Finite Group Databasebeta

← C2×#14271295 #16828666

Subgroups — C2×#14271295 #16828666

Order 3712.

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×D29C211yesmaximal, characteristic, complemented
2C22×#90782C211yesmaximal, characteristic, complemented
3C2×#11745300C211yesmaximal, characteristic, complemented
4#14271295C218yesmaximal, direct product factor
5C2×#11745169C211yesmaximal, characteristic, complemented
6C2×#11745180C211yesmaximal, characteristic, complemented
7C2×#11745168C211yesmaximal, characteristic, complemented
8C22×D8×C29C211yesmaximal, characteristic, complemented
9C24×D29—21no—
10C22×D4×C29C2211yescharacteristic
11C22×D4×C29C2211yescharacteristic, complemented
12C22×C22⋊D29—21no—
13C22×C22⋊D29—21no—
14C2×D4×D29—24no—
15C2×D4×D29—24no—
16C2×D4×D29—22no—
17C2×D4×D29C2214yescomplemented
18C22×D116C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C2×D8×C29—24no—
21C2×D8×C29C2214yescomplemented
22C22×C8×C29C2211yescharacteristic, complemented
23C2×C22⋊Dic29—21no—
24#11745300—22no—
25#11745300C2214yescomplemented
26C2×#90782—24no—
27C2×#90782C2214yescomplemented
28C2×C4⋊Dic29C2211yescharacteristic, complemented
29C22×#6597C2211yescharacteristic, complemented
30#11745180C2214yescomplemented
31#11745169C2214yescomplemented
32#11745168C2214yescomplemented
33C24×C29—21no—
34C24×C29—21no—
35C23×D29—28no—
36C23×D29—21no—
37C23×D29—24no—
38C23×D29—44no—
39C23×D29—44no—
40C23×D29D411yescharacteristic
41C2×D4×C29—21no—
42C2×D4×C29—22no—
43C2×D4×C29—22no—
44C2×D4×C29—22no—
45C2×D4×C29—21no—
46C2×D4×C29D414yescomplemented
47C2×D4×C29C2312yes—
48C2×D4×C29C2312yescomplemented
49C22×C4×C29C2311yescharacteristic
50C2×C22⋊D29—28no—
51C2×C22⋊D29—24no—
52C2×C22⋊D29—24no—
53C2×C22⋊D29—44no—
54C22×Dic29—21no—
55C22×Dic29D411yescharacteristic
56D4×D29—216no—
57D4×D29—28no—
58D4×D29—28no—
59C2×D116—21no—
60C2×D116—22no—
61C2×D116D414yescomplemented
62C2×D116C2312yes—
63C2×C4×D29—22no—
64C2×C4×D29—22no—
65C2×C4×D29D412yes—
66C2×C4×D29C2312yes—
67D8×C29—28no—
68C2×C8×C29—22no—
69C2×C8×C29C2312yescomplemented
70C22⋊Dic29—24no—
71#90782—28no—
72C4⋊Dic29—21no—
73C4⋊Dic29C2312yescomplemented
74C2×#6597—22no—
75C2×#6597C2312yescomplemented
76C23×C29—24no—
77C23×C29—22no—
78C23×C29—22no—
79C23×C29—42no—
80C23×C29—42no—
81C23×C29—42no—
82C23×C29—42no—
83C23×C29—42no—
84C23×C29C2×D411yescharacteristic
85C22×D29—22no—
86C22×D29—22no—
87C22×D29—216no—
88C22×D29—24no—
89C22×D29—22no—
90C22×D29—48no—
91C22×D29—432no—
92C22×D29—42no—
93C22×D29—42no—
94C22×D29—48no—
95C22×D29D814yescomplemented
96C22×D29C2×D412yes—
97D4×C29—24no—
98D4×C29—22no—
99D4×C29—22no—
100D4×C29—22no—
101D4×C29—42no—
102D4×C29C2×D414yescomplemented
103C2×C4×C29C2×D411yescharacteristic
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2×D412yes—
106C2×C4×C29C2411yescharacteristic
107C22⋊D29—216no—
108C22⋊D29—48no—
109C2×Dic29—22no—
110C2×Dic29—22no—
111C2×Dic29—42no—
112C2×Dic29C2×D412yes—
113D116—24no—
114D116—22no—
115D116C2×D414yescomplemented
116C4×D29—24no—
117C4×D29—22no—
118C4×D29—22no—
119C8×C29—22no—
120#6597—22no—
121C2×#447—291nomaximal, Sylow
122C22×C29—24no—
123C22×C29—42no—
124C22×C29—42no—
125C22×C29—48no—
126C22×C29—42no—
127C22×C29—42no—
128C22×C29—42no—
129C22×C29—84no—
130C22×C29#12012yes—
131C22×C29C2×D812yes—
132C22×C29AGL(1,ℤ/8)12yes—
133C22×C29C22×D411yescharacteristic
134C2×D29—22no—
135C2×D29—22no—
136C2×D29—28no—
137C2×D29—24no—
138C2×D29—432no—
139C2×D29—42no—
140C2×D29—42no—
141C2×D29—48no—
142C2×D29C2×D814yescomplemented
143C4×C29#12012yes—
144C4×C29C22×D411yescharacteristic
145C4×C29C22×D411yescharacteristic
146Dic29—22no—
147Dic29—42no—
148C23×D4—291no—
149C22×D8—291no—
150C22×D8D2911yescharacteristic, complemented
151C2×#121—291no—
152#447—298no—
153C2×#102—291no—
154C2×#102—291no—
155C2×C22⋊C8—291no—
156C2×C29—44no—
157C2×C29—82no—
158C2×C29#44714yescomplemented
159C2×C29C2×#12011yescharacteristic
160C2×C29C22×D811yescharacteristic
161C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
162D29—24no—
163D29—44no—
164C25—581no—
165C22×D4—294no—
166C22×D4—291no—
167C22×D4—582no—
168C22×D4—581no—
169C22×D4—584no—
170C22×D4—584no—
171C22×D4—581no—
172C22×D4C2×D2911yescharacteristic
173C22×D4C2×D2911yescharacteristic, complemented
174C23×C4—291no—
175C2×D8—24no—
176C2×D8—294no—
177C2×D8—584no—
178C2×D8C2×D2914yescomplemented
179C22×C8—291no—
180C22×C8C2×D2911yescharacteristic, complemented
181#121—294no—
182#121—582no—
183C2×C4⋊C4—291no—
184C2×C22⋊C4—581no—
185#102—294no—
186#102—294no—
187C22⋊C8—294no—
188C29C2×#44711yescharacteristic, complemented, Sylow
189C24—21no—
190C24—21no—
191C24—291no—
192C24—584no—
193C24—588no—
194C24—581no—
195C24—1164no—
196C24—1164no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—294no—
203C2×D4—292no—
204C2×D4—584no—
205C2×D4—588no—
206C2×D4—5816no—
207C2×D4—588no—
208C2×D4—588no—
209C2×D4—584no—
210C2×D4—581no—
211C2×D4—582no—
212C2×D4—1164no—
213C2×D4C22⋊D2914yescomplemented
214C2×D4C22×D2912yes—
215C2×D4C22×D2912yescomplemented
216C22×C4—291no—
217C22×C4—292no—
218C22×C4—292no—
219C22×C4—582no—
220C22×C4—582no—
221C22×C4—581no—
222C22×C4C22×D2911yescharacteristic
223D8—28no—
224D8—588no—
225C2×C8—22no—
226C2×C8—292no—
227C2×C8—582no—
228C2×C8C22×D2912yescomplemented
229C4⋊C4—292no—
230C4⋊C4—581no—
231C22⋊C4—584no—
232C23—24no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—292no—
241C23—294no—
242C23—5816no—
243C23—582no—
244C23—582no—
245C23—584no—
246C23—582no—
247C23—1168no—
248C23—11632no—
249C23—1168no—
250C23—1162no—
251C23—1162no—
252C23D4×D2911yescharacteristic
253D4—24no—
254D4—22no—
255D4—22no—
256D4—22no—
257D4—42no—
258D4—294no—
259D4—5816no—
260D4—584no—
261D4—582no—
262D4—1168no—
263D4C2×C22⋊D2914yescomplemented
264C2×C4—292no—
265C2×C4—582no—
266C2×C4—584no—
267C2×C4—582no—
268C2×C4—582no—
269C2×C4—582no—
270C2×C4—1162no—
271C2×C4D4×D2911yescharacteristic
272C2×C4D4×D2912yes—
273C2×C4C2×C22⋊D2912yes—
274C2×C4C23×D2911yescharacteristic
275C8—22no—
276C8—582no—
277C22—24no—
278C22—48no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—294no—
286C22—588no—
287C22—582no—
288C22—582no—
289C22—584no—
290C22—11632no—
291C22—1168no—
292C22—1162no—
293C22—1162no—
294C22D8×D2912yes—
295C22#1174527312yes—
296C22#1174529912yes—
297C22C2×D4×D2911yescharacteristic
298C4—582no—
299C4—1162no—
300C4#1174529912yes—
301C4C2×D4×D2911yescharacteristic
302C4C22×C22⋊D2911yescharacteristic
303C2—44no—
304C2—82no—
305C2—584no—
306C2—1164no—
307C2#1427129514yesdirect product factor
308C2C2×#1174529911yescharacteristic
309C2C2×D8×D2911yescharacteristic
310C2C2×#1174527311yescharacteristic
subgroupG/H
centerC23D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC22×D8×C29C2
socleC23×C29C2×D4
hypercenterC22×D8D29
hypocenterC29C2×#447
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×D4×D29C2semidirect
2. C22×#90782C2semidirect
3. C2×#11745300C2semidirect
4. #14271295C2direct (N × H)
5. C2×#11745169C2semidirect
6. C2×#11745180C2semidirect
7. C2×#11745168C2semidirect
8. C22×D8×C29C2semidirect
11. C22×D4×C29C22semidirect
17. C2×D4×D29C22semidirect
21. C2×D8×C29C22semidirect
22. C22×C8×C29C22semidirect
25. #11745300C22semidirect
27. C2×#90782C22semidirect
28. C2×C4⋊Dic29C22semidirect
29. C22×#6597C22semidirect
30. #11745180C22semidirect
31. #11745169C22semidirect
32. #11745168C22semidirect
46. C2×D4×C29D4semidirect
48. C2×D4×C29C23semidirect
61. C2×D116D4semidirect
69. C2×C8×C29C23semidirect
73. C4⋊Dic29C23semidirect
75. C2×#6597C23semidirect
95. C22×D29D8semidirect
102. D4×C29C2×D4semidirect
115. D116C2×D4semidirect
142. C2×D29C2×D8semidirect
150. C22×D8D29semidirect
158. C2×C29#447semidirect
173. C22×D4C2×D29semidirect
178. C2×D8C2×D29semidirect
180. C22×C8C2×D29semidirect
188. C29C2×#447semidirect
213. C2×D4C22⋊D29semidirect
215. C2×D4C22×D29semidirect
228. C2×C8C22×D29semidirect
263. D4C2×C22⋊D29semidirect
307. C2#14271295direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D292
C22×#907822
C2×#117453002
#142712952
C2×#117451692
C2×#117451802
C2×#117451682
C22×D8×C292
C2×#44729