Finite Group Databasebeta

← #34202556

Subgroups — #34202556

Order 20224.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#23765657C211yesmaximal, characteristic
2#23765305C211yesmaximal, characteristic, complemented
3#23762767C211yesmaximal, characteristic, complemented
4#23762761C211yesmaximal, characteristic
5#23761471C211yesmaximal, characteristic, complemented
6#23761447C211yesmaximal, characteristic
7C79×#1362C211yesmaximal, characteristic, complemented
8C2×#14579104—21no—
9C2×#14579104—21no—
10#19098940C2211yescharacteristic
11#19098845—21no—
12#19098764—21no—
13C8×Dic158C2211yescharacteristic
14Dic79⋊C16—21no—
15#19098338—21no—
16C4×C79⋊C16C2211yescharacteristic
17#19098316C2211yescharacteristic
18#19098309—21no—
19#19098306—21no—
20#494×C79C2211yescharacteristic
21C4⋊C16×C79C2211yescharacteristic
22#335×C79C2211yescharacteristic
23C2×#11929613—41no—
24C2×D632—21no—
25C2×D632—21no—
26#14579158—21no—
27C4×Dic158D411yescharacteristic
28#14579104—42no—
29#14579104—42no—
30C2×C79⋊C16D411yescharacteristic
31C2×C79⋊C16D411yescharacteristic
32#14579092—41no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)D411yescharacteristic
35Dic79⋊C8—21no—
36#14579084—41no—
37#14579083—41no—
38C8×Dic79—21no—
39C2×Q16×C79—21no—
40C2×Q16×C79—21no—
41C4⋊Q8×C79—21no—
42C2×C16×C79—21no—
43C4⋊C8×C79—21no—
44C4×C8×C79C2311yescharacteristic
45C2×Dic158D811yescharacteristic
46C2×D316—41no—
47C2×D316—41no—
48C2×D316—41no—
49#11929613—82no—
50D632—41no—
51D632—41no—
52D632—81no—
53C2×C79⋊C8—21no—
54C2×C79⋊C8—41no—
55C4⋊Dic79D811yescharacteristic
56Dic79⋊C4—21no—
57C4×Dic79—21no—
58C79⋊C16—21no—
59C79⋊C16SD1612yescomplemented
60C2×Q8×C79—41no—
61Q16×C79—41no—
62Q16×C79—41no—
63Q16×C79—81no—
64C2×C8×C79—41no—
65C2×C8×C79C2×D411yescharacteristic
66C2×C8×C79#4111yescharacteristic
67C4⋊C4×C79—41no—
68C4⋊C4×C79—41no—
69C42×C79C2×D411yescharacteristic
70C16×C79—41no—
71C22×D79—81no—
72Q8×C79—81no—
73Q8×C79—81no—
74C2×C4×C79—81no—
75C2×C4×C79#12111yescharacteristic
76C2×C4×C79C2×D811yescharacteristic
77C2×C4×C79#13511yescharacteristic
78C2×Dic79—21no—
79C2×Dic79—21no—
80D316—81no—
81D316—81no—
82D316—81no—
83D316—81no—
84D316—161no—
85Dic158—21no—
86Dic158—21no—
87C8×C79—21no—
88C8×C79—81no—
89C8×C79C2×SD1611yescharacteristic
90C8×C79AGL(1,ℤ/8)11yescharacteristic
91C79⋊C8—41no—
92C79⋊C8—81no—
93C22×C79#46611yescharacteristic
94C2×D79—161no—
95C2×D79—161no—
96C2×D79—161no—
97C4×C79—21no—
98C4×C79—161no—
99C4×C79#46011yescharacteristic
100C4×C79#47211yescharacteristic
101C4×C79#50911yescharacteristic
102C4×C79#51011yescharacteristic
103Dic79—41no—
104Dic79—41no—
105#12444—791nomaximal, Sylow
106C2×C79#166811yescharacteristic
107C2×C79#222011yescharacteristic
108C2×C79#222311yescharacteristic
109D79—321no—
110#2248—791no—
111#2183—791no—
112#1670—791no—
113#1668—791no—
114#1371—791no—
115#1362D7911yescharacteristic, complemented
116#1361—791no—
117C79#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1581no—
119C2×SD32—1581no—
120#494C2×D7911yescharacteristic
121#493—791no—
122#475—1581no—
123#460—1581no—
124C8×Q8—791no—
125C4⋊C16—1581no—
126C4⋊C16C2×D7911yescharacteristic
127#346—1581no—
128C4×C16—791no—
129#335—791no—
130#335C2×D7911yescharacteristic
131#332—1581no—
132#331—1581no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—3161no—
136C2×D8—1581no—
137C2×D8—1581no—
138C4⋊Q8—21no—
139#127—1581no—
140C4×Q8—791no—
141SD32—3162no—
142SD32—3162no—
143C2×C16—21no—
144C2×C16—791no—
145C2×C16—791no—
146#106—3161no—
147C4⋊C8—21no—
148C4⋊C8—791no—
149C4⋊C8—1581no—
150C4⋊C8—1581no—
151#103—3161no—
152#102—3161no—
153C4×C8—1581no—
154C4×C8C22×D7911yescharacteristic
155C2×Q8—41no—
156C2×Q8—791no—
157C2×D4—3161no—
158C2×D4—3161no—
159C2×D4—3161no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—6322no—
164D8—3161no—
165D8—3161no—
166D8—6321no—
167C2×C8—41no—
168C2×C8—3161no—
169C2×C8—1581no—
170C2×C8D4×D7911yescharacteristic
171C2×C8#1192962411yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—791no—
175C4⋊C4—1581no—
176C42—1581no—
177C42D4×D7911yescharacteristic
178C16—41no—
179C16—792no—
180C16—1581no—
181C23—6321no—
182Q8—81no—
183Q8—81no—
184Q8—1581no—
185Q8—1581no—
186D4—6321no—
187D4—6321no—
188D4—6321no—
189D4—6321no—
190D4—12641no—
191C2×C4—81no—
192C2×C4—1581no—
193C2×C4—1581no—
194C2×C4#1457913411yescharacteristic
195C2×C4D8×D7911yescharacteristic
196C2×C4#1457918211yescharacteristic
197C8—21no—
198C8—81no—
199C8—3161no—
200C8—6321no—
201C8SD16×D7911yescharacteristic
202C8#1457918511yescharacteristic
203C22—12641no—
204C22—12641no—
205C22—12641no—
206C22#1909879811yescharacteristic
207C4—21no—
208C4—161no—
209C4—3161no—
210C4—3161no—
211C4#1909876211yescharacteristic
212C4#1909883211yescharacteristic
213C4#1909900011yescharacteristic
214C4#1909900511yescharacteristic
215C2—25281no—
216C2#2376275911yescharacteristic
217C2#2376551611yescharacteristic
218C2#2376553311yescharacteristic
subgroupG/H
centerC22#19098798
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
FittingC79×#1362C2
socleC22×C79#466
prime radical#19098940C22
hypercenter#1362D79
hypocenterC79#12444
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
2. #23765305C2semidirect
3. #23762767C2semidirect
5. #23761471C2semidirect
7. C79×#1362C2semidirect
59. C79⋊C16SD16semidirect
115. #1362D79semidirect
117. C79#12444semidirect

Maximal subgroups

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

subgroupindex
#237656572
#237653052
#237627672
#237627612
#237614712
#237614472
C79×#13622
#1244479