Finite Group Databasebeta

← C2×#20295212 #28372415

Subgroups — C2×#20295212 #28372415

Order 13952.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 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×#16812744C211yesmaximal, characteristic, complemented
2C2×#16812668C212yesmaximal, complemented
3#20295212C218yesmaximal, direct product factor
4C2×#16812614C211yesmaximal, characteristic, complemented
5C2×#16812613C212yesmaximal, complemented
6C2×#120×C109C211yesmaximal, characteristic, complemented
7#16812744—24no—
8#16812744—22no—
9#16812744C2214yescomplemented
10C2×#13182029—21no—
11C2×#13182028—22no—
12C22×C22⋊D109—22no—
13C22×C22⋊D109—21no—
14C22×C22⋊D109C2212yes—
15C2×D4×D109—22no—
16#16812668—24no—
17#16812668C2218yescomplemented
18C2×C22⋊Dic109—22no—
19C2×C22⋊Dic109C2212yescomplemented
20#16812614C2214yescomplemented
21#16812613C2218yescomplemented
22C25×C109C2211yescharacteristic, complemented
23C22×D4×C109—21no—
24C22×D4×C109C2211yescharacteristic
25#120×C109—22no—
26#120×C109C2214yescomplemented
27C2×C22⋊C4×C109—21no—
28C2×C22⋊C4×C109C2211yescharacteristic, complemented
29C23×D109—21no—
30C23×D109D412yescomplemented
31#13182029—24no—
32#13182029—24no—
33#13182028—28no—
34#13182028—28no—
35C2×Dic218—21no—
36C2×D436—21no—
37C2×C22⋊D109—22no—
38C2×C22⋊D109—24no—
39C2×C22⋊D109—22no—
40C2×C22⋊D109—22no—
41C2×C22⋊D109—22no—
42C2×C22⋊D109—22no—
43C2×C22⋊D109—24no—
44C2×C22⋊D109—28no—
45C2×C22⋊D109—48no—
46C2×C22⋊D109D414yescomplemented
47C2×C22⋊D109C2314yes—
48D4×D109—28no—
49D4×D109—28no—
50C22×Dic109—21no—
51C22×Dic109—22no—
52C22×Dic109D412yescomplemented
53C2×C4×D109—22no—
54C22⋊Dic109—22no—
55C22⋊Dic109—28no—
56C22⋊Dic109C2314yescomplemented
57C24×C109—21no—
58C24×C109—22no—
59C24×C109—21no—
60C24×C109—22no—
61C24×C109—44no—
62C24×C109D411yescharacteristic, complemented
63C24×C109C2311yescharacteristic
64C24×C109C2312yescomplemented
65C2×D4×C109—22no—
66C2×D4×C109—22no—
67C2×D4×C109—24no—
68C2×D4×C109—44no—
69C2×D4×C109D412yescomplemented
70C2×D4×C109C2312yes—
71C22×C4×C109—21no—
72C22×C4×C109D411yescharacteristic, complemented
73C22⋊C4×C109—21no—
74C22⋊C4×C109—24no—
75C22⋊C4×C109C2312yescomplemented
76C23×C109—24no—
77C23×C109—28no—
78C23×C109—21no—
79C23×C109—22no—
80C23×C109—24no—
81C23×C109—21no—
82C23×C109—42no—
83C23×C109—44no—
84C23×C109—44no—
85C23×C109—42no—
86C23×C109—48no—
87C23×C109—44no—
88C23×C109—42no—
89C23×C109—42no—
90C23×C109—41no—
91C23×C109—41no—
92C23×C109C2×D412yes—
93C23×C109C2×D412yes—
94C23×C109C2×D411yescharacteristic
95C23×C109C2×D411yescharacteristic
96C23×C109C2×D412yescomplemented
97C23×C109C2411yescharacteristic
98C22×D109—22no—
99C22×D109—24no—
100C22×D109—22no—
101C22×D109—44no—
102C22×D109—82no—
103C22×D109C2×D414yescomplemented
104D4×C109—24no—
105D4×C109—24no—
106D4×C109—48no—
107C2×C4×C109—21no—
108C2×C4×C109—21no—
109C2×C4×C109—22no—
110C2×C4×C109—42no—
111C2×C4×C109C2×D412yescomplemented
112C22⋊D109—28no—
113C22⋊D109—24no—
114C22⋊D109—24no—
115C22⋊D109—24no—
116C22⋊D109—24no—
117C22⋊D109—416no—
118C2×Dic109—24no—
119C2×Dic109—22no—
120C2×Dic109—22no—
121C2×Dic109—24no—
122C2×Dic109—44no—
123C2×Dic109C2×D414yescomplemented
124D436—24no—
125C4×D109—28no—
126Dic218—24no—
127C22×C109—24no—
128C22×C109—22no—
129C22×C109—28no—
130C22×C109—24no—
131C22×C109—22no—
132C22×C109—42no—
133C22×C109—41no—
134C22×C109—41no—
135C22×C109—42no—
136C22×C109—44no—
137C22×C109—42no—
138C22×C109—48no—
139C22×C109—44no—
140C22×C109—42no—
141C22×C109—42no—
142C22×C109—44no—
143C22×C109—81no—
144C22×C109#12012yes—
145C22×C109#12011yescharacteristic
146C22×C109#12011yescharacteristic
147C22×C109C22×D412yes—
148C22×C109C22×D411yescharacteristic
149C2×D109—24no—
150C2×D109—44no—
151C2×D109—44no—
152C2×D109—82no—
153C2×D109—84no—
154C4×C109—22no—
155C4×C109—42no—
156Dic109—24no—
157Dic109—44no—
158C2×C109—22no—
159C2×C109—21no—
160C2×C109—22no—
161C2×C109—21no—
162C2×C109—42no—
163C2×C109—44no—
164C2×C109B(4,𝔽2)12yescomplemented
165C2×C109C2×#12011yescharacteristic
166D109—44no—
167C2×B(4,𝔽2)—1091nomaximal, Sylow
168C109C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
169C2×H(5,𝔽2)—1091no—
170C2×#120—1092no—
171C2×#120D10911yescharacteristic, complemented
172B(4,𝔽2)—1098no—
173C2×#99—1091no—
174C2×#99—1092no—
175C25C2×D10911yescharacteristic, complemented
176H(5,𝔽2)—1094no—
177H(5,𝔽2)—2184no—
178H(5,𝔽2)—2182no—
179C2×#41—2182no—
180C2×#41—2181no—
181C22×D4—21no—
182C22×D4—1092no—
183C22×D4—2182no—
184C22×D4—2182no—
185C22×D4—2181no—
186C22×D4C2×D10911yescharacteristic
187#120—22no—
188#120—1098no—
189#120—2184no—
190#120C2×D10914yescomplemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—1092no—
193C2×C22⋊C4—2182no—
194C2×C22⋊C4C2×D10911yescharacteristic, complemented
195#99—1094no—
196#99—1098no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—44no—
202C24—1092no—
203C24—2181no—
204C24C22⋊D10911yescharacteristic, complemented
205C24C22×D10911yescharacteristic
206C24C22×D10912yescomplemented
207#41—2184no—
208#41—2188no—
209#41—2188no—
210#41—2184no—
211C2×Q8—2181no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—44no—
216C2×D4—1094no—
217C2×D4—1094no—
218C2×D4—4368no—
219C2×D4—2181no—
220C2×D4—2188no—
221C2×D4—2188no—
222C2×D4—2188no—
223C2×D4—2182no—
224C2×D4—2184no—
225C2×D4—2182no—
226C2×D4—2182no—
227C2×D4—2182no—
228C2×D4—2182no—
229C2×D4—2184no—
230C2×D4C22⋊D10912yescomplemented
231C2×D4C22×D10912yes—
232C22×C4—21no—
233C22×C4—1092no—
234C22×C4—2182no—
235C22×C4—2182no—
236C22×C4—2181no—
237C22×C4C22⋊D10911yescharacteristic, complemented
238C22⋊C4—21no—
239C22⋊C4—24no—
240C22⋊C4—1094no—
241C22⋊C4—2188no—
242C22⋊C4—2182no—
243C22⋊C4C22×D10912yescomplemented
244C23—24no—
245C23—28no—
246C23—21no—
247C23—24no—
248C23—22no—
249C23—21no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—8722no—
261C23—1094no—
262C23—4364no—
263C23—2184no—
264C23—2182no—
265C23—2182no—
266C23D4×D10912yes—
267C23D4×D10912yes—
268C23C2×C22⋊D10911yescharacteristic
269C23C2×C22⋊D10911yescharacteristic
270C23C2×C22⋊D10912yescomplemented
271C23C23×D10911yescharacteristic
272Q8—2184no—
273D4—24no—
274D4—24no—
275D4—48no—
276D4—43616no—
277D4—2184no—
278D4—2188no—
279D4—2184no—
280D4—2184no—
281D4—2184no—
282D4—2184no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—1094no—
288C2×C4—4364no—
289C2×C4—2188no—
290C2×C4—2184no—
291C2×C4—2184no—
292C2×C4—2182no—
293C2×C4—2182no—
294C2×C4C2×C22⋊D10912yescomplemented
295C22—22no—
296C22—24no—
297C22—24no—
298C22—28no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—81no—
312C22—8722no—
313C22—8724no—
314C22—4364no—
315C22—4364no—
316C22—2184no—
317C22#1681266711yescharacteristic
318C22#1681266712yes—
319C22#1681266711yescharacteristic
320C22C2×D4×D10912yes—
321C22C22×C22⋊D10911yescharacteristic
322C4—22no—
323C4—42no—
324C4—4364no—
325C4—2184no—
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—4364no—
333C2#2029521212yesdirect product factor
334C2C2×#1681266711yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC23×C109C24
FrattiniC23C23×D109
FittingC2×#120×C109C2
socleC22×C109#120
hypercenterC2×#120D109
hypocenterC109C2×B(4,𝔽2)
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16812744C2semidirect
2. C2×#16812668C2semidirect
3. #20295212C2direct (N × H)
4. C2×#16812614C2semidirect
5. C2×#16812613C2semidirect
6. C2×#120×C109C2semidirect
9. #16812744C22semidirect
17. #16812668C22semidirect
19. C2×C22⋊Dic109C22semidirect
20. #16812614C22semidirect
21. #16812613C22semidirect
22. C25×C109C22semidirect
26. #120×C109C22semidirect
28. C2×C22⋊C4×C109C22semidirect
30. C23×D109D4semidirect
46. C2×C22⋊D109D4semidirect
52. C22×Dic109D4semidirect
56. C22⋊Dic109C23semidirect
62. C24×C109D4semidirect
64. C24×C109C23semidirect
69. C2×D4×C109D4semidirect
72. C22×C4×C109D4semidirect
75. C22⋊C4×C109C23semidirect
96. C23×C109C2×D4semidirect
103. C22×D109C2×D4semidirect
111. C2×C4×C109C2×D4semidirect
123. C2×Dic109C2×D4semidirect
164. C2×C109B(4,𝔽2)semidirect
168. C109C2×B(4,𝔽2)semidirect
171. C2×#120D109semidirect
175. C25C2×D109semidirect
190. #120C2×D109semidirect
194. C2×C22⋊C4C2×D109semidirect
204. C24C22⋊D109semidirect
206. C24C22×D109semidirect
230. C2×D4C22⋊D109semidirect
237. C22×C4C22⋊D109semidirect
243. C22⋊C4C22×D109semidirect
270. C23C2×C22⋊D109semidirect
294. C2×C4C2×C22⋊D109semidirect
333. C2#20295212direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168127442
C2×#168126682
#202952122
C2×#168126142
C2×#168126132
C2×#120×C1092
C2×B(4,𝔽2)109