Finite Group Databasebeta

← #37510970

Subgroups — #37510970

Order 22912.

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#26039240C211yesmaximal, characteristic
2#26038880C211yesmaximal, characteristic, complemented
3#26038711C211yesmaximal, characteristic, complemented
4#26038252C211yesmaximal, characteristic, complemented
5#26038247C211yesmaximal, characteristic, complemented
6#26038222C211yesmaximal, characteristic, complemented
7C2×SD32×C179C211yesmaximal, characteristic, complemented
8#19153158—21no—
9#19153144—21no—
10#19153139—21no—
11#19153137—22no—
12#19153135C2211yescharacteristic
13#19153132—21no—
14D8×D179—21no—
15#19153119C2211yescharacteristic
16#19153059—21no—
17#19153056—21no—
18#19153053C2211yescharacteristic, complemented
19#19153049C2211yescharacteristic, complemented
20#19153032—21no—
21C2×Q16×C179C2211yescharacteristic, complemented
22C2×D8×C179C2211yescharacteristic
23SD32×C179—22no—
24C2×C16×C179C2211yescharacteristic, complemented
25#15668259—21no—
26#15668259D411yescharacteristic
27#15668258—22no—
28#15668258—41no—
29C2×C22⋊D179—42no—
30D4×D179—22no—
31D4×D179—41no—
32Dic716D411yescharacteristic
33#15668249—41no—
34#15668248—21no—
35#15668247—22no—
36#15668246—41no—
37D1432D411yescharacteristic
38#15668244—22no—
39#15668243—21no—
40#15668243D411yescharacteristic
41#15668242—21no—
42C8×D179—21no—
43C179⋊C16—21no—
44C2×Q8×C179—21no—
45C2×D4×C179—21no—
46Q16×C179—21no—
47Q16×C179—21no—
48D8×C179—21no—
49D8×C179D412yes—
50C2×C8×C179C2311yescharacteristic
51C16×C179—21no—
52C22×D179—42no—
53Dic358—21no—
54Dic358D811yescharacteristic
55D716—21no—
56D716D811yescharacteristic
57C22⋊D179—21no—
58C22⋊D179—42no—
59C22⋊D179—42no—
60C22⋊D179—41no—
61C2×Dic179—42no—
62C4×D179—21no—
63C4×D179—41no—
64C179⋊C8—21no—
65C179⋊C8—41no—
66C23×C179—41no—
67Q8×C179—41no—
68Q8×C179—41no—
69D4×C179—22no—
70D4×C179—41no—
71C2×C4×C179—41no—
72C2×C4×C179C2×D411yescharacteristic
73C8×C179C2×D411yescharacteristic
74C8×C179C2×D411yescharacteristic
75C22×C179—42no—
76C22×C179—161no—
77C22×C179AGL(1,ℤ/8)11yescharacteristic
78C2×D179—21no—
79C2×D179—41no—
80C2×D179—162no—
81C4×C179—81no—
82C4×C179#12011yescharacteristic
83C4×C179C2×D811yescharacteristic
84Dic179—21no—
85Dic179—41no—
86C2×C179—21no—
87C2×C179—82no—
88C2×C179#44711yescharacteristic
89D179—41no—
90D179—81no—
91C179#219411yescharacteristic, complemented, Sylow
92#2194—1791nomaximal, Sylow
93#576—1791no—
94#510—1791no—
95C2×SD32D17911yescharacteristic, complemented
96#473—1791no—
97#361—1791no—
98#359—1791no—
99#350—1791no—
100H(5,𝔽2)—3581no—
101#137—3581no—
102AGL(1,ℤ/8)—3582no—
103AGL(1,ℤ/8)—3581no—
104#135—3581no—
105#135—1791no—
106C2×Q16C2×D17911yescharacteristic, complemented
107C2×D8—3581no—
108C2×D8C2×D17911yescharacteristic
109#131—1791no—
110Q32—3581no—
111SD32—22no—
112SD32—3581no—
113M32—1791no—
114C2×C16C2×D17911yescharacteristic, complemented
115#108—1791no—
116#101—3581no—
117#41—3582no—
118#41—3581no—
119#41—1791no—
120#41—7161no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—3582no—
124C2×D4—7161no—
125C2×D4—7162no—
126Q16—21no—
127Q16—21no—
128Q16—1791no—
129Q16—7161no—
130SD16—3582no—
131SD16—3581no—
132SD16—7161no—
133D8—21no—
134D8—3582no—
135D8—1791no—
136D8C22⋊D17912yes—
137M16—3581no—
138M16—3581no—
139M16—1791no—
140C2×C8—3581no—
141C2×C8C22×D17911yescharacteristic
142C16—21no—
143C16—3581no—
144C23—41no—
145C23—7162no—
146Q8—41no—
147Q8—41no—
148Q8—3581no—
149Q8—1791no—
150D4—22no—
151D4—41no—
152D4—3581no—
153D4—3581no—
154D4—1791no—
155D4—7162no—
156D4—7162no—
157D4—7161no—
158C2×C4—41no—
159C2×C4—3581no—
160C2×C4—7162no—
161C2×C4—7161no—
162C2×C4D4×D17911yescharacteristic
163C8—3581no—
164C8—7161no—
165C8D4×D17911yescharacteristic
166C8C2×C22⋊D17911yescharacteristic
167C22—42no—
168C22—161no—
169C22—28642no—
170C22—3581no—
171C22—7161no—
172C22#1915313711yescharacteristic
173C4—81no—
174C4—3581no—
175C4—7161no—
176C4#1915308111yescharacteristic
177C4D8×D17911yescharacteristic
178C2—21no—
179C2—82no—
180C2—14321no—
181C2—7161no—
182C2#2603857111yescharacteristic
subgroupG/H
centerC2#26038571
commutator, cosocleC2×C8×C179C23
FrattiniC2×C8C22×D179
FittingC2×SD32×C179C2
socleC2×C179#447
prime radical#26039240C2
hypercenterC2×SD32D179
hypocenterC179#2194
LCSG ▷ C2×C8×C179 ▷ C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #26038880C2semidirect
3. #26038711C2semidirect
4. #26038252C2semidirect
5. #26038247C2semidirect
6. #26038222C2semidirect
7. C2×SD32×C179C2semidirect
18. #19153053C22semidirect
19. #19153049C22semidirect
21. C2×Q16×C179C22semidirect
24. C2×C16×C179C22semidirect
91. C179#2194semidirect
95. C2×SD32D179semidirect
106. C2×Q16C2×D179semidirect
114. C2×C16C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260392402
#260388802
#260387112
#260382522
#260382472
#260382222
C2×SD32×C1792
#2194179