Finite Group Databasebeta

← #352×D101 #27172766

Subgroups — #352×D101 #27172766

Order 12928.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D101C211yesmaximal, characteristic
2#100×D101C211yesmaximal, characteristic
3#99×D101C211yesmaximal, characteristic
4#20264968C211yesmaximal, characteristic, complemented
5#20264967C211yesmaximal, characteristic, complemented
6#20264966C211yesmaximal, characteristic, complemented
7#352×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101C2211yescharacteristic
9C22×C4×D101C411yescharacteristic, complemented
10M16×D101—21no—
11#15692240—21no—
12#15692239C411yescharacteristic, complemented
13#15692238C411yescharacteristic, complemented
14#15692237—21no—
15#15692236C2211yescharacteristic
16C4⋊C4×D101—21no—
17C22⋊C4×D101—21no—
18C22⋊C4×D101—21no—
19C22⋊C4×D101C411yescharacteristic, complemented
20#15692172C2211yescharacteristic
21#15692171—21no—
22#15692170C2211yescharacteristic
23#15692169—21no—
24#123×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic
26#99×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C23×D101D411yescharacteristic
29C2×D404D411yescharacteristic
30C2×C22⋊D101C2×C411yescharacteristic
31C2×C22⋊D101D411yescharacteristic
32D4×D101—41no—
33D4×D101—41no—
34C22×Dic101C2×C411yescharacteristic, complemented
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37C2×C4×D101—21no—
38C2×C4×D101—41no—
39C2×C4×D101—41no—
40C2×C4×D101—41no—
41C2×C4×D101D411yescharacteristic
42#13108058—21no—
43#13108057—41no—
44C8×D101—41no—
45C4⋊Dic101—21no—
46#13108053—21no—
47#13108053—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101C2×C411yescharacteristic, complemented
51#13108051—21no—
52#13108051—21no—
53#13108051—21no—
54#13108051—41no—
55C2×D4×C101C2311yescharacteristic
56C22×C4×C101C2×C411yescharacteristic, complemented
57M16×C101—21no—
58C4⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101C2×C411yescharacteristic, complemented
62C23×C101C22×C411yescharacteristic
63C23×C101C2×D411yescharacteristic
64C22×D101—21no—
65C22×D101—21no—
66C22×D101—21no—
67C22×D101—21no—
68C22×D101—42no—
69C22×D101—42no—
70C22×D101C22⋊C411yescharacteristic
71C22×D101C22⋊C411yescharacteristic
72C22×D101C22⋊C411yescharacteristic
73D4×C101—41no—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—41no—
77C2×C4×C101—41no—
78C2×C4×C101C2×D411yescharacteristic
79C22⋊D101—41no—
80C22⋊D101—41no—
81C2×Dic101—21no—
82C2×Dic101—21no—
83C2×Dic101—41no—
84C2×Dic101—41no—
85C2×Dic101C22⋊C411yescharacteristic
86D404—41no—
87C4×D101—21no—
88C4×D101—21no—
89C4×D101—42no—
90C4×D101—42no—
91C4×D101—41no—
92C4×D101—41no—
93C4×D101—82no—
94C8×C101—41no—
95#11717579—41no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101C2×C22⋊C411yescharacteristic
101C2×D101—22no—
102C2×D101—21no—
103C2×D101—21no—
104C2×D101—42no—
105C2×D101—42no—
106C2×D101—42no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—81no—
110C2×D101#9911yescharacteristic
111C2×D101#9911yescharacteristic
112C4×C101—21no—
113C4×C101—41no—
114C4×C101—41no—
115C4×C101—81no—
116Dic101—21no—
117Dic101—41no—
118Dic101—41no—
119Dic101—81no—
120C2×C101—21no—
121C2×C101—41no—
122C2×C101—41no—
123C2×C101C2×#9911yescharacteristic
124D101—21no—
125D101—41no—
126D101—41no—
127D101#35212yesdirect product factor
128C2×#352—1011nomaximal, Sylow
129C101C2×#35211yescharacteristic, complemented, Sylow
130C2×#123—1011no—
131C2×#100—1011no—
132C2×#99—1011no—
133#352—1011no—
134#352—1011no—
135#352—1011no—
136#352D10111yescharacteristic, direct product factor
137C22×D4—1011no—
138C23×C4—1011no—
139C2×M16—2021no—
140#123—1011no—
141#123—1011no—
142#123—1011no—
143#123—2021no—
144#123—2021no—
145#123C2×D10111yescharacteristic
146C2×C4⋊C4—2021no—
147C2×C22⋊C4—1011no—
148C2×C22⋊C4—2021no—
149C2×C22⋊C4—2021no—
150#100—1011no—
151#100—2021no—
152#100C2×D10111yescharacteristic
153#99—1011no—
154#99—2021no—
155#99C2×D10111yescharacteristic
156C24—1011no—
157C24—1011no—
158C2×D4—1011no—
159C2×D4—1011no—
160C2×D4—1011no—
161C2×D4—4041no—
162C2×D4—4041no—
163C2×D4C22×D10111yescharacteristic
164C22×C4—1011no—
165C22×C4—1011no—
166C22×C4—4041no—
167C22×C4—4041no—
168C22×C4—4041no—
169C22×C4—2021no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4C4×D10111yescharacteristic, complemented
173M16—21no—
174M16—4041no—
175M16—2021no—
176C2×C8—4041no—
177C4⋊C4—21no—
178C4⋊C4—2021no—
179C4⋊C4—2021no—
180C4⋊C4—2021no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—1011no—
184C22⋊C4—4041no—
185C22⋊C4—2021no—
186C22⋊C4—2021no—
187C22⋊C4—2021no—
188C22⋊C4—2021no—
189C22⋊C4—2021no—
190C22⋊C4C4×D10111yescharacteristic, complemented
191C23—1011no—
192C23—1011no—
193C23—1011no—
194C23—4042no—
195C23—4042no—
196C23—2021no—
197C23—2021no—
198C23—2021no—
199C23—2021no—
200C23C2×C4×D10111yescharacteristic
201C23D4×D10111yescharacteristic
202D4—41no—
203D4—4041no—
204D4—4041no—
205D4—4041no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—8082no—
211C2×C4—1011no—
212C2×C4—4041no—
213C2×C4—4041no—
214C2×C4—4042no—
215C2×C4—4042no—
216C2×C4—4041no—
217C2×C4—4041no—
218C2×C4—2021no—
219C2×C4—2021no—
220C2×C4—2021no—
221C2×C4—2021no—
222C2×C4D4×D10111yescharacteristic
223C8—41no—
224C8—4041no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—8081no—
230C22—1011no—
231C22—1011no—
232C22—4041no—
233C22—4041no—
234C22—4042no—
235C22—4042no—
236C22—4042no—
237C22—2022no—
238C22—2021no—
239C22—2021no—
240C22C22⋊C4×D10111yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—8081no—
246C4—4041no—
247C4—4041no—
248C4—2021no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—1012no—
253C2—4041no—
254C2—4041no—
255C2—2021no—
256C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#352×C101C2
socleC2×C101C2×#99
prime radicalC2×D4×D101C22
cosocleC2×D4×C101C23
hypercenter#352D101
hypocenterC101C2×#352
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #20264968C2semidirect
5. #20264967C2semidirect
6. #20264966C2semidirect
7. #352×C101C2semidirect
9. C22×C4×D101C4semidirect
12. #15692239C4semidirect
13. #15692238C4semidirect
19. C22⋊C4×D101C4semidirect
34. C22×Dic101C2×C4semidirect
50. C22⋊Dic101C2×C4semidirect
56. C22×C4×C101C2×C4semidirect
61. C22⋊C4×C101C2×C4semidirect
127. D101#352direct (N × H)
129. C101C2×#352semidirect
136. #352D101direct (N × H)
172. C22×C4C4×D101semidirect
190. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
#123×D1012
#100×D1012
#99×D1012
#202649682
#202649672
#202649662
#352×C1012
C2×#352101