Finite Group Databasebeta

← #46836747

Subgroups — #46836747

Order 34688.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#31774443C211yesmaximal, characteristic
2#31773913C212yesmaximal, complemented
3#31773871C211yesmaximal, characteristic, complemented
4#31773658C212yesmaximal, complemented
5#568×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2211yescharacteristic
7#22576158C411yescharacteristic, complemented
8#22576156C411yescharacteristic, complemented
9#22576149C411yescharacteristic, complemented
10#22576147C411yescharacteristic, complemented
11#22576132—22no—
12C4×C22⋊D271—22no—
13#22576119C2212yes—
14C2×C4×Dic271C2211yescharacteristic
15#22576086—24no—
16#22576075—22no—
17C2×#41×C271C2211yescharacteristic
18#131×C271—24no—
19C2×M16×C271—22no—
20C2×M16×C271C2212yes—
21#16864654—22no—
22#16864654—22no—
23C2×Dic542C2×C411yescharacteristic
24C2×D1084C2×C411yescharacteristic
25C2×C22⋊D271—22no—
26C2×C22⋊D271—21no—
27C22×Dic271—21no—
28C2×C4×D271—21no—
29C4⋊Dic271—22no—
30Dic271⋊C4—22no—
31C22⋊Dic271—22no—
32C22⋊Dic271—22no—
33(C2×D271)⋊C4—22no—
34C4×Dic271—21no—
35C4×Dic271—22no—
36C4×Dic271C2×C411yescharacteristic
37C4×Dic271C2×C411yescharacteristic
38#41×C271—22no—
39#41×C271—22no—
40C2×Q8×C271C2×C411yescharacteristic
41C2×D4×C271C2×C411yescharacteristic
42C2×D4×C271D412yes—
43C22×C4×C271D412yes—
44C22×C4×C271C2311yescharacteristic
45M16×C271—24no—
46M16×C271—22no—
47M16×C271D414yescomplemented
48C2×C8×C271—24no—
49C2×C8×C271—22no—
50C22×D271—21no—
51Dic542—22no—
52D1084—22no—
53C22⋊D271—44no—
54C22⋊D271—42no—
55C2×Dic271—21no—
56C2×Dic271—21no—
57C2×Dic271—21no—
58C2×Dic271—42no—
59C2×Dic271—42no—
60C4×D271—41no—
61C4×D271—41no—
62C23×C271C22⋊C412yes—
63C23×C271C2×D411yescharacteristic
64Q8×C271—22no—
65D4×C271—24no—
66D4×C271—22no—
67C2×C4×C271—22no—
68C2×C4×C271—22no—
69C2×C4×C271C22⋊C412yes—
70C2×C4×C271C22×C411yescharacteristic
71C2×C4×C271C2×D412yes—
72C2×C4×C271C2×D411yescharacteristic
73C2×C4×C271#4112yes—
74C8×C271—24no—
75C22×C271—22no—
76C22×C271—41no—
77C22×C271—42no—
78C22×C271C2×C22⋊C411yescharacteristic
79C22×C271#12112yes—
80C2×D271—42no—
81C2×D271—41no—
82C4×C271—22no—
83C4×C271C4×D412yes—
84C4×C271#12011yescharacteristic
85C4×C271#12311yescharacteristic
86Dic271—22no—
87Dic271—41no—
88Dic271—41no—
89Dic271—82no—
90C2×C271—21no—
91C2×C271—22no—
92C2×C271—42no—
93C2×C271#38611yescharacteristic
94D271—81no—
95C271#188411yescharacteristic, complemented, Sylow
96#1884—2711nomaximal, Sylow
97#568D27111yescharacteristic, complemented
98#532—2711no—
99#421—2712no—
100#410—2711no—
101#339—2712no—
102C2×#41—2711no—
103C2×#41C2×D27111yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D27112yes—
107C4⋊Q8—2711no—
108#127—2711no—
109#124—2711no—
110#124—2711no—
111#121—5422no—
112C4×D4—5422no—
113#117—2712no—
114C2×C42—2711no—
115#104—5424no—
116#99—5422no—
117#41—22no—
118#41—22no—
119#41—5422no—
120#41—5422no—
121C2×Q8—2711no—
122C2×Q8C4×D27111yescharacteristic
123C2×D4—2711no—
124C2×D4—5422no—
125C2×D4—5421no—
126C2×D4C4×D27111yescharacteristic
127C2×D4D108412yes—
128C22×C4—5421no—
129C22×C4—5421no—
130C22×C4C22⋊D27112yes—
131C22×C4C22×D27111yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D27114yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—5422no—
138C4⋊C4—5422no—
139C22⋊C4—5422no—
140C22⋊C4—5422no—
141C22⋊C4—5422no—
142C42—2711no—
143C42—2711no—
144C42—5421no—
145C42—5422no—
146C23—5421no—
147C23(C2×D271)⋊C412yes—
148C23C2×D108411yescharacteristic
149Q8—22no—
150Q8—5422no—
151D4—24no—
152D4—22no—
153D4—5422no—
154D4—10844no—
155D4—10842no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—5421no—
159C2×C4—5421no—
160C2×C4—5421no—
161C2×C4—10842no—
162C2×C4—10841no—
163C2×C4—10841no—
164C2×C4—10842no—
165C2×C4(C2×D271)⋊C412yes—
166C2×C4C2×C4×D27111yescharacteristic
167C2×C4C2×C22⋊D27111yescharacteristic
168C2×C4C2×C22⋊D27112yes—
169C2×C4#1686465412yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—10842no—
175C22—10841no—
176C22C2×(C2×D271)⋊C411yescharacteristic
177C22#2257613412yes—
178C4—22no—
179C4—5422no—
180C4—10841no—
181C4—10841no—
182C4—21682no—
183C4C4×C22⋊D27112yes—
184C4#2257612911yescharacteristic
185C4#2257614311yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—21681no—
190C2#3177377611yescharacteristic
subgroupG/H
centerC4#22576129
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#568×C271C2
socleC2×C271#386
prime radical#31774443C2
cosocleC22×C4×C271C23
hypercenter#568D271
hypocenterC271#1884
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #31773913C2semidirect
3. #31773871C2semidirect
4. #31773658C2semidirect
5. #568×C271C2semidirect
7. #22576158C4semidirect
8. #22576156C4semidirect
9. #22576149C4semidirect
10. #22576147C4semidirect
47. M16×C271D4semidirect
95. C271#1884semidirect
97. #568D271semidirect
134. M16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317744432
#317739132
#317738712
#317736582
#568×C2712
#1884271