Finite Group Databasebeta

← #34205102

Subgroups — #34205102

Order 20224.

492 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 95 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#23766512C211yesmaximal, characteristic
2C2×#19098360C211yesmaximal, characteristic
3#23763615C212yesmaximal, complemented
4#23761526C212yesmaximal, complemented
5C79×#1888C211yesmaximal, characteristic
6#19098606—22no—
7C2×#14579080C2212yes—
8#19098470C2212yes—
9#19098456—22no—
10#19098443—21no—
11C22⋊C4×Dic79—22no—
12#19098409—21no—
13#19098360C414yescomplemented
14#19098322—22no—
15C2×#124×C79C2211yescharacteristic
16#519×C79C2211yescharacteristic
17#418×C79—21no—
18C2×#101×C79C2211yescharacteristic
19#339×C79—21no—
20C23×Dic79—21no—
21C2×#11929608—22no—
22C2×C4⋊Dic79D412yes—
23C2×C22⋊Dic79—22no—
24C2×C22⋊Dic79—21no—
25C2×C4×Dic79—22no—
26C2×C4×Dic79D412yes—
27#14579094—42no—
28#14579092—42no—
29#14579080—22no—
30#14579080C2×C414yescomplemented
31C79⋊#95—21no—
32C79⋊#95—21no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×Q8×C79C2311yescharacteristic
36C22×D4×C79D411yescharacteristic
37C2×M16×C79—21no—
38#124×C79—22no—
39#124×C79C2×C412yes—
40#124×C79Q812yes—
41C4×Q8×C79—42no—
42#117×C79—21no—
43#117×C79—22no—
44C2×C22⋊C4×C79—22no—
45C2×C42×C79D411yescharacteristic
46#102×C79—42no—
47#101×C79—22no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50C22×Dic79—44no—
51C22×Dic79—81no—
52#11929608—24no—
53#11929608—42no—
54C2×C79⋊C8—42no—
55C4⋊Dic79—42no—
56C22⋊Dic79—44no—
57C22⋊Dic79—81no—
58C4×Dic79—42no—
59C4×Dic79—44no—
60C24×C79—21no—
61C2×Q8×C79—41no—
62C2×Q8×C79—41no—
63C2×Q8×C79C22×C411yescharacteristic
64C2×Q8×C79C2×Q811yescharacteristic
65C2×Q8×C79#4112yes—
66C2×D4×C79—21no—
67C2×D4×C79—41no—
68C2×D4×C79—41no—
69C2×D4×C79C4⋊C412yes—
70C22×C4×C79—21no—
71C22×C4×C79C2×D411yescharacteristic
72C22×C4×C79C2×D412yes—
73M16×C79—41no—
74M16×C79—41no—
75C2×C8×C79—41no—
76C4⋊C4×C79—22no—
77C4⋊C4×C79—22no—
78C4⋊C4×C79—22no—
79C22⋊C4×C79—24no—
80C22⋊C4×C79—24no—
81C22⋊C4×C79—41no—
82C42×C79—21no—
83C42×C79—42no—
84C42×C79—41no—
85C42×C79C4⋊C412yes—
86C23×C79—22no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79#12011yescharacteristic
92Q8×C79—41no—
93Q8×C79—41no—
94Q8×C79—42no—
95D4×C79—42no—
96D4×C79—81no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79C2×C4⋊C411yescharacteristic
105C2×C4×C79C4×D412yes—
106C2×C4×C79C22⋊Q812yes—
107C2×C4×C79#12311yescharacteristic
108C2×Dic79—24no—
109C2×Dic79—42no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—84no—
115C8×C79—81no—
116C79⋊C8—44no—
117C22×C79—22no—
118C22×C79—44no—
119C22×C79—42no—
120C22×C79—81no—
121C22×C79—81no—
122C22×C79—82no—
123C22×C79#38511yescharacteristic
124C22×C79B(4,𝔽2)11yescharacteristic
125C22×C79#45811yescharacteristic
126C4×C79—21no—
127C4×C79—22no—
128C4×C79—22no—
129C4×C79—21no—
130C4×C79—42no—
131C4×C79—82no—
132Dic79—44no—
133Dic79—82no—
134#12972—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—82no—
138C2×C79#186711yescharacteristic
139C2×C79#220412yes—
140#2372—791no—
141C2×#355—791no—
142#1888D7911yescharacteristic
143#1857—792no—
144#1394—792no—
145C79#1297211yescharacteristic, complemented, Sylow
146C2×#124C2×D7911yescharacteristic
147#519C2×D7911yescharacteristic
148#428—1582no—
149#418—21no—
150C2×#101—792no—
151C2×#101C2×D7911yescharacteristic
152#391—792no—
153#388—1582no—
154#386—1581no—
155C4×C22⋊C4—1582no—
156#376—1581no—
157#355—794no—
158#339—21no—
159#339—1582no—
160C22×Q8C22×D7911yescharacteristic
161C22×D4C22⋊D7911yescharacteristic
162C23×C4—1581no—
163C2×M16—21no—
164C2×M16—1582no—
165#124—22no—
166#124Dic15812yes—
167#124C4×D7912yes—
168C4×Q8—42no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—792no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—1582no—
174C2×C22⋊C4—1581no—
175C2×C42—792no—
176C2×C42—1582no—
177C2×C42C22⋊D7911yescharacteristic
178#107—3162no—
179#106—3162no—
180#102—42no—
181#101—22no—
182#101—794no—
183#101—1582no—
184#95—1581no—
185#95—1581no—
186#95—1582no—
187#95—1582no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×C4×D7911yescharacteristic
192C2×Q8C2×Dic15811yescharacteristic
193C2×Q8#1192962312yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4Dic79⋊C412yes—
198C22×C4—21no—
199C22×C4—3164no—
200C22×C4—6321no—
201C22×C4—1582no—
202C22×C4—1582no—
203C22×C4D4×D7912yes—
204C22×C4C2×C22⋊D7911yescharacteristic
205M16—41no—
206M16—41no—
207M16—3162no—
208M16—1584no—
209C2×C8—41no—
210C2×C8—3162no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—3162no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—41no—
218C22⋊C4—3164no—
219C22⋊C4—6321no—
220C42—21no—
221C42—42no—
222C42—41no—
223C42—3162no—
224C42—3164no—
225C42Dic79⋊C412yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1457912911yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—3162no—
245C2×C4—3162no—
246C2×C4—3162no—
247C2×C4—3162no—
248C2×C4—3162no—
249C2×C4—6324no—
250C2×C4—1584no—
251C2×C4C2×Dic79⋊C411yescharacteristic
252C2×C4(C22⋊D79)⋊C412yes—
253C2×C4#1457913812yes—
254C2×C4#1457914211yescharacteristic
255C8—81no—
256C8—3164no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1909843511yescharacteristic
264C22#1909874911yescharacteristic
265C22#1909875211yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—82no—
272C4—3164no—
273C4—6322no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2376368011yescharacteristic
278C2#2376542412yes—
subgroupG/H
centerC22#19098749
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#1888C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
cosocleC22×Q8×C79C23
hypercenter#1888D79
hypocenterC79#12972
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1888

Complements

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

NH ≅ G/Ntype
3. #23763615C2semidirect
4. #23761526C2semidirect
13. #19098360C4semidirect
30. #14579080C2×C4semidirect
145. C79#12972semidirect

Maximal subgroups

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

subgroupindex
#237665122
C2×#190983602
#237636152
#237615262
C79×#18882
#1297279