Finite Group Databasebeta

← #27180786

Subgroups — #27180786

Order 12928.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#13108064C211yesmaximal, characteristic, complemented
2C2×#15692263C211yesmaximal, characteristic
3#20265753C211yesmaximal, characteristic
4#20265751C211yesmaximal, characteristic
5#20265434C214yesmaximal, complemented
6#20265417C214yesmaximal, complemented
7#20265213C211yesmaximal, characteristic, complemented
8#20265187C211yesmaximal, characteristic, complemented
9#419×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11C22×Dic202C2211yescharacteristic
12C2×#13108064—24no—
13C2×#13108064C2212yes—
14C2×#13108064C2212yes—
15#15692263—24no—
16C2×#13108057C2212yes—
17C22×#11717579C2211yescharacteristic
18C2×C8×D101C2212yes—
19#15692255—21no—
20#15692253—21no—
21#15692244—21no—
22#15692243—21no—
23#15692239—22no—
24#15692238—22no—
25#15692235C2214yes—
26#15692231C2214yes—
27#15692214C2211yescharacteristic
28#15692188C2212yes—
29#15692186C2212yes—
30#15692180C2214yes—
31C22×Q8×C101C2211yescharacteristic
32C2×M16×C101C2211yescharacteristic
33#117×C101C2211yescharacteristic
34#103×C101C2214yes—
35#13108074—22no—
36#13108074—22no—
37C2×Dic202—22no—
38C2×Dic202—22no—
39C2×Dic202—21no—
40C2×Dic202D411yescharacteristic
41C2×Dic202C2312yes—
42C2×D404D411yescharacteristic
43C2×C22⋊D101D412yes—
44C22×Dic101—21no—
45C2×C4×D101C2312yes—
46#13108064—24no—
47#13108064—24no—
48#13108057—24no—
49C2×#11717579D414yes—
50C2×#11717579C2311yescharacteristic
51C2×#11717579C2311yescharacteristic
52C8×D101—24no—
53C4⋊Dic101—24no—
54C4⋊Dic101C2312yes—
55#13108053—24no—
56C22⋊Dic101—21no—
57#13108051—24no—
58#13108051—24no—
59C4×Dic101—21no—
60C2×Q8×C101—21no—
61C2×Q8×C101—22no—
62C2×Q8×C101—22no—
63C2×Q8×C101C2312yes—
64C22×C4×C101—21no—
65C22×C4×C101C2311yescharacteristic
66M16×C101—22no—
67C2×C8×C101C2312yes—
68C4⋊C4×C101C2312yes—
69C22⋊C4×C101—21no—
70C42×C101—21no—
71C23×C101C2×D411yescharacteristic
72C22×D101C2×D412yes—
73Q8×C101—22no—
74Q8×C101—22no—
75Q8×C101—42no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101—42no—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101C2×D412yes—
81C2×C4×C101#4112yes—
82C2×C4×C101C2411yescharacteristic
83C22⋊D101—24no—
84C2×Dic101—22no—
85C2×Dic101—22no—
86C2×Dic101—42no—
87C2×Dic101C2×D412yes—
88D404—22no—
89C4×D101—22no—
90C4×D101—22no—
91Dic202—22no—
92Dic202—22no—
93Dic202—22no—
94Dic202—42no—
95C8×C101—22no—
96#11717579C2×D412yes—
97#11717579C2×D412yes—
98C22×C101—22no—
99C22×C101#12112yes—
100C22×C101C22×D411yescharacteristic
101C2×D101—22no—
102C2×D101—24no—
103C4×C101—42no—
104C4×C101—42no—
105C4×C101#12112yes—
106C4×C101C22×D411yescharacteristic
107C4×C101C2×#4111yescharacteristic
108Dic101—22no—
109Dic101—42no—
110Dic101—42no—
111C2×C101—22no—
112C2×C101C2×#12111yescharacteristic
113C2×C101#57812yes—
114D101—42no—
115#3059—1011nomaximal, Sylow
116C101#305911yescharacteristic, complemented, Sylow
117C22×Q16—1011no—
118C2×#131—1011no—
119#536—1011no—
120#536—1011no—
121#470—1014no—
122#467—1014no—
123#427—1011no—
124#419—1011no—
125#419D10111yescharacteristic, complemented
126C2×#41—1011no—
127C22×Q8—1011no—
128C22×Q8C2×D10111yescharacteristic
129C2×Q16—1012no—
130C2×Q16—1012no—
131C2×Q16—2024no—
132#131—2024no—
133C2×M16—1012no—
134C2×M16C2×D10111yescharacteristic
135C22×C8—1011no—
136C22×C8—1012no—
137C4⋊Q8—2021no—
138C4⋊Q8—2021no—
139#124—2021no—
140#124—2021no—
141#123—2022no—
142#123—2022no—
143C22⋊Q8—1014no—
144C22⋊Q8—1014no—
145#117—1011no—
146#117C2×D10111yescharacteristic
147#107—1012no—
148#106—1012no—
149#103—1014no—
150#103C2×D10114yes—
151#41—2022no—
152#41—2022no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1011no—
157C2×Q8—1012no—
158C2×Q8—2022no—
159C2×Q8—2022no—
160C2×Q8—2021no—
161C2×Q8C22×D10112yes—
162C2×D4—1012no—
163C2×D4—1011no—
164C22×C4—21no—
165C22×C4—1012no—
166C22×C4—2021no—
167C22×C4C22×D10111yescharacteristic
168Q16—2024no—
169Q16—2024no—
170M16—22no—
171M16—2024no—
172C2×C8—1014no—
173C2×C8—1011no—
174C2×C8—1011no—
175C2×C8—2024no—
176C2×C8C22×D10112yes—
177C4⋊C4—1012no—
178C4⋊C4—2024no—
179C4⋊C4—2024no—
180C4⋊C4C22×D10112yes—
181C22⋊C4—21no—
182C22⋊C4—2024no—
183C22⋊C4—2024no—
184C22⋊C4—2021no—
185C42—21no—
186C42—2021no—
187C23—1012no—
188C23D4×D10111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—4042no—
193Q8—2022no—
194Q8—2022no—
195Q8—2022no—
196D4—2024no—
197D4—2022no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—1012no—
202C2×C4—4042no—
203C2×C4—2022no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4—2022no—
207C2×C4D4×D10112yes—
208C2×C4D4×D10111yescharacteristic
209C2×C4#1310807412yes—
210C2×C4C23×D10111yescharacteristic
211C8—22no—
212C8—1012no—
213C8—1012no—
214C22—22no—
215C22—2024no—
216C22—2022no—
217C22#1569222812yes—
218C22C2×D4×D10111yescharacteristic
219C4—42no—
220C4—42no—
221C4—4042no—
222C4—4042no—
223C4—2022no—
224C4#1569222812yes—
225C4C2×D4×D10111yescharacteristic
226C4C2×#1310807411yescharacteristic
227C2—22no—
228C2—4042no—
229C2C2×#1569222811yescharacteristic
230C2#2026599312yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#419×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108074C22
hypercenter#419D101
hypocenterC101#3059
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#13108064C2semidirect
5. #20265434C2semidirect
6. #20265417C2semidirect
7. #20265213C2semidirect
8. #20265187C2semidirect
9. #419×C101C2semidirect
116. C101#3059semidirect
125. #419D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080642
C2×#156922632
#202657532
#202657512
#202654342
#202654172
#202652132
#202651872
#419×C1012
#3059101