Finite Group Databasebeta

← #46835304

Subgroups — #46835304

Order 34688.

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

normal subgroups only

This group has 266 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#31774047C212yesmaximal
2#31773814C211yesmaximal, characteristic
3#31773634C212yesmaximal, complemented
4#31773623C211yesmaximal, characteristic, complemented
5#544×C271C211yesmaximal, characteristic, complemented
6C22×D1084C2211yescharacteristic
7C2×#16864641—22no—
8C2×(C2×D271)⋊C4—21no—
9C2×(C2×D271)⋊C4—21no—
10#22576080—22no—
11C22⋊(C271⋊C8)C2212yes—
12C271⋊#95C2211yescharacteristic
13C4⋊Q8×C271—22no—
14#125×C271—22no—
15C22⋊Q8×C271—21no—
16C22⋊Q8×C271C2212yes—
17#117×C271—21no—
18C2×C4⋊C4×C271C2211yescharacteristic
19C23×D271—21no—
20C2×D1084—22no—
21C2×D1084—42no—
22C22×Dic271—21no—
23#16864641—44no—
24C2×C271⋊C8—22no—
25(C2×D271)⋊C4—42no—
26(C2×D271)⋊C4—42no—
27C2×Q8×C271—21no—
28C2×Q8×C271D412yes—
29C22×C4×C271D411yescharacteristic
30C22×C4×C271D411yescharacteristic
31C22×C4×C271C2311yescharacteristic
32C4⋊C4×C271—21no—
33C4⋊C4×C271—21no—
34C4⋊C4×C271—21no—
35C4⋊C4×C271—22no—
36C4⋊C4×C271—22no—
37C4⋊C4×C271—22no—
38C4⋊C4×C271D412yes—
39C22⋊C4×C271—22no—
40C42×C271—22no—
41C22×D271—22no—
42C22×D271—41no—
43C22×D271—41no—
44C22×D271—41no—
45C22×D271—41no—
46C22×D271—41no—
47C22×D271—41no—
48D1084—24no—
49D1084—42no—
50C2×Dic271—41no—
51C2×Dic271—41no—
52C2×Dic271—41no—
53C271⋊C8—42no—
54C23×C271C2×D411yescharacteristic
55Q8×C271—42no—
56Q8×C271—81no—
57C2×C4×C271—21no—
58C2×C4×C271—21no—
59C2×C4×C271—22no—
60C2×C4×C271—22no—
61C2×C4×C271—21no—
62C2×C4×C271—41no—
63C2×C4×C271—41no—
64C2×C4×C271SD1612yescomplemented
65C2×C4×C271C2×D412yes—
66C22×C271—21no—
67C22×C271—21no—
68C22×C271#12011yescharacteristic
69C22×C271C2×SD1611yescharacteristic
70C22×C271#13711yescharacteristic
71C2×D271—42no—
72C2×D271—42no—
73C2×D271—42no—
74C2×D271—81no—
75C2×D271—81no—
76C2×D271—81no—
77C2×D271—81no—
78C4×C271—22no—
79C4×C271—22no—
80C4×C271—42no—
81C4×C271—41no—
82C4×C271—42no—
83Dic271—81no—
84C2×C271—21no—
85C2×C271—21no—
86C2×C271#44811yescharacteristic
87C2×C271#45411yescharacteristic
88C2×C271B(4,𝔽2)11yescharacteristic
89D271—82no—
90C271#159911yescharacteristic, complemented, Sylow
91#1599—2711nomaximal, Sylow
92#544D27111yescharacteristic, complemented
93#450—2712no—
94#394—2711no—
95#328—2712no—
96#324—2711no—
97C22×D4—2711no—
98C2×SD16—5422no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D27112yes—
103#117—21no—
104C2×C4⋊C4C2×D27111yescharacteristic
105C2×C22⋊C4—5421no—
106C2×C22⋊C4—5421no—
107#102—5422no—
108C22⋊C8—2712no—
109#95—2711no—
110C24—5421no—
111C2×Q8—21no—
112C2×Q8C22⋊D27112yes—
113C2×D4—5422no—
114C2×D4—10842no—
115C22×C4—5421no—
116C22×C4C22⋊D27111yescharacteristic
117C22×C4C22⋊D27111yescharacteristic
118C22×C4C22×D27111yescharacteristic
119SD16—10844no—
120C2×C8—5422no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D27112yes—
128C22⋊C4—22no—
129C22⋊C4—10842no—
130C22⋊C4—10842no—
131C42—22no—
132C23—5422no—
133C23—10841no—
134C23—10841no—
135C23—10841no—
136C23—10841no—
137C23—10841no—
138C23—10841no—
139C23C2×C22⋊D27111yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—5424no—
143D4—10842no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—10841no—
152C2×C4—10841no—
153C2×C4—10841no—
154C2×C4#1686464112yescomplemented
155C2×C4C2×C22⋊D27112yes—
156C8—10842no—
157C22—21no—
158C22—21no—
159C22—10842no—
160C22—10842no—
161C22—10842no—
162C22—21681no—
163C22—21681no—
164C22—21681no—
165C22—21681no—
166C22#2257612911yescharacteristic
167C22C2×#1686464111yescharacteristic
168C22#2257619111yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—21681no—
175C2—21no—
176C2—21no—
177C2—21682no—
178C2#3177403611yescharacteristic
179C2#3177407011yescharacteristic
180C2#3177408411yescharacteristic
subgroupG/H
centerC22#22576129
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#544×C271C2
socleC22×C271#120
prime radicalC22×D1084C22
hypercenter#544D271
hypocenterC271#1599
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #31773634C2semidirect
4. #31773623C2semidirect
5. #544×C271C2semidirect
64. C2×C4×C271SD16semidirect
90. C271#1599semidirect
92. #544D271semidirect
154. C2×C4#16864641semidirect

Maximal subgroups

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

subgroupindex
#317740472
#317738142
#317736342
#317736232
#544×C2712
#1599271