Finite Group Databasebeta

← #36355519

Subgroups — #36355519

Order 22144.

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

normal subgroups only

This group has 220 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#26012292C211yesmaximal, characteristic, complemented
2#26011927C211yesmaximal, characteristic, complemented
3#26011758C211yesmaximal, characteristic, complemented
4#26011304C211yesmaximal, characteristic, complemented
5#26011299C211yesmaximal, characteristic, complemented
6#26011274C211yesmaximal, characteristic, complemented
7C2×D16×C173C211yesmaximal, characteristic, complemented
8#19149724—21no—
9#19149705—21no—
10#19149705—21no—
11#19149703—22no—
12#19149701C2211yescharacteristic
13#19149698—21no—
14D8×D173—21no—
15#19149685C2211yescharacteristic
16#19149623—21no—
17#19149620—21no—
18#19149619C2211yescharacteristic, complemented
19#19149615C2211yescharacteristic, complemented
20#19149596—21no—
21C2×D8×C173C2211yescharacteristic
22C2×D8×C173C2211yescharacteristic, complemented
23D16×C173—22no—
24C2×C16×C173C2211yescharacteristic, complemented
25#14583826—21no—
26#14583826D411yescharacteristic
27#14583825—22no—
28#14583825—41no—
29C2×C22⋊D173—42no—
30D4×D173—22no—
31D4×D173—41no—
32Dic692D411yescharacteristic
33#14583815—21no—
34#14583814—22no—
35#14583814—41no—
36D1384D411yescharacteristic
37#14583811—22no—
38#14583811—41no—
39#14583810—21no—
40#14583810D411yescharacteristic
41#14583809—21no—
42C8×D173—21no—
43#14583801—21no—
44C2×D4×C173—21no—
45C2×D4×C173—21no—
46D8×C173—21no—
47D8×C173—21no—
48D8×C173—21no—
49D8×C173D412yes—
50C2×C8×C173C2311yescharacteristic
51C16×C173—21no—
52C22×D173—42no—
53Dic346—21no—
54Dic346D811yescharacteristic
55D692—21no—
56D692D811yescharacteristic
57C22⋊D173—21no—
58C22⋊D173—42no—
59C22⋊D173—42no—
60C22⋊D173—41no—
61C2×Dic173—42no—
62C4×D173—21no—
63C4×D173—41no—
64#13063436—21no—
65#13063436—41no—
66C23×C173—41no—
67C23×C173—41no—
68D4×C173—22no—
69D4×C173—41no—
70D4×C173—41no—
71D4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C8×C173C2×D411yescharacteristic
74C8×C173C2×D411yescharacteristic
75C22×C173—42no—
76C22×C173—81no—
77C22×C173—161no—
78C22×C173—161no—
79C22×C173AGL(1,ℤ/8)11yescharacteristic
80C2×D173—21no—
81C2×D173—41no—
82C2×D173—162no—
83C4×C173#12011yescharacteristic
84C4×C173C2×D811yescharacteristic
85Dic173—21no—
86Dic173—41no—
87C2×C173—21no—
88C2×C173—82no—
89C2×C173—161no—
90C2×C173#44711yescharacteristic
91D173—41no—
92D173—81no—
93C173#219211yescharacteristic, complemented, Sylow
94#2192—1731nomaximal, Sylow
95#576—1731no—
96#509—1731no—
97C2×D16D17311yescharacteristic, complemented
98#472—1731no—
99#361—1731no—
100#359—1731no—
101#350—1731no—
102H(5,𝔽2)—3461no—
103AGL(1,ℤ/8)—3462no—
104AGL(1,ℤ/8)—3461no—
105AGL(1,ℤ/8)—3461no—
106#135—3461no—
107#135—1731no—
108C2×D8—3461no—
109C2×D8C2×D17311yescharacteristic
110C2×D8C2×D17311yescharacteristic, complemented
111#131—1731no—
112SD32—3461no—
113D16—22no—
114D16—3461no—
115M32—1731no—
116C2×C16C2×D17311yescharacteristic, complemented
117#108—1731no—
118#100—3461no—
119#41—3462no—
120#41—3461no—
121#41—1731no—
122#41—6921no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—3462no—
126C2×D4—6921no—
127C2×D4—6922no—
128Q16—1731no—
129SD16—3462no—
130SD16—3461no—
131SD16—6921no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—3462no—
136D8—1731no—
137D8—6921no—
138D8C22⋊D17312yes—
139M16—3461no—
140M16—3461no—
141M16—1731no—
142C2×C8—3461no—
143C2×C8C22×D17311yescharacteristic
144C16—21no—
145C16—3461no—
146C23—41no—
147C23—41no—
148C23—6922no—
149Q8—3461no—
150Q8—1731no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—3461no—
156D4—3461no—
157D4—1731no—
158D4—6922no—
159D4—6922no—
160D4—6921no—
161C2×C4—3461no—
162C2×C4—6922no—
163C2×C4—6921no—
164C2×C4D4×D17311yescharacteristic
165C8—3461no—
166C8—6921no—
167C8D4×D17311yescharacteristic
168C8C2×C22⋊D17311yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—3461no—
174C22—6921no—
175C22—27682no—
176C22#1914970311yescharacteristic
177C4—3461no—
178C4—6921no—
179C4#1914964711yescharacteristic
180C4D8×D17311yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—13841no—
185C2—6921no—
186C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
FittingC2×D16×C173C2
socleC2×C173#447
hypercenterC2×D16D173
hypocenterC173#2192
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #26012292C2semidirect
2. #26011927C2semidirect
3. #26011758C2semidirect
4. #26011304C2semidirect
5. #26011299C2semidirect
6. #26011274C2semidirect
7. C2×D16×C173C2semidirect
18. #19149619C22semidirect
19. #19149615C22semidirect
22. C2×D8×C173C22semidirect
24. C2×C16×C173C22semidirect
93. C173#2192semidirect
97. C2×D16D173semidirect
110. C2×D8C2×D173semidirect
116. C2×C16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122922
#260119272
#260117582
#260113042
#260112992
#260112742
C2×D16×C1732
#2192173