Finite Group Databasebeta

← #36352897

Subgroups — #36352897

Order 22144.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#26011901C211yesmaximal, characteristic, complemented
2#26011861C211yesmaximal, characteristic, complemented
3#26011850C211yesmaximal, characteristic
4#26011233C211yesmaximal, characteristic, complemented
5#26011233C211yesmaximal, characteristic, complemented
6#26011231C211yesmaximal, characteristic
7#563×C173C211yesmaximal, characteristic
8#19149679C2211yescharacteristic
9#19149612—21no—
10#19149610—21no—
11#19149603—21no—
12#19149602—21no—
13#19149602—21no—
14#19149600—21no—
15#19149590C2211yescharacteristic
16#19149590C2211yescharacteristic
17#19149590C2211yescharacteristic
18C4⋊Q8×C173—21no—
19#126×C173—21no—
20#126×C173—21no—
21#125×C173C2211yescharacteristic
22#125×C173C2211yescharacteristic
23#124×C173C2211yescharacteristic
24#123×C173—21no—
25C22⋊Q8×C173—21no—
26C22⋊Q8×C173—21no—
27C2×Dic346—21no—
28C2×#13063436—21no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33C2×Q8×C173—21no—
34C2×Q8×C173D411yescharacteristic
35C2×D4×C173D411yescharacteristic
36C22×C4×C173—21no—
37C4⋊C4×C173—21no—
38C4⋊C4×C173—21no—
39C4⋊C4×C173—21no—
40C4⋊C4×C173—21no—
41C4⋊C4×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173D411yescharacteristic
44C4⋊C4×C173D411yescharacteristic
45C4⋊C4×C173D411yescharacteristic
46C4⋊C4×C173D411yescharacteristic
47C22⋊C4×C173—21no—
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52C42×C173—21no—
53C42×C173C2311yescharacteristic
54Dic346—42no—
55C2×Dic173—41no—
56#13063436—22no—
57#13063436—41no—
58#13063436—41no—
59C23×C173—21no—
60Q8×C173—41no—
61Q8×C173—81no—
62D4×C173—41no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—81no—
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173#12011yescharacteristic
76C4×C173—21no—
77C4×C173—21no—
78C4×C173—21no—
79C4×C173—41no—
80C4×C173—41no—
81C4×C173—41no—
82C4×C173—41no—
83C4×C173—41no—
84Dic173—81no—
85C2×C173—81no—
86C2×C173#45411yescharacteristic
87C2×C173#45511yescharacteristic
88C2×C173#45611yescharacteristic
89C173#166411yescharacteristic, complemented, Sylow
90#1664—1731nomaximal, Sylow
91#563D17311yescharacteristic
92#501—1731no—
93#491—1731no—
94#489—1731no—
95#330—1731no—
96#330—1731no—
97#329—1731no—
98C4⋊Q8—21no—
99C4⋊Q8—1731no—
100#126—21no—
101#126—21no—
102#125C2×D17311yescharacteristic
103#125C2×D17311yescharacteristic
104#124C2×D17311yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—3461no—
109#106—3461no—
110#103—3461no—
111#103—3461no—
112#103—3461no—
113#102—3461no—
114#97—1731no—
115#97—1731no—
116#97—1731no—
117C2×Q8—21no—
118C2×Q8—3461no—
119C2×Q8C22⋊D17311yescharacteristic
120C2×D4C22⋊D17311yescharacteristic
121C22×C4—21no—
122C2×C8—3461no—
123C2×C8—3461no—
124C2×C8—3461no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4C22⋊D17311yescharacteristic
134C4⋊C4C22⋊D17311yescharacteristic
135C4⋊C4C22⋊D17311yescharacteristic
136C4⋊C4C22⋊D17311yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D17311yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—6922no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—6921no—
156C2×C4C2×C22⋊D17311yescharacteristic
157C2×C4C2×C22⋊D17311yescharacteristic
158C2×C4C2×C22⋊D17311yescharacteristic
159C8—3462no—
160C8—6921no—
161C8—6921no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1914964811yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—13841no—
175C2—81no—
176C2#2601166411yescharacteristic
177C2#2601166911yescharacteristic
178C2#2601167411yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#563×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenter#563D173
hypocenterC173#1664
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26011901C2semidirect
2. #26011861C2semidirect
4. #26011233C2semidirect
5. #26011233C2semidirect
89. C173#1664semidirect

Maximal subgroups

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

subgroupindex
#260119012
#260118612
#260118502
#260112332
#260112312
#563×C1732
#1664173