Finite Group Databasebeta

← #46835415

Subgroups — #46835415

Order 34688.

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

normal subgroups only

This group has 240 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#31774487C211yesmaximal, characteristic
2#31774230C211yesmaximal, characteristic, complemented
3#31774216C211yesmaximal, characteristic, complemented
4#31773834C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773620C211yesmaximal, characteristic, complemented
7#328×C271C211yesmaximal, characteristic, complemented
8#22576155—21no—
9#22576149—21no—
10#22576145—21no—
11#22576135—21no—
12#22576134C2211yescharacteristic
13#22576130—21no—
14(C22⋊D271)⋊C4—21no—
15C2×C4⋊Dic271—21no—
16C2×C4⋊Dic271C2211yescharacteristic
17C2×(C2×D271)⋊C4—21no—
18C2×(C2×D271)⋊C4C2211yescharacteristic
19#22576093—21no—
20#22576092—21no—
21#22576084—21no—
22#22576082—21no—
23C22⋊(C271⋊C8)C2211yescharacteristic
24C22⋊Q8×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27C23×D271D411yescharacteristic
28C2×D1084—21no—
29C2×D1084—21no—
30C2×D1084D411yescharacteristic
31C2×C22⋊D271—21no—
32C22×Dic271—21no—
33C22×Dic271D411yescharacteristic
34C2×C4×D271—21no—
35C2×C271⋊C8—21no—
36C4⋊Dic271—21no—
37C4⋊Dic271—21no—
38C4⋊Dic271—42no—
39C4⋊Dic271D411yescharacteristic
40Dic271⋊C4—21no—
41(C2×D271)⋊C4—21no—
42(C2×D271)⋊C4—21no—
43(C2×D271)⋊C4—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46(C2×D271)⋊C4—21no—
47(C2×D271)⋊C4—42no—
48C4×Dic271—21no—
49C2×Q8×C271D411yescharacteristic
50C22×C4×C271—21no—
51C22×C4×C271C2311yescharacteristic
52C2×C8×C271—21no—
53C4⋊C4×C271—21no—
54C4⋊C4×C271D411yescharacteristic
55C22⋊C4×C271—21no—
56C22×D271—21no—
57C22×D271—21no—
58C22×D271—21no—
59C22×D271—21no—
60C22×D271—41no—
61C22×D271—41no—
62D1084—41no—
63D1084—81no—
64D1084—81no—
65C22⋊D271—41no—
66C22⋊D271—41no—
67C2×Dic271—21no—
68C2×Dic271—21no—
69C2×Dic271—41no—
70C2×Dic271—41no—
71C2×Dic271—41no—
72C2×Dic271—41no—
73C4×D271—81no—
74C271⋊C8—41no—
75C23×C271C2×D411yescharacteristic
76Q8×C271—41no—
77C2×C4×C271—22no—
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271—41no—
83C2×C4×C271C2×D411yescharacteristic
84C2×C4×C271C2×D411yescharacteristic
85C8×C271—41no—
86C22×C271—21no—
87C22×C271—21no—
88C22×C271#12011yescharacteristic
89C22×C271#13511yescharacteristic
90C22×C271#13711yescharacteristic
91C2×D271—42no—
92C2×D271—41no—
93C2×D271—41no—
94C2×D271—41no—
95C2×D271—41no—
96C2×D271—41no—
97C2×D271—41no—
98C2×D271—81no—
99C4×C271—21no—
100C4×C271—21no—
101C4×C271—42no—
102C4×C271—41no—
103C4×C271—41no—
104Dic271—41no—
105Dic271—41no—
106Dic271—81no—
107C2×C271—21no—
108C2×C271—21no—
109C2×C271#45211yescharacteristic
110C2×C271#45311yescharacteristic
111C2×C271#45811yescharacteristic
112D271—81no—
113D271—81no—
114C271#161811yescharacteristic, complemented, Sylow
115#1618—2711nomaximal, Sylow
116#538—2711no—
117#484—2711no—
118#481—2711no—
119#399—2711no—
120#328D27111yescharacteristic, complemented
121#327—2711no—
122#323—2711no—
123#126—5421no—
124#124—5421no—
125#123—5421no—
126C22⋊Q8C2×D27111yescharacteristic
127#121—2711no—
128#121—5421no—
129#120—5421no—
130C4×D4—5421no—
131C2×C4⋊C4—2711no—
132C2×C4⋊C4—5421no—
133C2×C22⋊C4—2711no—
134C2×C22⋊C4—5421no—
135#107—5421no—
136#106—5421no—
137#103—5421no—
138#102—5421no—
139C22⋊C8—2711no—
140C22⋊C8C2×D27111yescharacteristic
141#95C2×D27111yescharacteristic
142C24—2711no—
143C2×Q8D108411yescharacteristic
144C2×D4—2711no—
145C2×D4—5421no—
146C2×D4—5421no—
147C2×D4—5421no—
148C22×C4—21no—
149C22×C4—2711no—
150C22×C4—5421no—
151C22×C4—5421no—
152C22×C4C22×D27111yescharacteristic
153C2×C8—21no—
154C2×C8—5421no—
155C4⋊C4—21no—
156C4⋊C4—2711no—
157C4⋊C4—5421no—
158C4⋊C4—5421no—
159C4⋊C4—5421no—
160C4⋊C4—10842no—
161C4⋊C4D108411yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—5421no—
164C22⋊C4—5421no—
165C22⋊C4—5421no—
166C22⋊C4—5421no—
167C22⋊C4—5421no—
168C22⋊C4—5421no—
169C22⋊C4—10842no—
170C42—5421no—
171C23—5421no—
172C23—5421no—
173C23—5421no—
174C23—5421no—
175C23—10841no—
176C23—10841no—
177C23D4×D27111yescharacteristic
178Q8—41no—
179D4—10841no—
180D4—10841no—
181D4—10841no—
182D4—21681no—
183D4—21681no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—5421no—
191C2×C4—5421no—
192C2×C4—10841no—
193C2×C4—10841no—
194C2×C4—10841no—
195C2×C4—10841no—
196C2×C4—21681no—
197C2×C4D4×D27111yescharacteristic
198C2×C4C2×D108411yescharacteristic
199C8—41no—
200C8—10841no—
201C22—21no—
202C22—21no—
203C22—10842no—
204C22—10841no—
205C22—10841no—
206C22—10841no—
207C22—10841no—
208C22—10841no—
209C22—10841no—
210C22—21681no—
211C22#2257613011yescharacteristic
212C22#2257618111yescharacteristic
213C22#2257619211yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—10841no—
220C4—10841no—
221C4—21681no—
222C2—21no—
223C2—21no—
224C2—21681no—
225C2—21681no—
226C2#3177406211yescharacteristic
227C2#3177406611yescharacteristic
228C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#328×C271C2
socleC22×C271#120
prime radical#31774487C2
hypercenter#328D271
hypocenterC271#1618
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #31774230C2semidirect
3. #31774216C2semidirect
4. #31773834C2semidirect
5. #31773630C2semidirect
6. #31773620C2semidirect
7. #328×C271C2semidirect
114. C271#1618semidirect
120. #328D271semidirect

Maximal subgroups

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

subgroupindex
#317744872
#317742302
#317742162
#317738342
#317736302
#317736202
#328×C2712
#1618271