Finite Group Databasebeta

← C22×#17984863 #31821131

Subgroups — C22×#17984863 #31821131

Order 17792.

1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 525 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×#14555786C211yesmaximal, characteristic, complemented
2C22×#14555784C211yesmaximal, characteristic, complemented
3C2×#17984863C2124yesmaximal, direct product factor
4C22×#14555775C211yesmaximal, characteristic, complemented
5C22×#14555774C211yesmaximal, characteristic, complemented
6C22×#14555770C211yesmaximal, characteristic, complemented
7C22×C8×D139C211yesmaximal, characteristic, complemented
8C22×SD16×C139C211yesmaximal, characteristic, complemented
9C2×#14555786—26no—
10C2×#14555786C22112yescomplemented
11C2×#14555784—26no—
12C2×#14555784C22112yescomplemented
13C22×Dic278C2211yescharacteristic
14C22×D556—21no—
15C22×D556C2211yescharacteristic
16C22×C22⋊D139—21no—
17C23×Dic139—21no—
18C22×C4×D139—21no—
19C22×C4×D139C2211yescharacteristic
20#17984863C22164yesdirect product factor
21C2×#14555775C22112yescomplemented
22C2×#14555774C22112yescomplemented
23C2×#14555770C22112yescomplemented
24C22×C139⋊C8C2211yescharacteristic
25C2×C8×D139C22112yescomplemented
26C22×Q8×C139C2211yescharacteristic
27C22×D4×C139C2211yescharacteristic
28C2×SD16×C139C22112yescomplemented
29C22×C8×C139C2211yescharacteristic
30C23×D139—21no—
31C23×D139D411yescharacteristic
32#14555786—224no—
33#14555786C23116yescomplemented
34#14555784—224no—
35#14555784C23116yescomplemented
36C2×Dic278—23no—
37C2×Dic278C2316yes—
38C2×D556—212no—
39C2×D556—23no—
40C2×D556C2316yes—
41C2×C22⋊D139—212no—
42C22×Dic139—212no—
43C22×Dic139—21no—
44C22×Dic139D411yescharacteristic
45C2×C4×D139—212no—
46C2×C4×D139D416yes—
47C2×C4×D139C2316yes—
48#14555775C23116yescomplemented
49#14555774C23116yescomplemented
50#14555770C23116yescomplemented
51C2×C139⋊C8C2316yes—
52C8×D139C23116yescomplemented
53C24×C139—21no—
54C2×Q8×C139—23no—
55C2×Q8×C139C2316yes—
56C2×D4×C139—23no—
57C2×D4×C139C2316yes—
58C22×C4×C139—21no—
59C22×C4×C139C2311yescharacteristic
60SD16×C139C23116yescomplemented
61C2×C8×C139C2316yes—
62C22×D139—24no—
63C22×D139—26no—
64C22×D139—44no—
65C22×D139C2×D416yes—
66Dic278—26no—
67Dic278C2414yes—
68D556—216no—
69D556—26no—
70D556C2414yes—
71C22⋊D139—216no—
72C2×Dic139—26no—
73C2×Dic139—216no—
74C2×Dic139C2×D416yes—
75C4×D139—216no—
76C4×D139C2×D4112yes—
77C4×D139C2414yes—
78C139⋊C8C2414yes—
79C23×C139—26no—
80C23×C139—44no—
81C23×C139C2×D411yescharacteristic
82Q8×C139—26no—
83Q8×C139C2414yes—
84D4×C139—26no—
85D4×C139C2414yes—
86C2×C4×C139—26no—
87C2×C4×C139C2×D413yes—
88C2×C4×C139C2413yes—
89C8×C139C2414yes—
90C22×C139—24no—
91C22×C139—412no—
92C22×C139#13514yescomplemented
93C22×C139C22×D413yes—
94C2×D139—212no—
95C2×D139—24no—
96C2×D139—412no—
97C2×D139C22×D414yes—
98C4×C139—24no—
99C4×C139C22×D413yes—
100C4×C139C2511yescharacteristic
101Dic139—24no—
102Dic139C22×D414yes—
103C2×C139—44no—
104C2×C139C2×#13516yescomplemented
105C2×C139C23×D411yescharacteristic
106D139—24no—
107D139—44no—
108C139C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—1391nomaximal, Sylow
110C22×#41—1391no—
111C22×#41—1391no—
112C2×#135—13924no—
113C22×Q16—1391no—
114C22×SD16—1391no—
115C22×SD16D13911yescharacteristic, complemented
116C22×D8—1391no—
117C23×C8—1391no—
118C2×#41—2786no—
119C2×#41—2786no—
120C2×#41—13912no—
121C2×#41—13912no—
122C22×Q8—1391no—
123C22×Q8C2×D13911yescharacteristic
124C22×D4—2781no—
125C22×D4—2781no—
126C22×D4—1391no—
127C22×D4C2×D13911yescharacteristic
128C23×C4—2781no—
129C23×C4—2781no—
130C23×C4—1391no—
131#135—13964no—
132C2×Q16—13912no—
133C2×SD16—13912no—
134C2×SD16C2×D139112yescomplemented
135C2×D8—13912no—
136C22×C8—1391no—
137C22×C8—13912no—
138C22×C8C2×D13911yescharacteristic
139C24—21no—
140C24—2781no—
141C24—1391no—
142#41—27824no—
143#41—27824no—
144#41—13916no—
145#41—13916no—
146C2×Q8—23no—
147C2×Q8—2783no—
148C2×Q8—1396no—
149C2×Q8C22×D13916yes—
150C2×D4—23no—
151C2×D4—27812no—
152C2×D4—27812no—
153C2×D4—2783no—
154C2×D4—1396no—
155C2×D4C22×D13916yes—
156C22×C4—21no—
157C22×C4—27812no—
158C22×C4—2781no—
159C22×C4—27812no—
160C22×C4—1391no—
161C22×C4—1396no—
162C22×C4—1396no—
163C22×C4C22×D13911yescharacteristic
164Q16—13916no—
165SD16—13916no—
166SD16C22×D139116yescomplemented
167D8—13916no—
168C2×C8—1396no—
169C2×C8—13916no—
170C2×C8C22×D13916yes—
171C23—26no—
172C23—44no—
173C23—2784no—
174C23—2786no—
175C23—5564no—
176C23—1396no—
177C23D4×D13911yescharacteristic
178Q8—26no—
179Q8—2786no—
180Q8—1394no—
181Q8C23×D13914yes—
182D4—26no—
183D4—27816no—
184D4—27816no—
185D4—2786no—
186D4—1394no—
187D4C23×D13914yes—
188C2×C4—26no—
189C2×C4—27816no—
190C2×C4—2786no—
191C2×C4—27816no—
192C2×C4—1396no—
193C2×C4—13912no—
194C2×C4—1394no—
195C2×C4D4×D13913yes—
196C2×C4C23×D13913yes—
197C8—1394no—
198C8C23×D13914yes—
199C22—24no—
200C22—412no—
201C22—27812no—
202C22—2784no—
203C22—55612no—
204C22—1394no—
205C22#1798486314yesdirect product factor
206C22C2×D4×D13913yes—
207C4—24no—
208C4—2784no—
209C4—1394no—
210C4C2×D4×D13913yes—
211C4C24×D13911yescharacteristic
212C2—44no—
213C2—2784no—
214C2—5564no—
215C2C2×#1798486316yesdirect product factor
216C2C22×D4×D13911yescharacteristic
subgroupG/H
centerC23D4×D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC22×SD16×C139C2
socleC23×C139C2×D4
hypercenterC22×SD16D139
hypocenterC139C22×#135
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14555786C2semidirect
2. C22×#14555784C2semidirect
3. C2×#17984863C2direct (N × H)
4. C22×#14555775C2semidirect
5. C22×#14555774C2semidirect
6. C22×#14555770C2semidirect
7. C22×C8×D139C2semidirect
8. C22×SD16×C139C2semidirect
10. C2×#14555786C22semidirect
12. C2×#14555784C22semidirect
20. #17984863C22direct (N × H)
21. C2×#14555775C22semidirect
22. C2×#14555774C22semidirect
23. C2×#14555770C22semidirect
25. C2×C8×D139C22semidirect
28. C2×SD16×C139C22semidirect
33. #14555786C23semidirect
35. #14555784C23semidirect
48. #14555775C23semidirect
49. #14555774C23semidirect
50. #14555770C23semidirect
52. C8×D139C23semidirect
60. SD16×C139C23semidirect
92. C22×C139#135semidirect
104. C2×C139C2×#135semidirect
108. C139C22×#135semidirect
115. C22×SD16D139semidirect
134. C2×SD16C2×D139semidirect
166. SD16C22×D139semidirect
205. C22#17984863direct (N × H)
215. C2C2×#17984863direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145557862
C22×#145557842
C2×#179848632
C22×#145557752
C22×#145557742
C22×#145557702
C22×C8×D1392
C22×SD16×C1392
C22×#135139