Finite Group Databasebeta

← #37510949

Subgroups — #37510949

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#26039236C211yesmaximal, characteristic
2#26038880C211yesmaximal, characteristic, complemented
3#26038694C211yesmaximal, characteristic, complemented
4#26038250C211yesmaximal, characteristic, complemented
5#26038249C211yesmaximal, characteristic, complemented
6#26038220C211yesmaximal, characteristic, complemented
7#510×C179C211yesmaximal, characteristic, complemented
8#41×D179—21no—
9#19153145—21no—
10#19153142—21no—
11#19153140—21no—
12#19153137—21no—
13#19153135C2211yescharacteristic
14#19153133—21no—
15C2×#15668242C2211yescharacteristic
16#19153089—21no—
17#19153059—21no—
18#19153056—21no—
19#19153053C2211yescharacteristic, complemented
20#19153045C2211yescharacteristic, complemented
21#19153037—21no—
22#135×C179C2211yescharacteristic
23C2×Q16×C179C2211yescharacteristic, complemented
24Q32×C179—21no—
25SD32×C179—21no—
26M32×C179C2211yescharacteristic, complemented
27#15668260—21no—
28#15668259—21no—
29#15668259—41no—
30#15668258—21no—
31#15668258—41no—
32Q8×D179—21no—
33D4×D179—21no—
34D4×D179—41no—
35C2×C4×D179—41no—
36C2×C4×D179D411yescharacteristic
37Dic716D411yescharacteristic
38#15668249—21no—
39#15668248—21no—
40#15668247—21no—
41#15668246—21no—
42D1432D411yescharacteristic
43#15668244—21no—
44#15668242—21no—
45#15668242—21no—
46C2×C179⋊C8D411yescharacteristic
47C4⋊Dic179—21no—
48Dic179⋊C4—41no—
49(C2×D179)⋊C4—41no—
50C179⋊C16—21no—
51#41×C179—21no—
52C2×Q8×C179—21no—
53Q16×C179—21no—
54Q16×C179—21no—
55Q16×C179D411yescharacteristic
56SD16×C179—21no—
57D8×C179D411yescharacteristic
58C2×C8×C179C2311yescharacteristic
59C16×C179—21no—
60C22×D179—41no—
61C22×D179D811yescharacteristic
62Dic358—21no—
63Dic358—41no—
64D716—21no—
65D716—41no—
66C22⋊D179—41no—
67C22⋊D179—41no—
68C22⋊D179—41no—
69C2×Dic179—41no—
70C2×Dic179—41no—
71C2×Dic179D811yescharacteristic
72C4×D179—21no—
73C4×D179—21no—
74C4×D179—41no—
75C4×D179—41no—
76C4×D179—41no—
77C179⋊C8—21no—
78Q8×C179—21no—
79Q8×C179—41no—
80Q8×C179—41no—
81D4×C179—21no—
82D4×C179—41no—
83C2×C4×C179—41no—
84C2×C4×C179—41no—
85C2×C4×C179C2×D411yescharacteristic
86C8×C179C2×D411yescharacteristic
87C8×C179C2×D411yescharacteristic
88C22×C179—41no—
89C22×C179C2×D811yescharacteristic
90C2×D179—21no—
91C2×D179—41no—
92C2×D179—41no—
93C2×D179—82no—
94C4×C179—41no—
95C4×C179—81no—
96C4×C179#12011yescharacteristic
97C4×C179AGL(1,ℤ/8)11yescharacteristic
98Dic179—21no—
99Dic179—41no—
100Dic179—81no—
101C2×C179—21no—
102C2×C179—81no—
103C2×C179#44711yescharacteristic
104D179—41no—
105D179—81no—
106C179#219111yescharacteristic, complemented, Sylow
107#2191—1791nomaximal, Sylow
108#575—1791no—
109#510—1791no—
110#510D17911yescharacteristic, complemented
111#470—1791no—
112#360—1791no—
113#360—1791no—
114#349—1791no—
115C2×#41—3581no—
116#137—3581no—
117#137—3581no—
118AGL(1,ℤ/8)—3581no—
119AGL(1,ℤ/8)—3581no—
120#135—3581no—
121#135—1791no—
122#135C2×D17911yescharacteristic
123C2×Q16C2×D17911yescharacteristic, complemented
124C2×M16—1791no—
125C22⋊Q8—3581no—
126Q32—21no—
127Q32—3581no—
128SD32—21no—
129SD32—3581no—
130M32—1791no—
131M32C2×D17911yescharacteristic, complemented
132#106—1791no—
133#103—3581no—
134#41—21no—
135#41—3581no—
136#41—3581no—
137#41—3581no—
138#41—7161no—
139#41—7161no—
140C2×Q8—21no—
141C2×Q8—3581no—
142C2×D4—3581no—
143C2×D4—7161no—
144C22×C4—1791no—
145C22×C4—7161no—
146Q16—21no—
147Q16—21no—
148Q16—3581no—
149Q16—1791no—
150Q16C22⋊D17911yescharacteristic
151SD16—21no—
152SD16—3581no—
153SD16—3581no—
154SD16—3581no—
155D8—3581no—
156D8—1791no—
157D8C22⋊D17911yescharacteristic
158M16—3581no—
159M16—3581no—
160C2×C8—1791no—
161C2×C8C22×D17911yescharacteristic
162C4⋊C4—3581no—
163C4⋊C4—7161no—
164C22⋊C4—7161no—
165C16—21no—
166C16—3581no—
167C23—1791no—
168C23—7161no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—3581no—
173Q8—7161no—
174D4—21no—
175D4—41no—
176D4—3581no—
177D4—7161no—
178D4—7161no—
179D4—7161no—
180D4—7161no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3581no—
184C2×C4—3581no—
185C2×C4—1791no—
186C2×C4—7161no—
187C2×C4—7161no—
188C2×C4—7161no—
189C2×C4—7161no—
190C2×C4—7161no—
191C2×C4D4×D17911yescharacteristic
192C8—3581no—
193C8D4×D17911yescharacteristic
194C8C2×C22⋊D17911yescharacteristic
195C22—41no—
196C22—3581no—
197C22—14322no—
198C22—7161no—
199C22—7161no—
200C22D8×D17911yescharacteristic
201C4—41no—
202C4—81no—
203C4—3581no—
204C4—14321no—
205C4—7161no—
206C4#1915308111yescharacteristic
207C4#1915313711yescharacteristic
208C2—21no—
209C2—81no—
210C2—14321no—
211C2—7161no—
212C2#2603857111yescharacteristic
subgroupG/H
centerC2#26038571
commutator, cosocleC2×C8×C179C23
FrattiniC2×C8C22×D179
Fitting#510×C179C2
socleC2×C179#447
prime radical#26039236C2
hypercenter#510D179
hypocenterC179#2191
LCSG ▷ C2×C8×C179 ▷ C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #26038880C2semidirect
3. #26038694C2semidirect
4. #26038250C2semidirect
5. #26038249C2semidirect
6. #26038220C2semidirect
7. #510×C179C2semidirect
19. #19153053C22semidirect
20. #19153045C22semidirect
23. C2×Q16×C179C22semidirect
26. M32×C179C22semidirect
106. C179#2191semidirect
110. #510D179semidirect
123. C2×Q16C2×D179semidirect
131. M32C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260392362
#260388802
#260386942
#260382502
#260382492
#260382202
#510×C1792
#2191179