Finite Group Databasebeta

← #27180784

Subgroups — #27180784

Order 12928.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15692262C211yesmaximal, characteristic
3#20265748C211yesmaximal, characteristic
4#20265433C214yesmaximal, complemented
5#20265416C214yesmaximal, complemented
6#20265213C211yesmaximal, characteristic, complemented
7#20265186C211yesmaximal, characteristic, complemented
8#20265185C211yesmaximal, characteristic, complemented
9#536×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11C2×#13108064—24no—
12C2×#13108064C2212yes—
13C2×#13108064C2212yes—
14#15692262—24no—
15C2×#13108058C2212yes—
16C2×#13108058C2211yescharacteristic
17C22×#11717579C2212yes—
18C22×#11717579C2211yescharacteristic
19#15692253—21no—
20#15692242—21no—
21#15692236—22no—
22#15692234C2214yes—
23#15692214C2211yescharacteristic
24#15692188C2212yes—
25#15692186C2212yes—
26#15692179C2214yes—
27#15692178C2214yes—
28C2×#41×C101C2211yescharacteristic
29C22×Q8×C101C2211yescharacteristic
30C4⋊Q8×C101—21no—
31#124×C101—21no—
32#123×C101—22no—
33C22⋊Q8×C101C2214yes—
34#117×C101C2211yescharacteristic
35C2×Dic202—22no—
36C2×Dic202—22no—
37C2×Dic202—21no—
38C2×Dic202C2312yes—
39C22×Dic101—21no—
40#13108064—24no—
41#13108064—24no—
42#13108058—22no—
43#13108058—24no—
44C2×#11717579—24no—
45C2×#11717579D414yes—
46C2×#11717579C2312yes—
47C2×#11717579C2311yescharacteristic
48C2×#11717579C2311yescharacteristic
49C4⋊Dic101C2312yes—
50#13108053—24no—
51C22⋊Dic101—21no—
52C22⋊Dic101—24no—
53C4×Dic101—21no—
54#41×C101—22no—
55#41×C101—22no—
56C2×Q8×C101—21no—
57C2×Q8×C101—22no—
58C2×Q8×C101—22no—
59C2×Q8×C101D411yescharacteristic
60C2×Q8×C101C2312yes—
61C2×D4×C101D412yes—
62C2×D4×C101D411yescharacteristic
63C22×C4×C101—21no—
64C22×C4×C101C2312yes—
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—24no—
67C4⋊C4×C101C2312yes—
68C22⋊C4×C101—21no—
69C22⋊C4×C101—24no—
70C42×C101—21no—
71C23×C101C2×D412yes—
72C23×C101C2×D411yescharacteristic
73Q8×C101—22no—
74Q8×C101—22no—
75Q8×C101—22no—
76Q8×C101—42no—
77D4×C101—24no—
78D4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—42no—
84C2×C4×C101C2×D412yes—
85C2×C4×C101C2×D412yes—
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101#4112yes—
88C2×C4×C101C2411yescharacteristic
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—42no—
92Dic202—22no—
93Dic202—22no—
94Dic202—42no—
95#11717579—22no—
96#11717579C2×D412yes—
97#11717579C2×D412yes—
98C22×C101—22no—
99C22×C101—22no—
100C22×C101—24no—
101C22×C101#12112yes—
102C22×C101C22×D411yescharacteristic
103C4×C101—22no—
104C4×C101—42no—
105C4×C101—42no—
106C4×C101#12112yes—
107C4×C101C22×D411yescharacteristic
108C4×C101C2×#4111yescharacteristic
109Dic101—42no—
110Dic101—42no—
111C2×C101—22no—
112C2×C101—42no—
113C2×C101C2×#12111yescharacteristic
114C2×C101#57812yes—
115#3059—1011nomaximal, Sylow
116C101#305911yescharacteristic, complemented, Sylow
117C22×Q16—1011no—
118C2×#131—1011no—
119#536—1011no—
120#536D10111yescharacteristic
121#470—1014no—
122#467—1014no—
123#427—1011no—
124#419—1011no—
125#419—1011no—
126C2×#41C2×D10111yescharacteristic
127C22×Q8—1011no—
128C22×Q8C2×D10111yescharacteristic
129C2×Q16—1012no—
130C2×Q16—1012no—
131C2×Q16—2024no—
132#131—2024no—
133C2×M16—1011no—
134C2×M16—1012no—
135C22×C8—1011no—
136C22×C8—1012no—
137C4⋊Q8—21no—
138C4⋊Q8—2021no—
139#124—21no—
140#124—2021no—
141#123—22no—
142#123—2022no—
143C22⋊Q8—1014no—
144C22⋊Q8C2×D10114yes—
145#117—1011no—
146#117C2×D10111yescharacteristic
147#107—1012no—
148#106—1012no—
149#103—1014no—
150#103—1014no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1012no—
157C2×Q8—2022no—
158C2×Q8—2022no—
159C2×Q8—2021no—
160C2×Q8C22⋊D10111yescharacteristic
161C2×Q8C22×D10112yes—
162C2×D4C22⋊D10112yes—
163C2×D4C22⋊D10111yescharacteristic
164C22×C4—21no—
165C22×C4—2021no—
166C22×C4C22×D10112yes—
167C22×C4C22×D10111yescharacteristic
168Q16—2024no—
169Q16—2024no—
170M16—2024no—
171M16—2022no—
172C2×C8—1014no—
173C2×C8—1011no—
174C2×C8—1011no—
175C2×C8—1012no—
176C2×C8—2024no—
177C4⋊C4—24no—
178C4⋊C4—1012no—
179C4⋊C4—2024no—
180C4⋊C4C22×D10112yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—2021no—
184C22⋊C4—2024no—
185C42—21no—
186C42—2021no—
187C23C2×C22⋊D10112yes—
188C23C2×C22⋊D10111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—4042no—
194Q8—2022no—
195Q8—2022no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—4042no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4D4×D10112yes—
207C2×C4C2×C22⋊D10112yes—
208C2×C4C2×C22⋊D10111yescharacteristic
209C2×C4#1310807312yes—
210C2×C4C23×D10111yescharacteristic
211C8—1012no—
212C8—1012no—
213C8—2022no—
214C22—22no—
215C22—24no—
216C22—22no—
217C22#1569222712yes—
218C22C22×C22⋊D10111yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—4042no—
223C4—4042no—
224C4#1569222712yes—
225C4C2×D4×D10111yescharacteristic
226C4C2×#1310807311yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1569222711yescharacteristic
230C2#2026599212yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#536×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×C101C22
hypercenter#536D101
hypocenterC101#3059
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13108064C2semidirect
4. #20265433C2semidirect
5. #20265416C2semidirect
6. #20265213C2semidirect
7. #20265186C2semidirect
8. #20265185C2semidirect
116. C101#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#131080642
C2×#156922622
#202657482
#202654332
#202654162
#202652132
#202651862
#202651852
#536×C1012
#3059101