Finite Group Databasebeta

← #46845893

Subgroups — #46845893

Order 34688.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#22576193C211yesmaximal, characteristic
2C22×#16864644C211yesmaximal, characteristic, complemented
3#31774471C211yesmaximal, characteristic
4#31774374C211yesmaximal, characteristic, complemented
5#31774122C214yesmaximal, complemented
6#31774106C214yesmaximal, complemented
7#31773925C211yesmaximal, characteristic, complemented
8#31773891C211yesmaximal, characteristic
9#419×C271C211yesmaximal, characteristic, complemented
10C2×#16864654C2211yescharacteristic
11C22×Dic542C2211yescharacteristic
12#22576193—24no—
13C2×Dic1084C2212yes—
14C2×#16864644—24no—
15C2×#16864644C2214yes—
16C2×#16864643C2212yes—
17C22×C271⋊C8C2211yescharacteristic
18#22576158C2212yes—
19#22576149C2212yes—
20#22576143—22no—
21#22576136—22no—
22C4×Dic542—22no—
23#22576125C2214yes—
24#22576119C2211yescharacteristic
25#22576116—22no—
26Dic271⋊C8C2214yes—
27#22576083C2212yes—
28#22576080C2212yes—
29C22×Q8×C271C2211yescharacteristic
30C2×M16×C271C2211yescharacteristic
31#117×C271C2211yescharacteristic
32#103×C271C2214yes—
33#16864654—22no—
34#16864654—22no—
35C2×Dic542—21no—
36C2×Dic542—22no—
37C2×Dic542D414yes—
38C2×Dic542C2311yescharacteristic
39C2×Dic542C2312yes—
40C2×D1084C2311yescharacteristic
41C2×C22⋊D271—21no—
42C22×Dic271—21no—
43C2×C4×D271—21no—
44Dic1084—24no—
45#16864644—28no—
46#16864643—24no—
47C2×C271⋊C8—22no—
48C2×C271⋊C8C2312yes—
49C4⋊Dic271D412yes—
50Dic271⋊C4—24no—
51Dic271⋊C4—24no—
52C22⋊Dic271D412yes—
53(C2×D271)⋊C4—24no—
54C4×Dic271—24no—
55C4×Dic271C2312yes—
56C2×Q8×C271—21no—
57C2×Q8×C271—22no—
58C2×Q8×C271—22no—
59C2×Q8×C271C2312yes—
60C22×C4×C271—21no—
61C22×C4×C271C2311yescharacteristic
62M16×C271—22no—
63C2×C8×C271C2312yes—
64C4⋊C4×C271C2312yes—
65C22⋊C4×C271—21no—
66C42×C271—21no—
67C22×D271—21no—
68Dic542—24no—
69Dic542—22no—
70Dic542—21no—
71Dic542—21no—
72Dic542C2×D414yes—
73D1084—21no—
74D1084—21no—
75C22⋊D271—42no—
76C2×Dic271—24no—
77C2×Dic271—22no—
78C2×Dic271—21no—
79C2×Dic271C2×D414yes—
80C4×D271—42no—
81C271⋊C8—22no—
82C23×C271C2×D411yescharacteristic
83Q8×C271—22no—
84Q8×C271—22no—
85Q8×C271—42no—
86C2×C4×C271—22no—
87C2×C4×C271—22no—
88C2×C4×C271—42no—
89C2×C4×C271C2×D411yescharacteristic
90C2×C4×C271C2×D412yes—
91C2×C4×C271#4112yes—
92C2×C4×C271C2411yescharacteristic
93C8×C271—22no—
94C22×C271—22no—
95C22×C271#12112yes—
96C22×C271C22×D411yescharacteristic
97C2×D271—41no—
98C2×D271—42no—
99C4×C271—42no—
100C4×C271—42no—
101C4×C271#12112yes—
102C4×C271C22×D411yescharacteristic
103C4×C271C2×#4111yescharacteristic
104Dic271—24no—
105Dic271—24no—
106Dic271—41no—
107C2×C271—22no—
108C2×C271C2×#12111yescharacteristic
109C2×C271#57812yes—
110D271—81no—
111C271#304411yescharacteristic, complemented, Sylow
112#3044—2711nomaximal, Sylow
113C2×#137—2711no—
114C22×Q16—2711no—
115#536—2711no—
116#519—2711no—
117#464—2714no—
118#462—2714no—
119#424—2711no—
120#419D27111yescharacteristic, complemented
121#416—2711no—
122C2×#41—2711no—
123C22×Q8—2711no—
124C22×Q8C2×D27111yescharacteristic
125#137—5424no—
126C2×Q16—2714no—
127C2×Q16—2712no—
128C2×Q16—5424no—
129C2×SD16—2712no—
130C2×M16C2×D27111yescharacteristic
131C22×C8—2711no—
132C4⋊Q8—2712no—
133#124—2712no—
134#123—5422no—
135C22⋊Q8—5422no—
136C4×Q8—2714no—
137C4×Q8—5422no—
138#117—2711no—
139#117—5422no—
140#117C2×D27111yescharacteristic
141C4⋊C8—2714no—
142#103—2712no—
143#103C2×D27114yes—
144#102—2712no—
145#41—5422no—
146#41—5422no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—2714no—
151C2×Q8—2712no—
152C2×Q8—2711no—
153C2×Q8—5421no—
154C2×Q8—5422no—
155C2×Q8C22×D27112yes—
156C2×D4—2711no—
157C2×D4—5421no—
158C22×C4—21no—
159C22×C4—5421no—
160C22×C4—5421no—
161C22×C4C22×D27111yescharacteristic
162Q16—5424no—
163Q16—5428no—
164SD16—5424no—
165M16—22no—
166C2×C8—2712no—
167C2×C8—5422no—
168C2×C8C22×D27112yes—
169C4⋊C4—2712no—
170C4⋊C4—5424no—
171C4⋊C4—5424no—
172C4⋊C4C22×D27112yes—
173C22⋊C4—21no—
174C22⋊C4—2712no—
175C22⋊C4—5424no—
176C42—21no—
177C42—2712no—
178C42—5424no—
179C23—5421no—
180C23D4×D27111yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—2714no—
185Q8—5424no—
186Q8—5422no—
187Q8—5421no—
188Q8—5421no—
189D4—5421no—
190D4—5421no—
191D4—10842no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—2714no—
196C2×C4—5424no—
197C2×C4—5422no—
198C2×C4—5421no—
199C2×C4—10842no—
200C2×C4D4×D27111yescharacteristic
201C2×C4D4×D27112yes—
202C2×C4#1686465412yes—
203C2×C4C23×D27111yescharacteristic
204C8—22no—
205C8—5422no—
206C22—22no—
207C22—10841no—
208C22—10842no—
209C22#2257613312yes—
210C22C2×D4×D27111yescharacteristic
211C4—42no—
212C4—42no—
213C4—5424no—
214C4—5424no—
215C4—10841no—
216C4#2257613312yes—
217C4C2×D4×D27111yescharacteristic
218C4C2×#1686465411yescharacteristic
219C2—22no—
220C2—21681no—
221C2C2×#2257613311yescharacteristic
222C2#3177471312yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#419×C271C2
socleC22×C271C22×D4
prime radicalC2×#16864654C22
hypercenter#419D271
hypocenterC271#3044
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#16864644C2semidirect
4. #31774374C2semidirect
5. #31774122C2semidirect
6. #31774106C2semidirect
7. #31773925C2semidirect
9. #419×C271C2semidirect
111. C271#3044semidirect
120. #419D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C22×#168646442
#317744712
#317743742
#317741222
#317741062
#317739252
#317738912
#419×C2712
#3044271