Finite Group Databasebeta

← C2×#14271352 #16828879

Subgroups — C2×#14271352 #16828879

Order 3712.

794 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11745359C211yesmaximal, characteristic
2C2×#11745291C212yesmaximal, complemented
3#14271352C218yesmaximal, direct product factor
4C2×#11745182C211yesmaximal, characteristic, complemented
5C2×#11745154C212yesmaximal, complemented
6C2×C29×#123C211yesmaximal, characteristic, complemented
7C22×D4×C29C2211yescharacteristic
8C23×C4×C29C2211yescharacteristic, complemented
9C22×C22⋊D29—21no—
10C22×C22⋊D29C2212yes—
11#11745359—22no—
12#11745359—24no—
13#11745359C2214yes—
14C2×#90807—22no—
15C2×D4×D29—22no—
16C2×#90805—21no—
17C29×#123—22no—
18C29×#123C2214yescomplemented
19C2×C4⋊C4×C29—21no—
20C2×C22⋊C4×C29—21no—
21C2×C22⋊C4×C29C2211yescharacteristic, complemented
22C2×C22⋊Dic29C2212yescomplemented
23#11745291—24no—
24#11745291C2218yescomplemented
25C2×#90781—22no—
26C2×#90779—22no—
27#11745182C2214yescomplemented
28#11745154C2218yescomplemented
29C24×C29D411yescharacteristic
30C24×C29C2311yescharacteristic
31C23×D29—21no—
32C23×D29D412yes—
33C2×D4×C29—22no—
34C2×D4×C29—22no—
35C2×D4×C29D412yes—
36C2×D4×C29C2312yes—
37C22×C4×C29—22no—
38C22×C4×C29—22no—
39C22×C4×C29—21no—
40C22×C4×C29—21no—
41C22×C4×C29—21no—
42C22×C4×C29D411yescharacteristic
43C22×C4×C29C2312yescomplemented
44C2×C22⋊D29—22no—
45C2×C22⋊D29—24no—
46C2×C22⋊D29—22no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—22no—
49C2×C22⋊D29—22no—
50C2×C22⋊D29—24no—
51C2×C22⋊D29D414yes—
52C2×C22⋊D29C2314yes—
53C22×Dic29—21no—
54C22×Dic29—22no—
55C22×Dic29D412yes—
56#90807—28no—
57#90807—28no—
58D4×D29—28no—
59D4×D29—28no—
60#90805—24no—
61#90805—24no—
62C2×D116—21no—
63C2×C4×D29—22no—
64C2×Dic58—21no—
65C4⋊C4×C29—24no—
66C22⋊C4×C29—24no—
67C22⋊C4×C29—21no—
68C22⋊C4×C29C2312yescomplemented
69C22⋊Dic29—22no—
70C22⋊Dic29C2314yescomplemented
71#90781—28no—
72#90779—28no—
73C23×C29—21no—
74C23×C29—21no—
75C23×C29—42no—
76C23×C29—42no—
77C23×C29C2×D412yes—
78C23×C29C2×D412yes—
79C23×C29C2×D411yescharacteristic
80C23×C29C2×D412yes—
81C23×C29C2×D411yescharacteristic
82C23×C29C2411yescharacteristic
83C22×D29—22no—
84C22×D29—24no—
85C22×D29—22no—
86C22×D29—44no—
87C22×D29—82no—
88C22×D29C2×D414yes—
89D4×C29—24no—
90D4×C29—24no—
91C2×C4×C29—21no—
92C2×C4×C29—21no—
93C2×C4×C29—24no—
94C2×C4×C29—22no—
95C2×C4×C29—22no—
96C2×C4×C29—42no—
97C2×C4×C29—42no—
98C2×C4×C29—41no—
99C2×C4×C29—41no—
100C2×C4×C29C2×D412yes—
101C22⋊D29—28no—
102C22⋊D29—24no—
103C22⋊D29—24no—
104C22⋊D29—24no—
105C22⋊D29—24no—
106C2×Dic29—24no—
107C2×Dic29—22no—
108C2×Dic29—22no—
109C2×Dic29—24no—
110C2×Dic29—44no—
111C2×Dic29C2×D414yes—
112D116—24no—
113C4×D29—28no—
114Dic58—24no—
115C2×#458—291nomaximal, Sylow
116C22×C29—24no—
117C22×C29—22no—
118C22×C29—22no—
119C22×C29—42no—
120C22×C29—41no—
121C22×C29—41no—
122C22×C29—42no—
123C22×C29—42no—
124C22×C29—81no—
125C22×C29#12012yes—
126C22×C29#12011yescharacteristic
127C22×C29#12011yescharacteristic
128C22×C29C22×D412yes—
129C22×C29C22×D411yescharacteristic
130C2×D29—24no—
131C2×D29—44no—
132C2×D29—44no—
133C2×D29—82no—
134C2×D29—84no—
135C4×C29—22no—
136C4×C29—42no—
137C4×C29—42no—
138Dic29—24no—
139Dic29—44no—
140C2×H(5,𝔽2)—291no—
141C2×#123—292no—
142C2×#123D2911yescharacteristic, complemented
143#458—298no—
144C2×#99—291no—
145C2×#99—292no—
146C2×C29—22no—
147C2×C29—21no—
148C2×C29—22no—
149C2×C29—21no—
150C2×C29—42no—
151C2×C29#45812yescomplemented
152C2×C29C2×#12011yescharacteristic
153D29—44no—
154H(5,𝔽2)—294no—
155H(5,𝔽2)—584no—
156H(5,𝔽2)—582no—
157C2×#41—582no—
158C2×#41—581no—
159C22×D4—292no—
160C22×D4—582no—
161C22×D4—581no—
162C22×D4C2×D2911yescharacteristic
163C23×C4C2×D2911yescharacteristic, complemented
164#123—22no—
165#123—298no—
166#123—584no—
167#123C2×D2914yescomplemented
168C2×C4⋊C4—21no—
169C2×C4⋊C4—582no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—292no—
172C2×C22⋊C4—582no—
173C2×C22⋊C4C2×D2911yescharacteristic, complemented
174#99—294no—
175#99—298no—
176C29C2×#45811yescharacteristic, complemented, Sylow
177C24—292no—
178C24—581no—
179C24C22⋊D2911yescharacteristic
180C24C22×D2911yescharacteristic
181#41—584no—
182#41—588no—
183#41—588no—
184#41—584no—
185C2×Q8—581no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—294no—
189C2×D4—294no—
190C2×D4—581no—
191C2×D4—588no—
192C2×D4—588no—
193C2×D4—582no—
194C2×D4—584no—
195C2×D4—582no—
196C2×D4—582no—
197C2×D4—582no—
198C2×D4—582no—
199C2×D4—584no—
200C2×D4C22⋊D2912yes—
201C2×D4C22×D2912yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—292no—
208C22×C4—582no—
209C22×C4—581no—
210C22×C4—582no—
211C22×C4C22⋊D2911yescharacteristic
212C22×C4C22×D2912yescomplemented
213C4⋊C4—24no—
214C4⋊C4—588no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—294no—
218C22⋊C4—588no—
219C22⋊C4—582no—
220C22⋊C4C22×D2912yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—294no—
226C23—584no—
227C23—582no—
228C23—582no—
229C23—1164no—
230C23—2322no—
231C23D4×D2912yes—
232C23D4×D2912yes—
233C23C2×C22⋊D2911yescharacteristic
234C23C2×C22⋊D2912yes—
235C23C2×C22⋊D2911yescharacteristic
236C23C23×D2911yescharacteristic
237Q8—584no—
238D4—24no—
239D4—24no—
240D4—584no—
241D4—588no—
242D4—584no—
243D4—584no—
244D4—584no—
245D4—584no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—294no—
256C2×C4—588no—
257C2×C4—584no—
258C2×C4—582no—
259C2×C4—582no—
260C2×C4—584no—
261C2×C4—1164no—
262C2×C4C2×C22⋊D2912yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—584no—
273C22—1164no—
274C22—1164no—
275C22—2322no—
276C22—2324no—
277C22#1174529911yescharacteristic
278C22#1174529912yes—
279C22#1174529911yescharacteristic
280C22C2×D4×D2912yes—
281C22C22×C22⋊D2911yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—584no—
286C4—1164no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—1164no—
293C2#1427135212yesdirect product factor
294C2C2×#1174529911yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC2×C29×#123C2
socleC22×C29#120
prime radicalC2×#11745359C2
hypercenterC2×#123D29
hypocenterC29C2×#458
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#11745291C2semidirect
3. #14271352C2direct (N × H)
4. C2×#11745182C2semidirect
5. C2×#11745154C2semidirect
6. C2×C29×#123C2semidirect
8. C23×C4×C29C22semidirect
18. C29×#123C22semidirect
21. C2×C22⋊C4×C29C22semidirect
22. C2×C22⋊Dic29C22semidirect
24. #11745291C22semidirect
27. #11745182C22semidirect
28. #11745154C22semidirect
43. C22×C4×C29C23semidirect
68. C22⋊C4×C29C23semidirect
70. C22⋊Dic29C23semidirect
142. C2×#123D29semidirect
151. C2×C29#458semidirect
163. C23×C4C2×D29semidirect
167. #123C2×D29semidirect
173. C2×C22⋊C4C2×D29semidirect
176. C29C2×#458semidirect
212. C22×C4C22×D29semidirect
220. C22⋊C4C22×D29semidirect
293. C2#14271352direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117453592
C2×#117452912
#142713522
C2×#117451822
C2×#117451542
C2×C29×#1232
C2×#45829