Finite Group Databasebeta

← #34202522

Subgroups — #34202522

Order 20224.

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

normal subgroups only

This group has 224 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#23765657C211yesmaximal, characteristic, complemented
2#23765478C211yesmaximal, characteristic, complemented
3#23765477C211yesmaximal, characteristic
4#23765236C211yesmaximal, characteristic, complemented
5#23761475C211yesmaximal, characteristic, complemented
6#23761436C211yesmaximal, characteristic, complemented
7C79×#1358C211yesmaximal, characteristic, complemented
8#19099004—21no—
9#19099001—21no—
10C2×#14579104—21no—
11C2×#14579104—21no—
12#19098981—21no—
13#19098940C2211yescharacteristic
14#19098681C2211yescharacteristic
15#19098384C2211yescharacteristic
16#19098379—21no—
17#19098377—21no—
18#19098348—21no—
19C4×C79⋊C16C2211yescharacteristic
20#494×C79C2211yescharacteristic
21#364×C79C2211yescharacteristic
22#364×C79C2211yescharacteristic
23#14579183D411yescharacteristic
24C2×D632—21no—
25C2×D632—21no—
26#14579167—21no—
27#14579158—21no—
28#14579107—41no—
29#14579106—41no—
30#14579104—42no—
31#14579104—41no—
32#14579104—41no—
33#14579104—41no—
34D1264—41no—
35#14579101—21no—
36#14579100—41no—
37C2×C79⋊C16D411yescharacteristic
38C2×C79⋊C16D411yescharacteristic
39C16×D79—41no—
40#14579097D411yescharacteristic
41#14579089—21no—
42C2×Q16×C79—21no—
43C2×Q16×C79—21no—
44C4⋊Q8×C79—21no—
45M32×C79—21no—
46M32×C79—21no—
47C4×C8×C79C2311yescharacteristic
48#11929624—21no—
49C2×D316—41no—
50C2×D316—81no—
51Dic316D811yescharacteristic
52#11929613—21no—
53D632—41no—
54D632—41no—
55D632—41no—
56D632—41no—
57D632D811yescharacteristic
58#11929608—21no—
59#11929607—41no—
60C8×D79—41no—
61C79⋊C16—21no—
62C79⋊C16—41no—
63C79⋊C16D811yescharacteristic
64C79⋊C16D811yescharacteristic
65C2×Q8×C79—41no—
66Q16×C79—41no—
67Q16×C79—41no—
68Q16×C79—41no—
69Q16×C79—41no—
70C2×C8×C79C2×D411yescharacteristic
71C2×C8×C79C2×D411yescharacteristic
72C4⋊C4×C79—81no—
73C42×C79#4111yescharacteristic
74C16×C79—41no—
75C16×C79—41no—
76C22×D79—81no—
77Q8×C79—81no—
78Q8×C79—81no—
79C2×C4×C79—21no—
80C2×C4×C79—81no—
81C2×C4×C79#12111yescharacteristic
82C22⋊D79—41no—
83D316—21no—
84D316—81no—
85D316—81no—
86D316—162no—
87C4×D79—41no—
88Dic158—21no—
89C8×C79C2×D811yescharacteristic
90C8×C79C2×D811yescharacteristic
91C8×C79#13511yescharacteristic
92C8×C79AGL(1,ℤ/8)11yescharacteristic
93C79⋊C8—41no—
94C22×C79#47211yescharacteristic
95C2×D79—41no—
96C2×D79—162no—
97C2×D79—161no—
98C4×C79—22no—
99C4×C79—161no—
100C4×C79#45911yescharacteristic
101C4×C79#46611yescharacteristic
102Dic79—41no—
103#12438—791nomaximal, Sylow
104C2×C79—21no—
105C2×C79#166711yescharacteristic
106D79—81no—
107D79—321no—
108#2248—791no—
109#2214—791no—
110#2213—791no—
111#2170—791no—
112#1373—791no—
113#1358D7911yescharacteristic, complemented
114#1357—791no—
115C79#1243811yescharacteristic, complemented, Sylow
116#510—1581no—
117#509—1581no—
118C2×SD32—1581no—
119C2×SD32—1581no—
120#504—1581no—
121#494C2×D7911yescharacteristic
122#493—791no—
123#443—791no—
124#364—791no—
125#364C2×D7911yescharacteristic
126#364C2×D7911yescharacteristic
127#362—1581no—
128#361—1581no—
129#350—1581no—
130C4×C16—791no—
131#135—791no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1581no—
135C2×D8—1581no—
136#131—1581no—
137C4⋊Q8—21no—
138#127—1581no—
139Q32—3161no—
140SD32—3161no—
141SD32—3161no—
142SD32—3161no—
143SD32—3161no—
144SD32—3162no—
145D16—3161no—
146M32—21no—
147M32—21no—
148M32—3161no—
149M32—1581no—
150C2×C16—3161no—
151C2×C16—791no—
152C2×C16—791no—
153#108—791no—
154#104—1581no—
155C4×C8C22×D7911yescharacteristic
156#41—1581no—
157C2×Q8—41no—
158C2×D4—3161no—
159C2×D4—6321no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16—791no—
165SD16—1581no—
166D8—3161no—
167D8—3161no—
168D8—3161no—
169D8—3161no—
170D8—791no—
171M16—3161no—
172M16—1581no—
173C2×C8—3161no—
174C2×C8D4×D7911yescharacteristic
175C2×C8D4×D7911yescharacteristic
176C4⋊C4—81no—
177C42#1192962411yescharacteristic
178C16—41no—
179C16—41no—
180C16—3161no—
181C16—791no—
182C16—791no—
183C16—1581no—
184C23—6321no—
185Q8—81no—
186Q8—81no—
187Q8—1581no—
188D4—3161no—
189D4—6321no—
190D4—6321no—
191D4—1581no—
192D4—12642no—
193C2×C4—21no—
194C2×C4—81no—
195C2×C4—3161no—
196C2×C4#1457913411yescharacteristic
197C8—3161no—
198C8D8×D7911yescharacteristic
199C8D8×D7911yescharacteristic
200C8#1457918211yescharacteristic
201C8#1457918811yescharacteristic
202C22—3161no—
203C22—12642no—
204C22—12641no—
205C22#1909883211yescharacteristic
206C4—22no—
207C4—161no—
208C4—3161no—
209C4#1909875711yescharacteristic
210C4#1909879811yescharacteristic
211C2—21no—
212C2—6321no—
213C2—25281no—
214C2#2376275411yescharacteristic
subgroupG/H
centerC2#23762754
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
FittingC79×#1358C2
socleC2×C79#1667
prime radical#23765477C2
hypercenter#1358D79
hypocenterC79#12438
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
1. #23765657C2semidirect
2. #23765478C2semidirect
4. #23765236C2semidirect
5. #23761475C2semidirect
6. #23761436C2semidirect
7. C79×#1358C2semidirect
113. #1358D79semidirect
115. C79#12438semidirect

Maximal subgroups

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

subgroupindex
#237656572
#237654782
#237654772
#237652362
#237614752
#237614362
C79×#13582
#1243879