Finite Group Databasebeta

← #46849509

Subgroups — #46849509

Order 34688.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#31774710C211yesmaximal, characteristic, complemented
2#31774699C211yesmaximal, characteristic, complemented
3#31774339C212yesmaximal, complemented
4#31774335C211yesmaximal, characteristic, complemented
5#31774334C212yesmaximal, complemented
6#31774330C212yesmaximal, complemented
7#31774327C211yesmaximal, characteristic, complemented
8C2×#22576103C211yesmaximal, characteristic, complemented
9SD32×D271C212yesmaximal, complemented
10#31774315C211yesmaximal, characteristic, complemented
11#510×C271C211yesmaximal, characteristic, complemented
12#22576208—21no—
13#22576206—21no—
14#22576192—22no—
15#22576191—21no—
16#22576187—22no—
17#22576185—21no—
18#22576183C2212yes—
19#22576183C2212yescomplemented
20#22576182C2211yescharacteristic, complemented
21Q16×D271C2212yescomplemented
22C2×D2168C2211yescharacteristic
23D8×D271C2212yes—
24#22576166C2211yescharacteristic
25#22576106C2211yescharacteristic, complemented
26#22576105C2212yescomplemented
27#22576104C2211yescharacteristic, complemented
28#22576103C2212yescomplemented
29#22576103C2212yescomplemented
30#22576103C2211yescharacteristic, complemented
31D4336C2212yescomplemented
32#22576101C2211yescharacteristic, complemented
33#22576099C2212yescomplemented
34C2×C271⋊C16C2211yescharacteristic, complemented
35C16×D271C2212yescomplemented
36#135×C271C2211yescharacteristic
37C2×Q16×C271C2211yescharacteristic, complemented
38Q32×C271C2212yescomplemented
39SD32×C271C2212yescomplemented
40M32×C271C2211yescharacteristic, complemented
41#16864655—22no—
42#16864655—22no—
43#16864655—41no—
44#16864654—21no—
45#16864654—41no—
46#16864654—42no—
47#16864654D411yescharacteristic
48Q8×D271—22no—
49Q8×D271—41no—
50C2×D1084—21no—
51C2×D1084—41no—
52D4×D271—22no—
53D4×D271—42no—
54Dic1084C2311yescharacteristic
55#16864644—22no—
56#16864643—21no—
57#16864641—22no—
58#16864641—22no—
59D2168—21no—
60D2168C2312yes—
61D2168C2311yescharacteristic
62#16864639—22no—
63#16864638D411yescharacteristic
64#16864637D412yes—
65C8×D271C2312yes—
66C271⋊C16C2311yescharacteristic
67C271⋊C16C2311yescharacteristic
68#41×C271—21no—
69C2×Q8×C271—21no—
70Q16×C271—21no—
71Q16×C271C2312yes—
72Q16×C271C2311yescharacteristic
73SD16×C271—21no—
74D8×C271C2311yescharacteristic
75C2×C8×C271C2311yescharacteristic
76C16×C271C2312yes—
77C22×D271—41no—
78C22×D271—42no—
79Dic542—21no—
80Dic542—42no—
81Dic542D811yescharacteristic
82D1084—22no—
83D1084—21no—
84D1084—42no—
85D1084—42no—
86D1084—41no—
87D1084D811yescharacteristic
88C22⋊D271—41no—
89C22⋊D271—42no—
90C22⋊D271D812yes—
91C4×D271—42no—
92C4×D271—41no—
93C4×D271—42no—
94C4×D271—42no—
95C4×D271C2×D412yes—
96C271⋊C8C2×D412yes—
97Q8×C271—21no—
98Q8×C271—22no—
99Q8×C271—41no—
100D4×C271—21no—
101D4×C271—41no—
102C2×C4×C271—41no—
103C2×C4×C271—41no—
104C2×C4×C271C2×D411yescharacteristic
105C8×C271C2×D411yescharacteristic
106C8×C271C2411yescharacteristic
107C22×C271—41no—
108C22×C271C2×D811yescharacteristic
109C2×D271—41no—
110C2×D271—42no—
111C2×D271—162no—
112C2×D271—161no—
113C2×D271C2×D812yes—
114C4×C271—41no—
115C4×C271—42no—
116C4×C271C2×D811yescharacteristic
117C4×C271C22×D411yescharacteristic
118Dic271—41no—
119Dic271C2×D812yes—
120C2×C271—21no—
121C2×C271—81no—
122C2×C271C22×D811yescharacteristic
123D271—22no—
124D271—81no—
125D271—82no—
126C271#341611yescharacteristic, complemented, Sylow
127#3416—2711nomaximal, Sylow
128#578—2711no—
129#576—2711no—
130#510—2712no—
131#510D27111yescharacteristic, complemented
132#509—2712no—
133#509—2711no—
134#508—2711no—
135#508—2712no—
136C2×SD32—2711no—
137C2×SD32—2712no—
138#504—2711no—
1392−1+4—5421no—
140H(5,𝔽2)—5421no—
141#137—5422no—
142#137—5421no—
143AGL(1,ℤ/8)—5422no—
144AGL(1,ℤ/8)—5421no—
145#135—2712no—
146#135—2712no—
147#135—2711no—
148#135C2×D27111yescharacteristic
149C2×Q16—2712no—
150C2×Q16C2×D27111yescharacteristic, complemented
151C2×D8—2712no—
152C2×D8—2711no—
153#131—2711no—
154Q32—2711no—
155Q32C2×D27112yescomplemented
156SD32—2712no—
157SD32—2712no—
158SD32—2711no—
159SD32—2711no—
160SD32—2712no—
161SD32C2×D27112yescomplemented
162D16—2711no—
163D16—2712no—
164M32—2712no—
165M32C2×D27111yescharacteristic, complemented
166C2×C16—2711no—
167C2×C16—2712no—
168#41—21no—
169#41—2711no—
170#41—5422no—
171#41—5422no—
172#41—5421no—
173#41—10842no—
174#41—10841no—
175#41—10841no—
176C2×Q8—21no—
177C2×Q8—5422no—
178C2×Q8—10841no—
179C2×D4—5422no—
180C2×D4—5421no—
181C2×D4—10841no—
182C2×D4—10842no—
183Q16—21no—
184Q16—2711no—
185Q16—5422no—
186Q16C22×D27112yes—
187Q16C22×D27111yescharacteristic
188SD16—21no—
189SD16—5422no—
190SD16—5422no—
191SD16—5421no—
192D8—2711no—
193D8—2712no—
194D8—5422no—
195D8—5421no—
196D8C22×D27111yescharacteristic
197M16—2711no—
198M16—2712no—
199C2×C8—2712no—
200C2×C8C22×D27111yescharacteristic
201C16—2711no—
202C16—2711no—
203C16C22×D27112yes—
204C23—10842no—
205C23—10841no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—2711no—
210Q8—5421no—
211Q8—10842no—
212D4—21no—
213D4—41no—
214D4—2712no—
215D4—2711no—
216D4—5422no—
217D4—5421no—
218D4—10842no—
219D4—10842no—
220D4—10842no—
221D4—10841no—
222D4—10841no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2712no—
226C2×C4—10842no—
227C2×C4—10842no—
228C2×C4—10842no—
229C2×C4—10841no—
230C2×C4D4×D27111yescharacteristic
231C8—2712no—
232C8D4×D27111yescharacteristic
233C8C23×D27111yescharacteristic
234C22—41no—
235C22—2712no—
236C22—10841no—
237C22—10842no—
238C22—43361no—
239C22—43362no—
240C22D8×D27111yescharacteristic
241C4—41no—
242C4—42no—
243C4—2712no—
244C4—10841no—
245C4D8×D27111yescharacteristic
246C4C2×D4×D27111yescharacteristic
247C2—21no—
248C2—81no—
249C2—5422no—
250C2—21682no—
251C2—21681no—
252C2C2×D8×D27111yescharacteristic
subgroupG/H
centerC2C2×D8×D271
commutator, cosocleC8×C271C24
FrattiniC8C23×D271
Fitting#510×C271C2
socleC2×C271C22×D8
hypercenter#510D271
hypocenterC271#3416
LCSG ▷ C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #31774710C2semidirect
2. #31774699C2semidirect
3. #31774339C2semidirect
4. #31774335C2semidirect
5. #31774334C2semidirect
6. #31774330C2semidirect
7. #31774327C2semidirect
8. C2×#22576103C2semidirect
9. SD32×D271C2semidirect
10. #31774315C2semidirect
11. #510×C271C2semidirect
19. #22576183C22semidirect
20. #22576182C22semidirect
21. Q16×D271C22semidirect
25. #22576106C22semidirect
26. #22576105C22semidirect
27. #22576104C22semidirect
28. #22576103C22semidirect
29. #22576103C22semidirect
30. #22576103C22semidirect
31. D4336C22semidirect
32. #22576101C22semidirect
33. #22576099C22semidirect
34. C2×C271⋊C16C22semidirect
35. C16×D271C22semidirect
37. C2×Q16×C271C22semidirect
38. Q32×C271C22semidirect
39. SD32×C271C22semidirect
40. M32×C271C22semidirect
126. C271#3416semidirect
131. #510D271semidirect
150. C2×Q16C2×D271semidirect
155. Q32C2×D271semidirect
161. SD32C2×D271semidirect
165. M32C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747102
#317746992
#317743392
#317743352
#317743342
#317743302
#317743272
C2×#225761032
SD32×D2712
#317743152
#510×C2712
#3416271