Finite Group Databasebeta

← #46846018

Subgroups — #46846018

Order 34688.

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

normal subgroups only

This group has 460 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
1C22×#16864644C211yesmaximal, characteristic, complemented
2C2×#22576168C211yesmaximal, characteristic
3#31774473C211yesmaximal, characteristic
4#31774471C211yesmaximal, characteristic
5#31774154C214yesmaximal, complemented
6#31774137C214yesmaximal, complemented
7#31773933C211yesmaximal, characteristic, complemented
8#31773907C211yesmaximal, characteristic, complemented
9#419×C271C211yesmaximal, characteristic, complemented
10C2×#16864654C2211yescharacteristic
11C22×Dic542C2211yescharacteristic
12C2×#16864644—24no—
13C2×#16864644C2212yes—
14C2×#16864644C2212yes—
15#22576168—24no—
16C2×#16864637C2212yes—
17C22×C271⋊C8C2211yescharacteristic
18C2×C8×D271C2212yes—
19#22576160—21no—
20#22576158—21no—
21#22576149—21no—
22#22576148—21no—
23#22576144—22no—
24#22576143—22no—
25#22576140C2214yes—
26#22576136C2214yes—
27#22576119C2211yescharacteristic
28#22576093C2212yes—
29#22576091C2212yes—
30#22576085C2214yes—
31C22×Q8×C271C2211yescharacteristic
32C2×M16×C271C2211yescharacteristic
33#117×C271C2211yescharacteristic
34#103×C271C2214yes—
35#16864654—22no—
36#16864654—22no—
37C2×Dic542—21no—
38C2×Dic542—22no—
39C2×Dic542—22no—
40C2×Dic542D411yescharacteristic
41C2×Dic542C2312yes—
42C2×D1084D411yescharacteristic
43C2×C22⋊D271D412yes—
44C22×Dic271—21no—
45C2×C4×D271C2312yes—
46#16864644—24no—
47#16864644—24no—
48#16864637—24no—
49C2×C271⋊C8D414yes—
50C2×C271⋊C8C2311yescharacteristic
51C2×C271⋊C8C2311yescharacteristic
52C8×D271—24no—
53C4⋊Dic271—24no—
54C4⋊Dic271C2312yes—
55Dic271⋊C4—24no—
56C22⋊Dic271—21no—
57(C2×D271)⋊C4—24no—
58(C2×D271)⋊C4—24no—
59C4×Dic271—21no—
60C2×Q8×C271—21no—
61C2×Q8×C271—22no—
62C2×Q8×C271—22no—
63C2×Q8×C271C2312yes—
64C22×C4×C271—21no—
65C22×C4×C271C2311yescharacteristic
66M16×C271—22no—
67C2×C8×C271C2312yes—
68C4⋊C4×C271C2312yes—
69C22⋊C4×C271—21no—
70C42×C271—21no—
71C22×D271C2×D412yes—
72Dic542—22no—
73Dic542—22no—
74Dic542—22no—
75Dic542—42no—
76D1084—22no—
77C22⋊D271—24no—
78C2×Dic271—22no—
79C2×Dic271—22no—
80C2×Dic271—42no—
81C2×Dic271C2×D412yes—
82C4×D271—22no—
83C4×D271—22no—
84C271⋊C8C2×D412yes—
85C271⋊C8C2×D412yes—
86C23×C271C2×D411yescharacteristic
87Q8×C271—22no—
88Q8×C271—22no—
89Q8×C271—42no—
90C2×C4×C271—22no—
91C2×C4×C271—22no—
92C2×C4×C271—42no—
93C2×C4×C271C2×D411yescharacteristic
94C2×C4×C271C2×D412yes—
95C2×C4×C271#4112yes—
96C2×C4×C271C2411yescharacteristic
97C8×C271—22no—
98C22×C271—22no—
99C22×C271#12112yes—
100C22×C271C22×D411yescharacteristic
101C2×D271—22no—
102C2×D271—24no—
103C4×C271—42no—
104C4×C271—42no—
105C4×C271#12112yes—
106C4×C271C22×D411yescharacteristic
107C4×C271C2×#4111yescharacteristic
108Dic271—22no—
109Dic271—42no—
110Dic271—42no—
111C2×C271—22no—
112C2×C271C2×#12111yescharacteristic
113C2×C271#57812yes—
114D271—42no—
115C271#305911yescharacteristic, complemented, Sylow
116#3059—2711nomaximal, Sylow
117C22×Q16—2711no—
118C2×#131—2711no—
119#536—2711no—
120#536—2711no—
121#470—2714no—
122#467—2714no—
123#427—2711no—
124#419—2711no—
125#419D27111yescharacteristic, complemented
126C2×#41—2711no—
127C22×Q8—2711no—
128C22×Q8C2×D27111yescharacteristic
129C2×Q16—2712no—
130C2×Q16—2712no—
131C2×Q16—5424no—
132#131—5424no—
133C2×M16—2712no—
134C2×M16C2×D27111yescharacteristic
135C22×C8—2711no—
136C22×C8—2712no—
137C4⋊Q8—5421no—
138C4⋊Q8—5421no—
139#124—5421no—
140#124—5421no—
141#123—5422no—
142#123—5422no—
143C22⋊Q8—2714no—
144C22⋊Q8—2714no—
145#117—2711no—
146#117C2×D27111yescharacteristic
147#107—2712no—
148#106—2712no—
149#103—2714no—
150#103C2×D27114yes—
151#41—5422no—
152#41—5422no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—2711no—
157C2×Q8—2712no—
158C2×Q8—5421no—
159C2×Q8—5422no—
160C2×Q8—5422no—
161C2×Q8C22×D27112yes—
162C2×D4—2712no—
163C2×D4—2711no—
164C22×C4—21no—
165C22×C4—2712no—
166C22×C4—5421no—
167C22×C4C22×D27111yescharacteristic
168Q16—5424no—
169Q16—5424no—
170M16—22no—
171M16—5424no—
172C2×C8—2714no—
173C2×C8—2711no—
174C2×C8—2711no—
175C2×C8—5424no—
176C2×C8C22×D27112yes—
177C4⋊C4—2712no—
178C4⋊C4—5424no—
179C4⋊C4—5424no—
180C4⋊C4C22×D27112yes—
181C22⋊C4—21no—
182C22⋊C4—5424no—
183C22⋊C4—5424no—
184C22⋊C4—5421no—
185C42—21no—
186C42—5421no—
187C23—2712no—
188C23D4×D27111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—5422no—
193Q8—5422no—
194Q8—5422no—
195Q8—10842no—
196D4—5424no—
197D4—5422no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—2712no—
202C2×C4—5422no—
203C2×C4—5422no—
204C2×C4—5422no—
205C2×C4—5422no—
206C2×C4—10842no—
207C2×C4D4×D27111yescharacteristic
208C2×C4D4×D27112yes—
209C2×C4#1686465412yes—
210C2×C4C23×D27111yescharacteristic
211C8—22no—
212C8—2712no—
213C8—2712no—
214C22—22no—
215C22—5422no—
216C22—5424no—
217C22#2257613312yes—
218C22C2×D4×D27111yescharacteristic
219C4—42no—
220C4—42no—
221C4—5422no—
222C4—10842no—
223C4—10842no—
224C4#2257613312yes—
225C4C2×D4×D27111yescharacteristic
226C4C2×#1686465411yescharacteristic
227C2—22no—
228C2—10842no—
229C2C2×#2257613311yescharacteristic
230C2#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#3059
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
1. C22×#16864644C2semidirect
5. #31774154C2semidirect
6. #31774137C2semidirect
7. #31773933C2semidirect
8. #31773907C2semidirect
9. #419×C271C2semidirect
115. C271#3059semidirect
125. #419D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646442
C2×#225761682
#317744732
#317744712
#317741542
#317741372
#317739332
#317739072
#419×C2712
#3059271