Finite Group Databasebeta

← #46835355

Subgroups — #46835355

Order 34688.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#31774225C212yesmaximal, complemented
2#31773822C211yesmaximal, characteristic
3#31773634C212yesmaximal, complemented
4#31773621C211yesmaximal, characteristic
5#543×C271C211yesmaximal, characteristic
6C2×C4⋊Dic271C2211yescharacteristic
7C2×C22⋊Dic271—21no—
8#22576091—22no—
9#22576084—22no—
10C22⋊(C271⋊C8)C2212yes—
11C271⋊#95—21no—
12C271⋊#95C2211yescharacteristic
13C22⋊Q8×C271—22no—
14C22⋊Q8×C271—21no—
15C22⋊Q8×C271—22no—
16C22⋊Q8×C271C2212yes—
17C2×C22⋊C4×C271—21no—
18C2×C22⋊C4×C271C2211yescharacteristic
19C22×Dic271—21no—
20C22×Dic271—21no—
21C2×C271⋊C8—22no—
22C4⋊Dic271—22no—
23C22⋊Dic271—42no—
24C24×C271D411yescharacteristic
25C2×Q8×C271—21no—
26C2×Q8×C271D412yes—
27C22×C4×C271—21no—
28C22×C4×C271—21no—
29C22×C4×C271D411yescharacteristic
30C22×C4×C271C2311yescharacteristic
31C4⋊C4×C271—21no—
32C4⋊C4×C271—22no—
33C4⋊C4×C271—22no—
34C4⋊C4×C271D412yes—
35C22⋊C4×C271—22no—
36C22⋊C4×C271—22no—
37C22⋊C4×C271—22no—
38C2×Dic271—41no—
39C2×Dic271—41no—
40C2×Dic271—41no—
41C2×Dic271—41no—
42C2×Dic271—41no—
43C2×Dic271—41no—
44C271⋊C8—42no—
45C23×C271—21no—
46C23×C271—21no—
47C23×C271—41no—
48C23×C271—41no—
49C23×C271SD1612yes—
50C23×C271C2×D411yescharacteristic
51Q8×C271—42no—
52Q8×C271—81no—
53C2×C4×C271—22no—
54C2×C4×C271—21no—
55C2×C4×C271—22no—
56C2×C4×C271—41no—
57C2×C4×C271—41no—
58C2×C4×C271—81no—
59C2×C4×C271—81no—
60C2×C4×C271C2×D412yes—
61C22×C271—21no—
62C22×C271—21no—
63C22×C271—22no—
64C22×C271—42no—
65C22×C271—41no—
66C22×C271—41no—
67C22×C271—81no—
68C22×C271#12011yescharacteristic
69C22×C271C2×SD1611yescharacteristic
70C22×C271#13711yescharacteristic
71C4×C271—22no—
72C4×C271—41no—
73C4×C271—42no—
74C4×C271—42no—
75Dic271—81no—
76Dic271—81no—
77C2×C271—21no—
78C2×C271—21no—
79C2×C271—42no—
80C2×C271#44811yescharacteristic
81C2×C271#45511yescharacteristic
82C2×C271#45811yescharacteristic
83C271#160811yescharacteristic, complemented, Sylow
84#1608—2711nomaximal, Sylow
85#543D27111yescharacteristic
86#483—2712no—
87#397—2711no—
88#328—2712no—
89#323—2711no—
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C2×D27112yes—
94C2×C4⋊C4—2711no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—5421no—
97C2×C22⋊C4C2×D27111yescharacteristic
98#106—5422no—
99#103—5422no—
100C22⋊C8—2712no—
101#95—2711no—
102#95—5421no—
103C24C22⋊D27111yescharacteristic
104C2×Q8—21no—
105C2×Q8C22⋊D27112yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—5421no—
109C22×C4—5421no—
110C22×C4C22⋊D27111yescharacteristic
111C22×C4C22×D27111yescharacteristic
112C2×C8—5422no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—5422no—
117C4⋊C4C22⋊D27112yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—10842no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1686464112yes—
127C23C2×C22⋊D27111yescharacteristic
128Q8—42no—
129Q8—81no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—10841no—
138C2×C4—10841no—
139C2×C4—10841no—
140C2×C4—10841no—
141C2×C4—10841no—
142C2×C4—10841no—
143C2×C4C2×C22⋊D27112yes—
144C8—10842no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#2257612911yescharacteristic
153C22C2×#1686464111yescharacteristic
154C22#2257619111yescharacteristic
155C4—22no—
156C4—41no—
157C4—42no—
158C4—42no—
159C4—21681no—
160C4—21681no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#3177403611yescharacteristic
165C2#3177407511yescharacteristic
166C2#3177408711yescharacteristic
subgroupG/H
centerC22#22576129
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#543×C271C2
socleC22×C271#120
prime radicalC24×C271D4
hypercenter#543D271
hypocenterC271#1608
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #31774225C2semidirect
3. #31773634C2semidirect
83. C271#1608semidirect

Maximal subgroups

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

subgroupindex
#317742252
#317738222
#317736342
#317736212
#543×C2712
#1608271