Finite Group Databasebeta

← #34201071

Subgroups — #34201071

Order 20224.

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

normal subgroups only

This group has 484 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#23764564C211yesmaximal, characteristic
2#23762356C212yesmaximal, complemented
3#23761421C212yesmaximal, complemented
4#23761328C211yesmaximal, characteristic
5C79×#2840C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098719—22no—
8#19098419—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13#393×C79—22no—
14#393×C79—21no—
15#393×C79—22no—
16#393×C79C2212yes—
17#379×C79—21no—
18#379×C79C2211yescharacteristic
19C2×#11929614—42no—
20#14579140—22no—
21#14579138—42no—
22C2×C22⋊Dic79—21no—
23C2×C22⋊Dic79—21no—
24C2×C22⋊Dic79—42no—
25#14579085—42no—
26C22⋊(C79⋊C8)—22no—
27C25×C79D411yescharacteristic
28C22×Q8×C79—21no—
29C22×Q8×C79D412yes—
30C2×C22⋊C4×C79—21no—
31C2×C22⋊C4×C79—22no—
32C2×C22⋊C4×C79—21no—
33C2×C22⋊C4×C79—22no—
34C2×C22⋊C4×C79—22no—
35C2×C22⋊C4×C79D411yescharacteristic
36C2×C22⋊C4×C79C2311yescharacteristic
37#95×C79—21no—
38#95×C79—22no—
39#95×C79—22no—
40#95×C79D412yes—
41C2×Dic158—22no—
42C22×Dic79—22no—
43C22×Dic79—41no—
44#11929614—84no—
45C2×C79⋊C8—42no—
46C4⋊Dic79—22no—
47Dic79⋊C4—42no—
48C22⋊Dic79—41no—
49C22⋊Dic79—41no—
50C22⋊Dic79—41no—
51C22⋊Dic79—41no—
52C22⋊Dic79—84no—
53C24×C79—21no—
54C24×C79—21no—
55C24×C79—41no—
56C24×C79—41no—
57C24×C79—82no—
58C24×C79Q1612yescomplemented
59C24×C79C2×D411yescharacteristic
60C2×Q8×C79—42no—
61C2×Q8×C79—82no—
62C2×Q8×C79—82no—
63C2×Q8×C79—81no—
64C22×C4×C79—22no—
65C22×C4×C79—21no—
66C22×C4×C79—22no—
67C22×C4×C79C2×D412yes—
68C22⋊C4×C79—41no—
69C22⋊C4×C79—41no—
70C22⋊C4×C79—82no—
71C22⋊C4×C79—82no—
72C22⋊C4×C79—82no—
73C22⋊C4×C79—81no—
74C22⋊C4×C79—81no—
75C23×C79—21no—
76C23×C79—21no—
77C23×C79—22no—
78C23×C79—42no—
79C23×C79—42no—
80C23×C79—42no—
81C23×C79—42no—
82C23×C79—44no—
83C23×C79—41no—
84C23×C79—41no—
85C23×C79—81no—
86C23×C79—82no—
87C23×C79—82no—
88C23×C79—82no—
89C23×C79—84no—
90C23×C79#12011yescharacteristic
91C23×C79C2×Q1611yescharacteristic
92C23×C79#13711yescharacteristic
93Q8×C79—44no—
94Q8×C79—82no—
95Q8×C79—84no—
96C2×C4×C79—22no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×Dic79—42no—
107C2×Dic79—82no—
108C2×Dic79—81no—
109C2×Dic79—82no—
110Dic158—82no—
111C79⋊C8—82no—
112C22×C79—22no—
113C22×C79—21no—
114C22×C79—21no—
115C22×C79—41no—
116C22×C79—41no—
117C22×C79—42no—
118C22×C79—42no—
119C22×C79—42no—
120C22×C79—44no—
121C22×C79—42no—
122C22×C79—81no—
123C22×C79—82no—
124C22×C79—82no—
125C22×C79—84no—
126C22×C79—82no—
127C22×C79#45111yescharacteristic
128C22×C79#45511yescharacteristic
129C22×C79B(4,𝔽2)11yescharacteristic
130C4×C79—42no—
131C4×C79—81no—
132C4×C79—82no—
133C4×C79—82no—
134Dic79—82no—
135Dic79—161no—
136#12217—791nomaximal, Sylow
137C2×C79—21no—
138C2×C79—21no—
139C2×C79—41no—
140C2×C79—41no—
141C2×C79—82no—
142C2×C79#160211yescharacteristic
143C2×C79#219912yes—
144#2840D7911yescharacteristic
145#2029—791no—
146#1602—792no—
147#1351—792no—
148#1316—791no—
149C79#1221711yescharacteristic, complemented, Sylow
150#543—791no—
151#451—1582no—
152#393—22no—
153#393—21no—
154#393—22no—
155#393C2×D7912yes—
156#379—21no—
157#379—1581no—
158#379C2×D7911yescharacteristic
159#342—791no—
160#342—1581no—
161#328—1582no—
162#323—792no—
163C25C22⋊D7911yescharacteristic
164C22×Q8—21no—
165C22×Q8C22⋊D7912yes—
166C2×Q16—3162no—
167C22⋊Q8—3162no—
168C22⋊Q8—1582no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—3162no—
175C2×C22⋊C4—1581no—
176C2×C22⋊C4—1581no—
177C2×C22⋊C4C22⋊D7911yescharacteristic
178C2×C22⋊C4C22×D7911yescharacteristic
179#103—3162no—
180C22⋊C8—1582no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95C22⋊D7912yes—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24#1192961412yescomplemented
191C24C2×C22⋊D7911yescharacteristic
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—81no—
196C2×Q8—1582no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—3161no—
201C22×C4—1582no—
202C22×C4C2×C22⋊D7912yes—
203Q16—6324no—
204C2×C8—3162no—
205C4⋊C4—3162no—
206C4⋊C4—1582no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—3161no—
215C22⋊C4—3161no—
216C22⋊C4—3161no—
217C22⋊C4—3161no—
218C22⋊C4—6324no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23#1457913011yescharacteristic
235C23C2×#1192961411yescharacteristic
236C23#1457919211yescharacteristic
237Q8—44no—
238Q8—82no—
239Q8—84no—
240Q8—6322no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—3162no—
252C2×C4—6322no—
253C2×C4—6321no—
254C2×C4—6322no—
255C8—6322no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22#1909871911yescharacteristic
272C22#1909874111yescharacteristic
273C22#1909875011yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—6322no—
279C4—12641no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2#2376235711yescharacteristic
286C2#2376540112yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2840C2
socleC22×C79B(4,𝔽2)
prime radicalC25×C79D4
hypercenter#2840D79
hypocenterC79#12217
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #23762356C2semidirect
3. #23761421C2semidirect
58. C24×C79Q16semidirect
149. C79#12217semidirect
190. C24#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645642
#237623562
#237614212
#237613282
C79×#28402
#1221779