Finite Group Databasebeta

← C2×C113×#1740 #43288373

Subgroups — C2×C113×#1740 #43288373

Order 28928.

2350 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C113C212yesmaximal
2C22×#117×C113C211yesmaximal, characteristic
3C113×#1740C218yesmaximal, direct product factor
4C2×#342×C113C214yesmaximal
5C24×C4×C113C2211yescharacteristic
6C2×#117×C113—28no—
7C2×#117×C113—22no—
8C2×#117×C113—24no—
9C2×#117×C113—22no—
10C2×#117×C113—24no—
11C2×#117×C113C4116yescomplemented
12C2×#117×C113C414yescomplemented
13C2×#117×C113C2218yes—
14C2×#117×C113C2214yes—
15C22×C4⋊C4×C113—22no—
16C22×C4⋊C4×C113C412yescomplemented
17C22×C22⋊C4×C113C2212yes—
18C22×C22⋊C4×C113C2214yes—
19C22×C42×C113—22no—
20C22×C42×C113C412yescomplemented
21#342×C113C22116yes—
22C25×C113C2311yescharacteristic
23C23×C4×C113—24no—
24C23×C4×C113C2×C414yes—
25C23×C4×C113C2×C411yescharacteristic
26C23×C4×C113C2×C418yes—
27C23×C4×C113C2×C412yes—
28C23×C4×C113C2×C411yescharacteristic
29C23×C4×C113D412yes—
30C23×C4×C113D414yes—
31C23×C4×C113D414yes—
32C23×C4×C113D412yes—
33C23×C4×C113Q814yes—
34C23×C4×C113C2312yes—
35#117×C113—232no—
36#117×C113—28no—
37#117×C113—28no—
38#117×C113—216no—
39#117×C113—28no—
40#117×C113—28no—
41#117×C113C2×C4132yescomplemented
42C2×C4⋊C4×C113—216no—
43C2×C4⋊C4×C113—24no—
44C2×C4⋊C4×C113—24no—
45C2×C4⋊C4×C113—28no—
46C2×C4⋊C4×C113C2×C418yescomplemented
47C2×C22⋊C4×C113—24no—
48C2×C22⋊C4×C113—24no—
49C2×C22⋊C4×C113—24no—
50C2×C22⋊C4×C113—28no—
51C2×C22⋊C4×C113C2×C414yes—
52C2×C22⋊C4×C113C2×C4116yes—
53C2×C22⋊C4×C113C2314yes—
54C2×C22⋊C4×C113C2318yes—
55C2×C42×C113—216no—
56C2×C42×C113—22no—
57C2×C42×C113—24no—
58C2×C42×C113—22no—
59C2×C42×C113—28no—
60C2×C42×C113C2×C418yescomplemented
61C24×C113—44no—
62C24×C113C22×C411yescharacteristic
63C24×C113C22×C414yes—
64C24×C113C22×C411yescharacteristic
65C24×C113C2×D412yes—
66C24×C113C2×D412yes—
67C24×C113C2×D412yes—
68C24×C113C2×Q812yes—
69C24×C113C2411yescharacteristic
70C22×C4×C113—24no—
71C22×C4×C113—24no—
72C22×C4×C113—216no—
73C22×C4×C113—28no—
74C22×C4×C113—416no—
75C22×C4×C113—48no—
76C22×C4×C113C4214yes—
77C22×C4×C113C4218yes—
78C22×C4×C113C4214yes—
79C22×C4×C113C22⋊C411yescharacteristic
80C22×C4×C113C22⋊C412yes—
81C22×C4×C113C22⋊C418yes—
82C22×C4×C113C22⋊C418yes—
83C22×C4×C113C22⋊C412yes—
84C22×C4×C113C22⋊C412yes—
85C22×C4×C113C22⋊C414yes—
86C22×C4×C113C22⋊C4116yes—
87C22×C4×C113C22⋊C414yes—
88C22×C4×C113C22⋊C411yescharacteristic
89C22×C4×C113C4⋊C414yes—
90C22×C4×C113C4⋊C418yes—
91C22×C4×C113C4⋊C418yes—
92C22×C4×C113C4⋊C4116yes—
93C22×C4×C113C4⋊C418yes—
94C22×C4×C113C4⋊C414yes—
95C22×C4×C113C22×C418yes—
96C22×C4×C113C22×C412yes—
97C22×C4×C113C22×C412yes—
98C22×C4×C113C22×C418yes—
99C22×C4×C113C2×D414yes—
100C22×C4×C113C2×D414yes—
101C22×C4×C113C2×D414yes—
102C22×C4×C113C2×Q814yes—
103C4⋊C4×C113—232no—
104C4⋊C4×C113—216no—
105C22⋊C4×C113—28no—
106C22⋊C4×C113—216no—
107C22⋊C4×C113—28no—
108C22⋊C4×C113—28no—
109C22⋊C4×C113C22×C4116yes—
110C42×C113—232no—
111C42×C113—28no—
112C42×C113—28no—
113C23×C113—24no—
114C23×C113—28no—
115C23×C113—42no—
116C23×C113—48no—
117C23×C113—44no—
118C23×C113—44no—
119C23×C113—44no—
120C23×C113—42no—
121C23×C113C2×C4212yes—
122C23×C113C2×C4212yes—
123C23×C113C2×C22⋊C412yes—
124C23×C113C2×C22⋊C411yescharacteristic
125C23×C113C2×C22⋊C414yes—
126C23×C113C2×C22⋊C414yes—
127C23×C113C2×C22⋊C411yescharacteristic
128C23×C113C2×C4⋊C412yes—
129C23×C113C2×C4⋊C414yes—
130C23×C113C2×C4⋊C412yes—
131C23×C113C2×C4⋊C414yes—
132C23×C113C23×C411yescharacteristic
133C23×C113C23×C412yes—
134C23×C113C22×D412yes—
135C23×C113C22×D411yescharacteristic
136C23×C113C22×Q811yescharacteristic
137C2×C4×C113—28no—
138C2×C4×C113—216no—
139C2×C4×C113—416no—
140C2×C4×C113—432no—
141C2×C4×C113—48no—
142C2×C4×C113—48no—
143C2×C4×C113—48no—
144C2×C4×C113#9518yes—
145C2×C4×C113#95116yes—
146C2×C4×C113#9518yes—
147C2×C4×C113#95116yes—
148C2×C4×C113#9518yes—
149C2×C4×C113#9514yes—
150C2×C4×C113#9514yes—
151C2×C4×C113C2×C4214yes—
152C2×C4×C113C2×C4214yes—
153C2×C4×C113C2×C22⋊C418yes—
154C2×C4×C113C2×C22⋊C412yes—
155C2×C4×C113C2×C22⋊C412yes—
156C2×C4×C113C2×C22⋊C418yes—
157C2×C4×C113C2×C22⋊C414yes—
158C2×C4×C113C2×C4⋊C414yes—
159C2×C4×C113C2×C4⋊C414yes—
160C2×C4×C113C2×C4⋊C418yes—
161C2×C4×C113C2×C4⋊C418yes—
162C22×C113—22no—
163C22×C113—22no—
164C22×C113—24no—
165C22×C113—24no—
166C22×C113—28no—
167C22×C113—24no—
168C22×C113—24no—
169C22×C113—28no—
170C22×C113—42no—
171C22×C113—44no—
172C22×C113—42no—
173C22×C113—44no—
174C22×C113—42no—
175C22×C113—41no—
176C22×C113—41no—
177C22×C113C2×#9512yes—
178C22×C113C2×#9514yes—
179C22×C113C2×#9512yes—
180C22×C113#41014yes—
181C22×C113C22×C4211yescharacteristic
182C22×C113C22×C22⋊C412yes—
183C22×C113C22×C22⋊C411yescharacteristic
184C22×C113C22×C4⋊C411yescharacteristic
185C22×C113C22×C4⋊C412yes—
186C4×C113—416no—
187C4×C113—48no—
188C4×C113C2×#9514yes—
189C4×C113C2×#9518yes—
190C4×C113C2×#9514yes—
191C2×#1740C11311yesmaximal, characteristic, direct product factor, Sylow
192C2×C113—22no—
193C2×C113—21no—
194C2×C113—22no—
195C2×C113—22no—
196C2×C113—24no—
197C2×C113—21no—
198C2×C113#174014yesdirect product factor
199C2×C113C22×#9511yescharacteristic
200C2×C113C2×#41012yes—
201C22×#117C2×C11312yes—
202C22×#117C2×C11311yescharacteristic
203#1740C2×C11318yesdirect product factor
204C2×#342C2×C11314yes—
205C113C2×#174011yescharacteristic, direct product factor, Sylow
206C24×C4C22×C11311yescharacteristic
207C2×#117—22no—
208C2×#117—24no—
209C2×#117—22no—
210C2×#117—28no—
211C2×#117—24no—
212C2×#117C4×C113116yescomplemented
213C2×#117C4×C11314yescomplemented
214C2×#117C22×C11318yes—
215C2×#117C22×C11314yes—
216C22×C4⋊C4—22no—
217C22×C4⋊C4C4×C11312yescomplemented
218C22×C22⋊C4C22×C11312yes—
219C22×C22⋊C4C22×C11314yes—
220C22×C42—22no—
221C22×C42C4×C11312yescomplemented
222#342C22×C113116yes—
223C25C23×C11311yescharacteristic
224C23×C4—24no—
225C23×C4C2×C4×C11314yes—
226C23×C4C2×C4×C11311yescharacteristic
227C23×C4C2×C4×C11318yes—
228C23×C4C2×C4×C11312yes—
229C23×C4C2×C4×C11311yescharacteristic
230C23×C4D4×C11312yes—
231C23×C4D4×C11314yes—
232C23×C4D4×C11314yes—
233C23×C4D4×C11312yes—
234C23×C4Q8×C11314yes—
235C23×C4C23×C11312yes—
236#117—232no—
237#117—28no—
238#117—28no—
239#117—216no—
240#117—28no—
241#117—28no—
242#117C2×C4×C113132yescomplemented
243C2×C4⋊C4—216no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4C2×C4×C11318yescomplemented
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4C2×C4×C11314yes—
253C2×C22⋊C4C2×C4×C113116yes—
254C2×C22⋊C4C23×C11314yes—
255C2×C22⋊C4C23×C11318yes—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—216no—
260C2×C42—28no—
261C2×C42C2×C4×C11318yescomplemented
262C24—44no—
263C24C22×C4×C11311yescharacteristic
264C24C22×C4×C11314yes—
265C24C22×C4×C11311yescharacteristic
266C24C2×D4×C11312yes—
267C24C2×D4×C11312yes—
268C24C2×D4×C11312yes—
269C24C2×Q8×C11312yes—
270C24C24×C11311yescharacteristic
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—216no—
274C22×C4—28no—
275C22×C4—416no—
276C22×C4—48no—
277C22×C4C42×C11314yes—
278C22×C4C42×C11314yes—
279C22×C4C42×C11318yes—
280C22×C4C22⋊C4×C11312yes—
281C22×C4C22⋊C4×C11318yes—
282C22×C4C22⋊C4×C11311yescharacteristic
283C22×C4C22⋊C4×C11311yescharacteristic
284C22×C4C22⋊C4×C11318yes—
285C22×C4C22⋊C4×C11312yes—
286C22×C4C22⋊C4×C11312yes—
287C22×C4C22⋊C4×C11314yes—
288C22×C4C22⋊C4×C11314yes—
289C22×C4C22⋊C4×C113116yes—
290C22×C4C4⋊C4×C11314yes—
291C22×C4C4⋊C4×C11318yes—
292C22×C4C4⋊C4×C11318yes—
293C22×C4C4⋊C4×C11314yes—
294C22×C4C4⋊C4×C11318yes—
295C22×C4C4⋊C4×C113116yes—
296C22×C4C22×C4×C11318yes—
297C22×C4C22×C4×C11312yes—
298C22×C4C22×C4×C11312yes—
299C22×C4C22×C4×C11318yes—
300C22×C4C2×D4×C11314yes—
301C22×C4C2×D4×C11314yes—
302C22×C4C2×D4×C11314yes—
303C22×C4C2×Q8×C11314yes—
304C4⋊C4—232no—
305C4⋊C4—216no—
306C22⋊C4—28no—
307C22⋊C4—216no—
308C22⋊C4—28no—
309C22⋊C4—28no—
310C22⋊C4C22×C4×C113116yes—
311C42—232no—
312C42—28no—
313C42—28no—
314C23—24no—
315C23—28no—
316C23—42no—
317C23—48no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23C2×C42×C11312yes—
323C23C2×C42×C11312yes—
324C23C2×C22⋊C4×C11312yes—
325C23C2×C22⋊C4×C11311yescharacteristic
326C23C2×C22⋊C4×C11314yes—
327C23C2×C22⋊C4×C11314yes—
328C23C2×C22⋊C4×C11311yescharacteristic
329C23C2×C4⋊C4×C11312yes—
330C23C2×C4⋊C4×C11314yes—
331C23C2×C4⋊C4×C11312yes—
332C23C2×C4⋊C4×C11314yes—
333C23C23×C4×C11311yescharacteristic
334C23C23×C4×C11312yes—
335C23C22×D4×C11312yes—
336C23C22×D4×C11311yescharacteristic
337C23C22×Q8×C11311yescharacteristic
338C2×C4—28no—
339C2×C4—216no—
340C2×C4—416no—
341C2×C4—432no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4#95×C11318yes—
346C2×C4#95×C113116yes—
347C2×C4#95×C11318yes—
348C2×C4#95×C113116yes—
349C2×C4#95×C11314yes—
350C2×C4#95×C11318yes—
351C2×C4#95×C11314yes—
352C2×C4C2×C42×C11314yes—
353C2×C4C2×C42×C11314yes—
354C2×C4C2×C22⋊C4×C11318yes—
355C2×C4C2×C22⋊C4×C11312yes—
356C2×C4C2×C22⋊C4×C11312yes—
357C2×C4C2×C22⋊C4×C11318yes—
358C2×C4C2×C22⋊C4×C11314yes—
359C2×C4C2×C4⋊C4×C11314yes—
360C2×C4C2×C4⋊C4×C11314yes—
361C2×C4C2×C4⋊C4×C11318yes—
362C2×C4C2×C4⋊C4×C11318yes—
363C22—22no—
364C22—22no—
365C22—24no—
366C22—24no—
367C22—28no—
368C22—24no—
369C22—24no—
370C22—28no—
371C22—42no—
372C22—44no—
373C22—42no—
374C22—44no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22C2×#95×C11312yes—
379C22C2×#95×C11314yes—
380C22C2×#95×C11312yes—
381C22#410×C11314yes—
382C22C22×C42×C11311yescharacteristic
383C22C22×C22⋊C4×C11312yes—
384C22C22×C22⋊C4×C11311yescharacteristic
385C22C22×C4⋊C4×C11311yescharacteristic
386C22C22×C4⋊C4×C11312yes—
387C4—416no—
388C4—48no—
389C4C2×#95×C11314yes—
390C4C2×#95×C11314yes—
391C4C2×#95×C11318yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—24no—
398C2C113×#174014yesdirect product factor
399C2C22×#95×C11311yescharacteristic
400C2C2×#410×C11312yes—
subgroupG/H
centerC22×C4×C113C22⋊C4
commutatorC22C22×C42×C113
Frattini, cosocleC24C24×C113
socleC23×C113C2×C22⋊C4
prime radicalC25×C113C23
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C113 ◁ C24×C4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
3. C113×#1740C2direct (N × H)
11. C2×#117×C113C4semidirect
12. C2×#117×C113C4semidirect
16. C22×C4⋊C4×C113C4semidirect
20. C22×C42×C113C4semidirect
41. #117×C113C2×C4semidirect
46. C2×C4⋊C4×C113C2×C4semidirect
60. C2×C42×C113C2×C4semidirect
191. C2×#1740C113direct (N × H)
198. C2×C113#1740direct (N × H)
203. #1740C2×C113direct (N × H)
205. C113C2×#1740direct (N × H)
212. C2×#117C4×C113semidirect
213. C2×#117C4×C113semidirect
217. C22×C4⋊C4C4×C113semidirect
221. C22×C42C4×C113semidirect
242. #117C2×C4×C113semidirect
247. C2×C4⋊C4C2×C4×C113semidirect
261. C2×C42C2×C4×C113semidirect
398. C2C113×#1740direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#117×C1132
C113×#17402
C2×#342×C1132
C2×#1740113