Finite Group Databasebeta

← #34205107

Subgroups — #34205107

Order 20224.

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

normal subgroups only

This group has 460 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#23766490C211yesmaximal, characteristic
2C2×#19098362C211yesmaximal, characteristic
3#23763615C212yesmaximal
4#23761526C212yesmaximal
5C79×#1886C211yesmaximal, characteristic
6#19098606—22no—
7C2×#14579080C2212yes—
8#19098470—21no—
9#19098470C2212yes—
10#19098462—22no—
11C4⋊C4×Dic79—22no—
12C4×C4⋊Dic79—21no—
13#19098362C414yescomplemented
14#19098322—22no—
15C2×C4⋊Q8×C79C2211yescharacteristic
16#519×C79C2211yescharacteristic
17#419×C79—21no—
18C2×#101×C79C2211yescharacteristic
19#339×C79—21no—
20C2×#11929608—22no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79D412yes—
25C2×C4×Dic79—22no—
26C2×C4×Dic79—21no—
27C2×C4×Dic79D412yes—
28#14579094—42no—
29#14579092—42no—
30#14579080—22no—
31#14579080C2×C414yes—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×Q8×C79D411yescharacteristic
36C22×Q8×C79C2311yescharacteristic
37C2×M16×C79—21no—
38C4⋊Q8×C79—22no—
39C4⋊Q8×C79C2×C412yes—
40C4⋊Q8×C79Q812yes—
41C4×Q8×C79—41no—
42C4×Q8×C79—41no—
43#117×C79—21no—
44#117×C79—22no—
45C2×C4⋊C4×C79—22no—
46C2×C42×C79D411yescharacteristic
47#103×C79—41no—
48#103×C79—41no—
49#101×C79—22no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52#11929608—24no—
53#11929608—42no—
54C2×C79⋊C8—42no—
55C4⋊Dic79—24no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—81no—
59C4×Dic79—42no—
60C4×Dic79—44no—
61C4×Dic79—81no—
62C2×Q8×C79—21no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79—41no—
66C2×Q8×C79—41no—
67C2×Q8×C79C4⋊C412yes—
68C2×Q8×C79C22×C411yescharacteristic
69C2×Q8×C79C2×Q811yescharacteristic
70C2×Q8×C79#4112yes—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D412yes—
75M16×C79—41no—
76M16×C79—41no—
77C2×C8×C79—41no—
78C4⋊C4×C79—24no—
79C4⋊C4×C79—24no—
80C4⋊C4×C79—21no—
81C4⋊C4×C79—21no—
82C4⋊C4×C79—22no—
83C4⋊C4×C79—22no—
84C22⋊C4×C79—41no—
85C42×C79—21no—
86C42×C79—42no—
87C42×C79—41no—
88C42×C79C4⋊C412yes—
89C23×C79#12011yescharacteristic
90Q8×C79—41no—
91Q8×C79—41no—
92Q8×C79—42no—
93Q8×C79—42no—
94Q8×C79—81no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79C2×C4⋊C411yescharacteristic
108C2×C4×C79C4×D412yes—
109C2×C4×C79C22⋊Q812yes—
110C2×C4×C79#12311yescharacteristic
111C2×Dic79—24no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C8×C79—81no—
118C79⋊C8—44no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79#38511yescharacteristic
122C22×C79B(4,𝔽2)11yescharacteristic
123C22×C79#45811yescharacteristic
124C4×C79—24no—
125C4×C79—21no—
126C4×C79—22no—
127C4×C79—22no—
128C4×C79—21no—
129C4×C79—42no—
130C4×C79—81no—
131C4×C79—81no—
132Dic79—44no—
133Dic79—82no—
134#12973—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79#186711yescharacteristic
138C2×C79#220411yescharacteristic
139C2×C79#220511yescharacteristic
140#2369—791no—
141C2×#356—791no—
142#1886D7911yescharacteristic
143#1857—792no—
144#1394—792no—
145C79#1297311yescharacteristic, complemented, Sylow
146C2×C4⋊Q8C2×D7911yescharacteristic
147#519C2×D7911yescharacteristic
148#428—1582no—
149#419—21no—
150C2×#101—792no—
151C2×#101C2×D7911yescharacteristic
152#391—792no—
153#391—1581no—
154#389—1582no—
155C4×C4⋊C4—1582no—
156C4×C4⋊C4—1581no—
157#356—794no—
158#339—21no—
159#339—1582no—
160C22×Q8C22⋊D7911yescharacteristic
161C22×Q8C22×D7911yescharacteristic
162C2×M16—21no—
163C2×M16—1582no—
164C4⋊Q8—22no—
165C4⋊Q8Dic15812yes—
166C4⋊Q8C4×D7912yes—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—792no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1581no—
175C2×C4⋊C4—1581no—
176C2×C42—792no—
177C2×C42—1582no—
178C2×C42—1581no—
179C2×C42C22⋊D7911yescharacteristic
180#107—3162no—
181#106—3162no—
182#103—41no—
183#103—41no—
184#101—22no—
185#101—794no—
186#101—1582no—
187#95—1582no—
188#95—1582no—
189#95—1582no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8Dic79⋊C412yes—
196C2×Q8C2×C4×D7911yescharacteristic
197C2×Q8C2×Dic15811yescharacteristic
198C2×Q8#1192962312yes—
199C22×C4—21no—
200C22×C4—21no—
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—24no—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—3162no—
218C4⋊C4—3162no—
219C4⋊C4—6321no—
220C4⋊C4—1584no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42—41no—
225C42—3162no—
226C42—3164no—
227C42—6321no—
228C42Dic79⋊C412yes—
229C23#1457912911yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—3162no—
248C2×C4—3162no—
249C2×C4—3162no—
250C2×C4—3162no—
251C2×C4—3162no—
252C2×C4—1584no—
253C2×C4C2×Dic79⋊C411yescharacteristic
254C2×C4(C22⋊D79)⋊C412yes—
255C2×C4#1457913812yes—
256C2×C4#1457914211yescharacteristic
257C8—81no—
258C8—3164no—
259C22—21no—
260C22—21no—
261C22#1909843511yescharacteristic
262C22#1909874911yescharacteristic
263C22#1909875211yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4—3164no—
273C4—6322no—
274C2—21no—
275C2—21no—
276C2#2376368011yescharacteristic
277C2#2376542511yescharacteristic
278C2#2376542911yescharacteristic
subgroupG/H
centerC22#19098749
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#1886C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
cosocleC22×Q8×C79C23
hypercenter#1886D79
hypocenterC79#12973
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
13. #19098362C4semidirect
145. C79#12973semidirect

Maximal subgroups

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

subgroupindex
#237664902
C2×#190983622
#237636152
#237615262
C79×#18862
#1297379