Finite Group Databasebeta

← #27170070

Subgroups — #27170070

Order 12928.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265806C211yesmaximal, characteristic, complemented
2#20265328C211yesmaximal, characteristic
3#20265091C211yesmaximal, characteristic
4#20264915C211yesmaximal, characteristic, complemented
5#20264914C211yesmaximal, characteristic, complemented
6#20264902C211yesmaximal, characteristic, complemented
7#450×C101C211yesmaximal, characteristic
8C2×#13108062—21no—
9#15692254—21no—
10#15692253—21no—
11#15692245—21no—
12#15692245—21no—
13#15692234C2211yescharacteristic
14#15692232—21no—
15#15692211—21no—
16C2×#13108053C2211yescharacteristic
17C2×C22⋊Dic101—21no—
18C2×C22⋊Dic101—21no—
19#15692176—21no—
20C22⋊#11717579C2211yescharacteristic
21#15692162C2211yescharacteristic
22C22×D4×C101C2211yescharacteristic
23C2×SD16×C101—21no—
24C22⋊Q8×C101C2211yescharacteristic
25#102×C101—21no—
26C22⋊C8×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202D411yescharacteristic
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C22×Dic101D411yescharacteristic
32#13108062—42no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36#13108053—21no—
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41#13108053—21no—
42#13108053—21no—
43C22⋊Dic101—21no—
44C22⋊Dic101—42no—
45C22⋊Dic101—42no—
46C4×Dic101—21no—
47C4×Dic101—21no—
48C24×C101—21no—
49C2×Q8×C101D411yescharacteristic
50C2×D4×C101—21no—
51C2×D4×C101—21no—
52C2×D4×C101—42no—
53C22×C4×C101C2311yescharacteristic
54SD16×C101—42no—
55C2×C8×C101—21no—
56C4⋊C4×C101—21no—
57C4⋊C4×C101D411yescharacteristic
58C22⋊C4×C101—21no—
59C23×C101—22no—
60C23×C101—41no—
61C23×C101—41no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101—41no—
65C23×C101—41no—
66C23×C101C2×D411yescharacteristic
67Q8×C101—41no—
68D4×C101—22no—
69D4×C101—22no—
70D4×C101—41no—
71D4×C101—41no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—41no—
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101—41no—
83C2×Dic101—41no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101SD1612yescomplemented
87Dic202—41no—
88Dic202—81no—
89C8×C101—41no—
90#11717579—41no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—42no—
94C22×C101—42no—
95C22×C101—42no—
96C22×C101—81no—
97C22×C101—81no—
98C22×C101—81no—
99C22×C101—81no—
100C22×C101#12011yescharacteristic
101C22×C101C2×SD1611yescharacteristic
102C22×C101#13711yescharacteristic
103C4×C101—21no—
104C4×C101—21no—
105C4×C101—41no—
106C4×C101—41no—
107Dic101—22no—
108Dic101—41no—
109Dic101—41no—
110Dic101—41no—
111Dic101—81no—
112C2×C101—21no—
113C2×C101—21no—
114C2×C101—82no—
115C2×C101#44811yescharacteristic
116C2×C101#45411yescharacteristic
117C2×C101B(4,𝔽2)11yescharacteristic
118#1599—1011nomaximal, Sylow
119C101#159911yescharacteristic, complemented, Sylow
120#544—1011no—
121#450—1011no—
122#450D10111yescharacteristic
123#394—1011no—
124#328—1011no—
125#328—1011no—
126#324—1011no—
127C22×D4C2×D10111yescharacteristic
128C2×SD16—21no—
129C2×SD16—2021no—
130C4⋊Q8—2021no—
131C4⋊Q8—2021no—
132#125—2021no—
133#125—2021no—
134C22⋊Q8—1011no—
135C22⋊Q8—2021no—
136C22⋊Q8C2×D10111yescharacteristic
137#117—2021no—
138C2×C4⋊C4—1011no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4—2021no—
141#102—21no—
142#102—2021no—
143C22⋊C8—1011no—
144C22⋊C8C2×D10111yescharacteristic
145#95—1011no—
146C24—21no—
147C2×Q8—1011no—
148C2×Q8—2021no—
149C2×Q8C22⋊D10111yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—2021no—
156C22×C4C22×D10111yescharacteristic
157SD16—42no—
158SD16—4042no—
159C2×C8—21no—
160C2×C8—2021no—
161C4⋊C4—21no—
162C4⋊C4—1011no—
163C4⋊C4—2021no—
164C4⋊C4—2021no—
165C4⋊C4—2021no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C4⋊C4—2021no—
169C4⋊C4—2021no—
170C4⋊C4—2021no—
171C4⋊C4C22⋊D10111yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—4042no—
174C22⋊C4—4042no—
175C22⋊C4—2021no—
176C42—2021no—
177C42—2021no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D10111yescharacteristic
186Q8—41no—
187Q8—8081no—
188Q8—4041no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—1012no—
197C2×C4—4041no—
198C2×C4—4041no—
199C2×C4—4041no—
200C2×C4—4041no—
201C2×C4—2021no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4D4×D10111yescharacteristic
207C2×C4C2×C22⋊D10111yescharacteristic
208C8—41no—
209C8—4041no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#1569222311yescharacteristic
220C22SD16×D10111yescharacteristic
221C22#1569228411yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—8081no—
227C4—4041no—
228C4—4041no—
229C4—4041no—
230C4—2022no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#2026531411yescharacteristic
235C2#2026534811yescharacteristic
236C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#450×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
hypercenter#450D101
hypocenterC101#1599
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20265806C2semidirect
4. #20264915C2semidirect
5. #20264914C2semidirect
6. #20264902C2semidirect
86. C2×Dic101SD16semidirect
119. C101#1599semidirect

Maximal subgroups

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

subgroupindex
#202658062
#202653282
#202650912
#202649152
#202649142
#202649022
#450×C1012
#1599101