Finite Group Databasebeta

← #46835645

Subgroups — #46835645

Order 34688.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#31774550C211yesmaximal, characteristic
2#31774174C211yesmaximal, characteristic, complemented
3#31774099C211yesmaximal, characteristic, complemented
4#31773643C211yesmaximal, characteristic, complemented
5#31773626C211yesmaximal, characteristic, complemented
6#31773626C211yesmaximal, characteristic, complemented
7#498×C271C211yesmaximal, characteristic, complemented
8C2×#16864655—21no—
9C2×#16864641—21no—
10#22576145—22no—
11#22576136—21no—
12#22576135—21no—
13C4×D1084C2211yescharacteristic
14C4×D1084C2211yescharacteristic
15C4⋊C4×D271—21no—
16#22576093—21no—
17C4⋊(C271⋊C8)C2211yescharacteristic
18C4⋊(C271⋊C8)C2211yescharacteristic
19#22576084—21no—
20#22576080—21no—
21C4⋊Q8×C271C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23#106×C271—22no—
24C4×C8×C271C2211yescharacteristic
25#16864655—42no—
26#16864655—42no—
27C2×D1084—22no—
28C2×D1084D411yescharacteristic
29C2×D1084D411yescharacteristic
30C2×C4×D271—22no—
31C2×C4×D271—21no—
32C2×C4×D271—21no—
33#16864641—42no—
34C2×C271⋊C8—21no—
35C2×C271⋊C8—21no—
36C4⋊Dic271D411yescharacteristic
37C4⋊Dic271D411yescharacteristic
38Dic271⋊C4—22no—
39(C2×D271)⋊C4—22no—
40(C2×D271)⋊C4—21no—
41(C2×D271)⋊C4—21no—
42C2×Q8×C271—21no—
43C2×Q8×C271—21no—
44C2×C8×C271—21no—
45C4⋊C4×C271—21no—
46C4⋊C4×C271—21no—
47C4⋊C4×C271—21no—
48C4⋊C4×C271D412yes—
49C42×C271C2311yescharacteristic
50C22×D271—21no—
51C22×D271—21no—
52D1084—21no—
53D1084—21no—
54D1084—21no—
55D1084—44no—
56D1084SD1612yescomplemented
57C2×Dic271—21no—
58C2×Dic271—21no—
59C4×D271—44no—
60C4×D271—41no—
61C4×D271—41no—
62C4×D271—41no—
63C4×D271—41no—
64C271⋊C8—41no—
65C271⋊C8—41no—
66Q8×C271—41no—
67Q8×C271—41no—
68Q8×C271—41no—
69Q8×C271—41no—
70C2×C4×C271—22no—
71C2×C4×C271—41no—
72C2×C4×C271C2×D411yescharacteristic
73C2×C4×C271C2×D411yescharacteristic
74C2×C4×C271C2×D411yescharacteristic
75C8×C271—22no—
76C22×C271#12011yescharacteristic
77C2×D271—22no—
78C2×D271—41no—
79C2×D271—41no—
80C2×D271—41no—
81C2×D271—41no—
82C2×D271—41no—
83C4×C271—21no—
84C4×C271—42no—
85C4×C271—81no—
86C4×C271C2×SD1611yescharacteristic
87C4×C271#13511yescharacteristic
88C4×C271AGL(1,ℤ/8)11yescharacteristic
89C4×C271AGL(1,ℤ/8)11yescharacteristic
90Dic271—41no—
91Dic271—41no—
92C2×C271#44911yescharacteristic
93C2×C271#45011yescharacteristic
94C2×C271#45611yescharacteristic
95D271—42no—
96D271—81no—
97C271#165411yescharacteristic, complemented, Sylow
98#1654—2711nomaximal, Sylow
99#547—2711no—
100#498D27111yescharacteristic, complemented
101#474—2711no—
102#461—2711no—
103#332—2711no—
104D4⋊C8—2711no—
105D4⋊C8—2711no—
106C2×#41—5421no—
107C2×SD16—5421no—
108C4⋊Q8C2×D27111yescharacteristic
109C4⋊Q8C2×D27111yescharacteristic
110#123—5422no—
111C22⋊Q8—5421no—
112#121—5421no—
113C4×D4—2711no—
114C4×D4—2711no—
115C2×C4⋊C4—5421no—
116#107—5421no—
117#106—22no—
118C4⋊C8—2711no—
119C4⋊C8—2711no—
120#103—5421no—
121#102—5421no—
122C4×C8C2×D27111yescharacteristic
123#41—10842no—
124#41—10842no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—2711no—
128C2×D4—2711no—
129C2×D4—5422no—
130C22×C4—5422no—
131C22×C4—5421no—
132C22×C4—5421no—
133SD16—10842no—
134C2×C8—21no—
135C2×C8—5421no—
136C2×C8—5421no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—2711no—
141C4⋊C4—2711no—
142C4⋊C4—5422no—
143C4⋊C4C22⋊D27112yes—
144C22⋊C4—5422no—
145C22⋊C4—5421no—
146C22⋊C4—5421no—
147C42C22×D27111yescharacteristic
148C23—5421no—
149C23—5421no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—2712no—
155D4—5421no—
156D4—5421no—
157D4—5421no—
158D4—10844no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—5421no—
162C2×C4—5421no—
163C2×C4—10844no—
164C2×C4—10841no—
165C2×C4—10841no—
166C2×C4—10841no—
167C2×C4—10841no—
168C2×C4D4×D27111yescharacteristic
169C2×C4D4×D27111yescharacteristic
170C2×C4C2×C22⋊D27111yescharacteristic
171C8—22no—
172C8—10841no—
173C8—10841no—
174C22—5422no—
175C22—10841no—
176C22—10841no—
177C22—10841no—
178C22—10841no—
179C22—10841no—
180C22#2257612811yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—10841no—
185C4—10841no—
186C4SD16×D27111yescharacteristic
187C4#2257618111yescharacteristic
188C4#2257618711yescharacteristic
189C4#2257618711yescharacteristic
190C2—10842no—
191C2—21681no—
192C2#3177404111yescharacteristic
193C2#3177404611yescharacteristic
194C2#3177407911yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#498×C271C2
socleC22×C271#120
prime radical#31774550C2
hypercenter#498D271
hypocenterC271#1654
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #31774174C2semidirect
3. #31774099C2semidirect
4. #31773643C2semidirect
5. #31773626C2semidirect
6. #31773626C2semidirect
7. #498×C271C2semidirect
56. D1084SD16semidirect
97. C271#1654semidirect
100. #498D271semidirect

Maximal subgroups

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

subgroupindex
#317745502
#317741742
#317740992
#317736432
#317736262
#498×C2712
#1654271