Finite Group Databasebeta

← C41×#12126 #23788327

Subgroups — C41×#12126 #23788327

Order 10496.

522 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2683C211yesmaximal, characteristic, complemented
2C41×#2006C211yesmaximal, characteristic
3C41×#1604C212yesmaximal, complemented
4C41×#1354C212yesmaximal, complemented
5C41×#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C41—21no—
7C41×#537C2211yescharacteristic
8C2×C22⋊Q8×C41—22no—
9C41×#448—22no—
10C2×#104×C41—21no—
11C41×#396—22no—
12C41×#395C2212yes—
13C41×#390C2211yescharacteristic
14C41×#389—21no—
15C41×#386—21no—
16C41×#343C2211yescharacteristic
17C41×#327—22no—
18C41×#324C2212yes—
19C2×#41×C41—21no—
20C22×Q8×C41D412yes—
21C22×D4×C41D411yescharacteristic
22C23×C4×C41—21no—
23AGL(1,ℤ/8)×C41—42no—
24AGL(1,ℤ/8)×C41—42no—
25C2×SD16×C41—41no—
26C2×SD16×C41—42no—
27C2×D8×C41—41no—
28C2×M16×C41—21no—
29C22⋊Q8×C41—41no—
30C22⋊Q8×C41—48no—
31#121×C41—22no—
32#121×C41—41no—
33C2×C4⋊C4×C41—21no—
34C2×C4⋊C4×C41—22no—
35C2×C4⋊C4×C41—22no—
36C2×C4⋊C4×C41—21no—
37C2×C4⋊C4×C41C2311yescharacteristic
38C2×C22⋊C4×C41—22no—
39C2×C22⋊C4×C41—22no—
40C2×C22⋊C4×C41—21no—
41C2×C42×C41D411yescharacteristic
42#104×C41—42no—
43#103×C41—42no—
44C22⋊C8×C41—22no—
45#95×C41—22no—
46#95×C41D412yes—
47C24×C41—21no—
48#41×C41—41no—
49#41×C41—41no—
50#41×C41—41no—
51#41×C41—41no—
52C2×Q8×C41—21no—
53C2×Q8×C41—44no—
54C2×Q8×C41—42no—
55C2×D4×C41—21no—
56C2×D4×C41—22no—
57C2×D4×C41—21no—
58C2×D4×C41—41no—
59C2×D4×C41—41no—
60C2×D4×C41—41no—
61C2×D4×C41SD1612yescomplemented
62C22×C4×C41—22no—
63C22×C4×C41—21no—
64C22×C4×C41—22no—
65C22×C4×C41—41no—
66C22×C4×C41—44no—
67C22×C4×C41—81no—
68C22×C4×C41C2×D411yescharacteristic
69C22×C4×C41C2×D412yes—
70SD16×C41—44no—
71SD16×C41—84no—
72D8×C41—44no—
73M16×C41—22no—
74M16×C41—41no—
75C2×C8×C41—41no—
76C2×C8×C41—42no—
77C4⋊C4×C41—24no—
78C4⋊C4×C41—22no—
79C4⋊C4×C41—41no—
80C4⋊C4×C41—44no—
81C4⋊C4×C41—44no—
82C4⋊C4×C41—81no—
83C22⋊C4×C41—44no—
84C22⋊C4×C41—44no—
85C22⋊C4×C41—42no—
86C22⋊C4×C41—81no—
87C42×C41—41no—
88C23×C41—22no—
89C23×C41—42no—
90C23×C41—42no—
91C23×C41—41no—
92C23×C41—41no—
93C23×C41—41no—
94C23×C41#12011yescharacteristic
95Q8×C41—41no—
96Q8×C41—41no—
97Q8×C41—44no—
98Q8×C41—82no—
99D4×C41—42no—
100D4×C41—41no—
101D4×C41—41no—
102D4×C41—81no—
103D4×C41—82no—
104D4×C41—81no—
105D4×C41—81no—
106C2×C4×C41—21no—
107C2×C4×C41—21no—
108C2×C4×C41—22no—
109C2×C4×C41—24no—
110C2×C4×C41—22no—
111C2×C4×C41—42no—
112C2×C4×C41—41no—
113C2×C4×C41—41no—
114C2×C4×C41—42no—
115C2×C4×C41—42no—
116C2×C4×C41—42no—
117C2×C4×C41—42no—
118C2×C4×C41—42no—
119C2×C4×C41—41no—
120C2×C4×C41—81no—
121C2×C4×C41—81no—
122C2×C4×C41—84no—
123C2×C4×C41C2×SD1611yescharacteristic
124C2×C4×C41#13711yescharacteristic
125C8×C41—42no—
126C8×C41—82no—
127#12126C4111yesmaximal, characteristic, direct product factor, Sylow
128C22×C41—21no—
129C22×C41—21no—
130C22×C41—44no—
131C22×C41—42no—
132C22×C41—82no—
133C22×C41—81no—
134C22×C41—81no—
135C22×C41—81no—
136C22×C41—81no—
137C22×C41—81no—
138C22×C41#44811yescharacteristic
139C22×C41#45311yescharacteristic
140C22×C41B(4,𝔽2)11yescharacteristic
141C4×C41—21no—
142C4×C41—21no—
143C4×C41—42no—
144C4×C41—42no—
145C4×C41—44no—
146C4×C41—82no—
147C4×C41—81no—
148#2683C2×C4111yescharacteristic, complemented
149#2006C2×C4111yescharacteristic
150#1604C2×C4112yescomplemented
151#1354C2×C4112yescomplemented
152#1318C2×C4111yescharacteristic, complemented
153C2×C41—21no—
154C2×C41—21no—
155C2×C41—82no—
156C2×C41—161no—
157C2×C41#159611yescharacteristic
158C2×C41#219611yescharacteristic
159C2×C41#220311yescharacteristic
160C2×AGL(1,ℤ/8)—21no—
161#537C22×C4111yescharacteristic
162C2×C22⋊Q8—22no—
163#448—22no—
164C2×#104—21no—
165#396—22no—
166#395C22×C4112yes—
167#390C22×C4111yescharacteristic
168#389—21no—
169#386—21no—
170#343C22×C4111yescharacteristic
171#327—22no—
172#324C22×C4112yes—
173C41#1212611yescharacteristic, direct product factor, Sylow
174C2×#41—21no—
175C22×Q8D4×C4112yes—
176C22×D4D4×C4111yescharacteristic
177C23×C4—21no—
178AGL(1,ℤ/8)—42no—
179AGL(1,ℤ/8)—42no—
180C2×SD16—41no—
181C2×SD16—42no—
182C2×D8—41no—
183C2×M16—21no—
184C22⋊Q8—41no—
185C22⋊Q8—48no—
186#121—22no—
187#121—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C23×C4111yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C42D4×C4111yescharacteristic
197#104—42no—
198#103—42no—
199C22⋊C8—22no—
200#95—22no—
201#95D4×C4112yes—
202C24—21no—
203#41—41no—
204#41—41no—
205#41—41no—
206#41—41no—
207C2×Q8—21no—
208C2×Q8—44no—
209C2×Q8—42no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4SD16×C4112yescomplemented
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—81no—
223C22×C4C2×D4×C4111yescharacteristic
224C22×C4C2×D4×C4112yes—
225SD16—44no—
226SD16—84no—
227D8—44no—
228M16—22no—
229M16—41no—
230C2×C8—41no—
231C2×C8—42no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—81no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C42—41no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#120×C4111yescharacteristic
250Q8—41no—
251Q8—41no—
252Q8—44no—
253Q8—82no—
254D4—42no—
255D4—41no—
256D4—41no—
257D4—81no—
258D4—82no—
259D4—81no—
260D4—81no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4C2×SD16×C4111yescharacteristic
279C2×C4#137×C4111yescharacteristic
280C8—42no—
281C8—82no—
282C22—21no—
283C22—21no—
284C22—44no—
285C22—42no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22C41×#44811yescharacteristic
293C22C41×#45311yescharacteristic
294C22C41×B(4,𝔽2)11yescharacteristic
295C4—21no—
296C4—21no—
297C4—42no—
298C4—42no—
299C4—44no—
300C4—82no—
301C4—81no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2—161no—
306C2C41×#159611yescharacteristic
307C2C41×#219611yescharacteristic
308C2C41×#220311yescharacteristic
subgroupG/H
center, socleC22×C41B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C41
prime radicalC41×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C23×C41 ◁ C2×C4⋊C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C41×#2683C2semidirect
3. C41×#1604C2semidirect
4. C41×#1354C2semidirect
5. C41×#1318C2semidirect
61. C2×D4×C41SD16semidirect
127. #12126C41direct (N × H)
148. #2683C2×C41semidirect
150. #1604C2×C41semidirect
151. #1354C2×C41semidirect
152. #1318C2×C41semidirect
173. C41#12126direct (N × H)
216. C2×D4SD16×C41semidirect

Maximal subgroups

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

subgroupindex
C41×#26832
C41×#20062
C41×#16042
C41×#13542
C41×#13182
#1212641