Finite Group Databasebeta

← #36352636

Subgroups — #36352636

Order 22144.

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

normal subgroups only

This group has 236 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#26012202C211yesmaximal, characteristic, complemented
2#26011865C211yesmaximal, characteristic, complemented
3#26011692C211yesmaximal, characteristic, complemented
4#26011687C211yesmaximal, characteristic
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#333×C173C211yesmaximal, characteristic, complemented
8C2×#14583815—21no—
9C2×#14583813—22no—
10C2×D1384—21no—
11#19149676C2211yescharacteristic
12#19149670—21no—
13#19149659—21no—
14#19149655—22no—
15C4×Dic346C2211yescharacteristic
16C4×D692C2211yescharacteristic
17#19149641—21no—
18C4⋊C4×D173—21no—
19#19149601—21no—
20#19149601—21no—
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—22no—
31C2×C4×D173—21no—
32#14583815—42no—
33#14583813—44no—
34D1384—42no—
35C2×#13063436—21no—
36C4⋊Dic173—22no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—22no—
40#14583805—21no—
41#14583803—22no—
42#14583803—21no—
43C4×Dic173—21no—
44C2×Q8×C173D412yes—
45C2×C8×C173—21no—
46C2×C8×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51C22×D173—41no—
52Dic346—21no—
53Dic346—21no—
54D692—21no—
55D692—42no—
56D692—41no—
57D692—41no—
58D692—41no—
59D692—41no—
60D692SD1612yescomplemented
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C4×D173—41no—
65C4×D173—41no—
66C4×D173—82no—
67#13063436—22no—
68Q8×C173—42no—
69C2×C4×C173—22no—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—41no—
74C8×C173—41no—
75C22×C173#12011yescharacteristic
76C2×D173—22no—
77C2×D173—41no—
78C2×D173—41no—
79C2×D173—81no—
80C2×D173—81no—
81C2×D173—81no—
82C4×C173—21no—
83C4×C173—42no—
84C4×C173C2×SD1611yescharacteristic
85C4×C173#13511yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87C4×C173#13711yescharacteristic
88Dic173—41no—
89Dic173—41no—
90Dic173—41no—
91C2×C173#44811yescharacteristic
92C2×C173#44911yescharacteristic
93C2×C173#45311yescharacteristic
94D173—42no—
95D173—161no—
96C173#162211yescharacteristic, complemented, Sylow
97#1622—1731nomaximal, Sylow
98#554—1731no—
99#492—1731no—
100#460—1731no—
101#459—1731no—
102#333D17311yescharacteristic, complemented
103Q8⋊C8—1731no—
104D4⋊C8—1731no—
105C2×SD16—3461no—
106C2×SD16—3462no—
107C2×D8—3461no—
108C4⋊Q8C2×D17311yescharacteristic
109#127—1731no—
110#125—3461no—
111C22⋊Q8—3461no—
112C22⋊Q8—3462no—
113C4×Q8—1731no—
114C4×D4—3461no—
115C4×D4—1731no—
116C2×C4⋊C4—3461no—
117C4⋊C8C2×D17311yescharacteristic
118C4⋊C8C2×D17311yescharacteristic
119#102—3461no—
120#102—3461no—
121C4×C8—1731no—
122C2×Q8—1731no—
123C2×Q8D69212yes—
124C2×D4—3461no—
125C2×D4—3461no—
126C2×D4—3461no—
127C2×D4—1731no—
128C22×C4—3462no—
129C22×C4—3461no—
130SD16—6922no—
131SD16—6924no—
132D8—6922no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3461no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3462no—
139C4⋊C4—3462no—
140C4⋊C4—3461no—
141C4⋊C4—1731no—
142C4⋊C4—1731no—
143C22⋊C4—3462no—
144C22⋊C4—3461no—
145C42—3461no—
146C42C22×D17311yescharacteristic
147C23—3461no—
148C23—6921no—
149Q8—42no—
150Q8—3461no—
151Q8—3461no—
152D4—3461no—
153D4—1732no—
154D4—6922no—
155D4—6921no—
156D4—6921no—
157D4—6921no—
158D4—6921no—
159C2×C4—22no—
160C2×C4—3461no—
161C2×C4—3461no—
162C2×C4—3461no—
163C2×C4—13842no—
164C2×C4—6921no—
165C2×C4—6921no—
166C2×C4D4×D17311yescharacteristic
167C2×C4D4×D17311yescharacteristic
168C2×C4C2×D69211yescharacteristic
169C8—41no—
170C8—41no—
171C8—3462no—
172C22—3462no—
173C22—13841no—
174C22—13841no—
175C22—13841no—
176C22—6921no—
177C22—6921no—
178C22#1914964911yescharacteristic
179C4—21no—
180C4—42no—
181C4—6921no—
182C4—6921no—
183C4—6921no—
184C4SD16×D17311yescharacteristic
185C4#1914970211yescharacteristic
186C4#1914970611yescharacteristic
187C4#1914971111yescharacteristic
188C2—6922no—
189C2—27681no—
190C2#2601163111yescharacteristic
191C2#2601163911yescharacteristic
192C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#333×C173C2
socleC22×C173#120
prime radical#26011687C2
hypercenter#333D173
hypocenterC173#1622
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #26012202C2semidirect
2. #26011865C2semidirect
3. #26011692C2semidirect
5. #26011221C2semidirect
6. #26011218C2semidirect
7. #333×C173C2semidirect
60. D692SD16semidirect
96. C173#1622semidirect
102. #333D173semidirect

Maximal subgroups

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

subgroupindex
#260122022
#260118652
#260116922
#260116872
#260112212
#260112182
#333×C1732
#1622173