Finite Group Databasebeta

← C41×#12207 #23788408

Subgroups — C41×#12207 #23788408

Order 10496.

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

normal subgroups only

This group has 388 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×#2817C211yesmaximal, characteristic
2C41×#2030C211yesmaximal, characteristic
3C41×#1608C212yesmaximal
4C41×#1344C212yesmaximal
5C41×#1317C211yesmaximal, characteristic
6C41×#543C2211yescharacteristic
7C2×C22⋊Q8×C41—21no—
8C41×#483—22no—
9C41×#400—22no—
10C41×#397—21no—
11C41×#397C2212yes—
12C41×#395—22no—
13C41×#381—21no—
14C41×#381C2211yescharacteristic
15C41×#342—21no—
16C41×#342C2211yescharacteristic
17C41×#328—22no—
18C41×#323C2212yes—
19C22×Q8×C41—21no—
20C23×C4×C41D411yescharacteristic
21C22⋊Q8×C41—22no—
22C22⋊Q8×C41—42no—
23C22⋊Q8×C41—42no—
24C22⋊Q8×C41—42no—
25C2×C4⋊C4×C41—22no—
26C2×C4⋊C4×C41—21no—
27C2×C4⋊C4×C41—21no—
28C2×C4⋊C4×C41D412yes—
29C2×C22⋊C4×C41—22no—
30C2×C22⋊C4×C41—21no—
31C2×C22⋊C4×C41—21no—
32C2×C22⋊C4×C41D411yescharacteristic
33C2×C22⋊C4×C41C2311yescharacteristic
34#106×C41—42no—
35#103×C41—42no—
36C22⋊C8×C41—22no—
37#95×C41—22no—
38#95×C41—21no—
39#95×C41—22no—
40#95×C41—22no—
41#95×C41—42no—
42#95×C41D412yes—
43C24×C41C2×D411yescharacteristic
44C2×Q8×C41—22no—
45C2×Q8×C41—42no—
46C2×Q8×C41—81no—
47C2×Q8×C41—81no—
48C22×C4×C41—21no—
49C22×C4×C41—21no—
50C22×C4×C41—21no—
51C22×C4×C41—22no—
52C22×C4×C41—22no—
53C22×C4×C41—22no—
54C22×C4×C41—41no—
55C22×C4×C41—41no—
56C22×C4×C41—41no—
57C22×C4×C41SD1612yes—
58C22×C4×C41C2×D412yes—
59C2×C8×C41—42no—
60C4⋊C4×C41—22no—
61C4⋊C4×C41—42no—
62C4⋊C4×C41—42no—
63C4⋊C4×C41—44no—
64C4⋊C4×C41—42no—
65C4⋊C4×C41—81no—
66C22⋊C4×C41—41no—
67C22⋊C4×C41—41no—
68C22⋊C4×C41—41no—
69C22⋊C4×C41—42no—
70C22⋊C4×C41—42no—
71C22⋊C4×C41—41no—
72C22⋊C4×C41—41no—
73C22⋊C4×C41—41no—
74C23×C41—21no—
75C23×C41—21no—
76C23×C41—42no—
77C23×C41#12011yescharacteristic
78C23×C41C2×SD1611yescharacteristic
79C23×C41AGL(1,ℤ/8)11yescharacteristic
80Q8×C41—82no—
81Q8×C41—84no—
82C2×C4×C41—22no—
83C2×C4×C41—22no—
84C2×C4×C41—41no—
85C2×C4×C41—41no—
86C2×C4×C41—42no—
87C2×C4×C41—42no—
88C2×C4×C41—42no—
89C2×C4×C41—44no—
90C2×C4×C41—42no—
91C2×C4×C41—42no—
92C2×C4×C41—41no—
93C2×C4×C41—42no—
94C2×C4×C41—42no—
95C2×C4×C41—42no—
96C2×C4×C41—81no—
97C2×C4×C41—82no—
98C2×C4×C41—81no—
99C2×C4×C41—82no—
100C8×C41—82no—
101#12207C4111yesmaximal, characteristic, direct product factor, Sylow
102C22×C41—22no—
103C22×C41—21no—
104C22×C41—21no—
105C22×C41—41no—
106C22×C41—41no—
107C22×C41—42no—
108C22×C41—81no—
109C22×C41#45011yescharacteristic
110C22×C41#45511yescharacteristic
111C22×C41B(4,𝔽2)11yescharacteristic
112C4×C41—42no—
113C4×C41—42no—
114C4×C41—82no—
115C4×C41—81no—
116C4×C41—82no—
117C4×C41—82no—
118C4×C41—161no—
119#2817C2×C4111yescharacteristic
120#2030C2×C4111yescharacteristic
121#1608C2×C4112yes—
122#1344C2×C4112yes—
123#1317C2×C4111yescharacteristic
124C2×C41—21no—
125C2×C41—21no—
126C2×C41—41no—
127C2×C41—41no—
128C2×C41#160111yescharacteristic
129C2×C41#220112yes—
130#543C22×C4111yescharacteristic
131C2×C22⋊Q8—21no—
132#483—22no—
133#400—22no—
134#397—21no—
135#397C22×C4112yes—
136#395—22no—
137#381—21no—
138#381C22×C4111yescharacteristic
139#342—21no—
140#342C22×C4111yescharacteristic
141#328—22no—
142#323C22×C4112yes—
143C41#1220711yescharacteristic, direct product factor, Sylow
144C22×Q8—21no—
145C23×C4D4×C4111yescharacteristic
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4D4×C4112yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4D4×C4111yescharacteristic
158C2×C22⋊C4C23×C4111yescharacteristic
159#106—42no—
160#103—42no—
161C22⋊C8—22no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—42no—
167#95D4×C4112yes—
168C24C2×D4×C4111yescharacteristic
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—81no—
172C2×Q8—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4SD16×C4112yes—
183C22×C4C2×D4×C4112yes—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#120×C4111yescharacteristic
203C23C2×SD16×C4111yescharacteristic
204C23AGL(1,ℤ/8)×C4111yescharacteristic
205Q8—82no—
206Q8—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—82no—
225C8—82no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22C41×#45011yescharacteristic
234C22C41×#45511yescharacteristic
235C22C41×B(4,𝔽2)11yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—82no—
242C4—161no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2C41×#160111yescharacteristic
248C2C41×#220112yes—
subgroupG/H
center, socleC22×C41B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C41
prime radicalC24×C41C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C23×C41 ◁ C2×C22⋊C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
101. #12207C41direct (N × H)
143. C41#12207direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#28172
C41×#20302
C41×#16082
C41×#13442
C41×#13172
#1220741