Finite Group Databasebeta

← #36352860

Subgroups — #36352860

Order 22144.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012245C211yesmaximal, characteristic, complemented
2#26011878C211yesmaximal, characteristic
3#26011855C211yesmaximal, characteristic, complemented
4#26011233C211yesmaximal, characteristic, complemented
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic, complemented
7#490×C173C211yesmaximal, characteristic, complemented
8C2×#14583814—21no—
9C2×#14583811—21no—
10#19149678—21no—
11#19149674—21no—
12#19149674—21no—
13#19149671C2211yescharacteristic
14#19149667C2211yescharacteristic
15#19149664—21no—
16#19149659—21no—
17#19149657—21no—
18#19149600—21no—
19#19149600—21no—
20#19149590C2211yescharacteristic
21#19149590C2211yescharacteristic
22#127×C173C2211yescharacteristic
23#125×C173C2211yescharacteristic
24#102×C173—21no—
25#102×C173—21no—
26#97×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—21no—
31#14583814—42no—
32#14583811—42no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×D4×C173—21no—
48C2×D4×C173—21no—
49C2×D4×C173—21no—
50C2×C8×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C4⋊C4×C173D411yescharacteristic
55C42×C173C2311yescharacteristic
56C22×D173—21no—
57Dic346—41no—
58Dic346—81no—
59D692—41no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C4×D173—81no—
64#13063436—22no—
65#13063436—41no—
66C23×C173—41no—
67D4×C173—41no—
68D4×C173—41no—
69D4×C173—42no—
70D4×C173—42no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—41no—
77C22×C173—81no—
78C22×C173—81no—
79C22×C173—81no—
80C22×C173#12011yescharacteristic
81C2×D173—41no—
82C2×D173—41no—
83C2×D173—41no—
84C4×C173—21no—
85C4×C173—21no—
86C4×C173—21no—
87C4×C173—41no—
88C4×C173—41no—
89Dic173—41no—
90Dic173—41no—
91Dic173—41no—
92C2×C173—161no—
93C2×C173#45311yescharacteristic
94C2×C173#45411yescharacteristic
95C2×C173#45411yescharacteristic
96D173—81no—
97C173#165711yescharacteristic, complemented, Sylow
98#1657—1731nomaximal, Sylow
99#563—1731no—
100#496—1731no—
101#490—1731no—
102#490D17311yescharacteristic, complemented
103#330—1731no—
104#330—1731no—
105#329—1731no—
106C2×SD16—3461no—
107C2×D8—3461no—
108C4⋊Q8—3461no—
109#127C2×D17311yescharacteristic
110#126—3461no—
111#126—3461no—
112#125—1731no—
113#125C2×D17311yescharacteristic
114#124—1731no—
115#123—3461no—
116C22⋊Q8—3461no—
117C22⋊Q8—3461no—
118#102—21no—
119#102—21no—
120#102—3461no—
121#102—3461no—
122#97—1731no—
123#97—1731no—
124#97C2×D17311yescharacteristic
125C2×Q8—3461no—
126C2×Q8—1731no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—1731no—
131C22×C4—3461no—
132SD16—6922no—
133D8—6922no—
134C2×C8—21no—
135C2×C8—3461no—
136C2×C8—3461no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—3461no—
140C4⋊C4—3461no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4—1731no—
144C4⋊C4—1731no—
145C4⋊C4C22⋊D17311yescharacteristic
146C4⋊C4C22⋊D17311yescharacteristic
147C22⋊C4—3461no—
148C22⋊C4—3461no—
149C22⋊C4—3461no—
150C22⋊C4—3461no—
151C22⋊C4—3461no—
152C42—3461no—
153C42C22×D17311yescharacteristic
154C23—41no—
155C23—3461no—
156Q8—13841no—
157Q8—6921no—
158D4—41no—
159D4—41no—
160D4—42no—
161D4—42no—
162D4—6921no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—3461no—
166C2×C4—3461no—
167C2×C4—3461no—
168C2×C4—13841no—
169C2×C4D4×D17311yescharacteristic
170C2×C4D4×D17311yescharacteristic
171C2×C4C2×C22⋊D17311yescharacteristic
172C8—41no—
173C8—3462no—
174C8—6921no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—6921no—
179C22—6921no—
180C22—6921no—
181C22#1914964711yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—6921no—
188C4—6921no—
189C4—6921no—
190C2—161no—
191C2—13841no—
192C2#2601165711yescharacteristic
193C2#2601166211yescharacteristic
194C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#490×C173C2
socleC22×C173#120
prime radical#26011878C2
hypercenter#490D173
hypocenterC173#1657
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #26012245C2semidirect
3. #26011855C2semidirect
4. #26011233C2semidirect
5. #26011232C2semidirect
6. #26011230C2semidirect
7. #490×C173C2semidirect
97. C173#1657semidirect
102. #490D173semidirect

Maximal subgroups

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

subgroupindex
#260122452
#260118782
#260118552
#260112332
#260112322
#260112302
#490×C1732
#1657173