Finite Group Databasebeta

← C29×#12256 #20307156

Subgroups — C29×#12256 #20307156

Order 7424.

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

normal subgroups only

This group has 442 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
1C29×#2551C211yesmaximal, characteristic
2C29×#2060C211yesmaximal, characteristic
3C29×#1605C212yesmaximal
4C29×#1354C212yesmaximal
5C29×#1325C211yesmaximal, characteristic
6C29×#544C2211yescharacteristic
7C2×C29×C4⋊Q8—22no—
8C29×#451—22no—
9C29×#428—21no—
10C29×#395—21no—
11C29×#395C2212yes—
12C29×#389—22no—
13C29×#384C2211yescharacteristic
14C4×C4⋊C4×C29—21no—
15C29×#343C2211yescharacteristic
16C29×#339—21no—
17C29×#328—22no—
18C29×#324C2212yes—
19C22×Q8×C29—21no—
20C22×Q8×C29D412yes—
21C2×Q16×C29—42no—
22C2×M16×C29—21no—
23C29×C4⋊Q8—41no—
24C29×C4⋊Q8—48no—
25C29×#125—41no—
26C29×C22⋊Q8—22no—
27C29×#117—21no—
28C2×C4⋊C4×C29—21no—
29C2×C4⋊C4×C29—21no—
30C2×C4⋊C4×C29—22no—
31C2×C4⋊C4×C29—22no—
32C2×C4⋊C4×C29—22no—
33C2×C4⋊C4×C29D411yescharacteristic
34C2×C4⋊C4×C29C2311yescharacteristic
35C2×C42×C29—22no—
36C2×C42×C29D411yescharacteristic
37C29×#107—41no—
38C29×#106—41no—
39C29×#103—42no—
40C29×C22⋊C8—22no—
41C29×#95—21no—
42C29×#95—22no—
43C29×#95D412yes—
44C2×Q8×C29—22no—
45C2×Q8×C29—44no—
46C2×Q8×C29—44no—
47C2×Q8×C29—42no—
48C2×Q8×C29—81no—
49C22×C4×C29—21no—
50C22×C4×C29—22no—
51C22×C4×C29—22no—
52C22×C4×C29—21no—
53C22×C4×C29C2×D411yescharacteristic
54C22×C4×C29C2×D412yes—
55Q16×C29—84no—
56M16×C29—22no—
57M16×C29—41no—
58C2×C8×C29—41no—
59C2×C8×C29—42no—
60C4⋊C4×C29—24no—
61C4⋊C4×C29—24no—
62C4⋊C4×C29—21no—
63C4⋊C4×C29—21no—
64C4⋊C4×C29—22no—
65C4⋊C4×C29—41no—
66C4⋊C4×C29—41no—
67C4⋊C4×C29—44no—
68C4⋊C4×C29—44no—
69C4⋊C4×C29—44no—
70C4⋊C4×C29—42no—
71C22⋊C4×C29—41no—
72C42×C29—21no—
73C42×C29—44no—
74C42×C29—41no—
75C42×C29Q1612yescomplemented
76#12256C2911yesmaximal, characteristic, direct product factor, Sylow
77C23×C29#12011yescharacteristic
78Q8×C29—44no—
79Q8×C29—82no—
80Q8×C29—84no—
81Q8×C29—82no—
82C2×C4×C29—22no—
83C2×C4×C29—22no—
84C2×C4×C29—22no—
85C2×C4×C29—21no—
86C2×C4×C29—21no—
87C2×C4×C29—24no—
88C2×C4×C29—24no—
89C2×C4×C29—22no—
90C2×C4×C29—42no—
91C2×C4×C29—42no—
92C2×C4×C29—42no—
93C2×C4×C29—42no—
94C2×C4×C29—42no—
95C2×C4×C29—42no—
96C2×C4×C29—41no—
97C2×C4×C29—41no—
98C2×C4×C29—41no—
99C2×C4×C29C2×Q1611yescharacteristic
100C2×C4×C29#13711yescharacteristic
101C8×C29—42no—
102C8×C29—82no—
103#2551C2×C2911yescharacteristic
104#2060C2×C2911yescharacteristic
105#1605C2×C2912yes—
106#1354C2×C2912yes—
107#1325C2×C2911yescharacteristic
108C22×C29—21no—
109C22×C29—21no—
110C22×C29#45111yescharacteristic
111C22×C29#45611yescharacteristic
112C22×C29B(4,𝔽2)11yescharacteristic
113C4×C29—24no—
114C4×C29—21no—
115C4×C29—21no—
116C4×C29—44no—
117C4×C29—42no—
118C4×C29—44no—
119C4×C29—81no—
120C4×C29—81no—
121C4×C29—81no—
122#544C22×C2911yescharacteristic
123C2×C4⋊Q8—22no—
124#451—22no—
125#428—21no—
126#395—21no—
127#395C22×C2912yes—
128#389—22no—
129#384C22×C2911yescharacteristic
130C4×C4⋊C4—21no—
131#343C22×C2911yescharacteristic
132#339—21no—
133#328—22no—
134#324C22×C2912yes—
135C2×C29—21no—
136C2×C29—21no—
137C2×C29#160511yescharacteristic
138C2×C29#219611yescharacteristic
139C2×C29#220511yescharacteristic
140C22×Q8—21no—
141C22×Q8D4×C2912yes—
142C2×Q16—42no—
143C2×M16—21no—
144C4⋊Q8—41no—
145C4⋊Q8—48no—
146#125—41no—
147C22⋊Q8—22no—
148#117—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4D4×C2911yescharacteristic
155C2×C4⋊C4C23×C2911yescharacteristic
156C2×C42—22no—
157C2×C42D4×C2911yescharacteristic
158#107—41no—
159#106—41no—
160#103—42no—
161C22⋊C8—22no—
162#95—21no—
163#95—22no—
164#95D4×C2912yes—
165C29#1225611yescharacteristic, direct product factor, Sylow
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—81no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×D4×C2911yescharacteristic
176C22×C4C2×D4×C2912yes—
177Q16—84no—
178M16—22no—
179M16—41no—
180C2×C8—41no—
181C2×C8—42no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C22⋊C4—41no—
194C42—21no—
195C42—44no—
196C42—41no—
197C42Q16×C2912yescomplemented
198C23C29×#12011yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—82no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4C2×Q16×C2911yescharacteristic
221C2×C4C29×#13711yescharacteristic
222C8—42no—
223C8—82no—
224C22—21no—
225C22—21no—
226C22C29×#45111yescharacteristic
227C22C29×#45611yescharacteristic
228C22C29×B(4,𝔽2)11yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2C29×#160511yescharacteristic
241C2C29×#219611yescharacteristic
242C2C29×#220511yescharacteristic
subgroupG/H
center, socleC22×C29B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C29
prime radicalC23×C29#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C29 ◁ C23×C29 ◁ C2×C4⋊C4×C29 ◁ G

Complements

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

NH ≅ G/Ntype
75. C42×C29Q16semidirect
76. #12256C29direct (N × H)
165. C29#12256direct (N × H)
197. C42Q16×C29semidirect

Maximal subgroups

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

subgroupindex
C29×#25512
C29×#20602
C29×#16052
C29×#13542
C29×#13252
#1225629