Finite Group Databasebeta

← #46838377

Subgroups — #46838377

Order 34688.

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

normal subgroups only

This group has 226 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
2#31774074C211yesmaximal, characteristic
3#31773692C211yesmaximal, characteristic, complemented
4#31773687C212yesmaximal, complemented
5#563×C271C211yesmaximal, characteristic, complemented
6#22576205C2211yescharacteristic
7#22576191—21no—
8#22576143—22no—
9#22576088—21no—
10#22576077C2211yescharacteristic
11#22576075C2212yes—
12#22576074—22no—
13C4⋊Q8×C271—21no—
14#126×C271—22no—
15#125×C271—21no—
16#124×C271C2211yescharacteristic
17#123×C271C2212yes—
18C22⋊Q8×C271—22no—
19#16864654—21no—
20#16864653—41no—
21C2×C22⋊D271—22no—
22D4×D271—41no—
23#16864644—41no—
24#16864641—41no—
25#16864638—21no—
26Dic271⋊C4—42no—
27C22⋊Dic271—22no—
28(C2×D271)⋊C4—42no—
29C2×Q8×C271—21no—
30C2×Q8×C271D411yescharacteristic
31C2×D4×C271C2311yescharacteristic
32C22×C4×C271D412yes—
33C4⋊C4×C271—21no—
34C4⋊C4×C271—22no—
35C4⋊C4×C271—22no—
36C4⋊C4×C271—21no—
37C4⋊C4×C271—22no—
38C22⋊C4×C271—22no—
39C22⋊C4×C271—22no—
40C22⋊C4×C271D412yes—
41C42×C271—21no—
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×C271C2×D412yes—
53Q8×C271—41no—
54Q8×C271—81no—
55D4×C271—41no—
56C2×C4×C271—21no—
57C2×C4×C271—22no—
58C2×C4×C271—21no—
59C2×C4×C271—22no—
60C2×C4×C271—42no—
61C2×C4×C271C2×D411yescharacteristic
62C22×C271—22no—
63C22×C271—42no—
64C22×C271#12011yescharacteristic
65C2×D271—41no—
66C2×D271—82no—
67C4×C271—21no—
68C4×C271—42no—
69C4×C271—41no—
70C4×C271—42no—
71C4×C271—41no—
72Dic271—41no—
73Dic271—82no—
74C2×C271—21no—
75C2×C271—42no—
76C2×C271B(4,𝔽2)11yescharacteristic
77D271—81no—
78C271#220111yescharacteristic, complemented, Sylow
79#2201—2711nomaximal, Sylow
80#563D27111yescharacteristic, complemented
81#458—2712no—
82#455—2711no—
83#354—2711no—
84#352—2712no—
85H(5,𝔽2)—2711no—
86#137—5421no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D27111yescharacteristic
91#123—5422no—
92#123C2×D27112yes—
93C22⋊Q8—22no—
94#104—5421no—
95#100—2711no—
96#99—2712no—
97#99—5422no—
98#41—5421no—
99#41—10841no—
100C2×Q8—21no—
101C2×Q8C22⋊D27111yescharacteristic
102C2×D4—5422no—
103C2×D4—10841no—
104C2×D4C22×D27111yescharacteristic
105C22×C4C22⋊D27112yes—
106Q16—10841no—
107SD16—10841no—
108M16—5421no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—10842no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—5422no—
118C22⋊C4—10842no—
119C22⋊C4C22⋊D27112yes—
120C42—21no—
121C42C22⋊D27111yescharacteristic
122C23—5422no—
123C23C2×C22⋊D27112yes—
124Q8—41no—
125Q8—81no—
126Q8—5421no—
127D4—41no—
128D4—5421no—
129D4—10842no—
130D4—10841no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5422no—
137C2×C4—10841no—
138C2×C4—10842no—
139C2×C4C2×C22⋊D27111yescharacteristic
140C8—10841no—
141C22—22no—
142C22—42no—
143C22—10841no—
144C22—21682no—
145C22#2257612911yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—10841no—
152C4—21682no—
153C2—21no—
154C2—42no—
155C2—21681no—
156C2#3177408411yescharacteristic
subgroupG/H
centerC2#31774084
commutator, cosocleC2×D4×C271C23
FrattiniC2×D4C22×D271
Fitting#563×C271C2
socleC2×C271B(4,𝔽2)
prime radical#22576205C22
hypercenter#563D271
hypocenterC271#2201
LCSG ▷ C2×D4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #31773692C2semidirect
4. #31773687C2semidirect
5. #563×C271C2semidirect
78. C271#2201semidirect
80. #563D271semidirect

Maximal subgroups

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

subgroupindex
#317740862
#317740742
#317736922
#317736872
#563×C2712
#2201271