Finite Group Databasebeta

← #27182534

Subgroups — #27182534

Order 12928.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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#20265713C212yesmaximal, complemented
2#20265482C212yesmaximal, complemented
3#20265476C212yesmaximal, complemented
4#20265430C211yesmaximal, characteristic
5#20265425C212yesmaximal, complemented
6#20265401C212yesmaximal, complemented
7#20265395C212yesmaximal, complemented
8#20265200C211yesmaximal, characteristic, complemented
9#470×C101C211yesmaximal, characteristic, complemented
10C2×#13108061C2212yes—
11C2×D808C2211yescharacteristic
12C2×#13108058C2211yescharacteristic
13#15692250—22no—
14#15692246C2212yes—
15#15692244C2212yes—
16#15692239—22no—
17#15692229C2212yes—
18Q8×Dic101C2212yes—
19#15692217C2212yes—
20#15692215—22no—
21#15692213—22no—
22C2×C4×Dic101C2211yescharacteristic
23#15692188C2212yes—
24#15692187C2211yescharacteristic
25#15692184C2212yes—
26#15692184C2212yes—
27#15692179C2212yes—
28#15692177C2212yes—
29#15692175C2212yes—
30C2×Q16×C101C2211yescharacteristic
31C2×M16×C101C2211yescharacteristic
32C22⋊Q8×C101C2212yes—
33#106×C101C2211yescharacteristic
34#103×C101C2212yes—
35C2×D404C2312yes—
36C2×C22⋊D101—22no—
37C22×Dic101D412yes—
38C2×C4×D101—22no—
39#13108061—42no—
40D808—41no—
41#13108058—41no—
42C2×#11717579C2312yes—
43C4⋊Dic101—22no—
44C4⋊Dic101—22no—
45C4⋊Dic101C2312yes—
46#13108053—22no—
47#13108051—22no—
48#13108051—22no—
49#13108051—22no—
50#13108051—22no—
51C4×Dic101—22no—
52C4×Dic101D411yescharacteristic
53C4×Dic101D411yescharacteristic
54C4×Dic101C2312yes—
55C2×Q8×C101C2312yes—
56C22×C4×C101C2311yescharacteristic
57Q16×C101—41no—
58M16×C101—41no—
59C2×C8×C101C2311yescharacteristic
60C2×C8×C101C2311yescharacteristic
61C4⋊C4×C101—22no—
62C4⋊C4×C101C2312yes—
63C22⋊C4×C101—22no—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66Q8×C101—22no—
67Q8×C101—22no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2411yescharacteristic
74C22⋊D101—44no—
75C2×Dic101—22no—
76C2×Dic101—42no—
77C2×Dic101C2×D412yes—
78C2×Dic101C2×D412yes—
79D404—22no—
80D404—22no—
81C4×D101—44no—
82C8×C101—21no—
83C8×C101—21no—
84#11717579—22no—
85C22×C101—21no—
86C22×C101—22no—
87C22×C101C22×D411yescharacteristic
88C2×D101—42no—
89C2×D101—42no—
90C2×D101—42no—
91C4×C101—21no—
92C4×C101—42no—
93C4×C101—42no—
94C4×C101H(5,𝔽2)11yescharacteristic
95C4×C1012−1+411yescharacteristic
96Dic101—22no—
97Dic101—22no—
98Dic101—42no—
99C2×C101—41no—
100C2×C101#53711yescharacteristic
101C2×C101#57512yes—
102D101—82no—
103#3256—1011nomaximal, Sylow
104C101#325611yescharacteristic, complemented, Sylow
105#529—1012no—
106#479—1012no—
107#478—1012no—
108#470D10111yescharacteristic, complemented
109#469—1011no—
110#468—1012no—
111#464—1012no—
112#463—1012no—
113#423—1011no—
114C2×Q16C2×D10111yescharacteristic
115C2×SD16—1012no—
116C2×D8—1011no—
117C2×M16—1011no—
118C2×M16C2×D10111yescharacteristic
119#126—2022no—
120#125—1012no—
121#124—1012no—
122#123—2022no—
123C22⋊Q8C2×D10112yes—
124#121—1012no—
125C4×Q8—1012no—
126C4×D4—1012no—
127C4×D4—2022no—
128#117—2022no—
129C2×C42—1011no—
130#107—1012no—
131#106—1011no—
132#106C2×D10111yescharacteristic
133C4⋊C8—1012no—
134C4⋊C8—1012no—
135#103—1012no—
136#103C2×D10112yes—
137#102—1012no—
138#102—1012no—
139C2×Q8C22×D10112yes—
140C2×D4—1012no—
141C2×D4—2022no—
142C22×C4—1012no—
143C22×C4—2022no—
144C22×C4C22×D10111yescharacteristic
145Q16—41no—
146SD16—4042no—
147D8—4041no—
148M16—41no—
149M16—4041no—
150C2×C8—1012no—
151C2×C8C22×D10111yescharacteristic
152C2×C8C22×D10111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—1012no—
155C4⋊C4—2022no—
156C4⋊C4—2022no—
157C4⋊C4—2022no—
158C4⋊C4C22×D10112yes—
159C22⋊C4—22no—
160C22⋊C4—2022no—
161C22⋊C4—2022no—
162C22⋊C4—2022no—
163C22⋊C4—2022no—
164C42—1011no—
165C42—1011no—
166C42—1012no—
167C42—2022no—
168C23—2022no—
169C23D4×D10111yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—4044no—
173D4—2022no—
174D4—2022no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—1012no—
180C2×C4—1012no—
181C2×C4—4044no—
182C2×C4—4042no—
183C2×C4—2022no—
184C2×C4D4×D10111yescharacteristic
185C2×C4C23×D10111yescharacteristic
186C8—21no—
187C8—21no—
188C8—2022no—
189C22—21no—
190C22—22no—
191C22—4042no—
192C22—4042no—
193C22—4042no—
194C22C2×D4×D10111yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—4042no—
199C4—2022no—
200C4—2022no—
201C4#1569230011yescharacteristic
202C4#1569230211yescharacteristic
203C2—41no—
204C2—8082no—
205C2#2026575411yescharacteristic
206C2#2026597612yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#470×C101C2
socleC22×C101C22×D4
prime radical#20265430C2
hypercenter#470D101
hypocenterC101#3256
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #20265713C2semidirect
2. #20265482C2semidirect
3. #20265476C2semidirect
5. #20265425C2semidirect
6. #20265401C2semidirect
7. #20265395C2semidirect
8. #20265200C2semidirect
9. #470×C101C2semidirect
104. C101#3256semidirect
108. #470D101semidirect

Maximal subgroups

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

subgroupindex
#202657132
#202654822
#202654762
#202654302
#202654252
#202654012
#202653952
#202652002
#470×C1012
#3256101