Finite Group Databasebeta

← C41×#11549 #23787750

Subgroups — C41×#11549 #23787750

Order 10496.

496 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C41×#2501C211yesmaximal, characteristic
2C41×#1463C211yesmaximal, characteristic, complemented
3C2×C41×#324C211yesmaximal, characteristic, complemented
4C41×#1325C212yesmaximal, complemented
5C41×#1318C212yesmaximal, complemented
6C2×#123×C41—21no—
7C2×#123×C41—21no—
8C2×#121×C41C411yescharacteristic
9C22×C4⋊C4×C41C2211yescharacteristic
10C41×#407—21no—
11C2×C22⋊C8×C41—21no—
12C41×#399—21no—
13C41×#399—21no—
14C41×#390C411yescharacteristic
15C41×#390C2211yescharacteristic
16C41×#384C411yescharacteristic
17C41×#384C2211yescharacteristic
18C4×C22⋊C4×C41C411yescharacteristic
19C41×#324—21no—
20C41×#324—21no—
21C41×#324C2212yes—
22C41×#324C2212yes—
23C22×D4×C41—21no—
24C22×D4×C41C2×C411yescharacteristic
25C23×C4×C41C2×C411yescharacteristic
26C23×C4×C41D411yescharacteristic
27C23×C4×C41D411yescharacteristic
28C2×M16×C41—41no—
29C22×C8×C41—41no—
30#123×C41—44no—
31#123×C41—44no—
32#121×C41—41no—
33#121×C41—41no—
34C2×C4⋊C4×C41—21no—
35C2×C4⋊C4×C41—21no—
36C2×C4⋊C4×C41—21no—
37C2×C4⋊C4×C41—21no—
38C2×C4⋊C4×C41—41no—
39C2×C4⋊C4×C41—41no—
40C2×C4⋊C4×C41C2×C411yescharacteristic
41C2×C4⋊C4×C41C2×C411yescharacteristic
42C2×C4⋊C4×C41D411yescharacteristic
43C2×C4⋊C4×C41D411yescharacteristic
44C2×C4⋊C4×C41C2311yescharacteristic
45C2×C22⋊C4×C41—21no—
46C2×C22⋊C4×C41—21no—
47C2×C22⋊C4×C41—21no—
48C2×C22⋊C4×C41—21no—
49C2×C22⋊C4×C41—21no—
50C2×C22⋊C4×C41—21no—
51C2×C42×C41C2×C411yescharacteristic
52C2×C42×C41C2×C411yescharacteristic
53C22⋊C8×C41—22no—
54C22⋊C8×C41—22no—
55C22⋊C8×C41—41no—
56C22⋊C8×C41—41no—
57#95×C41—21no—
58C24×C41—21no—
59C24×C41C22⋊C411yescharacteristic
60C2×D4×C41—22no—
61C2×D4×C41—41no—
62C2×D4×C41—41no—
63C2×D4×C41—42no—
64C2×D4×C41—42no—
65C2×D4×C41—41no—
66C2×D4×C41—41no—
67C22×C4×C41—21no—
68C22×C4×C41—21no—
69C22×C4×C41—21no—
70C22×C4×C41—21no—
71C22×C4×C41—21no—
72C22×C4×C41—21no—
73C22×C4×C41—21no—
74C22×C4×C41—21no—
75C22×C4×C41—22no—
76C22×C4×C41—21no—
77C22×C4×C41—41no—
78C22×C4×C41—42no—
79C22×C4×C41—42no—
80C22×C4×C41C22⋊C411yescharacteristic
81C22×C4×C41C22⋊C411yescharacteristic
82C22×C4×C41C22⋊C411yescharacteristic
83C22×C4×C41C22×C411yescharacteristic
84C22×C4×C41C2×D411yescharacteristic
85C22×C4×C41C2×D411yescharacteristic
86M16×C41—82no—
87C2×C8×C41—42no—
88C2×C8×C41—42no—
89C2×C8×C41—82no—
90C4⋊C4×C41—41no—
91C4⋊C4×C41—41no—
92C4⋊C4×C41—41no—
93C4⋊C4×C41—41no—
94C4⋊C4×C41—41no—
95C4⋊C4×C41—42no—
96C4⋊C4×C41—42no—
97C4⋊C4×C41—42no—
98C4⋊C4×C41—42no—
99C22⋊C4×C41—24no—
100C22⋊C4×C41—42no—
101C22⋊C4×C41—42no—
102C22⋊C4×C41—42no—
103C22⋊C4×C41—42no—
104C22⋊C4×C41—42no—
105C42×C41—22no—
106C42×C41—41no—
107C23×C41—22no—
108C23×C41—21no—
109C23×C41—21no—
110C23×C41—41no—
111C23×C41—41no—
112C23×C41—41no—
113C23×C41—41no—
114C23×C41—41no—
115C23×C41—41no—
116C23×C41—41no—
117C23×C41—41no—
118C23×C41#9911yescharacteristic
119C23×C41#9911yescharacteristic
120C23×C41C2×C22⋊C411yescharacteristic
121D4×C41—82no—
122D4×C41—82no—
123D4×C41—82no—
124C2×C4×C41—21no—
125C2×C4×C41—21no—
126C2×C4×C41—22no—
127C2×C4×C41—22no—
128C2×C4×C41—21no—
129C2×C4×C41—21no—
130C2×C4×C41—21no—
131C2×C4×C41—21no—
132C2×C4×C41—21no—
133C2×C4×C41—21no—
134C2×C4×C41—21no—
135C2×C4×C41—42no—
136C2×C4×C41—42no—
137C2×C4×C41—41no—
138C2×C4×C41—41no—
139C2×C4×C41—42no—
140C2×C4×C41—42no—
141C2×C4×C41—41no—
142C2×C4×C41—41no—
143C2×C4×C41—41no—
144C2×C4×C41—41no—
145C2×C4×C41—41no—
146C2×C4×C41—41no—
147C2×C4×C41—41no—
148C2×C4×C41—41no—
149C2×C4×C41—41no—
150C2×C4×C41—81no—
151C2×C4×C41—81no—
152C2×C4×C41#13112yes—
153C8×C41—82no—
154C8×C41—82no—
155#11549C4111yesmaximal, characteristic, direct product factor, Sylow
156C22×C41—21no—
157C22×C41—21no—
158C22×C41—21no—
159C22×C41—21no—
160C22×C41—21no—
161C22×C41—21no—
162C22×C41—21no—
163C22×C41—42no—
164C22×C41—42no—
165C22×C41—81no—
166C22×C41—81no—
167C22×C41—81no—
168C22×C41—81no—
169C22×C41—81no—
170C22×C41—81no—
171C22×C41—81no—
172C22×C41#35312yes—
173C22×C41#40811yescharacteristic
174C22×C41C2×#9911yescharacteristic
175C22×C41#41311yescharacteristic
176C4×C41—22no—
177C4×C41—42no—
178C4×C41—42no—
179C4×C41—41no—
180C4×C41—41no—
181C4×C41—41no—
182C4×C41—41no—
183C4×C41—41no—
184C4×C41—81no—
185#2501C2×C4111yescharacteristic
186#1463C2×C4111yescharacteristic, complemented
187C2×#324C2×C4111yescharacteristic, complemented
188#1325C2×C4112yescomplemented
189#1318C2×C4112yescomplemented
190C2×C41—21no—
191C2×C41—21no—
192C2×C41—22no—
193C2×C41—41no—
194C2×C41—161no—
195C2×C41#146811yescharacteristic
196C2×C41C2×#35311yescharacteristic
197C2×C41#213311yescharacteristic
198C2×#123—21no—
199C2×#123—21no—
200C2×#121C4×C4111yescharacteristic
201C22×C4⋊C4C22×C4111yescharacteristic
202#407—21no—
203C2×C22⋊C8—21no—
204#399—21no—
205#399—21no—
206#390C4×C4111yescharacteristic
207#390C22×C4111yescharacteristic
208#384C4×C4111yescharacteristic
209#384C22×C4111yescharacteristic
210C4×C22⋊C4C4×C4111yescharacteristic
211#324—21no—
212#324—21no—
213#324C22×C4112yes—
214#324C22×C4112yes—
215C41#1154911yescharacteristic, direct product factor, Sylow
216C22×D4—21no—
217C22×D4C2×C4×C4111yescharacteristic
218C23×C4C2×C4×C4111yescharacteristic
219C23×C4D4×C4111yescharacteristic
220C23×C4D4×C4111yescharacteristic
221C2×M16—41no—
222C22×C8—41no—
223#123—44no—
224#123—44no—
225#121—41no—
226#121—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C2×C4×C4111yescharacteristic
234C2×C4⋊C4C2×C4×C4111yescharacteristic
235C2×C4⋊C4D4×C4111yescharacteristic
236C2×C4⋊C4D4×C4111yescharacteristic
237C2×C4⋊C4C23×C4111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42C2×C4×C4111yescharacteristic
245C2×C42C2×C4×C4111yescharacteristic
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250#95—21no—
251C24—21no—
252C24C22⋊C4×C4111yescharacteristic
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C22⋊C4×C4111yescharacteristic
274C22×C4C22⋊C4×C4111yescharacteristic
275C22×C4C22⋊C4×C4111yescharacteristic
276C22×C4C22×C4×C4111yescharacteristic
277C22×C4C2×D4×C4111yescharacteristic
278C22×C4C2×D4×C4111yescharacteristic
279M16—82no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—82no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—22no—
299C42—41no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23#99×C4111yescharacteristic
312C23#99×C4111yescharacteristic
313C23C2×C22⋊C4×C4111yescharacteristic
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#131×C4112yes—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C41×#35312yes—
365C22C41×#40811yescharacteristic
366C22C2×#99×C4111yescharacteristic
367C22C41×#41311yescharacteristic
368C4—22no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2C41×#146811yescharacteristic
383C2C2×C41×#35311yescharacteristic
384C2C41×#213311yescharacteristic
subgroupG/H
center, socleC22×C41C2×#99
commutatorC22×C4C22×C4×C41
Frattini, cosocleC2×C4⋊C4C23×C41
prime radicalC2×#121×C41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C24×C41 ◁ C22×C4⋊C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#1463C2semidirect
3. C2×C41×#324C2semidirect
4. C41×#1325C2semidirect
5. C41×#1318C2semidirect
155. #11549C41direct (N × H)
186. #1463C2×C41semidirect
187. C2×#324C2×C41semidirect
188. #1325C2×C41semidirect
189. #1318C2×C41semidirect
215. C41#11549direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#25012
C41×#14632
C2×C41×#3242
C41×#13252
C41×#13182
#1154941