Finite Group Databasebeta

← #46836887

Subgroups — #46836887

Order 34688.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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#31774664C211yesmaximal, characteristic
2#31773906C212yesmaximal, complemented
3#31773871C211yesmaximal, characteristic
4#31773653C212yesmaximal, complemented
5#536×C271C211yesmaximal, characteristic
6#22576167—24no—
7C2×#16864638—22no—
8C2×#16864638C2212yes—
9#22576119C2212yes—
10#22576084—24no—
11#22576075—22no—
12C2×#41×C271C2211yescharacteristic
13C22×Q8×C271C2211yescharacteristic
14C4⋊Q8×C271C412yescomplemented
15#124×C271C412yescomplemented
16#123×C271—22no—
17C22⋊Q8×C271—22no—
18#117×C271C2211yescharacteristic
19#16864638—24no—
20#16864638—22no—
21#16864638—22no—
22C2×C271⋊C8—24no—
23C2×C271⋊C8D414yes—
24C4⋊Dic271—22no—
25C22⋊Dic271—22no—
26C4×Dic271—22no—
27#41×C271—22no—
28#41×C271—22no—
29C2×Q8×C271—21no—
30C2×Q8×C271—22no—
31C2×Q8×C271—22no—
32C2×Q8×C271C2×C412yes—
33C2×Q8×C271C2×C411yescharacteristic
34C2×D4×C271C2×C411yescharacteristic
35C2×D4×C271D412yes—
36C22×C4×C271—21no—
37C22×C4×C271D412yes—
38C22×C4×C271C2311yescharacteristic
39C4⋊C4×C271—24no—
40C4⋊C4×C271—21no—
41C22⋊C4×C271—24no—
42C22⋊C4×C271—21no—
43C42×C271C2×C412yes—
44C2×Dic271—42no—
45C271⋊C8—24no—
46C23×C271C22⋊C412yes—
47C23×C271C2×D411yescharacteristic
48Q8×C271—22no—
49Q8×C271—24no—
50Q8×C271—41no—
51Q8×C271—41no—
52D4×C271—24no—
53D4×C271—22no—
54C2×C4×C271—22no—
55C2×C4×C271—22no—
56C2×C4×C271—22no—
57C2×C4×C271—22no—
58C2×C4×C271—42no—
59C2×C4×C271C22⋊C412yes—
60C2×C4×C271C22×C411yescharacteristic
61C2×C4×C271C2×D412yes—
62C2×C4×C271C2×D411yescharacteristic
63C2×C4×C271#4112yes—
64C22×C271—22no—
65C22×C271—41no—
66C22×C271—42no—
67C22×C271C2×C22⋊C411yescharacteristic
68C22×C271#12112yes—
69C4×C271—22no—
70C4×C271—42no—
71C4×C271—42no—
72C4×C271C4×D412yes—
73C4×C271#12011yescharacteristic
74C4×C271#12311yescharacteristic
75Dic271—82no—
76C2×C271—22no—
77C2×C271—21no—
78C2×C271—42no—
79C2×C271#38611yescharacteristic
80C271#190911yescharacteristic, complemented, Sylow
81#1909—2711nomaximal, Sylow
82#568—2711no—
83#536D27111yescharacteristic
84#419—2712no—
85#410—2711no—
86H(3,ℤ/4)—2712no—
87C2×#41C2×D27111yescharacteristic
88C22×Q8C2×D27111yescharacteristic
89#131—5424no—
90C2×M16—2712no—
91C2×M16—5422no—
92C4⋊Q8Dic27112yescomplemented
93#124Dic27112yescomplemented
94#123—22no—
95C22⋊Q8—22no—
96#117—2712no—
97#117C2×D27111yescharacteristic
98#103—5424no—
99#99—5422no—
100#41—22no—
101#41—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic27112yes—
106C2×Q8C2×Dic27111yescharacteristic
107C2×D4C2×Dic27111yescharacteristic
108C2×D4C22⋊D27112yes—
109C22×C4—21no—
110C22×C4C22⋊D27112yes—
111C22×C4C22×D27111yescharacteristic
112M16—5424no—
113M16—5422no—
114M16—5422no—
115C2×C8—2714no—
116C2×C8—5424no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—5422no—
120C22⋊C4—24no—
121C22⋊C4—21no—
122C22⋊C4—5422no—
123C42—5422no—
124C42C2×Dic27112yes—
125C23C22⋊Dic27112yes—
126C23C2×C22⋊D27111yescharacteristic
127Q8—22no—
128Q8—24no—
129Q8—41no—
130Q8—41no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—10842no—
139C2×C4C22⋊Dic27112yes—
140C2×C4C22×Dic27111yescharacteristic
141C2×C4D4×D27112yes—
142C2×C4C2×C22⋊D27111yescharacteristic
143C2×C4#1686465312yes—
144C8—5424no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×C22⋊Dic27111yescharacteristic
149C22#2257613212yes—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—21682no—
154C4D4×Dic27112yes—
155C4#2257612811yescharacteristic
156C4#2257614111yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#3177377711yescharacteristic
subgroupG/H
centerC2#31773777
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#536×C271C2
socleC2×C271#386
prime radicalC2×#41×C271C22
cosocleC22×C4×C271C23
hypercenter#536D271
hypocenterC271#1909
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #31773906C2semidirect
4. #31773653C2semidirect
14. C4⋊Q8×C271C4semidirect
15. #124×C271C4semidirect
80. C271#1909semidirect
92. C4⋊Q8Dic271semidirect
93. #124Dic271semidirect

Maximal subgroups

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

subgroupindex
#317746642
#317739062
#317738712
#317736532
#536×C2712
#1909271