Finite Group Databasebeta

← #27181746

Subgroups — #27181746

Order 12928.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265796C212yesmaximal, complemented
2#20265575C211yesmaximal, characteristic, complemented
3#20265536C212yesmaximal, complemented
4#20265432C211yesmaximal, characteristic, complemented
5#20265427C211yesmaximal, characteristic
6#20265403C212yesmaximal, complemented
7#20265289C211yesmaximal, characteristic, complemented
8#20265284C211yesmaximal, characteristic, complemented
9SD16×Dic101C212yesmaximal, complemented
10#20265131C211yesmaximal, characteristic, complemented
11#495×C101C211yesmaximal, characteristic, complemented
12C2×#13108057C2211yescharacteristic
13C2×C8×D101C2211yescharacteristic
14#15692255C2212yes—
15#15692248—22no—
16#15692241—22no—
17#15692239—22no—
18#15692232—22no—
19#15692231C2212yes—
20#15692226C2212yes—
21Q8×Dic101C2212yes—
22D4×Dic101C2212yes—
23#15692212C2211yescharacteristic
24#15692187C2212yes—
25#15692187C2211yescharacteristic
26#15692187C2211yescharacteristic
27#15692179C2212yes—
28#15692179C2212yes—
29#15692176C2212yes—
30#15692176C2212yes—
31#15692166C2211yescharacteristic
32#15692165C2211yescharacteristic
33C8×Dic101C2211yescharacteristic
34C2×Q16×C101C2211yescharacteristic
35C2×SD16×C101C2212yes—
36C2×D8×C101C2211yescharacteristic
37#124×C101C2212yes—
38C4×C8×C101C2211yescharacteristic
39C2×Dic202—22no—
40C2×C22⋊D101—22no—
41C22×Dic101—22no—
42C2×C4×D101C2311yescharacteristic
43#13108057—22no—
44C2×#11717579C2311yescharacteristic
45C2×#11717579C2311yescharacteristic
46C8×D101—22no—
47C4⋊Dic101—22no—
48C4⋊Dic101—22no—
49C4⋊Dic101C2312yes—
50C4⋊Dic101C2312yes—
51#13108053—22no—
52#13108053D412yes—
53C22⋊Dic101—22no—
54C22⋊Dic101—22no—
55#13108051—22no—
56#13108051D412yes—
57C4×Dic101—22no—
58C4×Dic101C2311yescharacteristic
59C2×Q8×C101C2312yes—
60C2×D4×C101C2312yes—
61Q16×C101—22no—
62SD16×C101—24no—
63D8×C101—22no—
64C2×C8×C101C2311yescharacteristic
65C2×C8×C101C2311yescharacteristic
66C22⋊C4×C101—22no—
67C22⋊C4×C101—22no—
68C42×C101C2311yescharacteristic
69C23×C101—22no—
70C22×D101C2×D411yescharacteristic
71Q8×C101—22no—
72Q8×C101—22no—
73D4×C101—22no—
74D4×C101—22no—
75C2×C4×C101—22no—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—82no—
79C2×Dic101—22no—
80C2×Dic101—22no—
81C2×Dic101—82no—
82C2×Dic101C2×D412yes—
83C2×Dic101C2×D411yescharacteristic
84C4×D101—21no—
85C4×D101—21no—
86Dic202—82no—
87C8×C101#4112yes—
88C8×C101#4112yes—
89#11717579—21no—
90#11717579—21no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101—42no—
94C22×C101C22×D411yescharacteristic
95C2×D101—21no—
96C2×D101—22no—
97C4×C101—22no—
98C4×C101—42no—
99C4×C101C2×#4111yescharacteristic
100C4×C101C2×#4111yescharacteristic
101Dic101—21no—
102Dic101—22no—
103Dic101—42no—
104Dic101—42no—
105C2×C101—82no—
106C2×C101#53211yescharacteristic
107C2×C101#57712yes—
108D101—41no—
109#3173—1011nomaximal, Sylow
110C101#317311yescharacteristic, complemented, Sylow
111#542—1012no—
112#498—1011no—
113#495D10111yescharacteristic, complemented
114#489—1012no—
115#470—1011no—
116#469—1011no—
117#465—1012no—
118#442—1011no—
119#441—1011no—
120C4×SD16—1012no—
121#405—1011no—
122C2×Q16C2×D10111yescharacteristic
123C2×SD16C2×D10112yes—
124C2×D8C2×D10111yescharacteristic
125C2×M16—1011no—
126C22×C8—1011no—
127C4⋊Q8—1012no—
128#126—2022no—
129#124—2022no—
130#124C2×D10112yes—
131#123—2022no—
132C22⋊Q8—1012no—
133C22⋊Q8—2022no—
134#121—1012no—
135C4×Q8—1012no—
136C4×D4—1012no—
137#117—1011no—
138#106—1012no—
139#106—1011no—
140#106—1011no—
141#103—1012no—
142#103—1012no—
143#102—1012no—
144#102—1012no—
145#97—1011no—
146#97—1011no—
147C4×C8—1011no—
148C4×C8C2×D10111yescharacteristic
149C2×Q8—2022no—
150C2×Q8C22×D10112yes—
151C2×D4—2022no—
152C2×D4C22×D10112yes—
153C22×C4—1011no—
154C22×C4—2022no—
155Q16—22no—
156SD16—24no—
157D8—22no—
158M16—2022no—
159C2×C8—1011no—
160C2×C8—1011no—
161C2×C8—2022no—
162C2×C8C22×D10111yescharacteristic
163C2×C8C22×D10111yescharacteristic
164C4⋊C4—1012no—
165C4⋊C4—1012no—
166C4⋊C4—1012no—
167C4⋊C4—2022no—
168C4⋊C4—2022no—
169C4⋊C4—2022no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—1012no—
173C22⋊C4—2022no—
174C22⋊C4—2022no—
175C22⋊C4—2022no—
176C42—1011no—
177C42—2022no—
178C42C22×D10111yescharacteristic
179C23—22no—
180C23—1011no—
181Q8—22no—
182Q8—22no—
183Q8—8082no—
184D4—22no—
185D4—22no—
186D4—8082no—
187C2×C4—22no—
188C2×C4—8082no—
189C2×C4—1012no—
190C2×C4—1011no—
191C2×C4—2022no—
192C2×C4—2022no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4D4×D10112yes—
196C2×C4C23×D10111yescharacteristic
197C8—2021no—
198C8—2021no—
199C8#1310807312yes—
200C8#1310807312yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—2022no—
205C22—2021no—
206C22C2×D4×D10111yescharacteristic
207C4—22no—
208C4—42no—
209C4—4042no—
210C4—4042no—
211C4—2022no—
212C4—2021no—
213C4C2×#1310807311yescharacteristic
214C4C2×#1310807311yescharacteristic
215C2—82no—
216C2—4041no—
217C2#2026572511yescharacteristic
218C2#2026598412yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#495×C101C2
socleC22×C101C22×D4
prime radical#20265427C2
hypercenter#495D101
hypocenterC101#3173
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20265796C2semidirect
2. #20265575C2semidirect
3. #20265536C2semidirect
4. #20265432C2semidirect
6. #20265403C2semidirect
7. #20265289C2semidirect
8. #20265284C2semidirect
9. SD16×Dic101C2semidirect
10. #20265131C2semidirect
11. #495×C101C2semidirect
110. C101#3173semidirect
113. #495D101semidirect

Maximal subgroups

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

subgroupindex
#202657962
#202655752
#202655362
#202654322
#202654272
#202654032
#202652892
#202652842
SD16×Dic1012
#202651312
#495×C1012
#3173101