Finite Group Databasebeta

← C2×#22553447 #30675138

Subgroups — C2×#22553447 #30675138

Order 16768.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D131C211yesmaximal, characteristic, complemented
2C22×D1048C211yesmaximal, characteristic, complemented
3C22×#14523498C211yesmaximal, characteristic, complemented
4C2×#16862221C211yesmaximal, characteristic, complemented
5#22553447C218yesmaximal, direct product factor
6C2×#16862166C211yesmaximal, characteristic, complemented
7C2×(C2×D131)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C131C211yesmaximal, characteristic, complemented
9C24×D131—21no—
10C22×D524—21no—
11C22×D524C2211yescharacteristic
12C22×D524C2211yescharacteristic, complemented
13C22×C22⋊D131—21no—
14C2×D4×D131—24no—
15C2×D4×D131—22no—
16C2×D4×D131—24no—
17C2×D4×D131C2214yescomplemented
18C22×C4×D131C2211yescharacteristic
19C2×D1048—24no—
20C2×D1048C2214yescomplemented
21C2×#14523498—24no—
22C2×#14523498C2214yescomplemented
23C22×C131⋊C8C2211yescharacteristic, complemented
24#16862221—22no—
25#16862221C2214yescomplemented
26C2×(C2×D131)⋊C4—21no—
27#16862166C2214yescomplemented
28(C2×D131)⋊C8C2214yescomplemented
29C22×D4×C131C2211yescharacteristic
30C22×C8×C131C2211yescharacteristic, complemented
31C2×C4⋊C4×C131C2211yescharacteristic, complemented
32#102×C131C2214yescomplemented
33C23×D131—28no—
34C23×D131—24no—
35C23×D131—21no—
36C23×D131—21no—
37C23×D131—44no—
38C23×D131—44no—
39C23×D131D411yescharacteristic
40C2×D524—24no—
41C2×D524—21no—
42C2×D524—22no—
43C2×D524—22no—
44C2×D524—22no—
45C2×D524—21no—
46C2×D524—44no—
47C2×D524D414yescomplemented
48C2×D524C2312yes—
49C2×D524C2312yescomplemented
50C2×C22⋊D131—28no—
51C2×C22⋊D131—24no—
52D4×D131—216no—
53D4×D131—28no—
54D4×D131—28no—
55C22×Dic131D411yescharacteristic
56C2×C4×D131—22no—
57C2×C4×D131—22no—
58C2×C4×D131D412yes—
59C2×C4×D131C2312yes—
60D1048—28no—
61#14523498—28no—
62C2×C131⋊C8—22no—
63C2×C131⋊C8C2312yescomplemented
64(C2×D131)⋊C4—24no—
65C24×C131—21no—
66C2×D4×C131—21no—
67C2×D4×C131—22no—
68C2×D4×C131D414yescomplemented
69C2×D4×C131C2312yes—
70C22×C4×C131—21no—
71C22×C4×C131C2311yescharacteristic
72C2×C8×C131—22no—
73C2×C8×C131C2312yescomplemented
74C4⋊C4×C131—21no—
75C4⋊C4×C131C2312yescomplemented
76C22×D131—22no—
77C22×D131—22no—
78C22×D131—216no—
79C22×D131—24no—
80C22×D131—22no—
81C22×D131—22no—
82C22×D131—48no—
83C22×D131—432no—
84C22×D131—42no—
85C22×D131—42no—
86C22×D131—48no—
87C22×D131—42no—
88C22×D131—42no—
89C22×D131—42no—
90C22×D131D814yescomplemented
91C22×D131C2×D412yes—
92D524—24no—
93D524—22no—
94D524—22no—
95D524—22no—
96D524—48no—
97D524—42no—
98D524C2×D414yescomplemented
99C22⋊D131—216no—
100C2×Dic131—22no—
101C2×Dic131C2×D412yes—
102C4×D131—24no—
103C4×D131—22no—
104C4×D131—22no—
105C131⋊C8—22no—
106C23×C131—24no—
107C23×C131—22no—
108C23×C131—42no—
109C23×C131—42no—
110C23×C131C2×D411yescharacteristic
111D4×C131—24no—
112D4×C131—22no—
113D4×C131C2×D414yescomplemented
114C2×C4×C131—22no—
115C2×C4×C131—42no—
116C2×C4×C131C2×D411yescharacteristic
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2411yescharacteristic
120C8×C131—22no—
121C22×C131—24no—
122C22×C131—42no—
123C22×C131—42no—
124C22×C131—48no—
125C22×C131#12012yes—
126C22×C131C2×D812yes—
127C22×C131AGL(1,ℤ/8)12yes—
128C22×C131C22×D411yescharacteristic
129C2×D131—22no—
130C2×D131—22no—
131C2×D131—28no—
132C2×D131—24no—
133C2×D131—432no—
134C2×D131—42no—
135C2×D131—42no—
136C2×D131—48no—
137C2×D131—42no—
138C2×D131—42no—
139C2×D131—42no—
140C2×D131—84no—
141C2×D131C2×D814yescomplemented
142C4×C131—42no—
143C4×C131#12012yes—
144C4×C131C22×D411yescharacteristic
145C4×C131C22×D411yescharacteristic
146Dic131—22no—
147C2×C131—44no—
148C2×C131#44714yescomplemented
149C2×C131C2×#12011yescharacteristic
150C2×C131C22×D811yescharacteristic
151C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
152D131—24no—
153D131—44no—
154D131—82no—
155C131C2×#44711yescharacteristic, complemented, Sylow
156C2×#447—1311nomaximal, Sylow
157C23×D4—1311no—
158C22×D8—1311no—
159C22×D8—1311no—
160C2×#121—1311no—
161#447—1318no—
162C2×#102—1311no—
163C2×#102D13111yescharacteristic, complemented
164C2×C22⋊C8—1311no—
165C25—2621no—
166C22×D4—2621no—
167C22×D4—2624no—
168C22×D4—2624no—
169C22×D4—2622no—
170C22×D4—2621no—
171C22×D4—1314no—
172C22×D4—1311no—
173C22×D4—1311no—
174C22×D4C2×D13111yescharacteristic
175C23×C4—1311no—
176C2×D8—2624no—
177C2×D8—2624no—
178C2×D8—1314no—
179C2×D8—1314no—
180C22×C8—1311no—
181C22×C8C2×D13111yescharacteristic, complemented
182#121—2622no—
183#121—1314no—
184C2×C4⋊C4C2×D13111yescharacteristic, complemented
185C2×C22⋊C4—2621no—
186#102—1314no—
187#102C2×D13114yescomplemented
188C22⋊C8—1314no—
189C24—21no—
190C24—2628no—
191C24—2624no—
192C24—2621no—
193C24—2621no—
194C24—5244no—
195C24—5244no—
196C24—1311no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—2624no—
200C2×D4—26216no—
201C2×D4—2628no—
202C2×D4—2628no—
203C2×D4—2628no—
204C2×D4—2624no—
205C2×D4—2621no—
206C2×D4—2622no—
207C2×D4—2621no—
208C2×D4—2622no—
209C2×D4—2622no—
210C2×D4—5244no—
211C2×D4—1314no—
212C2×D4—1312no—
213C2×D4—1312no—
214C2×D4D52414yescomplemented
215C2×D4C22×D13112yes—
216C22×C4—21no—
217C22×C4—2622no—
218C22×C4—2622no—
219C22×C4—1311no—
220C22×C4—1312no—
221C22×C4—1312no—
222C22×C4C22×D13111yescharacteristic
223D8—2628no—
224D8—2628no—
225C2×C8—22no—
226C2×C8—2622no—
227C2×C8—1312no—
228C2×C8C22×D13112yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D13112yescomplemented
231C22⋊C4—2624no—
232C23—24no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—26216no—
237C23—2622no—
238C23—2622no—
239C23—2624no—
240C23—2622no—
241C23—2622no—
242C23—52432no—
243C23—5248no—
244C23—5248no—
245C23—5242no—
246C23—5242no—
247C23—5242no—
248C23—5242no—
249C23—5242no—
250C23—1314no—
251C23—1312no—
252C23D4×D13111yescharacteristic
253D4—24no—
254D4—22no—
255D4—26216no—
256D4—2624no—
257D4—2622no—
258D4—2622no—
259D4—2622no—
260D4—5248no—
261D4—5242no—
262D4—1314no—
263D4C2×D52414yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—2622no—
267C2×C4—2624no—
268C2×C4—2622no—
269C2×C4—2622no—
270C2×C4—1312no—
271C2×C4D4×D13111yescharacteristic
272C2×C4D4×D13112yes—
273C2×C4C2×D52412yes—
274C2×C4C23×D13111yescharacteristic
275C8—22no—
276C8—2622no—
277C22—24no—
278C22—48no—
279C22—42no—
280C22—42no—
281C22—2628no—
282C22—2622no—
283C22—2622no—
284C22—2624no—
285C22—52432no—
286C22—5248no—
287C22—5242no—
288C22—5242no—
289C22—5242no—
290C22—5242no—
291C22—5242no—
292C22—10484no—
293C22—1314no—
294C22#1686221612yes—
295C22D8×D13112yes—
296C22#1686227312yes—
297C22C2×D4×D13111yescharacteristic
298C4—42no—
299C4—2622no—
300C4#1686221612yes—
301C4C2×D4×D13111yescharacteristic
302C4C22×D52411yescharacteristic
303C2—44no—
304C2—2624no—
305C2—5244no—
306C2—10482no—
307C2#2255344714yesdirect product factor
308C2C2×#1686221611yescharacteristic
309C2C2×D8×D13111yescharacteristic
310C2C2×#1686227311yescharacteristic
subgroupG/H
centerC23D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#102×C131C2
socleC23×C131C2×D4
hypercenterC2×#102D131
hypocenterC131C2×#447
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C22×D1048C2semidirect
3. C22×#14523498C2semidirect
4. C2×#16862221C2semidirect
5. #22553447C2direct (N × H)
6. C2×#16862166C2semidirect
7. C2×(C2×D131)⋊C8C2semidirect
8. C2×#102×C131C2semidirect
12. C22×D524C22semidirect
17. C2×D4×D131C22semidirect
20. C2×D1048C22semidirect
22. C2×#14523498C22semidirect
23. C22×C131⋊C8C22semidirect
25. #16862221C22semidirect
27. #16862166C22semidirect
28. (C2×D131)⋊C8C22semidirect
30. C22×C8×C131C22semidirect
31. C2×C4⋊C4×C131C22semidirect
32. #102×C131C22semidirect
47. C2×D524D4semidirect
49. C2×D524C23semidirect
63. C2×C131⋊C8C23semidirect
68. C2×D4×C131D4semidirect
73. C2×C8×C131C23semidirect
75. C4⋊C4×C131C23semidirect
90. C22×D131D8semidirect
98. D524C2×D4semidirect
113. D4×C131C2×D4semidirect
141. C2×D131C2×D8semidirect
148. C2×C131#447semidirect
155. C131C2×#447semidirect
163. C2×#102D131semidirect
181. C22×C8C2×D131semidirect
184. C2×C4⋊C4C2×D131semidirect
187. #102C2×D131semidirect
214. C2×D4D524semidirect
228. C2×C8C22×D131semidirect
230. C4⋊C4C22×D131semidirect
263. D4C2×D524semidirect
307. C2#22553447direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C22×D10482
C22×#145234982
C2×#168622212
#225534472
C2×#168621662
C2×(C2×D131)⋊C82
C2×#102×C1312
C2×#447131