Finite Group Databasebeta

← #36352674

Subgroups — #36352674

Order 22144.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012215C211yesmaximal, characteristic, complemented
2#26011839C211yesmaximal, characteristic, complemented
3#26011692C211yesmaximal, characteristic, complemented
4#26011687C211yesmaximal, characteristic
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#332×C173C211yesmaximal, characteristic, complemented
8C2×#14583815—21no—
9C2×D1384—21no—
10#19149676C2211yescharacteristic
11#19149674—22no—
12#19149668—21no—
13#19149659—21no—
14Q8×Dic173—21no—
15C4×Dic346C2211yescharacteristic
16C4×D692C2211yescharacteristic
17#19149637—21no—
18#19149601—21no—
19#19149601—21no—
20#19149599—22no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×D692—21no—
27C2×D692—21no—
28C2×D692—21no—
29C2×D692D411yescharacteristic
30C2×C4×D173—21no—
31#14583815—42no—
32D1384—42no—
33C2×#13063436—21no—
34C4⋊Dic173—22no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38#14583803—22no—
39#14583803—22no—
40#14583803—21no—
41C4×Dic173—22no—
42C4×Dic173—21no—
43C2×Q8×C173—21no—
44C2×C8×C173—21no—
45C2×C8×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D412yes—
48C42×C173C2311yescharacteristic
49C22×D173—21no—
50C22×D173—41no—
51Dic346—21no—
52Dic346—21no—
53D692—21no—
54D692—21no—
55D692—42no—
56D692—41no—
57D692—41no—
58D692—41no—
59D692—41no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C4×D173—41no—
64C4×D173—41no—
65#13063436—41no—
66Q8×C173—41no—
67Q8×C173—41no—
68C2×C4×C173—22no—
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C8×C173—41no—
73C8×C173—41no—
74C22×C173#12011yescharacteristic
75C2×D173—41no—
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—81no—
79C2×D173—81no—
80C2×D173—81no—
81C4×C173—21no—
82C4×C173—42no—
83C4×C173#13511yescharacteristic
84C4×C173#13511yescharacteristic
85C4×C173AGL(1,ℤ/8)11yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87Dic173—41no—
88Dic173—41no—
89Dic173—41no—
90C2×C173#44911yescharacteristic
91C2×C173#44911yescharacteristic
92C2×C173#45411yescharacteristic
93D173—81no—
94D173—161no—
95C173#162811yescharacteristic, complemented, Sylow
96#1628—1731nomaximal, Sylow
97#556—1731no—
98#486—1731no—
99#460—1731no—
100#459—1731no—
101#332D17311yescharacteristic, complemented
102Q8⋊C8—1731no—
103D4⋊C8—1731no—
104C2×SD16—3461no—
105C2×D8—3461no—
106C4⋊Q8C2×D17311yescharacteristic
107#127—1731no—
108#126—3462no—
109#124—3461no—
110C22⋊Q8—3461no—
111C4×Q8—3461no—
112C4×Q8—1731no—
113C4×D4—1731no—
114#117—3461no—
115C4⋊C8C2×D17311yescharacteristic
116C4⋊C8C2×D17311yescharacteristic
117#102—3461no—
118#102—3461no—
119#102—3462no—
120C4×C8—1731no—
121C2×Q8—21no—
122C2×Q8—1731no—
123C2×D4—3461no—
124C2×D4—3461no—
125C2×D4—3461no—
126C2×D4—1731no—
127C22×C4—3461no—
128SD16—6922no—
129D8—6922no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—3461no—
133C4⋊C4—21no—
134C4⋊C4—3462no—
135C4⋊C4—3461no—
136C4⋊C4—1731no—
137C4⋊C4—1731no—
138C4⋊C4D69212yes—
139C22⋊C4—3462no—
140C22⋊C4—3462no—
141C22⋊C4—3461no—
142C42—3462no—
143C42—3461no—
144C42C22×D17311yescharacteristic
145C23—3461no—
146C23—6921no—
147Q8—41no—
148Q8—41no—
149Q8—3461no—
150Q8—3461no—
151D4—3461no—
152D4—3461no—
153D4—6922no—
154D4—6921no—
155D4—6921no—
156D4—6921no—
157D4—6921no—
158C2×C4—22no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—3461no—
162C2×C4—6921no—
163C2×C4—6921no—
164C2×C4D4×D17311yescharacteristic
165C2×C4D4×D17311yescharacteristic
166C2×C4C2×D69211yescharacteristic
167C8—41no—
168C8—41no—
169C8—6921no—
170C22—13841no—
171C22—13841no—
172C22—13841no—
173C22—6921no—
174C22—6921no—
175C22—6921no—
176C22#1914964911yescharacteristic
177C4—21no—
178C4—42no—
179C4—6921no—
180C4—6921no—
181C4—6921no—
182C4#1914970011yescharacteristic
183C4#1914970211yescharacteristic
184C4#1914970311yescharacteristic
185C4#1914970611yescharacteristic
186C2—13841no—
187C2—27681no—
188C2#2601163711yescharacteristic
189C2#2601163911yescharacteristic
190C2#2601166611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173#120
prime radical#26011687C2
hypercenter#332D173
hypocenterC173#1628
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #26012215C2semidirect
2. #26011839C2semidirect
3. #26011692C2semidirect
5. #26011221C2semidirect
6. #26011218C2semidirect
7. #332×C173C2semidirect
95. C173#1628semidirect
101. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260122152
#260118392
#260116922
#260116872
#260112212
#260112182
#332×C1732
#1628173