Finite Group Databasebeta

← #27171514

Subgroups — #27171514

Order 12928.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#20265943C211yesmaximal, characteristic, complemented
2#20265727C211yesmaximal, characteristic
3#20265195C211yesmaximal, characteristic, complemented
4#20265193C211yesmaximal, characteristic, complemented
5#20265150C211yesmaximal, characteristic, complemented
6#20264937C211yesmaximal, characteristic, complemented
7#339×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9C2×#13108074C2211yescharacteristic
10#15692263—22no—
11#15692261—22no—
12C2×#13108058C2211yescharacteristic
13M16×D101—22no—
14#15692255C411yescharacteristic, complemented
15#15692252C411yescharacteristic, complemented
16#15692243C411yescharacteristic, complemented
17#15692243C411yescharacteristic, complemented
18#15692229—22no—
19C4×D404—22no—
20#15692214C2211yescharacteristic
21#15692183—22no—
22#15692181—22no—
23#15692169—22no—
24C2×M16×C101C2211yescharacteristic
25#117×C101C2211yescharacteristic
26C2×C42×C101C2211yescharacteristic
27#13108074—21no—
28#13108074—21no—
29#13108074—21no—
30#13108074—21no—
31#13108074—21no—
32#13108074—21no—
33#13108074—21no—
34#13108074—21no—
35C2×Dic202C2×C411yescharacteristic
36C2×Dic202C2×C411yescharacteristic
37C2×D404—22no—
38C2×D404C2×C411yescharacteristic
39C2×D404C2×C411yescharacteristic
40C2×C22⋊D101—21no—
41C2×C22⋊D101D412yes—
42C2×C4×D101—21no—
43C2×C4×D101D412yes—
44#13108058—21no—
45#13108058D412yescomplemented
46#13108057—24no—
47C2×#11717579—21no—
48C8×D101—24no—
49C4⋊Dic101—22no—
50C4⋊Dic101—21no—
51C22⋊Dic101—21no—
52#13108051—22no—
53#13108051—22no—
54C4×Dic101—21no—
55C22×C4×C101—21no—
56C22×C4×C101C2311yescharacteristic
57M16×C101—21no—
58M16×C101D412yescomplemented
59C2×C8×C101—21no—
60C4⋊C4×C101—21no—
61C22⋊C4×C101—21no—
62C42×C101—21no—
63C42×C101—21no—
64C42×C101C2×C411yescharacteristic
65C42×C101C2×C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C22×D101C22⋊C412yes—
69C2×C4×C101—21no—
70C2×C4×C101—21no—
71C2×C4×C101—42no—
72C2×C4×C101—41no—
73C2×C4×C101C22×C411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101#4111yescharacteristic
78C2×C4×C101#4111yescharacteristic
79C22⋊D101—22no—
80C22⋊D101—22no—
81C22⋊D101—41no—
82C22⋊D101—41no—
83C2×Dic101—21no—
84C2×Dic101—41no—
85C2×Dic101C22⋊C412yes—
86D404—21no—
87D404—21no—
88D404—21no—
89D404—21no—
90D404—44no—
91C4×D101—22no—
92C4×D101—22no—
93C4×D101—41no—
94C4×D101—41no—
95Dic202—21no—
96Dic202—21no—
97Dic202—21no—
98Dic202—21no—
99C8×C101—22no—
100#11717579—22no—
101C22×C101—41no—
102C22×C101C2×C22⋊C411yescharacteristic
103C22×C101#12111yescharacteristic
104C22×C101#12111yescharacteristic
105C2×D101—22no—
106C2×D101—42no—
107C2×D101—42no—
108C2×D101—41no—
109C4×C101—22no—
110C4×C101—41no—
111C4×C101—81no—
112C4×C101C4×D411yescharacteristic
113C4×C101C4×D411yescharacteristic
114C4×C101#12011yescharacteristic
115C4×C101#12311yescharacteristic
116Dic101—22no—
117Dic101—41no—
118Dic101—81no—
119C2×C101—21no—
120C2×C101—21no—
121C2×C101—21no—
122C2×C101#38611yescharacteristic
123D101—42no—
124D101—81no—
125#1884—1011nomaximal, Sylow
126C101#188411yescharacteristic, complemented, Sylow
127#568—1011no—
128#532—1011no—
129#421—1011no—
130#421—1011no—
131#410—1011no—
132#339—1011no—
133#339D10111yescharacteristic, complemented
134C2×#41—1011no—
135C2×#41—1011no—
136#131—2022no—
137#131—2022no—
138C2×M16—1011no—
139C2×M16—2022no—
140C2×M16C2×D10111yescharacteristic
141C4⋊Q8—1011no—
142#127—1011no—
143#124—1011no—
144#124—1011no—
145#121—2022no—
146C4×D4—2022no—
147#117—1011no—
148#117C2×D10111yescharacteristic
149C2×C42C2×D10111yescharacteristic
150#104—2022no—
151#104—2022no—
152#99—2022no—
153#41—2021no—
154#41—2021no—
155#41—2021no—
156#41—2021no—
157#41—2021no—
158#41—2021no—
159#41—2021no—
160#41—2021no—
161C2×Q8—1011no—
162C2×Q8—1011no—
163C2×D4—1012no—
164C2×D4—1011no—
165C2×D4—1011no—
166C2×D4—2022no—
167C2×D4—2021no—
168C22×C4—21no—
169C22×C4—1012no—
170C22×C4—2021no—
171C22×C4C22×D10111yescharacteristic
172M16—21no—
173M16—1012no—
174M16—2024no—
175M16—2021no—
176M16D40412yescomplemented
177C2×C8—21no—
178C2×C8—2024no—
179C2×C8—2021no—
180C4⋊C4—21no—
181C4⋊C4—2022no—
182C4⋊C4—2021no—
183C22⋊C4—21no—
184C22⋊C4—2022no—
185C22⋊C4—2022no—
186C22⋊C4—2021no—
187C42—21no—
188C42—21no—
189C42—2021no—
190C42C4×D10111yescharacteristic
191C42C4×D10111yescharacteristic
192C23—1012no—
193C23—2021no—
194C23D4×D10111yescharacteristic
195Q8—2021no—
196Q8—2021no—
197Q8—2021no—
198Q8—2021no—
199D4—4044no—
200D4—4041no—
201D4—4041no—
202D4—2022no—
203D4—2022no—
204D4—2021no—
205D4—2021no—
206D4—2021no—
207D4—2021no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—1012no—
213C2×C4—4041no—
214C2×C4—4041no—
215C2×C4—4041no—
216C2×C4—2021no—
217C2×C4—2022no—
218C2×C4—2022no—
219C2×C4C2×C4×D10111yescharacteristic
220C2×C4D4×D10111yescharacteristic
221C2×C4D4×D10111yescharacteristic
222C2×C4C2×D40411yescharacteristic
223C2×C4#1310807411yescharacteristic
224C2×C4#1310807511yescharacteristic
225C8—22no—
226C8—2022no—
227C22—41no—
228C22—4042no—
229C22—4042no—
230C22—4041no—
231C22—2022no—
232C22C22⋊C4×D10111yescharacteristic
233C22#1569222811yescharacteristic
234C22#1569223011yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—8081no—
239C4—4041no—
240C4—2022no—
241C4#1569221711yescharacteristic
242C4C4×D40411yescharacteristic
243C4#1569222511yescharacteristic
244C4#1569224011yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—8081no—
249C2—4042no—
250C2#2026505511yescharacteristic
subgroupG/H
centerC4#15692225
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#339×C101C2
socleC2×C101#386
prime radical#20265727C2
cosocleC22×C4×C101C23
hypercenter#339D101
hypocenterC101#1884
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #20265943C2semidirect
3. #20265195C2semidirect
4. #20265193C2semidirect
5. #20265150C2semidirect
6. #20264937C2semidirect
7. #339×C101C2semidirect
14. #15692255C4semidirect
15. #15692252C4semidirect
16. #15692243C4semidirect
17. #15692243C4semidirect
45. #13108058D4semidirect
58. M16×C101D4semidirect
126. C101#1884semidirect
133. #339D101semidirect
176. M16D404semidirect

Maximal subgroups

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

subgroupindex
#202659432
#202657272
#202651952
#202651932
#202651502
#202649372
#339×C1012
#1884101