Finite Group Databasebeta

← C13×#12225 #15698564

Subgroups — C13×#12225 #15698564

Order 3328.

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

normal subgroups only

This group has 336 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
1C13×#2809C211yesmaximal, characteristic
2C13×#2028C211yesmaximal, characteristic
3C13×#1617C212yesmaximal
4C13×#1353C212yesmaximal
5C13×#1319C211yesmaximal, characteristic
6C13×#544C2211yescharacteristic
7C13×#484—22no—
8C13×#419—21no—
9C13×#400C2212yes—
10C13×#398—22no—
11C13×#391—21no—
12C13×#391C2211yescharacteristic
13C13×#389—21no—
14C13×#387—22no—
15C13×#343C2211yescharacteristic
16C13×#339—21no—
17C13×#328—22no—
18C13×#324C2212yes—
19C22×Q8×C13D411yescharacteristic
20C2×C13×M16—21no—
21C13×C4⋊Q8—41no—
22C13×#125—41no—
23C13×C22⋊Q8—22no—
24C13×#117—21no—
25C2×C13×C4⋊C4—22no—
26C2×C13×C4⋊C4—21no—
27C2×C13×C4⋊C4—21no—
28C2×C13×C4⋊C4—21no—
29C2×C13×C4⋊C4D412yes—
30C2×C13×C4⋊C4C2311yescharacteristic
31C2×C42×C13—21no—
32C2×C42×C13D411yescharacteristic
33C13×#107—42no—
34C13×#103—41no—
35C13×#103—41no—
36C13×#103—42no—
37C13×C22⋊C8—22no—
38C13×#95—22no—
39C13×#95—22no—
40C13×#95—22no—
41C13×#95—22no—
42C13×#95D412yes—
43#12225C1311yesmaximal, characteristic, direct product factor, Sylow
44C2×Q8×C13—21no—
45C2×Q8×C13—22no—
46C2×Q8×C13—41no—
47C2×Q8×C13—41no—
48C2×Q8×C13Q1612yes—
49C22×C4×C13—21no—
50C22×C4×C13—21no—
51C22×C4×C13—22no—
52C22×C4×C13—22no—
53C22×C4×C13C2×D411yescharacteristic
54C22×C4×C13C2×D412yes—
55C13×M16—41no—
56C13×M16—41no—
57C2×C8×C13—42no—
58C2×C8×C13—41no—
59C13×C4⋊C4—21no—
60C13×C4⋊C4—21no—
61C13×C4⋊C4—22no—
62C13×C4⋊C4—41no—
63C13×C4⋊C4—42no—
64C13×C4⋊C4—44no—
65C13×C4⋊C4—42no—
66C13×C4⋊C4—42no—
67C13×C4⋊C4—81no—
68C13×C4⋊C4—81no—
69C13×C22⋊C4—41no—
70C42×C13—41no—
71C42×C13—41no—
72C42×C13—81no—
73#2809C2×C1311yescharacteristic
74#2028C2×C1311yescharacteristic
75#1617C2×C1312yes—
76#1353C2×C1312yes—
77#1319C2×C1311yescharacteristic
78C23×C13#12011yescharacteristic
79Q8×C13—42no—
80Q8×C13—81no—
81Q8×C13—82no—
82C2×C4×C13—21no—
83C2×C4×C13—21no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—41no—
88C2×C4×C13—41no—
89C2×C4×C13—41no—
90C2×C4×C13—41no—
91C2×C4×C13—42no—
92C2×C4×C13—42no—
93C2×C4×C13—42no—
94C2×C4×C13—42no—
95C2×C4×C13—42no—
96C2×C4×C13—42no—
97C2×C4×C13—41no—
98C2×C4×C13—41no—
99C2×C4×C13—42no—
100C2×C4×C13C2×Q1611yescharacteristic
101C2×C4×C13#13711yescharacteristic
102C8×C13—81no—
103C8×C13—82no—
104#544C22×C1311yescharacteristic
105#484—22no—
106#419—21no—
107#400C22×C1312yes—
108#398—22no—
109#391—21no—
110#391C22×C1311yescharacteristic
111#389—21no—
112#387—22no—
113#343C22×C1311yescharacteristic
114#339—21no—
115#328—22no—
116#324C22×C1312yes—
117C22×C13—21no—
118C22×C13—21no—
119C22×C13#45111yescharacteristic
120C22×C13#45511yescharacteristic
121C22×C13B(4,𝔽2)11yescharacteristic
122C4×C13—21no—
123C4×C13—21no—
124C4×C13—42no—
125C4×C13—42no—
126C4×C13—42no—
127C4×C13—82no—
128C4×C13—81no—
129C4×C13—81no—
130C4×C13—82no—
131C22×Q8D4×C1311yescharacteristic
132C2×M16—21no—
133C4⋊Q8—41no—
134#125—41no—
135C22⋊Q8—22no—
136#117—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4D4×C1312yes—
142C2×C4⋊C4C23×C1311yescharacteristic
143C2×C42—21no—
144C2×C42D4×C1311yescharacteristic
145#107—42no—
146#103—41no—
147#103—41no—
148#103—42no—
149C22⋊C8—22no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95D4×C1312yes—
155C2×C13—21no—
156C2×C13—21no—
157C2×C13#160211yescharacteristic
158C2×C13#220111yescharacteristic
159C2×C13#220411yescharacteristic
160C2×Q8—21no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8C13×Q1612yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4C2×D4×C1311yescharacteristic
170C22×C4C2×D4×C1312yes—
171M16—41no—
172M16—41no—
173C2×C8—42no—
174C2×C8—41no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—41no—
186C42—41no—
187C42—41no—
188C42—81no—
189C13#1222511yescharacteristic, direct product factor, Sylow
190C23C13×#12011yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×C13×Q1611yescharacteristic
213C2×C4C13×#13711yescharacteristic
214C8—81no—
215C8—82no—
216C22—21no—
217C22—21no—
218C22C13×#45111yescharacteristic
219C22C13×#45511yescharacteristic
220C22C13×B(4,𝔽2)11yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—81no—
229C4—82no—
230C2—21no—
231C2—21no—
232C2C13×#160211yescharacteristic
233C2C13×#220111yescharacteristic
234C2C13×#220411yescharacteristic
subgroupG/H
center, socleC22×C13B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C13
prime radicalC23×C13#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C13 ◁ C23×C13 ◁ C2×C13×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
43. #12225C13direct (N × H)
189. C13#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C13×#28092
C13×#20282
C13×#16172
C13×#13532
C13×#13192
#1222513