Finite Group Databasebeta

← #36352689

Subgroups — #36352689

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#26012162C211yesmaximal, characteristic
2#26011865C211yesmaximal, characteristic, complemented
3#26011783C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#476×C173C211yesmaximal, characteristic, complemented
8C2×#14583826—21no—
9C2×#14583813—22no—
10#19149676C2211yescharacteristic
11#19149675—21no—
12#19149653—21no—
13#19149652—22no—
14C4×Dic346C2211yescharacteristic
15#19149641—21no—
16#19149612—21no—
17C4⋊#13063436C2211yescharacteristic
18#19149602—21no—
19C4×#13063436C2211yescharacteristic
20C4⋊Q8×C173C2211yescharacteristic
21C4×D4×C173C2211yescharacteristic
22#106×C173—21no—
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25#14583826—42no—
26#14583826—42no—
27C2×Dic346D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692D412yes—
31C2×C22⋊D173—22no—
32C2×C22⋊D173—22no—
33C2×C4×D173—22no—
34#14583813—44no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583803—22no—
40C4×Dic173—21no—
41C2×Q8×C173—21no—
42C2×D4×C173D411yescharacteristic
43C22×C4×C173—21no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—22no—
51Dic346—21no—
52Dic346SD1612yescomplemented
53D692—42no—
54D692—41no—
55D692—41no—
56D692—41no—
57D692—41no—
58C22⋊D173—44no—
59C22⋊D173—82no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C4×D173—44no—
63#13063436—22no—
64#13063436—41no—
65C23×C173—21no—
66Q8×C173—42no—
67D4×C173—21no—
68D4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173—41no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—41no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173#12011yescharacteristic
81C2×D173—42no—
82C2×D173—42no—
83C2×D173—42no—
84C4×C173—21no—
85C4×C173—41no—
86C4×C173—81no—
87C4×C173C2×SD1611yescharacteristic
88C4×C173#13511yescharacteristic
89C4×C173AGL(1,ℤ/8)11yescharacteristic
90C4×C173#13711yescharacteristic
91Dic173—22no—
92Dic173—41no—
93C2×C173—81no—
94C2×C173#45011yescharacteristic
95C2×C173#45211yescharacteristic
96C2×C173#45311yescharacteristic
97D173—82no—
98C173#163111yescharacteristic, complemented, Sylow
99#1631—1731nomaximal, Sylow
100#550—1731no—
101#492—1731no—
102#477—1731no—
103#476D17311yescharacteristic, complemented
104#331—1731no—
105Q8⋊C8—1731no—
106D4⋊C8—1731no—
107C2×#41—3461no—
108C2×SD16—3462no—
109C4⋊Q8C2×D17311yescharacteristic
110#127—3461no—
111#127—1731no—
112#121—3462no—
113#121—3461no—
114C4×Q8—1731no—
115C4×D4—3461no—
116C4×D4C2×D17311yescharacteristic
117#107—3461no—
118#106—21no—
119C4⋊C8—1731no—
120C4⋊C8C2×D17311yescharacteristic
121#103—3461no—
122#102—21no—
123C4×C8—1731no—
124#41—6922no—
125#41—6922no—
126C2×Q8—21no—
127C2×Q8—1731no—
128C2×D4—3462no—
129C2×D4—3462no—
130C2×D4—3461no—
131C2×D4—3461no—
132C2×D4—1732no—
133C2×D4C22⋊D17311yescharacteristic
134C22×C4—21no—
135C22×C4—3462no—
136SD16—6924no—
137C2×C8—21no—
138C2×C8—3461no—
139C2×C8—3461no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—3461no—
143C4⋊C4—1731no—
144C4⋊C4C22⋊D17311yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—3462no—
147C42—3461no—
148C42C22×D17311yescharacteristic
149C23—21no—
150C23—3462no—
151Q8—42no—
152Q8—3461no—
153Q8—1732no—
154D4—21no—
155D4—21no—
156D4—13842no—
157D4—6922no—
158D4—6924no—
159D4—6921no—
160D4—6921no—
161D4—6921no—
162D4—6921no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3461no—
168C2×C4—3461no—
169C2×C4—6924no—
170C2×C4D4×D17311yescharacteristic
171C2×C4D4×D17311yescharacteristic
172C2×C4C2×C22⋊D17311yescharacteristic
173C8—41no—
174C8—3462no—
175C8—6921no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—6922no—
180C22—6922no—
181C22—6922no—
182C22#1914964711yescharacteristic
183C4—21no—
184C4—41no—
185C4—81no—
186C4—3462no—
187C4—6921no—
188C4SD16×D17311yescharacteristic
189C4#1914969911yescharacteristic
190C4#1914970611yescharacteristic
191C4#1914971011yescharacteristic
192C2—81no—
193C2—13842no—
194C2#2601164011yescharacteristic
195C2#2601165111yescharacteristic
196C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#476×C173C2
socleC22×C173#120
prime radical#26012162C2
hypercenter#476D173
hypocenterC173#1631
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #26011865C2semidirect
3. #26011783C2semidirect
4. #26011234C2semidirect
5. #26011221C2semidirect
6. D4⋊#13063436C2semidirect
7. #476×C173C2semidirect
52. Dic346SD16semidirect
98. C173#1631semidirect
103. #476D173semidirect

Maximal subgroups

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

subgroupindex
#260121622
#260118652
#260117832
#260112342
#260112212
D4⋊#130634362
#476×C1732
#1631173