Finite Group Databasebeta

← #34202537

Subgroups — #34202537

Order 20224.

232 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 69 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#23765656C211yesmaximal, characteristic
2C16×Dic158C211yesmaximal, characteristic, complemented
3#23762771C211yesmaximal, characteristic, complemented
4#23762761C211yesmaximal, characteristic
5#23761448C211yesmaximal, characteristic, complemented
6#23761445C211yesmaximal, characteristic
7C79×#1372C211yesmaximal, characteristic, complemented
8C2×#14579106—21no—
9C2×#14579106—21no—
10#19098943C2211yescharacteristic
11#19098940C2211yescharacteristic
12#19098849—21no—
13#19098764—21no—
14C8×Dic158C2211yescharacteristic
15C4⋊(C79⋊C16)C2211yescharacteristic
16Dic79⋊C16—21no—
17C16×Dic79—21no—
18#19098308—21no—
19#19098306—21no—
20C4×C16×C79C2211yescharacteristic
21#335×C79C2211yescharacteristic
22#335×C79C2211yescharacteristic
23C2×Dic316—21no—
24C2×Dic316—21no—
25C2×#11929613—41no—
26C2×D632—21no—
27C2×D632—21no—
28#14579161—21no—
29#14579158—21no—
30C4×Dic158D411yescharacteristic
31#14579106—42no—
32#14579106—42no—
33C2×C79⋊C16—21no—
34#14579093—41no—
35C4⋊(C79⋊C8)D411yescharacteristic
36Dic79⋊C8—21no—
37#14579086—41no—
38#14579083—41no—
39C8×Dic79—21no—
40C2×C16×C79D411yescharacteristic
41C2×C16×C79D411yescharacteristic
42C4⋊C8×C79—21no—
43C4⋊C8×C79—21no—
44C4×C8×C79C2311yescharacteristic
45C2×Dic158—41no—
46C2×Dic158D811yescharacteristic
47C2×D316—41no—
48C2×D316—41no—
49C2×D316—41no—
50Dic316—41no—
51Dic316—41no—
52Dic316—81no—
53#11929613—82no—
54D632—41no—
55D632—41no—
56D632—81no—
57C2×C79⋊C8—21no—
58C4⋊Dic79—41no—
59C4⋊Dic79—41no—
60C4⋊Dic79D811yescharacteristic
61Dic79⋊C4—21no—
62C4×Dic79—21no—
63C79⋊C16—41no—
64C2×C8×C79—41no—
65C2×C8×C79—41no—
66C2×C8×C79C2×D411yescharacteristic
67C2×C8×C79#4111yescharacteristic
68C42×C79C2×D411yescharacteristic
69C16×C79—21no—
70C16×C79SD1612yescomplemented
71C22×D79—81no—
72C2×C4×C79#12111yescharacteristic
73C2×C4×C79C2×D811yescharacteristic
74C2×C4×C79#13511yescharacteristic
75C2×Dic79—21no—
76C2×Dic79—21no—
77C2×Dic79—81no—
78D316—81no—
79D316—81no—
80D316—81no—
81D316—81no—
82D316—161no—
83Dic158—21no—
84Dic158—81no—
85Dic158—81no—
86Dic158SD3212yescomplemented
87C8×C79—21no—
88C8×C79—81no—
89C8×C79—81no—
90C8×C79C2×SD1611yescharacteristic
91C8×C79AGL(1,ℤ/8)11yescharacteristic
92C79⋊C8—41no—
93C22×C79#46611yescharacteristic
94C2×D79—161no—
95C2×D79—161no—
96C2×D79—161no—
97C4×C79—21no—
98C4×C79#46011yescharacteristic
99C4×C79#47211yescharacteristic
100C4×C79C2×SD3211yescharacteristic
101C4×C79#50811yescharacteristic
102Dic79—22no—
103Dic79—41no—
104Dic79—161no—
105#12441—791nomaximal, Sylow
106C2×C79#166811yescharacteristic
107C2×C79#221711yescharacteristic
108C2×C79#222311yescharacteristic
109D79—321no—
110#2248—791no—
111Q8×C16—791no—
112#1670—791no—
113#1668—791no—
114#1372D7911yescharacteristic, complemented
115#1362—791no—
116#1361—791no—
117C79#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1581no—
119C2×SD32—1581no—
120#494—791no—
121#493—791no—
122#475—1581no—
123#460—1581no—
124C8×Q8—791no—
125C4⋊C16—791no—
126C4⋊C16—1581no—
127C4×C16—1581no—
128C4×C16C2×D7911yescharacteristic
129#335C2×D7911yescharacteristic
130#335C2×D7911yescharacteristic
131#332—1581no—
132#331—1581no—
133C2×Q16—1581no—
134C2×Q16—1581no—
135C2×SD16—3161no—
136C2×D8—1581no—
137C2×D8—1581no—
138C4⋊Q8—1581no—
139#127—1581no—
140C4×Q8—791no—
141SD32—3162no—
142SD32—3162no—
143C2×C16—1581no—
144C2×C16D31611yescharacteristic
145C2×C16D31611yescharacteristic
146#106—3161no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—791no—
150C4⋊C8—1581no—
151#103—3161no—
152#102—3161no—
153C4×C8—1581no—
154C4×C8C22×D7911yescharacteristic
155C2×Q8—3161no—
156C2×Q8—791no—
157C2×D4—3161no—
158C2×D4—3161no—
159C2×D4—3161no—
160Q16—3161no—
161Q16—3161no—
162Q16—6321no—
163SD16—6322no—
164D8—3161no—
165D8—3161no—
166D8—6321no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1581no—
170C2×C8C2×D31611yescharacteristic
171C2×C8#1192962511yescharacteristic
172C4⋊C4—3161no—
173C4⋊C4—3161no—
174C4⋊C4—791no—
175C4⋊C4—1581no—
176C42—1581no—
177C42D4×D7911yescharacteristic
178C16—21no—
179C16—3161no—
180C16#1192961312yescomplemented
181C23—6321no—
182Q8—792no—
183Q8—6321no—
184Q8—6321no—
185Q8—1581no—
186D4—6321no—
187D4—6321no—
188D4—6321no—
189D4—6321no—
190D4—12641no—
191C2×C4—6321no—
192C2×C4—1581no—
193C2×C4—1581no—
194C2×C4#1457913611yescharacteristic
195C2×C4D8×D7911yescharacteristic
196C2×C4#1457918411yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—3161no—
201C8C2×#1192961311yescharacteristic
202C8#1457918911yescharacteristic
203C22—12641no—
204C22—12641no—
205C22—12641no—
206C22#1909880011yescharacteristic
207C4—21no—
208C4—3161no—
209C4—1582no—
210C4—12641no—
211C4#1909876411yescharacteristic
212C4#1909883411yescharacteristic
213C4SD32×D7911yescharacteristic
214C4#1909899411yescharacteristic
215C2—25281no—
216C2#2376276111yescharacteristic
217C2#2376549711yescharacteristic
218C2#2376553111yescharacteristic
subgroupG/H
centerC22#19098800
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
FittingC79×#1372C2
socleC22×C79#466
prime radical#19098940C22
hypercenter#1372D79
hypocenterC79#12441
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
2. C16×Dic158C2semidirect
3. #23762771C2semidirect
5. #23761448C2semidirect
7. C79×#1372C2semidirect
70. C16×C79SD16semidirect
86. Dic158SD32semidirect
114. #1372D79semidirect
117. C79#12441semidirect
180. C16#11929613semidirect

Maximal subgroups

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

subgroupindex
#237656562
C16×Dic1582
#237627712
#237627612
#237614482
#237614452
C79×#13722
#1244179