Finite Group Databasebeta

← #46838316

Subgroups — #46838316

Order 34688.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#31774694C211yesmaximal, characteristic
2#31774340C211yesmaximal, characteristic, complemented
3#31774337C211yesmaximal, characteristic, complemented
4#31774156C211yesmaximal, characteristic, complemented
5#31773706C211yesmaximal, characteristic, complemented
6#31773678C211yesmaximal, characteristic, complemented
7#360×C271C211yesmaximal, characteristic, complemented
8#41×D271—21no—
9#22576192—21no—
10#22576189—21no—
11#22576187—21no—
12#22576184—21no—
13#22576182C2211yescharacteristic
14#22576180—21no—
15C2×Dic1084C2211yescharacteristic, complemented
16C2×#16864637C2211yescharacteristic
17#22576140—21no—
18Dic2168—21no—
19#22576106—21no—
20#22576105—21no—
21#22576104—21no—
22#22576100C2211yescharacteristic, complemented
23#22576083—21no—
24#135×C271C2211yescharacteristic
25M32×C271C2211yescharacteristic, complemented
26#106×C271C2211yescharacteristic, complemented
27#16864655—21no—
28#16864654—21no—
29#16864654—41no—
30#16864653—21no—
31#16864653—41no—
32C2×Dic542—21no—
33Q8×D271—21no—
34D4×D271—21no—
35D4×D271—41no—
36C2×C4×D271—41no—
37C2×C4×D271D411yescharacteristic
38Dic1084—21no—
39Dic1084—21no—
40Dic1084D411yescharacteristic
41#16864644—21no—
42#16864643—21no—
43#16864642—21no—
44#16864641—21no—
45D2168D411yescharacteristic
46#16864639—21no—
47#16864637—21no—
48#16864637—21no—
49C2×C271⋊C8D411yescharacteristic
50C4⋊Dic271—41no—
51(C2×D271)⋊C4—41no—
52C271⋊C16—21no—
53#41×C271—21no—
54Q16×C271D411yescharacteristic
55SD16×C271—21no—
56D8×C271D411yescharacteristic
57C2×C8×C271C2311yescharacteristic
58C4⋊C4×C271—21no—
59C16×C271—21no—
60C22×D271—41no—
61C22×D271D811yescharacteristic
62Dic542—21no—
63Dic542—41no—
64Dic542—41no—
65Dic542—41no—
66D1084—21no—
67D1084—41no—
68C22⋊D271—41no—
69C22⋊D271—41no—
70C22⋊D271—41no—
71C2×Dic271—41no—
72C2×Dic271—41no—
73C2×Dic271D811yescharacteristic
74C4×D271—21no—
75C4×D271—21no—
76C4×D271—41no—
77C4×D271—41no—
78C4×D271—41no—
79C271⋊C8—21no—
80Q8×C271—21no—
81D4×C271—21no—
82D4×C271—41no—
83C2×C4×C271—41no—
84C2×C4×C271—41no—
85C2×C4×C271C2×D411yescharacteristic
86C8×C271C2×D411yescharacteristic
87C8×C271C2×D411yescharacteristic
88C22×C271—41no—
89C22×C271C2×D811yescharacteristic
90C2×D271—21no—
91C2×D271—41no—
92C2×D271—41no—
93C2×D271—82no—
94C4×C271—41no—
95C4×C271—81no—
96C4×C271#12011yescharacteristic
97C4×C271AGL(1,ℤ/8)11yescharacteristic
98Dic271—21no—
99Dic271—41no—
100Dic271—81no—
101C2×C271—21no—
102C2×C271—81no—
103C2×C271#44711yescharacteristic
104D271—41no—
105D271—81no—
106C271#219111yescharacteristic, complemented, Sylow
107#2191—2711nomaximal, Sylow
108#575—2711no—
109#510—2711no—
110#510—2711no—
111#470—2711no—
112#360—2711no—
113#360D27111yescharacteristic, complemented
114#349—2711no—
115C2×#41—5421no—
116#137—5421no—
117#137—5421no—
118AGL(1,ℤ/8)—5421no—
119AGL(1,ℤ/8)—5421no—
120#135—2711no—
121#135—5421no—
122#135C2×D27111yescharacteristic
123C2×Q16—2711no—
124C2×M16—2711no—
125C22⋊Q8—5421no—
126Q32—5421no—
127Q32—5421no—
128SD32—5421no—
129SD32—5421no—
130M32—2711no—
131M32C2×D27111yescharacteristic, complemented
132#106C2×D27111yescharacteristic, complemented
133#103—5421no—
134#41—21no—
135#41—5421no—
136#41—5421no—
137#41—5421no—
138#41—10841no—
139#41—10841no—
140C2×Q8—5421no—
141C2×Q8—5421no—
142C2×D4—5421no—
143C2×D4—10841no—
144C22×C4—2711no—
145C22×C4—10841no—
146Q16—2711no—
147Q16—5421no—
148Q16—5421no—
149Q16—5421no—
150Q16D108411yescharacteristic
151SD16—21no—
152SD16—5421no—
153SD16—5421no—
154SD16—5421no—
155D8—2711no—
156D8—5421no—
157D8D108411yescharacteristic
158M16—5421no—
159M16—5421no—
160C2×C8—2711no—
161C2×C8C22×D27111yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—10841no—
164C22⋊C4—10841no—
165C16—21no—
166C16—5421no—
167C23—2711no—
168C23—10841no—
169Q8—21no—
170Q8—5421no—
171Q8—10841no—
172Q8—10841no—
173Q8—10841no—
174D4—21no—
175D4—41no—
176D4—5421no—
177D4—10841no—
178D4—10841no—
179D4—10841no—
180D4—10841no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—2711no—
184C2×C4—5421no—
185C2×C4—5421no—
186C2×C4—10841no—
187C2×C4—10841no—
188C2×C4—10841no—
189C2×C4—10841no—
190C2×C4—10841no—
191C2×C4D4×D27111yescharacteristic
192C8—5421no—
193C8D4×D27111yescharacteristic
194C8C2×D108411yescharacteristic
195C22—41no—
196C22—5421no—
197C22—10841no—
198C22—10841no—
199C22—21682no—
200C22D8×D27111yescharacteristic
201C4—41no—
202C4—81no—
203C4—5421no—
204C4—10841no—
205C4—21681no—
206C4#2257613011yescharacteristic
207C4#2257618711yescharacteristic
208C2—21no—
209C2—81no—
210C2—10841no—
211C2—21681no—
212C2#3177403211yescharacteristic
subgroupG/H
centerC2#31774032
commutator, cosocleC2×C8×C271C23
FrattiniC2×C8C22×D271
Fitting#360×C271C2
socleC2×C271#447
prime radical#31774694C2
hypercenter#360D271
hypocenterC271#2191
LCSG ▷ C2×C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #31774340C2semidirect
3. #31774337C2semidirect
4. #31774156C2semidirect
5. #31773706C2semidirect
6. #31773678C2semidirect
7. #360×C271C2semidirect
15. C2×Dic1084C22semidirect
22. #22576100C22semidirect
25. M32×C271C22semidirect
26. #106×C271C22semidirect
106. C271#2191semidirect
113. #360D271semidirect
131. M32C2×D271semidirect
132. #106C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746942
#317743402
#317743372
#317741562
#317737062
#317736782
#360×C2712
#2191271