Finite Group Databasebeta

← #46836780

Subgroups — #46836780

Order 34688.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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, complemented
2#31774404C211yesmaximal, characteristic
3#31773860C212yesmaximal, complemented
4#31773669C212yesmaximal, complemented
5#413×C271C211yesmaximal, characteristic, complemented
6C22×C4×D271C2211yescharacteristic
7#22576167—24no—
8C2×#16864638—22no—
9C2×#16864638C2212yes—
10#22576141C412yes—
11#22576136—21no—
12#22576131—21no—
13#22576128C412yes—
14C4×C22⋊D271—22no—
15#22576119C2211yescharacteristic
16(C2×D271)⋊C8—24no—
17C2×#41×C271C2211yescharacteristic
18C2×M16×C271C2212yes—
19#101×C271—21no—
20#100×C271—21no—
21C23×D271C2×C411yescharacteristic
22C2×C22⋊D271—22no—
23C22×Dic271C2×C411yescharacteristic
24C2×C4×D271—21no—
25C2×C4×D271—22no—
26C2×C4×D271—21no—
27C2×C4×D271—22no—
28#16864638—24no—
29#16864638—22no—
30#16864638—22no—
31C2×C271⋊C8—24no—
32C2×C271⋊C8D414yescomplemented
33C4⋊Dic271—21no—
34Dic271⋊C4—22no—
35C22⋊Dic271—22no—
36C22⋊Dic271C2×C412yes—
37(C2×D271)⋊C4—22no—
38C4×Dic271—21no—
39#41×C271—22no—
40#41×C271—22no—
41C2×Q8×C271D411yescharacteristic
42C2×D4×C271C2×C412yes—
43C2×D4×C271D411yescharacteristic
44C22×C4×C271D412yes—
45C22×C4×C271C2311yescharacteristic
46M16×C271—22no—
47M16×C271—22no—
48C2×C8×C271—22no—
49C22×D271—21no—
50C22×D271—22no—
51C22×D271—42no—
52C22⋊D271—44no—
53C2×Dic271—22no—
54C2×Dic271—21no—
55C2×Dic271—22no—
56C4×D271—41no—
57C4×D271—42no—
58C4×D271—41no—
59C271⋊C8—24no—
60C23×C271C22⋊C412yes—
61C23×C271C22×C411yescharacteristic
62Q8×C271—22no—
63D4×C271—24no—
64D4×C271—22no—
65C2×C4×C271—22no—
66C2×C4×C271—22no—
67C2×C4×C271C22⋊C412yes—
68C2×C4×C271C2×D412yes—
69C2×C4×C271C2×D411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C2×C4×C271#4112yes—
72C8×C271—42no—
73C22×C271—22no—
74C22×C271—41no—
75C22×C271—42no—
76C22×C271C2×C22⋊C411yescharacteristic
77C22×C271C4×D412yes—
78C2×D271—22no—
79C2×D271—24no—
80C2×D271—41no—
81C2×D271—42no—
82C2×D271—41no—
83C4×C271—22no—
84C4×C271#12011yescharacteristic
85C4×C271#12112yes—
86C4×C271#12311yescharacteristic
87Dic271—42no—
88Dic271—41no—
89C2×C271—21no—
90C2×C271—22no—
91C2×C271—42no—
92C2×C271#38611yescharacteristic
93D271—42no—
94C271#189111yescharacteristic, complemented, Sylow
95#1891—2711nomaximal, Sylow
96#568—2711no—
97#525—2711no—
98#413D27111yescharacteristic, complemented
99#407—2712no—
100#344—2712no—
101C2×#41C2×D27111yescharacteristic
102C23×C4—2711no—
103#131—5424no—
104C2×M16—2712no—
105C2×M16—5422no—
106C2×M16C2×D27112yes—
107#123—2712no—
108C22⋊Q8—5421no—
109#121—5421no—
110#120—2712no—
111C4×D4—5422no—
112#117—2711no—
113#101—21no—
114#100—21no—
115C22⋊C8—5424no—
116C24—2711no—
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D27111yescharacteristic
120C2×D4—5422no—
121C2×D4C4×D27112yes—
122C2×D4C22⋊D27111yescharacteristic
123C22×C4—2711no—
124C22×C4—5422no—
125C22×C4—5421no—
126C22×C4—5421no—
127C22×C4—5422no—
128C22×C4D108412yes—
129C22×C4C22×D27111yescharacteristic
130M16—22no—
131M16—22no—
132M16—5424no—
133M16—5422no—
134M16—5422no—
135C2×C8—22no—
136C2×C8—2714no—
137C2×C8—5424no—
138C4⋊C4—5422no—
139C4⋊C4—5421no—
140C22⋊C4—2712no—
141C22⋊C4—5422no—
142C22⋊C4—5422no—
143C42—5421no—
144C23—5421no—
145C23—5422no—
146C23—10842no—
147C23(C2×D271)⋊C412yes—
148C23C2×C4×D27111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—10844no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—5422no—
156C2×C4—5421no—
157C2×C4—5422no—
158C2×C4—10841no—
159C2×C4—10842no—
160C2×C4—10841no—
161C2×C4(C2×D271)⋊C412yes—
162C2×C4D4×D27112yes—
163C2×C4C2×C22⋊D27111yescharacteristic
164C2×C4C2×D108411yescharacteristic
165C2×C4#1686465312yes—
166C8—42no—
167C8—5424no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—5422no—
172C22—5424no—
173C22—10841no—
174C22—10842no—
175C22—10841no—
176C22C2×(C2×D271)⋊C411yescharacteristic
177C22(C22⋊D271)⋊C412yes—
178C4—22no—
179C4—10842no—
180C4—10841no—
181C4#2257613011yescharacteristic
182C4#2257613212yes—
183C4#2257614411yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—10842no—
188C2#3177377411yescharacteristic
subgroupG/H
centerC4#22576130
commutatorC23×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#413×C271C2
socleC2×C271#386
prime radical#31774404C2
cosocleC22×C4×C271C23
hypercenter#413D271
hypocenterC271#1891
LCSG ▷ C23×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #31774664C2semidirect
3. #31773860C2semidirect
4. #31773669C2semidirect
5. #413×C271C2semidirect
32. C2×C271⋊C8D4semidirect
94. C271#1891semidirect
98. #413D271semidirect

Maximal subgroups

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

subgroupindex
#317746642
#317744042
#317738602
#317736692
#413×C2712
#1891271