Finite Group Databasebeta

← #27170112

Subgroups — #27170112

Order 12928.

260 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#20265332C211yesmaximal, characteristic, complemented
2#20265315C211yesmaximal, characteristic, complemented
3#20265089C211yesmaximal, characteristic, complemented
4#20264914C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264901C211yesmaximal, characteristic, complemented
7#538×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×#13108064—21no—
10C2×#13108062—21no—
11C2×C22⋊Dic101—21no—
12#15692178—21no—
13#15692178—21no—
14C22⋊#11717579C2211yescharacteristic
15C22⋊#11717579C2211yescharacteristic
16#15692162—21no—
17#15692162C2211yescharacteristic
18#126×C101—21no—
19#124×C101—21no—
20#123×C101—21no—
21C22⋊Q8×C101C2211yescharacteristic
22#121×C101—21no—
23#121×C101C2211yescharacteristic
24#120×C101—21no—
25C4×D4×C101—21no—
26C2×C22⋊C4×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—42no—
30C22×Dic101—21no—
31C22×Dic101—21no—
32#13108064—42no—
33#13108062—42no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C22⋊Dic101—42no—
37C24×C101D411yescharacteristic
38C2×Q8×C101D411yescharacteristic
39C2×D4×C101—21no—
40C2×D4×C101—21no—
41C2×D4×C101—21no—
42C2×D4×C101D411yescharacteristic
43C22×C4×C101—21no—
44C22×C4×C101D411yescharacteristic
45C22×C4×C101C2311yescharacteristic
46C4⋊C4×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101D411yescharacteristic
49C4⋊C4×C101D411yescharacteristic
50C22⋊C4×C101—21no—
51C22⋊C4×C101—21no—
52C22⋊C4×C101—21no—
53C22⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101—21no—
56C22⋊C4×C101—21no—
57C42×C101—21no—
58C23×C101—21no—
59C23×C101—21no—
60C23×C101—21no—
61C23×C101—21no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101C2×D411yescharacteristic
65Q8×C101—41no—
66D4×C101—41no—
67D4×C101—41no—
68D4×C101—41no—
69D4×C101—81no—
70D4×C101—81no—
71C2×C4×C101—21no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101—41no—
76C2×C4×C101—41no—
77C2×C4×C101—81no—
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101C2×D411yescharacteristic
80C2×Dic101—22no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101—41no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86Dic202—22no—
87Dic202—22no—
88Dic202—41no—
89Dic202—41no—
90#11717579—41no—
91#11717579—41no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—42no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101—41no—
101C22×C101—81no—
102C22×C101#12011yescharacteristic
103C22×C101#13511yescharacteristic
104C22×C101#13711yescharacteristic
105C4×C101—21no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—41no—
110C4×C101—41no—
111Dic101—42no—
112Dic101—81no—
113C2×C101—21no—
114C2×C101—21no—
115C2×C101—81no—
116C2×C101—81no—
117C2×C101#45211yescharacteristic
118C2×C101#45511yescharacteristic
119C2×C101B(4,𝔽2)11yescharacteristic
120#1606—1011nomaximal, Sylow
121C101#160611yescharacteristic, complemented, Sylow
122#538D10111yescharacteristic
123#451—1011no—
124#448—1011no—
125#393—1011no—
126#328—1011no—
127#327—1011no—
128#323—1011no—
129C22×Q8—1011no—
130C2×Q16—2021no—
131C2×SD16—2021no—
132#126—21no—
133#124—21no—
134#123—21no—
135C22⋊Q8C2×D10111yescharacteristic
136#121—21no—
137#121C2×D10111yescharacteristic
138#120—21no—
139C4×D4—21no—
140C2×C22⋊C4—2021no—
141C2×C22⋊C4C2×D10111yescharacteristic
142#103—2021no—
143#103—2021no—
144C22⋊C8—1011no—
145C22⋊C8—1011no—
146#95—1011no—
147#95—2021no—
148C24C22⋊D10111yescharacteristic
149C2×Q8—4042no—
150C2×Q8—2021no—
151C2×Q8—2021no—
152C2×Q8C22⋊D10111yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C22⋊D10111yescharacteristic
157C22×C4—21no—
158C22×C4—2021no—
159C22×C4—2021no—
160C22×C4C22⋊D10111yescharacteristic
161C22×C4C22×D10111yescharacteristic
162Q16—4042no—
163SD16—4042no—
164C2×C8—2021no—
165C2×C8—2021no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4C22⋊D10111yescharacteristic
169C4⋊C4C22⋊D10111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—4042no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×C22⋊D10111yescharacteristic
186Q8—41no—
187Q8—4041no—
188Q8—4041no—
189Q8—2022no—
190Q8—2022no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—4041no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—4041no—
207C2×C4—4041no—
208C2×C4—2022no—
209C2×C4C2×C22⋊D10111yescharacteristic
210C2×C4C2×C22⋊D10111yescharacteristic
211C8—4041no—
212C8—4041no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22#1569222411yescharacteristic
224C22#1569227511yescharacteristic
225C22#1569228511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—8081no—
233C4—4042no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#2026533911yescharacteristic
239C2#2026535511yescharacteristic
240C2#2026536411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#538×C101C2
socleC22×C101#120
hypercenter#538D101
hypocenterC101#1606
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #20265332C2semidirect
2. #20265315C2semidirect
3. #20265089C2semidirect
4. #20264914C2semidirect
5. #20264911C2semidirect
6. #20264901C2semidirect
121. C101#1606semidirect

Maximal subgroups

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

subgroupindex
#202653322
#202653152
#202650892
#202649142
#202649112
#202649012
#538×C1012
#1606101