C2×Dic271—21no—
54C2×Dic271—42no—
55C2×Dic271C22⋊C412yes—
56C2×Dic271C22⋊C411yescharacteristic
57C4×D271—22no—
58C23×C271—21no—
59C23×C271C2×D412yes—
60D4×C271—41no—
61D4×C271—41no—
62D4×C271—82no—
63C2×C4×C271—21no—
64C2×C4×C271—42no—
65C2×C4×C271C22×C411yescharacteristic
66C22×C271—22no—
67C22×C271—42no—
68C22×C271—42no—
69C22×C271—41no—
70C22×C271C2×C22⋊C411yescharacteristic
71C2×D271—21no—
72C2×D271—41no—
73C2×D271—42no—
74C2×D271—41no—
75C2×D271—84no—
76C4×C271—21no—
77C4×C271—41no—
78C4×C271—82no—
79Dic271—22no—
80Dic271—41no—
81Dic271—82no—
82Dic271#9912yes—
83C2×C271—21no—
84C2×C271—42no—
85C2×C271—81no—
86C2×C271C2×#9911yescharacteristic
87D271—41no—
88D271—81no—
89C271#212811yescharacteristic, complemented, Sylow
90#2128—2711nomaximal, Sylow
91#535—2711no—
92#410—2712no—
93#353—2711no—
94#353—2712no—
95#353D27111yescharacteristic, complemented
96C2×#41—2711no—
97C22×D4—2711no—
98#127—2711no—
99#127C2×D27111yescharacteristic
100#124—2711no—
101#124—2711no—
102#121—5422no—
103#120—5422no—
104#117—2711no—
105#117—5422no—
106#99—2712no—
107#99—5422no—
108#99C2×D27112yescomplemented
109C24—5421no—
110#41—10842no—
111C2×Q8—2711no—
112C2×D4—22no—
113C2×D4—2712no—
114C2×D4—2711no—
115C2×D4—5421no—
116C2×D4—5422no—
117C2×D4—10842no—
118C2×D4C4×D27111yescharacteristic
119C2×D4C22×D27111yescharacteristic
120C22×C4—2712no—
121C22×C4—2711no—
122C4⋊C4—5421no—
123C4⋊C4—10842no—
124C22⋊C4—22no—
125C22⋊C4—5422no—
126C22⋊C4—5421no—
127C22⋊C4—5422no—
128C22⋊C4—5422no—
129C42—2711no—
130C42—10842no—
131C42C4×D27111yescharacteristic
132C23—21no—
133C23—2711no—
134C23—5421no—
135C23—10842no—
136C23—21682no—
137C23D4×D27112yes—
138Q8—10841no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—10841no—
143D4—10842no—
144D4—10842no—
145D4—21682no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—2711no—
149C2×C4—2712no—
150C2×C4—5422no—
151C2×C4—5422no—
152C2×C4—5422no—
153C2×C4—5421no—
154C2×C4—10842no—
155C2×C4C2×C4×D27111yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—5421no—
161C22—10841no—
162C22—10842no—
163C22—10841no—
164C22—21684no—
165C22C22⋊C4×D27111yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—2712no—
170C4—5422no—
171C4—10841no—
172C4—21682no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—10841no—
177C2—21681no—
178C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC2×C4×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#353×C271C2
socleC2×C271C2×#99
prime radical#31774460C2
cosocleC2×D4×C271C23
hypercenter#353D271
hypocenterC271#2128
LCSG ▷ C2×C4×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
2. #31773869C2semidirect
3. #31773690C2semidirect
4. #31773689C2semidirect
5. #353×C271C2semidirect
7. C2×D4×D271C4semidirect
9. #22576148C4semidirect
10. #22576147C4semidirect
13. #22576117C4semidirect
15. #22576075C22semidirect
18. #99×C271C22semidirect
89. C271#2128semidirect
95. #353D271semidirect
108. #99C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744602
#317738692
#317736902
#317736892
#353×C2712
#2128271