Finite Group Databasebeta

← #34201599

Subgroups — #34201599

Order 20224.

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

normal subgroups only

This group has 324 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#23764916C211yesmaximal, characteristic
3#23761394C212yesmaximal, complemented
4#23761336C211yesmaximal, characteristic
5C79×#3195C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098611—21no—
8#19098527—21no—
9#19098385—22no—
10#19098379—22no—
11#19098345C2212yes—
12C79⋊#338C2211yescharacteristic
13C2×C22⋊Q8×C79—21no—
14#483×C79—22no—
15#470×C79—21no—
16#470×C79C2212yes—
17#451×C79—22no—
18#407×C79C2211yescharacteristic
19C2×#11929608—21no—
20C2×#11929608—21no—
21#14579101—22no—
22#14579097—22no—
23#14579096—42no—
24C22⋊(C79⋊C8)—42no—
25C22×Q8×C79—21no—
26C23×C4×C79D411yescharacteristic
27C2×Q16×C79—21no—
28C2×Q16×C79D412yes—
29C2×M16×C79D411yescharacteristic
30C2×M16×C79C2311yescharacteristic
31C22⋊Q8×C79—22no—
32C22⋊Q8×C79—22no—
33C22⋊Q8×C79—41no—
34C22⋊Q8×C79—41no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—41no—
37C2×C22⋊C4×C79—41no—
38#106×C79—21no—
39#106×C79D412yes—
40#103×C79—22no—
41#103×C79—22no—
42C22⋊C8×C79—22no—
43#11929608—41no—
44#11929608—41no—
45#11929608—41no—
46#11929608—41no—
47C2×C79⋊C8—22no—
48C2×C79⋊C8—41no—
49C79⋊C16—42no—
50C24×C79SD1611yescharacteristic
51C2×Q8×C79—22no—
52C2×Q8×C79—41no—
53C2×Q8×C79—81no—
54C2×Q8×C79—81no—
55C22×C4×C79—21no—
56C22×C4×C79—21no—
57C22×C4×C79—41no—
58C22×C4×C79—41no—
59C22×C4×C79—41no—
60C22×C4×C79—41no—
61C22×C4×C79SD1611yescharacteristic
62C22×C4×C79C2×D411yescharacteristic
63Q16×C79—42no—
64Q16×C79—81no—
65M16×C79—41no—
66M16×C79—41no—
67C2×C8×C79—21no—
68C2×C8×C79C2×D412yes—
69C4⋊C4×C79—22no—
70C4⋊C4×C79—41no—
71C4⋊C4×C79—42no—
72C4⋊C4×C79—42no—
73C22⋊C4×C79—42no—
74C22⋊C4×C79—42no—
75C23×C79—21no—
76C23×C79—41no—
77C23×C79—42no—
78C23×C79C2×SD1611yescharacteristic
79Q8×C79—42no—
80Q8×C79—42no—
81Q8×C79—82no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—41no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—81no—
91C2×C4×C79—82no—
92C2×C4×C79—82no—
93C2×C4×C79#12011yescharacteristic
94C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
95C8×C79—22no—
96C8×C79—41no—
97C79⋊C8—42no—
98C79⋊C8—81no—
99C22×C79—22no—
100C22×C79—21no—
101C22×C79—41no—
102C22×C79—41no—
103C22×C79—42no—
104C22×C79—41no—
105C22×C79—81no—
106C22×C79#45011yescharacteristic
107C4×C79—21no—
108C4×C79—41no—
109C4×C79—82no—
110C4×C79—82no—
111C4×C79#45311yescharacteristic
112C4×C79#45811yescharacteristic
113#12299—791nomaximal, Sylow
114C2×C79—21no—
115C2×C79—42no—
116C2×C79—41no—
117C2×C79#161311yescharacteristic
118#3195D7911yescharacteristic
119#2239—792no—
120#2091—791no—
121#1342—792no—
122#1320—791no—
123C79#1229911yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—21no—
125#483—22no—
126#470—21no—
127#470C2×D7912yes—
128#451—22no—
129#430—791no—
130#430—1581no—
131#407—1581no—
132#407C2×D7911yescharacteristic
133#365—1582no—
134#362—1582no—
135#349—792no—
136#338—791no—
137C22×Q8—21no—
138C23×C4C22⋊D7911yescharacteristic
139C2×Q16—21no—
140C2×Q16C22⋊D7912yes—
141C2×M16—1581no—
142C2×M16—1581no—
143C2×M16C22⋊D7911yescharacteristic
144C2×M16C22×D7911yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152M32—1582no—
153#108—3162no—
154#108—1582no—
155#106—21no—
156#106C22⋊D7912yes—
157#103—22no—
158#103—22no—
159C22⋊C8—22no—
160C22⋊C8—3162no—
161C24#1192961211yescharacteristic
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4#1192961211yescharacteristic
173C22×C4C2×C22⋊D7911yescharacteristic
174Q16—42no—
175Q16—81no—
176M16—41no—
177M16—41no—
178M16—3161no—
179M16—3161no—
180M16—3161no—
181M16—3161no—
182C2×C8—21no—
183C2×C8—3161no—
184C2×C8—1582no—
185C2×C8C2×C22⋊D7912yes—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C16—3162no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1192961211yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#1457913011yescharacteristic
212C2×C4#1457918711yescharacteristic
213C8—22no—
214C8—41no—
215C8—3162no—
216C8—6321no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1909871411yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4#1909873111yescharacteristic
230C4#1909875311yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2376242011yescharacteristic
subgroupG/H
centerC2#23762420
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#3195C2
socleC2×C79#1613
prime radicalC24×C79SD16
hypercenter#3195D79
hypocenterC79#12299
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. #23765617C2semidirect
3. #23761394C2semidirect
123. C79#12299semidirect

Maximal subgroups

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

subgroupindex
#237656172
#237649162
#237613942
#237613362
C79×#31952
#1229979