Finite Group Databasebeta

← #46835376

Subgroups — #46835376

Order 34688.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774485C211yesmaximal, characteristic
2#31774227C211yesmaximal, characteristic, complemented
3#31774210C211yesmaximal, characteristic, complemented
4#31773633C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773621C211yesmaximal, characteristic, complemented
7#397×C271C211yesmaximal, characteristic, complemented
8#22576153—21no—
9#22576146—21no—
10#22576142—21no—
11#22576139C2211yescharacteristic
12#22576134C2211yescharacteristic
13#22576133—21no—
14#22576129—21no—
15C4×C22⋊D271—21no—
16#22576093—21no—
17#22576091—21no—
18#22576083—21no—
19#22576080—21no—
20C22⋊(C271⋊C8)C2211yescharacteristic
21C22⋊(C271⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C271C2211yescharacteristic
23C2×C22⋊C4×C271—21no—
24C2×C22⋊C4×C271C2211yescharacteristic
25#95×C271—21no—
26#95×C271C2211yescharacteristic
27C2×Dic542D411yescharacteristic
28C2×D1084D411yescharacteristic
29C2×C22⋊D271—21no—
30C2×C22⋊D271—21no—
31C2×C22⋊D271—21no—
32C2×C4×D271—21no—
33C2×C271⋊C8—21no—
34C2×C271⋊C8—21no—
35C4⋊Dic271D411yescharacteristic
36C4⋊Dic271D411yescharacteristic
37Dic271⋊C4—21no—
38Dic271⋊C4—21no—
39C22⋊Dic271—21no—
40C22⋊Dic271—21no—
41C22⋊Dic271—21no—
42(C2×D271)⋊C4—21no—
43(C2×D271)⋊C4—21no—
44C4×Dic271—21no—
45C24×C271D411yescharacteristic
46C22×C4×C271—21no—
47C22×C4×C271—21no—
48C22×C4×C271D411yescharacteristic
49C22×C4×C271C2311yescharacteristic
50C4⋊C4×C271—21no—
51C4⋊C4×C271—21no—
52C22⋊C4×C271—21no—
53C22⋊C4×C271—21no—
54C22⋊C4×C271—42no—
55C22×D271—21no—
56Dic542—41no—
57D1084—41no—
58C22⋊D271—41no—
59C22⋊D271—41no—
60C22⋊D271—81no—
61C22⋊D271—81no—
62C2×Dic271—21no—
63C2×Dic271—21no—
64C2×Dic271—21no—
65C4×D271—81no—
66C271⋊C8—41no—
67C271⋊C8—41no—
68C23×C271—21no—
69C23×C271—21no—
70C23×C271—21no—
71C23×C271—41no—
72C23×C271—41no—
73C23×C271C2×D411yescharacteristic
74C2×C4×C271—21no—
75C2×C4×C271—41no—
76C2×C4×C271—41no—
77C2×C4×C271—41no—
78C2×C4×C271—41no—
79C2×C4×C271—41no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271—41no—
83C2×C4×C271C2×D411yescharacteristic
84C2×C4×C271C2×D411yescharacteristic
85C22×C271—21no—
86C22×C271—21no—
87C22×C271—42no—
88C22×C271—41no—
89C22×C271—41no—
90C22×C271—41no—
91C22×C271—81no—
92C22×C271#12011yescharacteristic
93C22×C271#13511yescharacteristic
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—
102C4×C271—81no—
103Dic271—41no—
104Dic271—41no—
105Dic271—41no—
106C2×C271—21no—
107C2×C271—21no—
108C2×C271—81no—
109C2×C271#44911yescharacteristic
110C2×C271#45511yescharacteristic
111C2×C271#45811yescharacteristic
112D271—81no—
113C271#161211yescharacteristic, complemented, Sylow
114#1612—2711nomaximal, Sylow
115#538—2711no—
116#483—2711no—
117#480—2711no—
118#397D27111yescharacteristic, complemented
119#328—2711no—
120#327—2711no—
121#323—2711no—
122#126—5421no—
123#124—5421no—
124#123—5421no—
125C22⋊Q8—2711no—
126#121—2711no—
127#121—5421no—
128#120—5421no—
129C4×D4—5421no—
130C2×C4⋊C4C2×D27111yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4C2×D27111yescharacteristic
133#107—5421no—
134#106—5421no—
135#103—5421no—
136#102—5421no—
137C22⋊C8—2711no—
138C22⋊C8—2711no—
139#95—21no—
140#95C2×D27111yescharacteristic
141C24C22⋊D27111yescharacteristic
142C2×Q8—2711no—
143C2×D4—2711no—
144C2×D4—5421no—
145C2×D4—5421no—
146C2×D4—5421no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—5421no—
150C22×C4C22⋊D27111yescharacteristic
151C22×C4C22×D27111yescharacteristic
152C2×C8—5421no—
153C2×C8—5421no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2711no—
157C4⋊C4—2711no—
158C4⋊C4—5421no—
159C4⋊C4—5421no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—42no—
163C22⋊C4—5421no—
164C22⋊C4—5421no—
165C22⋊C4—5421no—
166C22⋊C4—5421no—
167C22⋊C4—5421no—
168C42—5421no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—5421no—
175C23C2×C22⋊D27111yescharacteristic
176Q8—10841no—
177D4—10841no—
178D4—10841no—
179D4—10841no—
180D4—21681no—
181D4—21681no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—5421no—
192C2×C4—5421no—
193C2×C4—5421no—
194C2×C4—21681no—
195C2×C4D4×D27111yescharacteristic
196C2×C4D4×D27111yescharacteristic
197C8—10841no—
198C8—10841no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—10841no—
207C22—10841no—
208C22—10841no—
209C22#2257612811yescharacteristic
210C22#2257618011yescharacteristic
211C22#2257618611yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—81no—
217C4—10841no—
218C4—10841no—
219C4—10841no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—21681no—
224C2#3177404011yescharacteristic
225C2#3177407411yescharacteristic
226C2#3177408611yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#397×C271C2
socleC22×C271#120
prime radical#31774485C2
hypercenter#397D271
hypocenterC271#1612
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #31774227C2semidirect
3. #31774210C2semidirect
4. #31773633C2semidirect
5. #31773630C2semidirect
6. #31773621C2semidirect
7. #397×C271C2semidirect
113. C271#1612semidirect
118. #397D271semidirect

Maximal subgroups

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

subgroupindex
#317744852
#317742272
#317742102
#317736332
#317736302
#317736212
#397×C2712
#1612271