Finite Group Databasebeta

← #34201616

Subgroups — #34201616

Order 20224.

310 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765617C212yesmaximal, complemented
2#23764953C211yesmaximal, characteristic
3#23761388C212yesmaximal, complemented
4#23761338C211yesmaximal, characteristic
5C79×#3202C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098589—21no—
8#19098580—21no—
9#19098385—22no—
10#19098379—22no—
11#19098345C2212yes—
12#19098320C2211yescharacteristic
13#536×C79—21no—
14#485×C79—22no—
15#470×C79C2212yes—
16#467×C79—21no—
17#451×C79—22no—
18#408×C79C2211yescharacteristic
19C2×#11929608—21no—
20#14579120—21no—
21#14579101—22no—
22#14579097—22no—
23Dic79⋊C8—42no—
24#14579088—42no—
25C2×#41×C79D411yescharacteristic
26C22×Q8×C79—21no—
27C2×Q16×C79—21no—
28C2×Q16×C79D412yes—
29C2×M16×C79C2311yescharacteristic
30C22×C8×C79D411yescharacteristic
31C4⋊Q8×C79—41no—
32#124×C79—41no—
33#123×C79—41no—
34#123×C79—41no—
35C22⋊Q8×C79—22no—
36C22⋊Q8×C79—22no—
37#117×C79—21no—
38#107×C79—21no—
39#106×C79D412yes—
40#103×C79—22no—
41#103×C79—22no—
42C22⋊C8×C79—22no—
43#11929608—41no—
44#11929608—41no—
45C2×C79⋊C8—41no—
46C4⋊Dic79—41no—
47C22⋊Dic79—41no—
48C4×Dic79—22no—
49C79⋊C16—42no—
50#41×C79—41no—
51#41×C79—41no—
52C2×Q8×C79—22no—
53C2×Q8×C79—41no—
54C2×Q8×C79—81no—
55C2×Q8×C79—81no—
56C2×Q8×C79SD1611yescharacteristic
57C2×D4×C79—21no—
58C2×D4×C79SD1611yescharacteristic
59C22×C4×C79—21no—
60C22×C4×C79—41no—
61C22×C4×C79C2×D411yescharacteristic
62Q16×C79—42no—
63Q16×C79—81no—
64M16×C79—41no—
65C2×C8×C79—21no—
66C2×C8×C79—41no—
67C2×C8×C79C2×D412yes—
68C4⋊C4×C79—22no—
69C4⋊C4×C79—42no—
70C4⋊C4×C79—42no—
71C22⋊C4×C79—41no—
72C22⋊C4×C79—42no—
73C22⋊C4×C79—42no—
74C42×C79—41no—
75C23×C79—21no—
76C23×C79AGL(1,ℤ/8)11yescharacteristic
77Q8×C79—41no—
78Q8×C79—42no—
79Q8×C79—42no—
80Q8×C79—82no—
81D4×C79—41no—
82D4×C79—81no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—82no—
91C2×C4×C79#12011yescharacteristic
92C2×C4×C79C2×SD1611yescharacteristic
93C2×Dic79—42no—
94C8×C79—22no—
95C8×C79—41no—
96C79⋊C8—81no—
97C22×C79—21no—
98C22×C79—41no—
99C22×C79—42no—
100C22×C79—41no—
101C22×C79#45011yescharacteristic
102C4×C79—21no—
103C4×C79—41no—
104C4×C79—82no—
105C4×C79—82no—
106C4×C79#45311yescharacteristic
107C4×C79#45811yescharacteristic
108Dic79—82no—
109#12302—791nomaximal, Sylow
110C2×C79—21no—
111C2×C79—41no—
112C2×C79—81no—
113C2×C79#161311yescharacteristic
114#3202D7911yescharacteristic
115#2239—792no—
116#2098—791no—
117#1340—792no—
118#1321—791no—
119C79#1230211yescharacteristic, complemented, Sylow
120#536—21no—
121#485—22no—
122#470C2×D7912yes—
123#467—21no—
124#451—22no—
125#430—791no—
126#424—1581no—
127#421—1581no—
128#408C2×D7911yescharacteristic
129#365—1582no—
130#362—1582no—
131#349—792no—
132H(3,ℤ/4)—791no—
133C2×#41C22⋊D7911yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16C22⋊D7912yes—
137C2×M16—1581no—
138C2×M16C22×D7911yescharacteristic
139C22×C8C22⋊D7911yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—1581no—
148M32—1582no—
149#108—1582no—
150#107—21no—
151#106C22⋊D7912yes—
152C4⋊C8—3162no—
153#104—3162no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#1192961211yescharacteristic
164C2×D4—21no—
165C2×D4#1192961211yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D7911yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—3161no—
173M16—3161no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—3161no—
177C2×C8C2×C22⋊D7912yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—3161no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—3161no—
186C42—41no—
187C42—1582no—
188C16—3162no—
189C23—21no—
190C23#1457918711yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—3162no—
206C2×C4#1457913011yescharacteristic
207C2×C4C2×#1192961211yescharacteristic
208C8—22no—
209C8—41no—
210C8—6321no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1909871411yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—6322no—
221C4#1909873111yescharacteristic
222C4#1909875311yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#2376242011yescharacteristic
subgroupG/H
centerC2#23762420
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#3202C2
socleC2×C79#1613
prime radicalC2×#41×C79D4
hypercenter#3202D79
hypocenterC79#12302
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #23765617C2semidirect
3. #23761388C2semidirect
119. C79#12302semidirect

Maximal subgroups

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

subgroupindex
#237656172
#237649532
#237613882
#237613382
C79×#32022
#1230279