Finite Group Databasebeta

← #36355512

Subgroups — #36355512

Order 22144.

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#26012288C211yesmaximal, characteristic
2#26011932C211yesmaximal, characteristic, complemented
3#26011746C211yesmaximal, characteristic, complemented
4#26011302C211yesmaximal, characteristic, complemented
5#26011301C211yesmaximal, characteristic, complemented
6#26011272C211yesmaximal, characteristic, complemented
7#510×C173C211yesmaximal, characteristic, complemented
8#41×D173—21no—
9#19149711—21no—
10#19149708—21no—
11#19149706—21no—
12#19149703—21no—
13#19149701C2211yescharacteristic
14#19149699—21no—
15C2×#14583809C2211yescharacteristic
16#19149655—21no—
17#19149625—21no—
18#19149622—21no—
19#19149619C2211yescharacteristic, complemented
20#19149611C2211yescharacteristic, complemented
21#19149603—21no—
22#135×C173C2211yescharacteristic
23C2×Q16×C173C2211yescharacteristic, complemented
24Q32×C173—21no—
25SD32×C173—21no—
26M32×C173C2211yescharacteristic, complemented
27#14583827—21no—
28#14583826—21no—
29#14583826—41no—
30#14583825—21no—
31#14583825—41no—
32Q8×D173—21no—
33D4×D173—21no—
34D4×D173—41no—
35C2×C4×D173—41no—
36C2×C4×D173D411yescharacteristic
37Dic692D411yescharacteristic
38#14583816—21no—
39#14583815—21no—
40#14583814—21no—
41#14583813—21no—
42D1384D411yescharacteristic
43#14583811—21no—
44#14583809—21no—
45#14583809—21no—
46C2×#13063436D411yescharacteristic
47C4⋊Dic173—21no—
48#14583805—41no—
49#14583803—41no—
50#14583801—21no—
51#41×C173—21no—
52C2×Q8×C173—21no—
53Q16×C173—21no—
54Q16×C173—21no—
55Q16×C173D411yescharacteristic
56SD16×C173—21no—
57D8×C173D411yescharacteristic
58C2×C8×C173C2311yescharacteristic
59C16×C173—21no—
60C22×D173—41no—
61C22×D173D811yescharacteristic
62Dic346—21no—
63Dic346—41no—
64D692—21no—
65D692—41no—
66C22⋊D173—41no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173D811yescharacteristic
72C4×D173—21no—
73C4×D173—21no—
74C4×D173—41no—
75C4×D173—41no—
76C4×D173—41no—
77#13063436—21no—
78Q8×C173—21no—
79Q8×C173—41no—
80Q8×C173—41no—
81D4×C173—21no—
82D4×C173—41no—
83C2×C4×C173—41no—
84C2×C4×C173—41no—
85C2×C4×C173C2×D411yescharacteristic
86C8×C173C2×D411yescharacteristic
87C8×C173C2×D411yescharacteristic
88C22×C173—41no—
89C22×C173C2×D811yescharacteristic
90C2×D173—21no—
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—82no—
94C4×C173—41no—
95C4×C173—81no—
96C4×C173#12011yescharacteristic
97C4×C173AGL(1,ℤ/8)11yescharacteristic
98Dic173—21no—
99Dic173—41no—
100Dic173—81no—
101C2×C173—21no—
102C2×C173—81no—
103C2×C173#44711yescharacteristic
104D173—41no—
105D173—81no—
106C173#219111yescharacteristic, complemented, Sylow
107#2191—1731nomaximal, Sylow
108#575—1731no—
109#510—1731no—
110#510D17311yescharacteristic, complemented
111#470—1731no—
112#360—1731no—
113#360—1731no—
114#349—1731no—
115C2×#41—3461no—
116#137—3461no—
117#137—3461no—
118AGL(1,ℤ/8)—3461no—
119AGL(1,ℤ/8)—3461no—
120#135—3461no—
121#135—1731no—
122#135C2×D17311yescharacteristic
123C2×Q16C2×D17311yescharacteristic, complemented
124C2×M16—1731no—
125C22⋊Q8—3461no—
126Q32—21no—
127Q32—3461no—
128SD32—21no—
129SD32—3461no—
130M32—1731no—
131M32C2×D17311yescharacteristic, complemented
132#106—1731no—
133#103—3461no—
134#41—21no—
135#41—3461no—
136#41—3461no—
137#41—3461no—
138#41—6921no—
139#41—6921no—
140C2×Q8—21no—
141C2×Q8—3461no—
142C2×D4—3461no—
143C2×D4—6921no—
144C22×C4—1731no—
145C22×C4—6921no—
146Q16—21no—
147Q16—21no—
148Q16—3461no—
149Q16—1731no—
150Q16C22⋊D17311yescharacteristic
151SD16—21no—
152SD16—3461no—
153SD16—3461no—
154SD16—3461no—
155D8—3461no—
156D8—1731no—
157D8C22⋊D17311yescharacteristic
158M16—3461no—
159M16—3461no—
160C2×C8—1731no—
161C2×C8C22×D17311yescharacteristic
162C4⋊C4—3461no—
163C4⋊C4—6921no—
164C22⋊C4—6921no—
165C16—21no—
166C16—3461no—
167C23—1731no—
168C23—6921no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—3461no—
173Q8—6921no—
174D4—21no—
175D4—41no—
176D4—3461no—
177D4—6921no—
178D4—6921no—
179D4—6921no—
180D4—6921no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—1731no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4—6921no—
191C2×C4D4×D17311yescharacteristic
192C8—3461no—
193C8D4×D17311yescharacteristic
194C8C2×C22⋊D17311yescharacteristic
195C22—41no—
196C22—3461no—
197C22—13842no—
198C22—6921no—
199C22—6921no—
200C22D8×D17311yescharacteristic
201C4—41no—
202C4—81no—
203C4—3461no—
204C4—13841no—
205C4—6921no—
206C4#1914964711yescharacteristic
207C4#1914970311yescharacteristic
208C2—21no—
209C2—81no—
210C2—13841no—
211C2—6921no—
212C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#510×C173C2
socleC2×C173#447
prime radical#26012288C2
hypercenter#510D173
hypocenterC173#2191
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #26011932C2semidirect
3. #26011746C2semidirect
4. #26011302C2semidirect
5. #26011301C2semidirect
6. #26011272C2semidirect
7. #510×C173C2semidirect
19. #19149619C22semidirect
20. #19149611C22semidirect
23. C2×Q16×C173C22semidirect
26. M32×C173C22semidirect
106. C173#2191semidirect
110. #510D173semidirect
123. C2×Q16C2×D173semidirect
131. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122882
#260119322
#260117462
#260113022
#260113012
#260112722
#510×C1732
#2191173