Finite Group Databasebeta

← #46835368

Subgroups — #46835368

Order 34688.

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

normal subgroups only

This group has 254 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#31774211C211yesmaximal, characteristic, complemented
3#31774208C211yesmaximal, characteristic, complemented
4#31773829C211yesmaximal, 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⋊Dic271—21no—
16C2×C4⋊Dic271C2211yescharacteristic
17C2×Dic271⋊C4C2211yescharacteristic
18#22576094—21no—
19#22576093—21no—
20#22576082—21no—
21#22576081—21no—
22C22⋊(C271⋊C8)C2211yescharacteristic
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⋊Dic271—42no—
43C4⋊Dic271D411yescharacteristic
44Dic271⋊C4—21no—
45Dic271⋊C4—21no—
46Dic271⋊C4—21no—
47C22⋊Dic271—21no—
48(C2×D271)⋊C4—21no—
49(C2×D271)⋊C4—21no—
50C2×D4×C271—21no—
51C2×D4×C271D411yescharacteristic
52C22×C4×C271—21no—
53C22×C4×C271C2311yescharacteristic
54C2×C8×C271—21no—
55C4⋊C4×C271D411yescharacteristic
56C22⋊C4×C271—21no—
57C22×D271—21no—
58Dic542—42no—
59D1084—41no—
60C22⋊D271—42no—
61C22⋊D271—41no—
62C22⋊D271—41no—
63C22⋊D271—81no—
64C2×Dic271—21no—
65C2×Dic271—21no—
66C2×Dic271—21no—
67C2×Dic271—21no—
68C2×Dic271—42no—
69C2×Dic271—41no—
70C2×Dic271—41no—
71C2×Dic271—41no—
72C2×Dic271—41no—
73C2×Dic271D812yes—
74C4×D271—42no—
75C271⋊C8—41no—
76C23×C271—21no—
77C23×C271C2×D411yescharacteristic
78D4×C271—41no—
79D4×C271—41no—
80D4×C271—41no—
81C2×C4×C271—22no—
82C2×C4×C271—21no—
83C2×C4×C271—41no—
84C2×C4×C271—41no—
85C2×C4×C271—41no—
86C2×C4×C271C2×D411yescharacteristic
87C2×C4×C271C2×D411yescharacteristic
88C8×C271—41no—
89C22×C271—21no—
90C22×C271—21no—
91C22×C271—41no—
92C22×C271—41no—
93C22×C271—41no—
94C22×C271#12011yescharacteristic
95C22×C271C2×D811yescharacteristic
96C22×C271AGL(1,ℤ/8)11yescharacteristic
97C2×D271—41no—
98C2×D271—41no—
99C2×D271—41no—
100C4×C271—21no—
101C4×C271—21no—
102C4×C271—42no—
103C4×C271—41no—
104Dic271—22no—
105Dic271—41no—
106Dic271—41no—
107Dic271—81no—
108C2×C271—21no—
109C2×C271—21no—
110C2×C271—81no—
111C2×C271#44711yescharacteristic
112C2×C271#45611yescharacteristic
113C2×C271#45811yescharacteristic
114D271—81no—
115C271#161011yescharacteristic, complemented, Sylow
116#1610—2711nomaximal, Sylow
117#537—2711no—
118#480—2711no—
119#480—2711no—
120#398—2711no—
121#327—2711no—
122#327D27111yescharacteristic, complemented
123#324—2711no—
124C2×#41—5421no—
125C22⋊Q8—5421no—
126C22⋊Q8—5421no—
127#121—2711no—
128#121—5421no—
129#121—5421no—
130#121—5421no—
131#121C2×D27111yescharacteristic
132C2×C4⋊C4—2711no—
133C2×C4⋊C4—2711no—
134C2×C4⋊C4—5421no—
135#107—5421no—
136#107—5421no—
137#102—5421no—
138#102—5421no—
139C22⋊C8—2711no—
140C22⋊C8C2×D27111yescharacteristic
141#95—5421no—
142#95C2×D27111yescharacteristic
143#41—10842no—
144#41—10842no—
145C2×Q8—5421no—
146C2×D4—21no—
147C2×D4—2711no—
148C2×D4—5421no—
149C2×D4—5421no—
150C2×D4—5421no—
151C2×D4D108411yescharacteristic
152C22×C4—21no—
153C22×C4—2711no—
154C22×C4—2711no—
155C22×C4—5421no—
156C22×C4—5421no—
157C22×C4—5421no—
158C22×C4C22×D27111yescharacteristic
159C2×C8—21no—
160C2×C8—5421no—
161C4⋊C4—2711no—
162C4⋊C4—5421no—
163C4⋊C4—5421no—
164C4⋊C4—5421no—
165C4⋊C4—5421no—
166C4⋊C4—5421no—
167C4⋊C4—10842no—
168C4⋊C4D108411yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—5421no—
171C22⋊C4—5421no—
172C22⋊C4—5421no—
173C23—21no—
174C23—5421no—
175C23D4×D27111yescharacteristic
176Q8—10842no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—10841no—
181D4—10842no—
182D4—10841no—
183D4—10841no—
184D4—21681no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—2712no—
191C2×C4—5421no—
192C2×C4—5421no—
193C2×C4—5421no—
194C2×C4—5421no—
195C2×C4—10842no—
196C2×C4—10842no—
197C2×C4—10841no—
198C2×C4—10841no—
199C2×C4—10841no—
200C2×C4—10841no—
201C2×C4D4×D27111yescharacteristic
202C2×C4C2×D108411yescharacteristic
203C8—41no—
204C8—10841no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—10841no—
211C22—10841no—
212C22—10841no—
213C22#2257613011yescharacteristic
214C22D8×D27111yescharacteristic
215C22#2257618711yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—5422no—
221C4—10841no—
222C4—10841no—
223C4—21681no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—21681no—
228C2#3177403211yescharacteristic
229C2#3177408111yescharacteristic
230C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#327×C271C2
socleC22×C271#120
prime radical#31774474C2
hypercenter#327D271
hypocenterC271#1610
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. #31774211C2semidirect
3. #31774208C2semidirect
4. #31773829C2semidirect
5. #31773630C2semidirect
6. #31773622C2semidirect
7. #327×C271C2semidirect
115. C271#1610semidirect
122. #327D271semidirect

Maximal subgroups

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

subgroupindex
#317744742
#317742112
#317742082
#317738292
#317736302
#317736222
#327×C2712
#1610271