Finite Group Databasebeta

← #46835402

Subgroups — #46835402

Order 34688.

246 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 51 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#31774474C211yesmaximal, characteristic
2#31774216C211yesmaximal, characteristic, complemented
3#31774213C211yesmaximal, characteristic, complemented
4#31773837C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773622C211yesmaximal, characteristic, complemented
7#327×C271C211yesmaximal, characteristic, complemented
8C2×#16864653—21no—
9#22576138—21no—
10#22576137—21no—
11#22576134C2211yescharacteristic
12#22576133—21no—
13#22576132—21no—
14#22576132—21no—
15C2×C4⋊Dic271C2211yescharacteristic
16C2×Dic271⋊C4C2211yescharacteristic
17#22576092—21no—
18#22576091—21no—
19#22576082—21no—
20#22576081—21no—
21C22⋊(C271⋊C8)C2211yescharacteristic
22C271⋊#95—21no—
23C271⋊#95—21no—
24#121×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27#16864653—42no—
28#16864653—42no—
29C2×Dic542—21no—
30C2×D1084D411yescharacteristic
31C2×C22⋊D271—21no—
32C2×C22⋊D271—21no—
33C2×C22⋊D271—21no—
34C22×Dic271—21no—
35C22×Dic271—21no—
36C22×Dic271D411yescharacteristic
37C22×Dic271D411yescharacteristic
38C2×C4×D271—21no—
39C2×C271⋊C8—21no—
40C4⋊Dic271—21no—
41C4⋊Dic271—21no—
42C4⋊Dic271D411yescharacteristic
43Dic271⋊C4—21no—
44Dic271⋊C4—21no—
45Dic271⋊C4—21no—
46C22⋊Dic271—21no—
47(C2×D271)⋊C4—21no—
48(C2×D271)⋊C4—21no—
49C2×D4×C271—21no—
50C2×D4×C271D411yescharacteristic
51C22×C4×C271—21no—
52C22×C4×C271C2311yescharacteristic
53C2×C8×C271—21no—
54C4⋊C4×C271D411yescharacteristic
55C22⋊C4×C271—21no—
56C22×D271—21no—
57Dic542—42no—
58D1084—41no—
59C22⋊D271—42no—
60C22⋊D271—41no—
61C22⋊D271—41no—
62C22⋊D271—81no—
63C2×Dic271—21no—
64C2×Dic271—21no—
65C2×Dic271—21no—
66C2×Dic271—21no—
67C2×Dic271—42no—
68C2×Dic271—41no—
69C2×Dic271—42no—
70C2×Dic271—41no—
71C2×Dic271SD1612yes—
72C4×D271—42no—
73C271⋊C8—41no—
74C23×C271—21no—
75C23×C271C2×D411yescharacteristic
76D4×C271—41no—
77D4×C271—41no—
78D4×C271—41no—
79C2×C4×C271—21no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271—41no—
83C2×C4×C271—41no—
84C2×C4×C271C2×D411yescharacteristic
85C2×C4×C271C2×D411yescharacteristic
86C8×C271—41no—
87C22×C271—21no—
88C22×C271—21no—
89C22×C271—41no—
90C22×C271—41no—
91C22×C271—41no—
92C22×C271#12011yescharacteristic
93C22×C271C2×SD1611yescharacteristic
94C22×C271AGL(1,ℤ/8)11yescharacteristic
95C2×D271—41no—
96C2×D271—41no—
97C2×D271—41no—
98C4×C271—21no—
99C4×C271—21no—
100C4×C271—41no—
101C4×C271—81no—
102Dic271—22no—
103Dic271—41no—
104Dic271—41no—
105Dic271—81no—
106C2×C271—21no—
107C2×C271—21no—
108C2×C271—81no—
109C2×C271#45011yescharacteristic
110C2×C271#45411yescharacteristic
111C2×C271#45811yescharacteristic
112D271—81no—
113C271#161611yescharacteristic, complemented, Sylow
114#1616—2711nomaximal, Sylow
115#537—2711no—
116#481—2711no—
117#481—2711no—
118#400—2711no—
119#327—2711no—
120#327D27111yescharacteristic, complemented
121#324—2711no—
122C2×#41—5421no—
123C22⋊Q8—5421no—
124C22⋊Q8—5421no—
125#121—2711no—
126#121—5421no—
127#121—5421no—
128#121—5421no—
129#121C2×D27111yescharacteristic
130C2×C4⋊C4—2711no—
131C2×C4⋊C4—2711no—
132#106—5421no—
133#106—5421no—
134#102—5421no—
135#102—5421no—
136C22⋊C8—2711no—
137C22⋊C8C2×D27111yescharacteristic
138#95—5421no—
139#95—5421no—
140#95C2×D27111yescharacteristic
141#41—10842no—
142#41—10842no—
143C2×Q8—5421no—
144C2×D4—21no—
145C2×D4—2711no—
146C2×D4—5421no—
147C2×D4—5421no—
148C2×D4—5421no—
149C2×D4D108411yescharacteristic
150C22×C4—21no—
151C22×C4—2711no—
152C22×C4—2711no—
153C22×C4—5421no—
154C22×C4—5421no—
155C22×C4—5421no—
156C22×C4C22×D27111yescharacteristic
157C2×C8—21no—
158C2×C8—5421no—
159C4⋊C4—2711no—
160C4⋊C4—5421no—
161C4⋊C4—5421no—
162C4⋊C4—5421no—
163C4⋊C4—5421no—
164C4⋊C4—5421no—
165C4⋊C4D108411yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—5421no—
168C22⋊C4—5421no—
169C22⋊C4—5421no—
170C23—21no—
171C23—5421no—
172C23D4×D27111yescharacteristic
173Q8—10842no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—10841no—
178D4—10842no—
179D4—10841no—
180D4—10841no—
181D4—21681no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2712no—
188C2×C4—5421no—
189C2×C4—5421no—
190C2×C4—5421no—
191C2×C4—5421no—
192C2×C4—10842no—
193C2×C4—10842no—
194C2×C4—10841no—
195C2×C4—10842no—
196C2×C4—10841no—
197C2×C4D4×D27111yescharacteristic
198C2×C4C2×D108411yescharacteristic
199C8—41no—
200C8—10841no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—10841no—
207C22—10841no—
208C22—10841no—
209C22#2257613011yescharacteristic
210C22SD16×D27111yescharacteristic
211C22#2257618411yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—5422no—
217C4—10841no—
218C4—10841no—
219C4—21681no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—21681no—
224C2#3177404911yescharacteristic
225C2#3177407111yescharacteristic
226C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#327×C271C2
socleC22×C271#120
prime radical#31774474C2
hypercenter#327D271
hypocenterC271#1616
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #31774216C2semidirect
3. #31774213C2semidirect
4. #31773837C2semidirect
5. #31773630C2semidirect
6. #31773622C2semidirect
7. #327×C271C2semidirect
113. C271#1616semidirect
120. #327D271semidirect

Maximal subgroups

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

subgroupindex
#317744742
#317742162
#317742132
#317738372
#317736302
#317736222
#327×C2712
#1616271