Finite Group Databasebeta

← #46846973

Subgroups — #46846973

Order 34688.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#31774452C211yesmaximal, characteristic, complemented
2#31774426C211yesmaximal, characteristic
3#31774303C211yesmaximal, characteristic, complemented
4#31774292C211yesmaximal, characteristic, complemented
5#31774266C211yesmaximal, characteristic, complemented
6#31774154C212yesmaximal, complemented
7#31774142C212yesmaximal
8#31773989C212yesmaximal, complemented
9#31773988C212yesmaximal, complemented
10#31773849C211yesmaximal, characteristic, complemented
11#489×C271C211yesmaximal, characteristic, complemented
12C2×#16864644C2212yes—
13C2×#16864642C2212yes—
14C2×#16864637C2212yes—
15#22576160C2211yescharacteristic
16#22576153—21no—
17#22576152C2211yescharacteristic
18#22576151—21no—
19#22576142—21no—
20#22576140C2212yes—
21#22576136—21no—
22#22576136C2211yescharacteristic
23#22576131C2211yescharacteristic
24Q8×Dic271—21no—
25Q8×Dic271C2211yescharacteristic
26#22576125C2212yes—
27D4×Dic271C2211yescharacteristic
28(C22⋊D271)⋊C4—21no—
29#22576118—21no—
30#22576116—21no—
31C42×D271C2211yescharacteristic
32#22576093C2212yes—
33#22576091C2212yes—
34#22576085C2212yes—
35#22576085C2212yes—
36#22576071C2211yescharacteristic
37C8×Dic271C2212yes—
38C4⋊Q8×C271C2211yescharacteristic
39#124×C271C2211yescharacteristic
40#103×C271C2212yes—
41#102×C271C2212yes—
42#97×C271C2211yescharacteristic
43C2×Dic542C2312yes—
44C2×C22⋊D271—21no—
45C22×Dic271—21no—
46C2×C4×D271D411yescharacteristic
47C2×C4×D271D411yescharacteristic
48C2×C4×D271C2311yescharacteristic
49#16864644—24no—
50#16864642—24no—
51#16864637—24no—
52C2×C271⋊C8C2312yes—
53C4⋊Dic271—22no—
54C4⋊Dic271—21no—
55C4⋊Dic271—21no—
56C4⋊Dic271—21no—
57C4⋊Dic271C2311yescharacteristic
58C4⋊Dic271C2311yescharacteristic
59Dic271⋊C4—22no—
60Dic271⋊C4—21no—
61Dic271⋊C4—21no—
62Dic271⋊C4—21no—
63C22⋊Dic271—21no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—22no—
66(C2×D271)⋊C4—21no—
67C4×Dic271—21no—
68C4×Dic271—22no—
69C4×Dic271D411yescharacteristic
70C4×Dic271D411yescharacteristic
71C4×Dic271C2311yescharacteristic
72C2×Q8×C271—21no—
73C2×Q8×C271C2311yescharacteristic
74C2×D4×C271C2311yescharacteristic
75C2×C8×C271C2312yes—
76C4⋊C4×C271—21no—
77C4⋊C4×C271C2312yes—
78C22⋊C4×C271—21no—
79C22⋊C4×C271—21no—
80C42×C271C2311yescharacteristic
81C22×D271C2×D411yescharacteristic
82Dic542—22no—
83Dic542—22no—
84C22⋊D271—42no—
85C2×Dic271—22no—
86C2×Dic271—21no—
87C2×Dic271—21no—
88C2×Dic271—42no—
89C2×Dic271C2×D411yescharacteristic
90C2×Dic271C2×D411yescharacteristic
91C2×Dic271C2×D411yescharacteristic
92C4×D271—22no—
93C4×D271—22no—
94C4×D271—41no—
95C271⋊C8#4114yes—
96C23×C271—21no—
97Q8×C271—22no—
98Q8×C271—42no—
99D4×C271—22no—
100C2×C4×C271—21no—
101C2×C4×C271—22no—
102C2×C4×C271C2×D411yescharacteristic
103C2×C4×C271C2×D411yescharacteristic
104C2×C4×C271C2411yescharacteristic
105C8×C271—22no—
106C22×C271—41no—
107C22×C271—41no—
108C22×C271—41no—
109C22×C271C22×D411yescharacteristic
110C2×D271—21no—
111C2×D271—21no—
112C2×D271—21no—
113C4×C271—21no—
114C4×C271—21no—
115C4×C271—41no—
116C4×C271—42no—
117C4×C271C2×#4112yes—
118Dic271—21no—
119Dic271—21no—
120Dic271—42no—
121Dic271—41no—
122Dic271—41no—
123Dic271#13712yes—
124C2×C271—81no—
125C2×C271#53211yescharacteristic
126C2×C271C2×#13711yescharacteristic
127C2×C271#57511yescharacteristic
128D271—41no—
129C271#317211yescharacteristic, complemented, Sylow
130#3172—2711nomaximal, Sylow
131#533—2711no—
132#529—2711no—
133#501—2711no—
134#497—2711no—
135#491—2711no—
136#489D27111yescharacteristic, complemented
137#470—2712no—
138#468—2712no—
139C4×Q16—2712no—
140C4×SD16—2712no—
141C4×M16—2711no—
142C2×Q16—2712no—
143C2×SD16—2712no—
144C2×M16—2712no—
145C4⋊Q8—2711no—
146C4⋊Q8C2×D27111yescharacteristic
147#126—5421no—
148#125—2711no—
149#125—5421no—
150#124C2×D27111yescharacteristic
151#123—5421no—
152C22⋊Q8—2711no—
153C22⋊Q8—2712no—
154C22⋊Q8—5421no—
155#121—2711no—
156C4×Q8—2711no—
157C4×Q8—2712no—
158C4×Q8—5421no—
159C4×D4—2711no—
160C4×D4—5421no—
161#117—5421no—
162#117—5421no—
163C2×C42—2711no—
164#107—2712no—
165#106—2712no—
166#103—2712no—
167#103—2712no—
168#103C2×D27112yes—
169#102C2×D27112yes—
170#97—2711no—
171#97C2×D27111yescharacteristic
172C4×C8—2712no—
173C2×Q8—21no—
174C2×Q8—2712no—
175C2×Q8C22×D27111yescharacteristic
176C2×D4—5421no—
177C2×D4C22×D27111yescharacteristic
178C22×C4—2711no—
179C22×C4—2711no—
180C22×C4—2711no—
181C22×C4—5421no—
182Q16—5424no—
183SD16—5424no—
184M16—5424no—
185C2×C8—2712no—
186C2×C8C22×D27112yes—
187C4⋊C4—21no—
188C4⋊C4—2711no—
189C4⋊C4—2711no—
190C4⋊C4—5421no—
191C4⋊C4—5422no—
192C4⋊C4—5421no—
193C4⋊C4—5422no—
194C4⋊C4—5421no—
195C4⋊C4—5421no—
196C4⋊C4—5421no—
197C4⋊C4—5421no—
198C4⋊C4C22×D27112yes—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—5422no—
202C22⋊C4—5421no—
203C22⋊C4—5421no—
204C22⋊C4—5421no—
205C42—2711no—
206C42—2711no—
207C42—2711no—
208C42—5421no—
209C42—5422no—
210C42C22×D27111yescharacteristic
211C23—21no—
212C23—2711no—
213Q8—22no—
214Q8—42no—
215Q8—5422no—
216Q8—5422no—
217D4—22no—
218D4—10842no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—2711no—
222C2×C4—2711no—
223C2×C4—2711no—
224C2×C4—5422no—
225C2×C4—5422no—
226C2×C4—5421no—
227C2×C4—5422no—
228C2×C4—5421no—
229C2×C4—10841no—
230C2×C4—10842no—
231C2×C4D4×D27111yescharacteristic
232C2×C4D4×D27111yescharacteristic
233C2×C4C23×D27111yescharacteristic
234C8—22no—
235C8—2714no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—5421no—
240C22—5421no—
241C22—5421no—
242C22C2×D4×D27111yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—2712no—
248C4—5421no—
249C4—5421no—
250C4—10841no—
251C4—10841no—
252C4—10842no—
253C4#41×D27112yes—
254C2—81no—
255C2—10841no—
256C2#3177444611yescharacteristic
257C2#137×D27111yescharacteristic
258C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
prime radical#22576131C22
hypercenter#489D271
hypocenterC271#3172
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #31774452C2semidirect
3. #31774303C2semidirect
4. #31774292C2semidirect
5. #31774266C2semidirect
6. #31774154C2semidirect
8. #31773989C2semidirect
9. #31773988C2semidirect
10. #31773849C2semidirect
11. #489×C271C2semidirect
129. C271#3172semidirect
136. #489D271semidirect

Maximal subgroups

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

subgroupindex
#317744522
#317744262
#317743032
#317742922
#317742662
#317741542
#317741422
#317739892
#317739882
#317738492
#489×C2712
#3172271