Finite Group Databasebeta

← #27173105

Subgroups — #27173105

Order 12928.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265982C211yesmaximal, characteristic
2#20265620C211yesmaximal, characteristic, complemented
3C2×#15692200C211yesmaximal, characteristic, complemented
4#20265452C211yesmaximal, characteristic, complemented
5#20264991C211yesmaximal, characteristic, complemented
6#20264985C211yesmaximal, characteristic, complemented
7#350×C101C211yesmaximal, characteristic, complemented
8#15692301—21no—
9#15692285—21no—
10#15692283—22no—
11#15692283—21no—
12#15692277—21no—
13#15692277C2211yescharacteristic
14C2×Dic404C2211yescharacteristic, complemented
15C2×D808—21no—
16C2×D808C2211yescharacteristic
17Dic808—21no—
18#15692200—22no—
19#15692200—21no—
20#15692191C2211yescharacteristic, complemented
21#15692173—21no—
22#131×C101C2211yescharacteristic
23M32×C101C2211yescharacteristic, complemented
24C2×C16×C101C2211yescharacteristic, complemented
25#13108075—41no—
26#13108074—22no—
27#13108074—21no—
28C2×Dic202—21no—
29C2×D404—22no—
30C2×D404—21no—
31D4×D101—42no—
32D4×D101—41no—
33Dic404—21no—
34Dic404—21no—
35Dic404D411yescharacteristic
36#13108064—41no—
37#13108063—22no—
38#13108063—21no—
39#13108062—41no—
40D808—22no—
41D808—21no—
42D808D411yescharacteristic
43D808D412yes—
44#13108058—21no—
45#41×C101D411yescharacteristic
46M16×C101—21no—
47M16×C101D411yescharacteristic
48C2×C8×C101—21no—
49C2×C8×C101C2311yescharacteristic
50C16×C101—21no—
51C16×C101—21no—
52C22×D101—42no—
53C22×D101—41no—
54Q8×C101D811yescharacteristic
55D4×C101—21no—
56D4×C101D811yescharacteristic
57C2×C4×C101—21no—
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—42no—
60C22⋊D101—41no—
61C2×Dic101—41no—
62D404—21no—
63D404—22no—
64D404—42no—
65D404—41no—
66C4×D101—42no—
67C4×D101—41no—
68Dic202—21no—
69Dic202—41no—
70Dic202—41no—
71C8×C101—21no—
72C8×C101C2×D411yescharacteristic
73C8×C101C2×D411yescharacteristic
74#11717579—41no—
75C22×C101—21no—
76C22×C101AGL(1,ℤ/8)11yescharacteristic
77C2×D101—42no—
78C2×D101—41no—
79C2×D101—161no—
80C2×D101—162no—
81C4×C101—21no—
82C4×C101#12011yescharacteristic
83C4×C101C2×D811yescharacteristic
84Dic101—41no—
85Dic101—81no—
86C2×C101—21no—
87C2×C101—41no—
88C2×C101#44711yescharacteristic
89D101—82no—
90D101—81no—
91#2194—1011nomaximal, Sylow
92C101#219411yescharacteristic, complemented, Sylow
93#576—1011no—
94#510—1011no—
95C2×SD32—1011no—
96#473—1011no—
97#361—1011no—
98#359—1011no—
99#350D10111yescharacteristic, complemented
100H(5,𝔽2)—2021no—
101#137—2021no—
102AGL(1,ℤ/8)—2022no—
103AGL(1,ℤ/8)—2021no—
104#135—1011no—
105#135—2021no—
106C2×Q16—1011no—
107C2×D8—1011no—
108C2×D8—2021no—
109#131C2×D10111yescharacteristic
110Q32—2021no—
111SD32—2022no—
112SD32—2021no—
113M32C2×D10111yescharacteristic, complemented
114C2×C16C2×D10111yescharacteristic, complemented
115#108—1011no—
116#101—2021no—
117#41—4041no—
118#41—2022no—
119#41—2021no—
120#41D40411yescharacteristic
121C2×Q8—2021no—
122C2×D4—4042no—
123C2×D4—4041no—
124C2×D4—2022no—
125C2×D4—2021no—
126Q16—1011no—
127Q16—4041no—
128Q16—2021no—
129Q16—2021no—
130SD16—4041no—
131SD16—2022no—
132SD16—2021no—
133D8—1012no—
134D8—1011no—
135D8—2022no—
136D8—2021no—
137M16—21no—
138M16—2021no—
139M16D40411yescharacteristic
140C2×C8—21no—
141C2×C8C22×D10111yescharacteristic
142C16—21no—
143C16—21no—
144C23—4042no—
145C23—4041no—
146Q8—4041no—
147Q8—4041no—
148Q8—2021no—
149Q8D80811yescharacteristic
150D4—21no—
151D4—4042no—
152D4—4042no—
153D4—4041no—
154D4—4041no—
155D4—2022no—
156D4—2021no—
157D4D80811yescharacteristic
158C2×C4—21no—
159C2×C4—4042no—
160C2×C4—4041no—
161C2×C4—4041no—
162C2×C4C2×D40411yescharacteristic
163C8—21no—
164C8—4041no—
165C8D4×D10111yescharacteristic
166C8D4×D10111yescharacteristic
167C22—21no—
168C22—16161no—
169C22—16162no—
170C22—4042no—
171C22—4041no—
172C22#1569228311yescharacteristic
173C4—21no—
174C4—8081no—
175C4—4041no—
176C4#1569222511yescharacteristic
177C4C2×D80811yescharacteristic
178C2—21no—
179C2—41no—
180C2—8082no—
181C2—8081no—
182C2#2026531311yescharacteristic
subgroupG/H
centerC2#20265313
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#350×C101C2
socleC2×C101#447
prime radical#20265982C2
hypercenter#350D101
hypocenterC101#2194
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #20265620C2semidirect
3. C2×#15692200C2semidirect
4. #20265452C2semidirect
5. #20264991C2semidirect
6. #20264985C2semidirect
7. #350×C101C2semidirect
14. C2×Dic404C22semidirect
20. #15692191C22semidirect
23. M32×C101C22semidirect
24. C2×C16×C101C22semidirect
92. C101#2194semidirect
99. #350D101semidirect
113. M32C2×D101semidirect
114. C2×C16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659822
#202656202
C2×#156922002
#202654522
#202649912
#202649852
#350×C1012
#2194101