Finite Group Databasebeta

← #36352649

Subgroups — #36352649

Order 22144.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012144C211yesmaximal, characteristic, complemented
2#26011683C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5#26011220C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#486×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9C2×#14583811—21no—
10C2×#14583811—21no—
11#19149664—22no—
12#19149655—21no—
13#19149654—21no—
14C4×D692C2211yescharacteristic
15C4×D692C2211yescharacteristic
16C4⋊C4×D173—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149600—21no—
20#19149600—21no—
21C4⋊Q8×C173C2211yescharacteristic
22#127×C173C2211yescharacteristic
23#102×C173—22no—
24C4×C8×C173C2211yescharacteristic
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33#14583811—42no—
34#14583811—42no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—22no—
40#14583803—22no—
41#14583803—21no—
42#14583803—21no—
43C2×Q8×C173—21no—
44C2×D4×C173—21no—
45C2×D4×C173—21no—
46C2×D4×C173—21no—
47C2×C8×C173—21no—
48C4⋊C4×C173—21no—
49C4⋊C4×C173D412yes—
50C42×C173C2311yescharacteristic
51C22×D173—21no—
52C22×D173—21no—
53D692—21no—
54D692—21no—
55D692—21no—
56D692—44no—
57D692D812yescomplemented
58C2×Dic173—21no—
59C2×Dic173—21no—
60C4×D173—44no—
61C4×D173—41no—
62C4×D173—41no—
63C4×D173—41no—
64C4×D173—41no—
65#13063436—41no—
66#13063436—41no—
67C23×C173—41no—
68Q8×C173—41no—
69Q8×C173—41no—
70D4×C173—42no—
71D4×C173—41no—
72D4×C173—41no—
73D4×C173—41no—
74D4×C173—41no—
75C2×C4×C173—22no—
76C2×C4×C173C2×D411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C8×C173—41no—
80C22×C173—81no—
81C22×C173—81no—
82C22×C173—81no—
83C22×C173#12011yescharacteristic
84C2×D173—22no—
85C2×D173—41no—
86C2×D173—41no—
87C2×D173—41no—
88C2×D173—41no—
89C2×D173—41no—
90C4×C173—21no—
91C4×C173—42no—
92C4×C173C2×D811yescharacteristic
93C4×C173#13511yescharacteristic
94C4×C173AGL(1,ℤ/8)11yescharacteristic
95C4×C173AGL(1,ℤ/8)11yescharacteristic
96Dic173—41no—
97Dic173—41no—
98C2×C173—161no—
99C2×C173#44711yescharacteristic
100C2×C173#44911yescharacteristic
101C2×C173#45411yescharacteristic
102D173—42no—
103D173—81no—
104C173#162511yescharacteristic, complemented, Sylow
105#1625—1731nomaximal, Sylow
106#547—1731no—
107#486D17311yescharacteristic, complemented
108#459—1731no—
109#459—1731no—
110#332—1731no—
111D4⋊C8—1731no—
112D4⋊C8—1731no—
113C2×#41—3461no—
114C2×D8—3461no—
115C2×D8—3461no—
116C4⋊Q8C2×D17311yescharacteristic
117#127C2×D17311yescharacteristic
118#123—3462no—
119C22⋊Q8—3461no—
120#121—3461no—
121C4×D4—1731no—
122C4×D4—1731no—
123C2×C4⋊C4—3461no—
124C4⋊C8—1731no—
125C4⋊C8—1731no—
126#102—22no—
127#102—3461no—
128#102—3461no—
129C4×C8C2×D17311yescharacteristic
130#41—6922no—
131#41—6922no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3462no—
137C2×D4—1731no—
138C2×D4—1731no—
139C22×C4—3462no—
140C22×C4—3461no—
141C22×C4—3461no—
142D8—6922no—
143D8—6922no—
144C2×C8—21no—
145C2×C8—3461no—
146C2×C8—3461no—
147C4⋊C4—21no—
148C4⋊C4—3462no—
149C4⋊C4—1731no—
150C4⋊C4—1731no—
151C4⋊C4C22⋊D17312yes—
152C22⋊C4—3462no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C42C22×D17311yescharacteristic
156C23—41no—
157C23—3461no—
158C23—3461no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—3461no—
167D4—3461no—
168D4—3461no—
169D4—1732no—
170D4—6924no—
171C2×C4—22no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—6924no—
175C2×C4—6921no—
176C2×C4—6921no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4D4×D17311yescharacteristic
180C2×C4D4×D17311yescharacteristic
181C2×C4C2×C22⋊D17311yescharacteristic
182C8—41no—
183C8—6921no—
184C8—6921no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—3462no—
189C22—6921no—
190C22—6921no—
191C22—6921no—
192C22—6921no—
193C22—6921no—
194C22#1914964711yescharacteristic
195C4—21no—
196C4—42no—
197C4—6921no—
198C4—6921no—
199C4D8×D17311yescharacteristic
200C4#1914970011yescharacteristic
201C4#1914970311yescharacteristic
202C4#1914970611yescharacteristic
203C2—161no—
204C2—13841no—
205C2—6922no—
206C2#2601162311yescharacteristic
207C2#2601163511yescharacteristic
208C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#486×C173C2
socleC22×C173#120
hypercenter#486D173
hypocenterC173#1625
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #26012144C2semidirect
2. #26011683C2semidirect
3. #26011683C2semidirect
4. #26011237C2semidirect
5. #26011220C2semidirect
6. #26011220C2semidirect
7. #486×C173C2semidirect
57. D692D8semidirect
104. C173#1625semidirect
107. #486D173semidirect

Maximal subgroups

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

subgroupindex
#260121442
#260116832
#260112372
#260112202
#486×C1732
#1625173