Finite Group Databasebeta

← #36352651

Subgroups — #36352651

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#26012146C211yesmaximal, characteristic, complemented
2#26011837C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic, complemented
4#26011236C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#459×C173C211yesmaximal, characteristic, complemented
8C2×#14583826—21no—
9C2×#14583811—21no—
10#19149679C2211yescharacteristic
11#19149661—22no—
12#19149658—21no—
13#19149650—21no—
14C4×D692C2211yescharacteristic
15C4⋊C4×D173—21no—
16C4⋊#13063436C2211yescharacteristic
17#19149600—21no—
18#19149600—22no—
19C4×#13063436C2211yescharacteristic
20C2×D8×C173—21no—
21#127×C173C2211yescharacteristic
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25#14583826—42no—
26#14583826—42no—
27C2×Dic346—21no—
28C2×D692D411yescharacteristic
29C2×C22⋊D173—22no—
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32#14583811—42no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37C4⋊Dic173D412yes—
38#14583805—22no—
39C22⋊Dic173—22no—
40#14583803—21no—
41C2×D4×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173D411yescharacteristic
45C22×C4×C173—21no—
46D8×C173—42no—
47C2×C8×C173—21no—
48C4⋊C4×C173D411yescharacteristic
49C22⋊C4×C173—21no—
50C42×C173C2311yescharacteristic
51C22×D173—21no—
52Dic346—41no—
53Dic346—41no—
54D692—21no—
55D692D812yescomplemented
56C22⋊D173—44no—
57C2×Dic173—22no—
58C2×Dic173—21no—
59C4×D173—44no—
60C4×D173—41no—
61C4×D173—41no—
62#13063436—41no—
63#13063436—41no—
64C23×C173—21no—
65C23×C173—41no—
66D4×C173—21no—
67D4×C173—21no—
68D4×C173—42no—
69D4×C173—41no—
70D4×C173—41no—
71D4×C173—41no—
72D4×C173—41no—
73C2×C4×C173—21no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173C2×D411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C8×C173—41no—
80C22×C173—41no—
81C22×C173—41no—
82C22×C173—41no—
83C22×C173—81no—
84C22×C173—81no—
85C22×C173—81no—
86C22×C173#12011yescharacteristic
87C2×D173—22no—
88C2×D173—41no—
89C2×D173—41no—
90C4×C173—21no—
91C4×C173—41no—
92C4×C173C2×D811yescharacteristic
93C4×C173#13511yescharacteristic
94C4×C173AGL(1,ℤ/8)11yescharacteristic
95C4×C173AGL(1,ℤ/8)11yescharacteristic
96Dic173—42no—
97Dic173—41no—
98C2×C173—81no—
99C2×C173—161no—
100C2×C173#44711yescharacteristic
101C2×C173#44911yescharacteristic
102C2×C173#45411yescharacteristic
103D173—42no—
104C173#162511yescharacteristic, complemented, Sylow
105#1625—1731nomaximal, Sylow
106#547—1731no—
107#486—1731no—
108#459—1731no—
109#459D17311yescharacteristic, complemented
110#332—1731no—
111D4⋊C8—1731no—
112D4⋊C8—1731no—
113C2×#41—3461no—
114C2×D8—21no—
115C2×D8—3461no—
116C4⋊Q8—1731no—
117#127C2×D17311yescharacteristic
118#123—3462no—
119C22⋊Q8—3461no—
120#121—3461no—
121C4×D4—1731no—
122C4×D4C2×D17311yescharacteristic
123C2×C4⋊C4—3461no—
124C4⋊C8—1731no—
125C4⋊C8C2×D17311yescharacteristic
126#102—21no—
127#102—3461no—
128#102—3462no—
129C4×C8—1731no—
130#41—6922no—
131#41—6922no—
132C2×Q8—3461no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3462no—
137C2×D4—1731no—
138C2×D4C22⋊D17311yescharacteristic
139C22×C4—21no—
140C22×C4—3462no—
141C22×C4—3461no—
142D8—42no—
143D8—6922no—
144C2×C8—21no—
145C2×C8—3461no—
146C2×C8—3461no—
147C4⋊C4—3462no—
148C4⋊C4—3461no—
149C4⋊C4—1731no—
150C4⋊C4—1732no—
151C4⋊C4C22⋊D17311yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3462no—
154C22⋊C4—3461no—
155C42C22×D17311yescharacteristic
156C23—21no—
157C23—41no—
158C23—3461no—
159Q8—6921no—
160Q8—6921no—
161D4—21no—
162D4—21no—
163D4—42no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—3461no—
169D4—1732no—
170D4—6924no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—3461no—
175C2×C4—3462no—
176C2×C4—6924no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4D4×D17311yescharacteristic
180C2×C4D4×D17311yescharacteristic
181C2×C4C2×C22⋊D17311yescharacteristic
182C8—41no—
183C8—6921no—
184C8—6921no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—3462no—
192C22—6921no—
193C22—6921no—
194C22#1914964711yescharacteristic
195C4—21no—
196C4—41no—
197C4—6922no—
198C4—6921no—
199C4D8×D17311yescharacteristic
200C4#1914969911yescharacteristic
201C4#1914970311yescharacteristic
202C4#1914970511yescharacteristic
203C2—81no—
204C2—161no—
205C2—6922no—
206C2#2601162311yescharacteristic
207C2#2601163411yescharacteristic
208C2#2601166211yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#459×C173C2
socleC22×C173#120
hypercenter#459D173
hypocenterC173#1625
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012146C2semidirect
2. #26011837C2semidirect
3. #26011683C2semidirect
4. #26011236C2semidirect
5. D4⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #459×C173C2semidirect
55. D692D8semidirect
104. C173#1625semidirect
109. #459D173semidirect

Maximal subgroups

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

subgroupindex
#260121462
#260118372
#260116832
#260112362
D4⋊#130634362
#260112182
#459×C1732
#1625173