Finite Group Databasebeta

← #27170048

Subgroups — #27170048

Order 12928.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265735C211yesmaximal, characteristic, complemented
2#20265311C211yesmaximal, characteristic, complemented
3#20265085C211yesmaximal, characteristic, complemented
4#20264912C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#447×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101—21no—
9C2×D4×D101—21no—
10C2×#13108059—21no—
11#15692240—21no—
12#15692237—21no—
13#15692229C2211yescharacteristic
14#15692227—21no—
15#15692225—21no—
16#15692223—21no—
17C2×C22⋊Dic101—21no—
18C2×#13108051C2211yescharacteristic
19#15692176—21no—
20C22⋊#11717579C2211yescharacteristic, complemented
21#15692162C2211yescharacteristic, complemented
22C22×D4×C101C2211yescharacteristic, complemented
23C2×D8×C101—21no—
24#121×C101C2211yescharacteristic
25#102×C101—21no—
26C22⋊C8×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C23×D101D411yescharacteristic
29C2×D404—21no—
30C2×D404D411yescharacteristic, complemented
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—42no—
35C2×C22⋊D101—42no—
36C2×C22⋊D101—42no—
37D4×D101—42no—
38D4×D101—42no—
39C22×Dic101—21no—
40C22×Dic101D411yescharacteristic
41C2×C4×D101—21no—
42#13108059—42no—
43C2×#11717579—21no—
44C4⋊Dic101D411yescharacteristic, complemented
45#13108053—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101—42no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51#13108051—21no—
52C24×C101—21no—
53C2×D4×C101—21no—
54C2×D4×C101—21no—
55C2×D4×C101—21no—
56C2×D4×C101—42no—
57C2×D4×C101D411yescharacteristic, complemented
58C22×C4×C101C2311yescharacteristic
59D8×C101—42no—
60C2×C8×C101—21no—
61C4⋊C4×C101D411yescharacteristic, complemented
62C22⋊C4×C101—21no—
63C23×C101—22no—
64C23×C101—21no—
65C23×C101—41no—
66C23×C101—41no—
67C23×C101—41no—
68C23×C101—41no—
69C23×C101—41no—
70C23×C101—41no—
71C23×C101C2×D411yescharacteristic
72C22×D101—21no—
73C22×D101—21no—
74C22×D101—21no—
75C22×D101—42no—
76C22×D101—41no—
77C22×D101—41no—
78C22×D101—81no—
79C22×D101—81no—
80C22×D101D812yescomplemented
81D4×C101—22no—
82D4×C101—22no—
83D4×C101—41no—
84D4×C101—41no—
85D4×C101—41no—
86D4×C101—41no—
87D4×C101—41no—
88C2×C4×C101—21no—
89C2×C4×C101—41no—
90C2×C4×C101C2×D411yescharacteristic
91C2×C4×C101C2×D411yescharacteristic
92C22⋊D101—42no—
93C22⋊D101—48no—
94C22⋊D101—41no—
95C22⋊D101—41no—
96C22⋊D101—81no—
97C2×Dic101—22no—
98C2×Dic101—21no—
99C2×Dic101—21no—
100C2×Dic101—41no—
101C2×Dic101—41no—
102C2×Dic101—41no—
103D404—41no—
104D404—42no—
105C4×D101—42no—
106C8×C101—41no—
107#11717579—41no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—42no—
111C22×C101—42no—
112C22×C101—41no—
113C22×C101—41no—
114C22×C101—42no—
115C22×C101—41no—
116C22×C101—81no—
117C22×C101—81no—
118C22×C101—81no—
119C22×C101—81no—
120C22×C101#12011yescharacteristic
121C22×C101C2×D811yescharacteristic
122C22×C101AGL(1,ℤ/8)11yescharacteristic
123C2×D101—22no—
124C2×D101—41no—
125C2×D101—41no—
126C2×D101—42no—
127C2×D101—41no—
128C2×D101—41no—
129C2×D101—41no—
130C2×D101—81no—
131C2×D101—84no—
132C4×C101—21no—
133C4×C101—21no—
134C4×C101—41no—
135Dic101—42no—
136Dic101—41no—
137Dic101—41no—
138C2×C101—21no—
139C2×C101—21no—
140C2×C101—81no—
141C2×C101—82no—
142C2×C101#44711yescharacteristic
143C2×C101#45311yescharacteristic
144C2×C101B(4,𝔽2)11yescharacteristic
145D101—42no—
146D101—81no—
147#1595—1011nomaximal, Sylow
148C101#159511yescharacteristic, complemented, Sylow
149#534—1011no—
150#447—1011no—
151#447D10111yescharacteristic, complemented
152#392—1011no—
153#327—1011no—
154#327—1011no—
155#323—1011no—
156C22×D4—2021no—
157C22×D4—2021no—
158C22×D4C2×D10111yescharacteristic, complemented
159C2×D8—21no—
160C2×D8—2021no—
161#123—2021no—
162#123—2021no—
163#121—1011no—
164#121—2021no—
165#121C2×D10111yescharacteristic
166#120—2021no—
167#120—2021no—
168C2×C22⋊C4—1011no—
169C2×C22⋊C4—2021no—
170#102—21no—
171#102—2021no—
172C22⋊C8—1011no—
173C22⋊C8C2×D10111yescharacteristic, complemented
174#95—1011no—
175C24—21no—
176C24—1011no—
177C24—2021no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—1011no—
183C2×D4—4042no—
184C2×D4—4042no—
185C2×D4—4042no—
186C2×D4—4042no—
187C2×D4—4042no—
188C2×D4—2021no—
189C2×D4—2021no—
190C2×D4—2021no—
191C2×D4—2021no—
192C2×D4C22⋊D10111yescharacteristic, complemented
193C22×C4—1011no—
194C22×C4—2021no—
195C22×C4—2021no—
196C22×C4C22×D10111yescharacteristic
197D8—42no—
198D8—4042no—
199C2×C8—21no—
200C2×C8—2021no—
201C4⋊C4—1011no—
202C4⋊C4—2021no—
203C4⋊C4C22⋊D10111yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—4042no—
206C22⋊C4—2021no—
207C22⋊C4—2021no—
208C22⋊C4—2021no—
209C22⋊C4—2021no—
210C22⋊C4—2021no—
211C23—22no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—8081no—
220C23—8081no—
221C23—1012no—
222C23—4041no—
223C23—4041no—
224C23—4042no—
225C23—2021no—
226C23—2021no—
227C23—2021no—
228C23D4×D10111yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—8081no—
237D4—4041no—
238D4—4042no—
239D4—4048no—
240D4—4042no—
241D4—4041no—
242D4—4041no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—4042no—
246C2×C4—4041no—
247C2×C4—4041no—
248C2×C4—4041no—
249C2×C4—2022no—
250C2×C4—2021no—
251C2×C4—2021no—
252C2×C4D4×D10111yescharacteristic
253C2×C4C2×C22⋊D10111yescharacteristic
254C8—41no—
255C8—4041no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—8081no—
269C22—8084no—
270C22—4042no—
271C22—4041no—
272C22—4041no—
273C22—4041no—
274C22—4041no—
275C22—4041no—
276C22—2022no—
277C22#1569222311yescharacteristic
278C22D8×D10111yescharacteristic
279C22#1569227911yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—4042no—
284C4—4041no—
285C4—4041no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—81no—
290C2—8081no—
291C2—4042no—
292C2#2026530911yescharacteristic
293C2#2026534311yescharacteristic
294C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#447×C101C2
socleC22×C101#120
hypercenter#447D101
hypocenterC101#1595
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20265735C2semidirect
2. #20265311C2semidirect
3. #20265085C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264900C2semidirect
7. #447×C101C2semidirect
20. C22⋊#11717579C22semidirect
21. #15692162C22semidirect
22. C22×D4×C101C22semidirect
26. C22⋊C8×C101C22semidirect
30. C2×D404D4semidirect
44. C4⋊Dic101D4semidirect
57. C2×D4×C101D4semidirect
61. C4⋊C4×C101D4semidirect
80. C22×D101D8semidirect
148. C101#1595semidirect
151. #447D101semidirect
158. C22×D4C2×D101semidirect
173. C22⋊C8C2×D101semidirect
192. C2×D4C22⋊D101semidirect
203. C4⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657352
#202653112
#202650852
#202649122
#202649112
#202649002
#447×C1012
#1595101