Finite Group Databasebeta

← #508×D139 #31819518

Subgroups — #508×D139 #31819518

Order 17792.

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

normal subgroups only

This group has 368 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#135×D139C212yesmaximal, complemented
2#22581386C211yesmaximal, characteristic, complemented
3#22581385C211yesmaximal, characteristic, complemented
4#22581384C212yesmaximal, complemented
5#22581383C212yesmaximal, complemented
6#22581382C211yesmaximal, characteristic, complemented
7Q32×D139C211yesmaximal, characteristic, complemented
8SD32×D139C212yesmaximal, complemented
9D16×D139C211yesmaximal, characteristic, complemented
10C2×C16×D139C211yesmaximal, characteristic, complemented
11#508×C139C211yesmaximal, characteristic, complemented
12#41×D139—22no—
13#17984865C2212yescomplemented
14#17984864C2212yescomplemented
15#17984863—22no—
16#17984862—22no—
17#17984861C2212yescomplemented
18Q16×D139C2212yes—
19SD16×D139—22no—
20D8×D139C2212yes—
21C2×C8×D139C2211yescharacteristic
22Dic1112C2211yescharacteristic, complemented
23#17984788C2212yescomplemented
24#17984787C2212yescomplemented
25#17984786C2212yescomplemented
26#17984785C2212yescomplemented
27D2224C2211yescharacteristic, complemented
28#17984783C2212yescomplemented
29C2×C139⋊C16C2211yescharacteristic, complemented
30C16×D139C2211yescharacteristic
31C16×D139C2211yescharacteristic
32C16×D139C2211yescharacteristic, complemented
33C16×D139C2211yescharacteristic, complemented
34#135×C139C2212yescomplemented
35Q32×C139C2211yescharacteristic, complemented
36SD32×C139C2212yescomplemented
37D16×C139C2211yescharacteristic, complemented
38C2×C16×C139C2211yescharacteristic, complemented
39#14555786—22no—
40#14555785—22no—
41#14555785—42no—
42#14555784—22no—
43#14555784—42no—
44Q8×D139—22no—
45D4×D139—22no—
46D4×D139—42no—
47C2×C4×D139—42no—
48C2×C4×D139D411yescharacteristic
49Dic556C2312yes—
50#14555775—22no—
51#14555774—22no—
52#14555773—22no—
53#14555772—22no—
54D1112C2312yes—
55#14555770—22no—
56C2×C139⋊C8D411yescharacteristic
57C8×D139D411yescharacteristic
58C8×D139D411yescharacteristic
59C8×D139C2311yescharacteristic
60C8×D139C2311yescharacteristic
61C139⋊C16C2311yescharacteristic
62C139⋊C16C2311yescharacteristic
63#41×C139—22no—
64Q16×C139C2312yes—
65SD16×C139—22no—
66D8×C139C2312yes—
67C2×C8×C139C2311yescharacteristic
68C16×C139C2311yescharacteristic
69C16×C139C2311yescharacteristic
70C22×D139—42no—
71C22×D139D811yescharacteristic
72Dic278—22no—
73Dic278—42no—
74D556—22no—
75D556—42no—
76C22⋊D139—42no—
77C22⋊D139—42no—
78C22⋊D139—42no—
79C2×Dic139—42no—
80C2×Dic139D811yescharacteristic
81C4×D139—42no—
82C4×D139—42no—
83C4×D139—42no—
84C4×D139D811yescharacteristic
85C4×D139D811yescharacteristic
86C4×D139C2×D411yescharacteristic
87C4×D139C2×D411yescharacteristic
88C139⋊C8C2×D411yescharacteristic
89C139⋊C8C2×D411yescharacteristic
90Q8×C139—22no—
91D4×C139—22no—
92D4×C139—42no—
93C2×C4×C139—42no—
94C2×C4×C139C2×D411yescharacteristic
95C8×C139C2×D411yescharacteristic
96C8×C139C2411yescharacteristic
97C22×C139—42no—
98C22×C139C2×D811yescharacteristic
99C2×D139—22no—
100C2×D139—42no—
101C2×D139—84no—
102C2×D139C2×D811yescharacteristic
103C2×D139C2×D811yescharacteristic
104C4×C139—42no—
105C4×C139C2×D811yescharacteristic
106C4×C139C22×D411yescharacteristic
107Dic139—42no—
108Dic139C2×D811yescharacteristic
109Dic139C2×D811yescharacteristic
110C2×C139—21no—
111C2×C139—82no—
112C2×C139C22×D811yescharacteristic
113D139—21no—
114D139—82no—
115D139#50812yesdirect product factor
116C139C2×#50811yescharacteristic, complemented, Sylow
117C2×#508—1391nomaximal, Sylow
118C2×#135—1392no—
119#508—1392no—
120#508—1391no—
121#508—1392no—
122#508—1391no—
123#508—1391no—
124#508D13911yescharacteristic, direct product factor
125C2×Q32—1391no—
126C2×SD32—1392no—
127C2×D16—1391no—
128C22×C16—1391no—
129C2×#41—2782no—
130#135—2782no—
131#135—2782no—
132#135—1392no—
133#135—1392no—
134#135—1392no—
135#135C2×D13912yescomplemented
136C2×Q16—1392no—
137C2×SD16—2782no—
138C2×D8—1392no—
139C22×C8—1391no—
140Q32—1392no—
141Q32—1391no—
142Q32C2×D13911yescharacteristic, complemented
143SD32—1392no—
144SD32—1392no—
145SD32—1392no—
146SD32C2×D13912yescomplemented
147D16—1392no—
148D16—1391no—
149D16C2×D13911yescharacteristic, complemented
150C2×C16—1391no—
151C2×C16—1391no—
152C2×C16—1391no—
153C2×C16—1391no—
154C2×C16—1391no—
155C2×C16C2×D13911yescharacteristic, complemented
156#41—22no—
157#41—2782no—
158#41—2782no—
159#41—2782no—
160#41—5562no—
161#41—5562no—
162C2×Q8—2782no—
163C2×D4—2782no—
164C2×D4—5562no—
165C22×C4—5562no—
166C22×C4—1391no—
167Q16—2782no—
168Q16—1392no—
169Q16C22×D13912yes—
170SD16—22no—
171SD16—2782no—
172SD16—2782no—
173SD16—2782no—
174D8—2782no—
175D8—1392no—
176D8C22×D13912yes—
177C2×C8—1391no—
178C2×C8—1391no—
179C2×C8—1391no—
180C2×C8—1391no—
181C2×C8—1391no—
182C2×C8C22×D13911yescharacteristic
183C16—1391no—
184C16—1391no—
185C16C22×D13911yescharacteristic
186C16C22×D13911yescharacteristic
187C23—5562no—
188C23—1391no—
189Q8—22no—
190Q8—2782no—
191Q8—5562no—
192D4—22no—
193D4—42no—
194D4—2782no—
195D4—5562no—
196D4—5562no—
197D4—5562no—
198D4—5562no—
199C2×C4—42no—
200C2×C4—5562no—
201C2×C4—5562no—
202C2×C4—5562no—
203C2×C4—5562no—
204C2×C4—1391no—
205C2×C4—1391no—
206C2×C4—1391no—
207C2×C4—1391no—
208C2×C4—1391no—
209C2×C4D4×D13911yescharacteristic
210C8—1391no—
211C8—1391no—
212C8D4×D13911yescharacteristic
213C8C23×D13911yescharacteristic
214C22—42no—
215C22—2782no—
216C22—5562no—
217C22—11124no—
218C22—1391no—
219C22—1391no—
220C22D8×D13911yescharacteristic
221C4—42no—
222C4—5562no—
223C4—1391no—
224C4—1391no—
225C4D8×D13911yescharacteristic
226C4C2×D4×D13911yescharacteristic
227C2—21no—
228C2—82no—
229C2—2781no—
230C2—11122no—
231C2—1392no—
232C2C2×D8×D13911yescharacteristic
subgroupG/H
centerC4D8×D139
commutator, cosocleC8×C139C24
FrattiniC8C23×D139
Fitting#508×C139C2
socleC2×C139C22×D8
hypercenter#508D139
hypocenterC139C2×#508
LCSG ▷ C8×C139 ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D139C2semidirect
2. #22581386C2semidirect
3. #22581385C2semidirect
4. #22581384C2semidirect
5. #22581383C2semidirect
6. #22581382C2semidirect
7. Q32×D139C2semidirect
8. SD32×D139C2semidirect
9. D16×D139C2semidirect
10. C2×C16×D139C2semidirect
11. #508×C139C2semidirect
13. #17984865C22semidirect
14. #17984864C22semidirect
17. #17984861C22semidirect
22. Dic1112C22semidirect
23. #17984788C22semidirect
24. #17984787C22semidirect
25. #17984786C22semidirect
26. #17984785C22semidirect
27. D2224C22semidirect
28. #17984783C22semidirect
29. C2×C139⋊C16C22semidirect
32. C16×D139C22semidirect
33. C16×D139C22semidirect
34. #135×C139C22semidirect
35. Q32×C139C22semidirect
36. SD32×C139C22semidirect
37. D16×C139C22semidirect
38. C2×C16×C139C22semidirect
115. D139#508direct (N × H)
116. C139C2×#508semidirect
124. #508D139direct (N × H)
135. #135C2×D139semidirect
142. Q32C2×D139semidirect
146. SD32C2×D139semidirect
149. D16C2×D139semidirect
155. C2×C16C2×D139semidirect

Maximal subgroups

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

subgroupindex
#135×D1392
#225813862
#225813852
#225813842
#225813832
#225813822
Q32×D1392
SD32×D1392
D16×D1392
C2×C16×D1392
#508×C1392
C2×#508139