Finite Group Databasebeta

← #27169672

Subgroups — #27169672

Order 12928.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#20265687C211yesmaximal, characteristic
2#20265139C211yesmaximal, characteristic, complemented
3#20265036C211yesmaximal, characteristic, complemented
4#20264915C211yesmaximal, characteristic, complemented
5#20264913C211yesmaximal, characteristic, complemented
6#20264911C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×#13108073C411yescharacteristic, complemented
9C22×C4×D101C2211yescharacteristic
10C2×#13108057—21no—
11#15692237—21no—
12#15692235C411yescharacteristic, complemented
13#15692234C2211yescharacteristic
14#15692226C411yescharacteristic, complemented
15#15692223—21no—
16D4×Dic101—21no—
17#15692215—21no—
18#15692213C411yescharacteristic, complemented
19C2×#13108053—21no—
20C2×#13108051—21no—
21C22⋊#11717579C2211yescharacteristic
22#15692167—21no—
23#15692162C2211yescharacteristic
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28#13108073—42no—
29C2×Dic202C2×C411yescharacteristic
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101—21no—
33C22×Dic101—21no—
34C22×Dic101D411yescharacteristic
35C2×C4×D101—42no—
36C2×C4×D101C2×C411yescharacteristic
37C2×C4×D101C2×C411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101D411yescharacteristic
40#13108057—42no—
41C2×#11717579—21no—
42C4⋊Dic101—21no—
43C4⋊Dic101C2×C411yescharacteristic
44#13108053—21no—
45#13108053—42no—
46C22⋊Dic101—21no—
47C22⋊Dic101—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—42no—
51C4×Dic101—21no—
52C2×D4×C101—21no—
53C2×D4×C101C2×C411yescharacteristic
54C22×C4×C101—21no—
55C22×C4×C101C2311yescharacteristic
56C2×C8×C101—21no—
57C4⋊C4×C101C2×C411yescharacteristic
58C22⋊C4×C101—21no—
59C23×C101—21no—
60C23×C101C2×D411yescharacteristic
61C22×D101—21no—
62C22×D101—21no—
63C22×D101—41no—
64C22×D101—41no—
65C22×D101C22⋊C411yescharacteristic
66C22×D101C22⋊C411yescharacteristic
67D4×C101—41no—
68D4×C101—41no—
69D4×C101—41no—
70C2×C4×C101—21no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101—41no—
74C2×C4×C101—41no—
75C2×C4×C101C22×C411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C22⋊D101—42no—
78C22⋊D101—42no—
79C2×Dic101—22no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—42no—
83C2×Dic101—41no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101C22⋊C411yescharacteristic
87C2×Dic101C22⋊C411yescharacteristic
88C4×D101—22no—
89C4×D101—22no—
90C4×D101—41no—
91C4×D101—41no—
92Dic202—41no—
93C8×C101—41no—
94#11717579—41no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101C2×C22⋊C411yescharacteristic
101C22×C101AGL(1,ℤ/8)11yescharacteristic
102C22×C101#13711yescharacteristic
103C2×D101—21no—
104C2×D101—21no—
105C2×D101—21no—
106C2×D101—21no—
107C2×D101—21no—
108C2×D101—21no—
109C2×D101—42no—
110C2×D101—42no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—41no—
114C4×C101—81no—
115Dic101—21no—
116Dic101—41no—
117Dic101—41no—
118Dic101—81no—
119Dic101#9912yes—
120C2×C101—21no—
121C2×C101—21no—
122C2×C101—81no—
123C2×C101C2×#9911yescharacteristic
124C2×C101#41711yescharacteristic
125C2×C101#42111yescharacteristic
126D101—41no—
127D101—41no—
128#1513—1011nomaximal, Sylow
129C101#151311yescharacteristic, complemented, Sylow
130#525—1011no—
131#407—1011no—
132#380—1011no—
133#328—1011no—
134#328—1011no—
135#327—1011no—
136#327D10111yescharacteristic, complemented
137C2×#41—1011no—
138C23×C4—1011no—
139C2×M16—2021no—
140#123—2021no—
141C22⋊Q8—1011no—
142C22⋊Q8—1011no—
143#121—1011no—
144#121C2×D10111yescharacteristic
145#120—2021no—
146C4×D4—2021no—
147C4×D4—2021no—
148#117—1011no—
149C2×C4⋊C4—2021no—
150C2×C22⋊C4—2021no—
151C22⋊C8—1011no—
152C22⋊C8—2021no—
153C22⋊C8C2×D10111yescharacteristic
154#95—1011no—
155#95C2×D10111yescharacteristic
156C24—1011no—
157#41—4042no—
158C2×Q8—1011no—
159C2×D4—21no—
160C2×D4—2021no—
161C2×D4—2021no—
162C2×D4C4×D10111yescharacteristic
163C22×C4—21no—
164C22×C4—1011no—
165C22×C4—1011no—
166C22×C4—1011no—
167C22×C4—1011no—
168C22×C4—1011no—
169C22×C4—4042no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4C22×D10111yescharacteristic
173M16—4042no—
174C2×C8—21no—
175C2×C8—2021no—
176C4⋊C4—1011no—
177C4⋊C4—4042no—
178C4⋊C4—2021no—
179C4⋊C4—2021no—
180C4⋊C4C4×D10111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—4042no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C22⋊C4—2021no—
186C22⋊C4—2021no—
187C42—2021no—
188C23—21no—
189C23—1011no—
190C23—1011no—
191C23—4041no—
192C23—4041no—
193C23—2021no—
194C23—2021no—
195C23D4×D10111yescharacteristic
196Q8—4041no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—4042no—
201D4—4042no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—1011no—
208C2×C4—1011no—
209C2×C4—4041no—
210C2×C4—4041no—
211C2×C4—4041no—
212C2×C4—4041no—
213C2×C4—4041no—
214C2×C4—4042no—
215C2×C4—2022no—
216C2×C4—2022no—
217C2×C4—2022no—
218C2×C4—2021no—
219C2×C4—2021no—
220C2×C4C2×C4×D10111yescharacteristic
221C2×C4D4×D10111yescharacteristic
222C8—41no—
223C8—4041no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—4042no—
230C22—4042no—
231C22—2021no—
232C22—2021no—
233C22—2021no—
234C22—2021no—
235C22—2021no—
236C22—2021no—
237C22C22⋊C4×D10111yescharacteristic
238C22#1569227911yescharacteristic
239C22#1569228411yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—81no—
244C4—8081no—
245C4—1012no—
246C4—4041no—
247C4—4041no—
248C4—2021no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—4041no—
253C2—4041no—
254C2#99×D10111yescharacteristic
255C2#2026517511yescharacteristic
256C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265687C2
cosocleC22×C4×C101C23
hypercenter#327D101
hypocenterC101#1513
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265139C2semidirect
3. #20265036C2semidirect
4. #20264915C2semidirect
5. #20264913C2semidirect
6. #20264911C2semidirect
7. #327×C101C2semidirect
8. C2×#13108073C4semidirect
12. #15692235C4semidirect
14. #15692226C4semidirect
18. #15692213C4semidirect
129. C101#1513semidirect
136. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202656872
#202651392
#202650362
#202649152
#202649132
#202649112
#327×C1012
#1513101