Finite Group Databasebeta

← C2×#27132854 #37581909

Subgroups — C2×#27132854 #37581909

Order 24448.

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×#20259367C211yesmaximal, characteristic
2C2×#20259306C212yesmaximal, complemented
3C2×#20259303C211yesmaximal, characteristic, complemented
4#27132854C218yesmaximal, direct product factor
5C2×#20259236C212yesmaximal, complemented
6C2×#99×C191C211yesmaximal, characteristic, complemented
7#20259367—24no—
8#20259367—22no—
9#20259367C2214yes—
10C2×#15690287—21no—
11C2×#15690286—22no—
12C22×C22⋊D191—21no—
13C22×C22⋊D191C2212yes—
14C2×D4×D191—22no—
15C23×Dic191C2211yescharacteristic, complemented
16#20259306—24no—
17#20259306C2218yescomplemented
18#20259303—22no—
19#20259303C2214yescomplemented
20C2×C4⋊Dic191—22no—
21C2×Dic191⋊C4—21no—
22C2×C22⋊Dic191—21no—
23C2×C22⋊Dic191C2211yescharacteristic, complemented
24C2×(C2×D191)⋊C4—22no—
25#20259236C2218yescomplemented
26C22×D4×C191C2211yescharacteristic
27C2×C22⋊C4×C191C2212yescomplemented
28#99×C191C2214yescomplemented
29C23×D191—21no—
30C23×D191D412yes—
31#15690287—24no—
32#15690287—24no—
33#15690286—28no—
34#15690286—28no—
35C2×Dic382—21no—
36C2×D764—21no—
37C2×C22⋊D191—22no—
38C2×C22⋊D191—24no—
39C2×C22⋊D191—22no—
40C2×C22⋊D191—22no—
41C2×C22⋊D191—24no—
42C2×C22⋊D191—22no—
43C2×C22⋊D191—22no—
44C2×C22⋊D191D414yes—
45C2×C22⋊D191C2314yes—
46D4×D191—28no—
47D4×D191—28no—
48C22×Dic191—21no—
49C22×Dic191—22no—
50C22×Dic191—21no—
51C22×Dic191—21no—
52C22×Dic191—22no—
53C22×Dic191—21no—
54C22×Dic191D412yes—
55C22×Dic191C2312yescomplemented
56C2×C4×D191—22no—
57C4⋊Dic191—28no—
58Dic191⋊C4—24no—
59C22⋊Dic191—24no—
60C22⋊Dic191—21no—
61C22⋊Dic191C2312yescomplemented
62(C2×D191)⋊C4—28no—
63C24×C191D411yescharacteristic
64C24×C191C2311yescharacteristic
65C2×D4×C191—22no—
66C2×D4×C191—22no—
67C2×D4×C191D412yes—
68C2×D4×C191C2312yes—
69C22×C4×C191—22no—
70C22×C4×C191D411yescharacteristic
71C22⋊C4×C191—22no—
72C22⋊C4×C191C2314yescomplemented
73C22×D191—22no—
74C22×D191—24no—
75C22×D191—22no—
76C22×D191—44no—
77C22×D191—82no—
78C22×D191C2×D414yes—
79Dic382—24no—
80D764—24no—
81C22⋊D191—28no—
82C22⋊D191—24no—
83C22⋊D191—24no—
84C22⋊D191—24no—
85C22⋊D191—24no—
86C2×Dic191—24no—
87C2×Dic191—22no—
88C2×Dic191—22no—
89C2×Dic191—24no—
90C2×Dic191—22no—
91C2×Dic191—22no—
92C2×Dic191—42no—
93C2×Dic191—41no—
94C2×Dic191—42no—
95C2×Dic191—41no—
96C2×Dic191C2×D414yes—
97C4×D191—28no—
98C23×C191—21no—
99C23×C191—21no—
100C23×C191—42no—
101C23×C191—42no—
102C23×C191C2×D412yes—
103C23×C191C2×D412yes—
104C23×C191C2×D411yescharacteristic
105C23×C191C2×D412yes—
106C23×C191C2×D411yescharacteristic
107C23×C191C2411yescharacteristic
108D4×C191—24no—
109D4×C191—24no—
110C2×C4×C191—21no—
111C2×C4×C191—21no—
112C2×C4×C191—24no—
113C2×C4×C191—44no—
114C2×C4×C191C2×D412yes—
115C22×C191—24no—
116C22×C191—22no—
117C22×C191—22no—
118C22×C191—42no—
119C22×C191—41no—
120C22×C191—41no—
121C22×C191—42no—
122C22×C191—42no—
123C22×C191—81no—
124C22×C191#12011yescharacteristic
125C22×C191#12012yes—
126C22×C191#12011yescharacteristic
127C22×C191C22×D412yes—
128C22×C191C22×D411yescharacteristic
129C2×D191—24no—
130C2×D191—44no—
131C2×D191—44no—
132C2×D191—82no—
133C2×D191—84no—
134C4×C191—22no—
135C4×C191—44no—
136Dic191—24no—
137Dic191—42no—
138Dic191—42no—
139C2×C191—21no—
140C2×C191—22no—
141C2×C191—22no—
142C2×C191—21no—
143C2×C191—42no—
144C2×C191#45812yescomplemented
145C2×C191C2×#12011yescharacteristic
146D191—44no—
147C191C2×#45811yescharacteristic, complemented, Sylow
148C2×#458—1911nomaximal, Sylow
149C2×H(5,𝔽2)—1911no—
150C2×#123—1911no—
151C2×#123—1912no—
152#458—1918no—
153C2×#99—1912no—
154C2×#99D19111yescharacteristic, complemented
155H(5,𝔽2)—3824no—
156H(5,𝔽2)—3822no—
157H(5,𝔽2)—1914no—
158C2×#41—3822no—
159C2×#41—3821no—
160C22×D4—3822no—
161C22×D4—3821no—
162C22×D4—1912no—
163C22×D4C2×D19111yescharacteristic
164C23×C4—1911no—
165#123—3824no—
166#123—3822no—
167#123—1914no—
168#123—1918no—
169C2×C4⋊C4—3822no—
170C2×C4⋊C4—3821no—
171C2×C22⋊C4—3822no—
172C2×C22⋊C4—3821no—
173C2×C22⋊C4—1911no—
174C2×C22⋊C4C2×D19112yescomplemented
175#99—1918no—
176#99C2×D19114yescomplemented
177C24—3821no—
178C24—1912no—
179C24D76411yescharacteristic
180C24C22×D19111yescharacteristic
181#41—3824no—
182#41—3828no—
183#41—3828no—
184#41—3824no—
185C2×Q8—3821no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—3821no—
189C2×D4—3828no—
190C2×D4—3828no—
191C2×D4—3822no—
192C2×D4—3824no—
193C2×D4—3822no—
194C2×D4—3822no—
195C2×D4—3822no—
196C2×D4—3824no—
197C2×D4—3822no—
198C2×D4—1914no—
199C2×D4—1914no—
200C2×D4D76412yes—
201C2×D4C22×D19112yes—
202C22×C4—22no—
203C22×C4—3822no—
204C22×C4—3821no—
205C22×C4—3822no—
206C22×C4—3822no—
207C22×C4—3821no—
208C22×C4—3821no—
209C22×C4—3821no—
210C22×C4—1912no—
211C22×C4—1912no—
212C22×C4D76411yescharacteristic
213C4⋊C4—3828no—
214C4⋊C4—3824no—
215C22⋊C4—22no—
216C22⋊C4—3828no—
217C22⋊C4—3824no—
218C22⋊C4—3821no—
219C22⋊C4—1912no—
220C22⋊C4C22×D19114yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—3824no—
226C23—3822no—
227C23—3822no—
228C23—1914no—
229C23—15282no—
230C23—7644no—
231C23D4×D19112yes—
232C23D4×D19112yes—
233C23C2×D76411yescharacteristic
234C23C2×D76412yes—
235C23C2×D76411yescharacteristic
236C23C23×D19111yescharacteristic
237Q8—3824no—
238D4—24no—
239D4—24no—
240D4—3824no—
241D4—3828no—
242D4—3824no—
243D4—3824no—
244D4—3824no—
245D4—3824no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—44no—
250C2×C4—3828no—
251C2×C4—3824no—
252C2×C4—3822no—
253C2×C4—3822no—
254C2×C4—3824no—
255C2×C4—3822no—
256C2×C4—3822no—
257C2×C4—1914no—
258C2×C4—7642no—
259C2×C4—7641no—
260C2×C4—7642no—
261C2×C4—7641no—
262C2×C4C2×D76412yes—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—3824no—
273C22—15282no—
274C22—15284no—
275C22—7644no—
276C22—7644no—
277C22#2025929211yescharacteristic
278C22#2025929212yes—
279C22#2025929211yescharacteristic
280C22C2×D4×D19112yes—
281C22C22×D76411yescharacteristic
282C4—22no—
283C4—44no—
284C4—3824no—
285C4—7642no—
286C4—7642no—
287C2—22no—
288C2—21no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—7644no—
293C2#2713285412yesdirect product factor
294C2C2×#2025929211yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC23×C191C24
FrattiniC23C23×D191
FittingC2×#99×C191C2
socleC22×C191#120
prime radicalC2×#20259367C2
hypercenterC2×#99D191
hypocenterC191C2×#458
LCSG ▷ C23×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#20259306C2semidirect
3. C2×#20259303C2semidirect
4. #27132854C2direct (N × H)
5. C2×#20259236C2semidirect
6. C2×#99×C191C2semidirect
15. C23×Dic191C22semidirect
17. #20259306C22semidirect
19. #20259303C22semidirect
23. C2×C22⋊Dic191C22semidirect
25. #20259236C22semidirect
27. C2×C22⋊C4×C191C22semidirect
28. #99×C191C22semidirect
55. C22×Dic191C23semidirect
61. C22⋊Dic191C23semidirect
72. C22⋊C4×C191C23semidirect
144. C2×C191#458semidirect
147. C191C2×#458semidirect
154. C2×#99D191semidirect
174. C2×C22⋊C4C2×D191semidirect
176. #99C2×D191semidirect
220. C22⋊C4C22×D191semidirect
293. C2#27132854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202593672
C2×#202593062
C2×#202593032
#271328542
C2×#202592362
C2×#99×C1912
C2×#458191