Finite Group Databasebeta

← #27173104

Subgroups — #27173104

Order 12928.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#20265979C211yesmaximal, characteristic
2#20265617C211yesmaximal, characteristic, complemented
3C2×#15692200C211yesmaximal, characteristic, complemented
4#20265453C211yesmaximal, characteristic, complemented
5#20264989C211yesmaximal, characteristic, complemented
6#20264960C211yesmaximal, characteristic, complemented
7#359×C101C211yesmaximal, characteristic, complemented
8#15692301—21no—
9#15692288—21no—
10#15692282—22no—
11#15692280—21no—
12#15692278—21no—
13C2×Dic404C2211yescharacteristic, complemented
14C2×D808C2211yescharacteristic
15D8×D101—21no—
16#15692261C2211yescharacteristic
17#15692201—21no—
18#15692200—22no—
19#15692199—21no—
20#15692195C2211yescharacteristic, complemented
21#15692173—21no—
22#135×C101C2211yescharacteristic
23C2×C16×C101C2211yescharacteristic, complemented
24#108×C101C2211yescharacteristic, complemented
25#13108075—22no—
26#13108074—41no—
27#13108074D411yescharacteristic
28C2×Dic202—21no—
29C2×D404—21no—
30C2×D404—41no—
31D4×D101—22no—
32D4×D101—42no—
33Dic404—21no—
34Dic404—21no—
35Dic404—41no—
36#13108063—41no—
37#13108061—22no—
38D808—21no—
39D808D412yes—
40#13108059—22no—
41#13108058D411yescharacteristic
42#13108057—21no—
43C8×D101—21no—
44#13108049—21no—
45#41×C101—21no—
46Q16×C101D411yescharacteristic
47SD16×C101—21no—
48D8×C101D411yescharacteristic
49M16×C101—21no—
50C2×C8×C101C2311yescharacteristic
51C16×C101—21no—
52C22×D101—42no—
53C22×D101—41no—
54Q8×C101—21no—
55D4×C101—21no—
56D4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—21no—
60C22⋊D101—42no—
61C2×Dic101—41no—
62D404—22no—
63D404—42no—
64D404—41no—
65D404D811yescharacteristic
66C4×D101—21no—
67C4×D101—42no—
68Dic202—41no—
69Dic202—41no—
70Dic202D811yescharacteristic
71C8×C101—41no—
72C8×C101C2×D411yescharacteristic
73C8×C101C2×D411yescharacteristic
74#11717579—21no—
75C22×C101—41no—
76C22×C101AGL(1,ℤ/8)11yescharacteristic
77C2×D101—21no—
78C2×D101—42no—
79C2×D101—162no—
80C2×D101—161no—
81C4×C101—41no—
82C4×C101#12011yescharacteristic
83C4×C101C2×D811yescharacteristic
84Dic101—21no—
85Dic101—81no—
86C2×C101—21no—
87C2×C101—81no—
88C2×C101#44711yescharacteristic
89D101—41no—
90D101—82no—
91#2194—1011nomaximal, Sylow
92C101#219411yescharacteristic, complemented, Sylow
93#576—1011no—
94#510—1011no—
95C2×SD32—1011no—
96#473—1011no—
97#361—1011no—
98#359D10111yescharacteristic, complemented
99#350—1011no—
100H(5,𝔽2)—2021no—
101#137—2021no—
102AGL(1,ℤ/8)—2021no—
103AGL(1,ℤ/8)—2022no—
104#135—2021no—
105#135C2×D10111yescharacteristic
106C2×Q16—1011no—
107C2×D8—1011no—
108C2×D8—2021no—
109#131—1011no—
110Q32—2021no—
111SD32—2022no—
112SD32—2021no—
113M32—1011no—
114C2×C16C2×D10111yescharacteristic, complemented
115#108C2×D10111yescharacteristic, complemented
116#101—2021no—
117#41—21no—
118#41—1011no—
119#41—4041no—
120#41—2022no—
121C2×Q8—2021no—
122C2×D4—4041no—
123C2×D4—4042no—
124C2×D4—2022no—
125C2×D4—2021no—
126Q16—4041no—
127Q16—2021no—
128Q16—2021no—
129Q16D40411yescharacteristic
130SD16—21no—
131SD16—4041no—
132SD16—2022no—
133D8—1012no—
134D8—2022no—
135D8—2021no—
136D8D40411yescharacteristic
137M16—21no—
138M16—1011no—
139M16—2021no—
140C2×C8—2021no—
141C2×C8C22×D10111yescharacteristic
142C16—21no—
143C16—2021no—
144C23—4042no—
145C23—4041no—
146Q8—21no—
147Q8—1011no—
148Q8—4041no—
149Q8—4041no—
150D4—21no—
151D4—41no—
152D4—1011no—
153D4—4042no—
154D4—4042no—
155D4—4041no—
156D4—2021no—
157D4—2022no—
158C2×C4—41no—
159C2×C4—4042no—
160C2×C4—4041no—
161C2×C4—2021no—
162C2×C4D4×D10111yescharacteristic
163C8—41no—
164C8—2021no—
165C8D4×D10111yescharacteristic
166C8C2×D40411yescharacteristic
167C22—41no—
168C22—16161no—
169C22—16162no—
170C22—4042no—
171C22—2021no—
172C22#1569228211yescharacteristic
173C4—41no—
174C4—8081no—
175C4—2021no—
176C4#1569222511yescharacteristic
177C4D8×D10111yescharacteristic
178C2—21no—
179C2—81no—
180C2—8082no—
181C2—4041no—
182C2#2026531211yescharacteristic
subgroupG/H
centerC2#20265312
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#359×C101C2
socleC2×C101#447
prime radical#20265979C2
hypercenter#359D101
hypocenterC101#2194
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #20265617C2semidirect
3. C2×#15692200C2semidirect
4. #20265453C2semidirect
5. #20264989C2semidirect
6. #20264960C2semidirect
7. #359×C101C2semidirect
13. C2×Dic404C22semidirect
20. #15692195C22semidirect
23. C2×C16×C101C22semidirect
24. #108×C101C22semidirect
92. C101#2194semidirect
98. #359D101semidirect
114. C2×C16C2×D101semidirect
115. #108C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659792
#202656172
C2×#156922002
#202654532
#202649892
#202649602
#359×C1012
#2194101