Finite Group Databasebeta

← #27172784

Subgroups — #27172784

Order 12928.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265749C211yesmaximal, characteristic
2#20265158C211yesmaximal, characteristic
3#20265149C211yesmaximal, characteristic
4#20264968C211yesmaximal, characteristic, complemented
5#20264967C211yesmaximal, characteristic, complemented
6#20264966C211yesmaximal, characteristic, complemented
7#352×C101C211yesmaximal, characteristic, complemented
8C2×#13108073C2211yescharacteristic
9C22×Dic202C411yescharacteristic, complemented
10M16×D101—21no—
11#15692254—21no—
12#15692241—21no—
13#15692239C411yescharacteristic, complemented
14#15692238C411yescharacteristic, complemented
15#15692236C2211yescharacteristic
16#15692233—21no—
17#15692232—21no—
18#15692211—21no—
19#15692211C411yescharacteristic, complemented
20#15692173—21no—
21#15692172C2211yescharacteristic
22#15692170C2211yescharacteristic
23#15692169—21no—
24#123×C101C2211yescharacteristic
25#100×C101C2211yescharacteristic
26#99×C101C2211yescharacteristic
27#13108073—41no—
28#13108073—41no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202—41no—
32C2×Dic202—41no—
33C2×Dic202D411yescharacteristic
34C2×C22⋊D101C2×C411yescharacteristic
35C2×C22⋊D101D411yescharacteristic
36C22×Dic101C2×C411yescharacteristic
37C22×Dic101C2×C411yescharacteristic
38C22×Dic101D411yescharacteristic
39C2×C4×D101D411yescharacteristic
40#13108058—21no—
41#13108057—41no—
42C8×D101—41no—
43C4⋊Dic101—21no—
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47#13108053—41no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101C2×C411yescharacteristic
51#13108051—21no—
52#13108051—21no—
53C4×Dic101—21no—
54C4×Dic101—41no—
55C2×D4×C101C2311yescharacteristic
56C22×C4×C101C2×C411yescharacteristic
57M16×C101—21no—
58C4⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101C2×C411yescharacteristic
62C23×C101C22×C411yescharacteristic
63C23×C101C2×D411yescharacteristic
64C22×D101C22⋊C411yescharacteristic
65D4×C101—41no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C2×D411yescharacteristic
71C22⋊D101—41no—
72C22⋊D101—41no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C4×D101—21no—
85C4×D101—21no—
86Dic202—41no—
87Dic202—42no—
88Dic202—41no—
89Dic202—41no—
90C8×C101—41no—
91#11717579—41no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101C2×C22⋊C411yescharacteristic
97C2×D101—21no—
98C2×D101—41no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101—41no—
102C4×C101—81no—
103Dic101—21no—
104Dic101—21no—
105Dic101—41no—
106Dic101—41no—
107Dic101—81no—
108Dic101#9911yescharacteristic
109Dic101#9911yescharacteristic
110C2×C101—21no—
111C2×C101—41no—
112C2×C101—41no—
113C2×C101C2×#9911yescharacteristic
114D101—41no—
115#2123—1011nomaximal, Sylow
116C101#212311yescharacteristic, complemented, Sylow
117#536—1011no—
118#413—1011no—
119#410—1011no—
120#352—1011no—
121#352—1011no—
122#352—1011no—
123#352D10111yescharacteristic, complemented
124C2×#41—1011no—
125C22×Q8—1011no—
126C2×M16—2021no—
127C4⋊Q8—2021no—
128#124—2021no—
129#123—1011no—
130#123—1011no—
131#123—1011no—
132#123C2×D10111yescharacteristic
133C22⋊Q8—2021no—
134C22⋊Q8—2021no—
135#117—1011no—
136#117—2021no—
137#101—2021no—
138#100—1011no—
139#100C2×D10111yescharacteristic
140#99—1011no—
141#99—2021no—
142#99C2×D10111yescharacteristic
143#41—4041no—
144#41—4041no—
145C2×Q8—1011no—
146C2×Q8—4041no—
147C2×Q8—4041no—
148C2×Q8—2021no—
149C2×Q8—2021no—
150C2×D4—1011no—
151C2×D4—1011no—
152C2×D4C22×D10111yescharacteristic
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4C4×D10111yescharacteristic
158M16—21no—
159M16—4041no—
160M16—2021no—
161C2×C8—4041no—
162C4⋊C4—21no—
163C4⋊C4—4041no—
164C4⋊C4—2021no—
165C4⋊C4—2021no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—1011no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C22⋊C4—2021no—
175C22⋊C4C4×D10111yescharacteristic
176C42—4041no—
177C42—2021no—
178C23—1011no—
179C23C2×C4×D10111yescharacteristic
180C23D4×D10111yescharacteristic
181Q8—4041no—
182Q8—4041no—
183Q8—4041no—
184Q8—4042no—
185D4—41no—
186D4—4041no—
187D4—4041no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—1011no—
193C2×C4—1011no—
194C2×C4—1011no—
195C2×C4—4041no—
196C2×C4—4041no—
197C2×C4—2021no—
198C2×C4—2021no—
199C2×C4—2021no—
200C2×C4—2021no—
201C2×C4—2021no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4D4×D10111yescharacteristic
206C8—41no—
207C8—4041no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—4041no—
213C22—2021no—
214C22C22⋊C4×D10111yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—8081no—
220C4—1011no—
221C4—1011no—
222C4—4041no—
223C4—4041no—
224C4—2021no—
225C4—2021no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—4041no—
230C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#352×C101C2
socleC2×C101C2×#99
prime radicalC2×#13108073C22
cosocleC2×D4×C101C23
hypercenter#352D101
hypocenterC101#2123
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×Dic202C4semidirect
13. #15692239C4semidirect
14. #15692238C4semidirect
19. #15692211C4semidirect
116. C101#2123semidirect
123. #352D101semidirect

Maximal subgroups

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

subgroupindex
#202657492
#202651582
#202651492
#202649682
#202649672
#202649662
#352×C1012
#2123101