Finite Group Databasebeta

← #34176027

Subgroups — #34176027

Order 20224.

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

normal subgroups only

This group has 276 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
1C2×#19098362C211yesmaximal, characteristic
2#23763794C211yesmaximal, characteristic
3C2×#333×C79C211yesmaximal, characteristic
4#19099041C411yescharacteristic
5#19098572—21no—
6C2×#14579080—21no—
7C2×#14579080C411yescharacteristic
8#19098362—21no—
9#19098362C412yescomplemented
10#19098323—21no—
11C2×C4⋊Q8×C79C2211yescharacteristic
12C2×C4⋊C8×C79—21no—
13#333×C79—21no—
14#333×C79C412yescomplemented
15C2×#11929608—21no—
16C2×#11929608—41no—
17Q8×Dic79—42no—
18#14579120—21no—
19#14579120—41no—
20#14579085—42no—
21#14579080—22no—
22#14579080—41no—
23#14579080—41no—
24C22×Q8×C79C2×C411yescharacteristic
25C22×Q8×C79D411yescharacteristic
26C22×C8×C79—41no—
27C4⋊Q8×C79—41no—
28C4⋊Q8×C79C2×C411yescharacteristic
29C4⋊Q8×C79C2×C411yescharacteristic
30C4⋊Q8×C79D411yescharacteristic
31C4⋊Q8×C79Q811yescharacteristic
32C2×C4⋊C4×C79—41no—
33C2×C42×C79D411yescharacteristic
34C4⋊C8×C79—22no—
35C4⋊C8×C79—41no—
36#11929608—42no—
37#11929608—41no—
38#11929608—41no—
39#11929608—81no—
40C2×C79⋊C8—41no—
41C2×C79⋊C8—81no—
42C4⋊Dic79—22no—
43C4⋊Dic79—42no—
44C22⋊Dic79—41no—
45C4×Dic79—41no—
46C4×Dic79—42no—
47C2×Q8×C79—21no—
48C2×Q8×C79—21no—
49C2×Q8×C79—41no—
50C2×Q8×C79—41no—
51C2×Q8×C79—41no—
52C2×Q8×C79—41no—
53C2×Q8×C79C4211yescharacteristic
54C2×Q8×C79C22⋊C411yescharacteristic
55C2×Q8×C79C4⋊C411yescharacteristic
56C2×Q8×C79C4⋊C411yescharacteristic
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79C22⋊C411yescharacteristic
61C2×C8×C79—42no—
62C2×C8×C79—82no—
63C4⋊C4×C79—41no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—41no—
66C4⋊C4×C79—41no—
67C42×C79—21no—
68C42×C79C22⋊C411yescharacteristic
69C42×C79C4⋊C411yescharacteristic
70C23×C79#9911yescharacteristic
71Q8×C79—41no—
72Q8×C79—41no—
73Q8×C79—41no—
74Q8×C79—41no—
75Q8×C79—42no—
76Q8×C79—81no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—41no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—41no—
91C2×C4×C79#9511yescharacteristic
92C2×C4×C79#9911yescharacteristic
93C2×Dic79—42no—
94C8×C79—82no—
95C79⋊C8—81no—
96C79⋊C8—82no—
97C22×C79—22no—
98C22×C79#34211yescharacteristic
99C22×C79AΣL(1,𝔽16)11yescharacteristic
100C22×C79#35211yescharacteristic
101C4×C79—22no—
102C4×C79—21no—
103C4×C79—21no—
104C4×C79—21no—
105C4×C79—21no—
106C4×C79—41no—
107C4×C79—41no—
108C4×C79—41no—
109C4×C79—41no—
110Dic79—82no—
111#7491—791nomaximal, Sylow
112C2×C79—21no—
113C2×C79—21no—
114C2×C79#139011yescharacteristic
115C2×C79#140512yes—
116C2×#356—791no—
117#1886—791no—
118C2×#333D7911yescharacteristic
119C79#749111yescharacteristic, complemented, Sylow
120C2×C4⋊Q8C2×D7911yescharacteristic
121#519—791no—
122C2×C4⋊C8—21no—
123#419—1581no—
124C2×#101—791no—
125C2×#101—1581no—
126#356—792no—
127#356—1581no—
128#339—1581no—
129#333—21no—
130#333Dic7912yescomplemented
131C22×Q8C4×D7911yescharacteristic
132C22×Q8D31611yescharacteristic
133C2×M16—3161no—
134C2×M16—1581no—
135C22×C8—41no—
136C4⋊Q8—41no—
137C4⋊Q8Dic15811yescharacteristic
138C4⋊Q8C4×D7911yescharacteristic
139C4⋊Q8C2×Dic7911yescharacteristic
140C4⋊Q8C22⋊D7911yescharacteristic
141C4×Q8—3162no—
142#117—3161no—
143#117—1581no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D7911yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—41no—
148#103—3162no—
149#101—3161no—
150#101—3161no—
151#101—1582no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C4×Dic7911yescharacteristic
159C2×Q8(C2×D79)⋊C411yescharacteristic
160C2×Q8Dic79⋊C411yescharacteristic
161C2×Q8C4⋊Dic7911yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4(C2×D79)⋊C411yescharacteristic
166M16—3162no—
167M16—3161no—
168M16—3161no—
169M16—6321no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8—3161no—
173C2×C8—6321no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—3162no—
179C4⋊C4—1582no—
180C22⋊C4—3161no—
181C42—21no—
182C42—3161no—
183C42—3162no—
184C42C22⋊Dic7911yescharacteristic
185C42Dic79⋊C411yescharacteristic
186C23#1457907511yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3162no—
208C2×C4C79⋊#9511yescharacteristic
209C2×C4#1457907511yescharacteristic
210C8—82no—
211C8—6321no—
212C8—6322no—
213C22—22no—
214C22#1909832911yescharacteristic
215C22#1909835111yescharacteristic
216C22#1909835411yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—6322no—
227C2—21no—
228C2—21no—
229C2#2376151911yescharacteristic
230C2#2376155512yes—
subgroupG/H
centerC22#19098351
commutatorC2×Q8×C79C42
FrattiniC2×C4⋊Q8C2×D79
FittingC2×#333×C79C2
socleC22×C79AΣL(1,𝔽16)
prime radicalC23×C79#99
cosocleC2×C4⋊Q8×C79C22
hypercenterC2×#333D79
hypocenterC79#7491
LCSG ▷ C2×Q8×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
9. #19098362C4semidirect
14. #333×C79C4semidirect
119. C79#7491semidirect
130. #333Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983622
#237637942
C2×#333×C792
#749179