Finite Group Databasebeta

← #36352652

Subgroups — #36352652

Order 22144.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#26011839C211yesmaximal, characteristic, complemented
2#26011686C211yesmaximal, characteristic, complemented
3#26011686C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#547×C173C211yesmaximal, characteristic, complemented
8C2×#14583811—21no—
9C2×#14583811—21no—
10#19149676C2211yescharacteristic
11C4⋊#13063436C2211yescharacteristic
12C4⋊#13063436C2211yescharacteristic
13#19149599—21no—
14#19149599—21no—
15#19149599—22no—
16C4×#13063436C2211yescharacteristic
17C2×#41×C173—21no—
18C4⋊Q8×C173C2211yescharacteristic
19#123×C173—22no—
20C22⋊Q8×C173—21no—
21#121×C173—21no—
22C4×D4×C173C2211yescharacteristic
23C4×D4×C173C2211yescharacteristic
24C2×C4⋊C4×C173—21no—
25C2×D692—21no—
26C2×D692—21no—
27C2×D692—21no—
28#14583811—42no—
29#14583811—42no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33#41×C173—42no—
34#41×C173—42no—
35C2×Q8×C173—21no—
36C2×D4×C173—22no—
37C2×D4×C173D411yescharacteristic
38C2×D4×C173D411yescharacteristic
39C22×C4×C173—22no—
40C22×C4×C173—21no—
41C22×C4×C173—21no—
42C4⋊C4×C173—22no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173D412yes—
45C4⋊C4×C173D411yescharacteristic
46C4⋊C4×C173D411yescharacteristic
47C22⋊C4×C173—22no—
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C42×C173C2311yescharacteristic
51C22×D173—41no—
52D692—42no—
53D692—41no—
54D692—41no—
55D692—41no—
56D692—41no—
57#13063436—41no—
58#13063436—41no—
59#13063436—41no—
60C23×C173—21no—
61C23×C173—21no—
62Q8×C173—41no—
63Q8×C173—41no—
64D4×C173—21no—
65D4×C173—21no—
66D4×C173—21no—
67D4×C173—44no—
68D4×C173D812yescomplemented
69C2×C4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—22no—
72C2×C4×C173—44no—
73C2×C4×C173—41no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C22×C173—22no—
81C22×C173—41no—
82C22×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173#12011yescharacteristic
87C2×D173—81no—
88C2×D173—81no—
89C2×D173—81no—
90C4×C173—21no—
91C4×C173—41no—
92C4×C173—42no—
93C4×C173—41no—
94C4×C173C2×D811yescharacteristic
95C4×C173#13511yescharacteristic
96C4×C173AGL(1,ℤ/8)11yescharacteristic
97C4×C173AGL(1,ℤ/8)11yescharacteristic
98C2×C173—42no—
99C2×C173—81no—
100C2×C173#44711yescharacteristic
101C2×C173#44911yescharacteristic
102C2×C173#45411yescharacteristic
103D173—161no—
104C173#162511yescharacteristic, complemented, Sylow
105#1625—1731nomaximal, Sylow
106#547D17311yescharacteristic, complemented
107#486—1731no—
108#459—1731no—
109#459—1731no—
110#332—1731no—
111D4⋊C8—1731no—
112D4⋊C8—1731no—
113C2×#41—21no—
114C2×D8—3461no—
115C2×D8—3461no—
116C4⋊Q8C2×D17311yescharacteristic
117#127—1731no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D17311yescharacteristic
122C4×D4C2×D17311yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—1731no—
125C4⋊C8—1731no—
126#102—3461no—
127#102—3461no—
128#102—3462no—
129C4×C8—1731no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—3461no—
135C2×D4—3461no—
136C2×D4—3461no—
137C2×D4C22⋊D17311yescharacteristic
138C2×D4C22⋊D17311yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—6922no—
143D8—6922no—
144C2×C8—3461no—
145C2×C8—3461no—
146C2×C8—3461no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D17311yescharacteristic
150C4⋊C4C22⋊D17311yescharacteristic
151C4⋊C4C22⋊D17312yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D17311yescharacteristic
156C23—21no—
157C23—21no—
158C23—6921no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—6922no—
166D4—6921no—
167D4—6921no—
168D4—6921no—
169D4—6921no—
170D4#1458381112yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D17311yescharacteristic
180C2×C4C2×C22⋊D17311yescharacteristic
181C2×C4C2×C22⋊D17311yescharacteristic
182C8—6921no—
183C8—6921no—
184C8—6921no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—13841no—
192C22—13841no—
193C22—13841no—
194C22#1914964811yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1458381111yescharacteristic
200C4#1914969911yescharacteristic
201C4#1914970411yescharacteristic
202C4#1914970511yescharacteristic
203C2—42no—
204C2—81no—
205C2—27681no—
206C2#2601162511yescharacteristic
207C2#2601163611yescharacteristic
208C2#2601166411yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#547×C173C2
socleC22×C173#120
hypercenter#547D173
hypocenterC173#1625
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #26011839C2semidirect
2. #26011686C2semidirect
3. #26011686C2semidirect
4. #26011237C2semidirect
5. D4⋊#13063436C2semidirect
6. D4⋊#13063436C2semidirect
7. #547×C173C2semidirect
68. D4×C173D8semidirect
104. C173#1625semidirect
106. #547D173semidirect
170. D4#14583811semidirect

Maximal subgroups

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

subgroupindex
#260118392
#260116862
#260112372
D4⋊#130634362
#547×C1732
#1625173