Finite Group Databasebeta

← #36355533

Subgroups — #36355533

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#26012292C211yesmaximal, characteristic
2#26011932C211yesmaximal, characteristic, complemented
3#26011763C211yesmaximal, characteristic, complemented
4#26011304C211yesmaximal, characteristic, complemented
5#26011299C211yesmaximal, characteristic, complemented
6#26011274C211yesmaximal, characteristic, complemented
7C2×SD32×C173C211yesmaximal, characteristic, complemented
8#19149724—21no—
9#19149710—21no—
10#19149705—21no—
11#19149703—22no—
12#19149701C2211yescharacteristic
13#19149698—21no—
14D8×D173—21no—
15#19149685C2211yescharacteristic
16#19149625—21no—
17#19149622—21no—
18#19149619C2211yescharacteristic, complemented
19#19149615C2211yescharacteristic, complemented
20#19149598—21no—
21C2×Q16×C173C2211yescharacteristic, complemented
22C2×D8×C173C2211yescharacteristic
23SD32×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#14583816—41no—
34#14583815—21no—
35#14583814—22no—
36#14583813—41no—
37D1384D411yescharacteristic
38#14583811—22no—
39#14583810—21no—
40#14583810D411yescharacteristic
41#14583809—21no—
42C8×D173—21no—
43#14583801—21no—
44C2×Q8×C173—21no—
45C2×D4×C173—21no—
46Q16×C173—21no—
47Q16×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—
67Q8×C173—41no—
68Q8×C173—41no—
69D4×C173—22no—
70D4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C8×C173C2×D411yescharacteristic
74C8×C173C2×D411yescharacteristic
75C22×C173—42no—
76C22×C173—161no—
77C22×C173AGL(1,ℤ/8)11yescharacteristic
78C2×D173—21no—
79C2×D173—41no—
80C2×D173—162no—
81C4×C173—81no—
82C4×C173#12011yescharacteristic
83C4×C173C2×D811yescharacteristic
84Dic173—21no—
85Dic173—41no—
86C2×C173—21no—
87C2×C173—82no—
88C2×C173#44711yescharacteristic
89D173—41no—
90D173—81no—
91C173#219411yescharacteristic, complemented, Sylow
92#2194—1731nomaximal, Sylow
93#576—1731no—
94#510—1731no—
95C2×SD32D17311yescharacteristic, complemented
96#473—1731no—
97#361—1731no—
98#359—1731no—
99#350—1731no—
100H(5,𝔽2)—3461no—
101#137—3461no—
102AGL(1,ℤ/8)—3462no—
103AGL(1,ℤ/8)—3461no—
104#135—3461no—
105#135—1731no—
106C2×Q16C2×D17311yescharacteristic, complemented
107C2×D8—3461no—
108C2×D8C2×D17311yescharacteristic
109#131—1731no—
110Q32—3461no—
111SD32—22no—
112SD32—3461no—
113M32—1731no—
114C2×C16C2×D17311yescharacteristic, complemented
115#108—1731no—
116#101—3461no—
117#41—3462no—
118#41—3461no—
119#41—1731no—
120#41—6921no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—3462no—
124C2×D4—6921no—
125C2×D4—6922no—
126Q16—21no—
127Q16—21no—
128Q16—1731no—
129Q16—6921no—
130SD16—3462no—
131SD16—3461no—
132SD16—6921no—
133D8—21no—
134D8—3462no—
135D8—1731no—
136D8C22⋊D17312yes—
137M16—3461no—
138M16—3461no—
139M16—1731no—
140C2×C8—3461no—
141C2×C8C22×D17311yescharacteristic
142C16—21no—
143C16—3461no—
144C23—41no—
145C23—6922no—
146Q8—41no—
147Q8—41no—
148Q8—3461no—
149Q8—1731no—
150D4—22no—
151D4—41no—
152D4—3461no—
153D4—3461no—
154D4—1731no—
155D4—6922no—
156D4—6922no—
157D4—6921no—
158C2×C4—41no—
159C2×C4—3461no—
160C2×C4—6922no—
161C2×C4—6921no—
162C2×C4D4×D17311yescharacteristic
163C8—3461no—
164C8—6921no—
165C8D4×D17311yescharacteristic
166C8C2×C22⋊D17311yescharacteristic
167C22—42no—
168C22—161no—
169C22—3461no—
170C22—6921no—
171C22—27682no—
172C22#1914970311yescharacteristic
173C4—81no—
174C4—3461no—
175C4—6921no—
176C4#1914964711yescharacteristic
177C4D8×D17311yescharacteristic
178C2—21no—
179C2—82no—
180C2—13841no—
181C2—6921no—
182C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
FittingC2×SD32×C173C2
socleC2×C173#447
prime radical#26012292C2
hypercenterC2×SD32D173
hypocenterC173#2194
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #26011932C2semidirect
3. #26011763C2semidirect
4. #26011304C2semidirect
5. #26011299C2semidirect
6. #26011274C2semidirect
7. C2×SD32×C173C2semidirect
18. #19149619C22semidirect
19. #19149615C22semidirect
21. C2×Q16×C173C22semidirect
24. C2×C16×C173C22semidirect
91. C173#2194semidirect
95. C2×SD32D173semidirect
106. C2×Q16C2×D173semidirect
114. C2×C16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122922
#260119322
#260117632
#260113042
#260112992
#260112742
C2×SD32×C1732
#2194173