Finite Group Databasebeta

← #34182672

Subgroups — #34182672

Order 20224.

392 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 99 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#23764233C211yesmaximal, characteristic
2#23762168C212yesmaximal, complemented
3#23761992C211yesmaximal, characteristic
4#23761292C212yesmaximal, complemented
5C79×#2980C211yesmaximal, characteristic
6#19098587C2211yescharacteristic
7C2×#14579088—21no—
8C2×#14579088—21no—
9C4×#11929608C2212yes—
10#19098409C2211yescharacteristic
11#19098315—22no—
12#19098313—22no—
13#19098303C411yescharacteristic, complemented
14#19098303C412yescomplemented
15#19098303C411yescharacteristic, complemented
16#532×C79C2211yescharacteristic
17#435×C79—22no—
18#435×C79—22no—
19#432×C79C2212yes—
20#408×C79—21no—
21#408×C79—21no—
22C2×#11929608—22no—
23C2×#11929608—21no—
24C2×C4×Dic79—21no—
25C4⋊(C79⋊C8)—22no—
26#14579088—42no—
27#14579088—42no—
28#14579072C2×C412yes—
29#14579072C2×C412yes—
30C4×C79⋊C8—22no—
31C79⋊#95—22no—
32C2×#41×C79D411yescharacteristic
33C2×#41×C79D411yescharacteristic
34C2×M16×C79—21no—
35C2×M16×C79—21no—
36C22×C8×C79—21no—
37C22×C8×C79—21no—
38C4⋊Q8×C79Q811yescharacteristic
39#127×C79Q811yescharacteristic
40#124×C79C2×C411yescharacteristic
41#124×C79C2×C411yescharacteristic
42#121×C79—22no—
43C4×D4×C79—22no—
44C2×C42×C79C2311yescharacteristic
45C4⋊C8×C79D412yes—
46C4⋊C8×C79D412yes—
47C22⋊C8×C79—22no—
48C22⋊C8×C79—22no—
49C22⋊C8×C79—22no—
50#97×C79—22no—
51C22×Dic79—41no—
52#11929608—41no—
53#11929608—41no—
54#11929608—44no—
55C2×C79⋊C8—22no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—41no—
58C4×Dic79—22no—
59C4×Dic79—41no—
60#41×C79—41no—
61#41×C79—41no—
62#41×C79—41no—
63#41×C79—41no—
64C2×Q8×C79C4⋊C411yescharacteristic
65C2×Q8×C79C4⋊C411yescharacteristic
66C2×D4×C79—21no—
67C2×D4×C79—21no—
68C2×D4×C79—22no—
69C2×D4×C79C4⋊C411yescharacteristic
70C2×D4×C79C4⋊C411yescharacteristic
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79#4112yes—
75M16×C79—81no—
76M16×C79—81no—
77C2×C8×C79—22no—
78C2×C8×C79—22no—
79C2×C8×C79—81no—
80C2×C8×C79—81no—
81C4⋊C4×C79—22no—
82C22⋊C4×C79—22no—
83C22⋊C4×C79—22no—
84C42×C79C22×C411yescharacteristic
85C42×C79C2×D412yes—
86C42×C79C2×Q811yescharacteristic
87C23×C79—21no—
88C23×C79—21no—
89C23×C79#12311yescharacteristic
90Q8×C79—41no—
91Q8×C79—41no—
92D4×C79—41no—
93D4×C79—41no—
94D4×C79—82no—
95D4×C79—81no—
96D4×C79—81no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—42no—
102C2×C4×C79—41no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79C2×C4⋊C411yescharacteristic
107C2×C4×C79C4×D412yes—
108C2×C4×C79#12011yescharacteristic
109C2×C4×C79C22⋊Q812yes—
110C2×Dic79—42no—
111C2×Dic79—82no—
112C8×C79—42no—
113C8×C79—42no—
114C79⋊C8—24no—
115C79⋊C8—42no—
116C79⋊C8—81no—
117C22×C79—21no—
118C22×C79—22no—
119C22×C79—41no—
120C22×C79—41no—
121C22×C79—42no—
122C22×C79—42no—
123C22×C79#38511yescharacteristic
124C4×C79—21no—
125C4×C79—22no—
126C4×C79—22no—
127C4×C79—41no—
128C4×C79—41no—
129C4×C79#45311yescharacteristic
130C4×C79#45511yescharacteristic
131Dic79—82no—
132#8918—791nomaximal, Sylow
133C2×C79—41no—
134C2×C79—81no—
135C2×C79—81no—
136C2×C79#185611yescharacteristic
137C2×C79#186112yes—
138#2980D7911yescharacteristic
139#1963—791no—
140#1569—792no—
141#1525—791no—
142#1300—792no—
143C79#891811yescharacteristic, complemented, Sylow
144#532C2×D7911yescharacteristic
145#435—22no—
146#435—22no—
147#432C2×D7912yes—
148#423—791no—
149C2×#104—1581no—
150C2×#104—1581no—
151#408—21no—
152#408—21no—
153C4×M16—792no—
154#376—791no—
155#335—1582no—
156#334—1582no—
157#329—791no—
158#329—792no—
159#329—791no—
160C2×#41C22⋊D7911yescharacteristic
161C2×#41C22⋊D7911yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—1581no—
165C2×M16—1582no—
166C22×C8—21no—
167C22×C8—21no—
168C4⋊Q8Dic15811yescharacteristic
169#127Dic15811yescharacteristic
170#124C4×D7911yescharacteristic
171#124C4×D7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174C2×C42—1581no—
175C2×C42C22×D7911yescharacteristic
176C4⋊C8—1582no—
177C4⋊C8C22⋊D7912yes—
178C4⋊C8C22⋊D7912yes—
179#104—3162no—
180#104—3162no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184#97—22no—
185#97—792no—
186#97—792no—
187C4×C8—1582no—
188#95—1582no—
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8Dic79⋊C411yescharacteristic
194C2×Q8Dic79⋊C411yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4Dic79⋊C411yescharacteristic
199C2×D4Dic79⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—3161no—
203C22×C4C2×C22⋊D7911yescharacteristic
204C22×C4#1192962412yes—
205M16—81no—
206M16—81no—
207M16—3164no—
208M16—3161no—
209M16—3161no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—3161no—
215C2×C8—1582no—
216C2×C8—1582no—
217C4⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C42—3161no—
221C42—1582no—
222C42C2×C4×D7911yescharacteristic
223C42C2×C22⋊D7912yes—
224C42C2×Dic15811yescharacteristic
225C23—21no—
226C23—21no—
227C23#1457914411yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—3162no—
245C2×C4—6322no—
246C2×C4C2×Dic79⋊C411yescharacteristic
247C2×C4C4×C22⋊D7912yes—
248C2×C4#1457913011yescharacteristic
249C2×C4#1457914012yes—
250C8—42no—
251C8—42no—
252C8—3162no—
253C8—6321no—
254C8—1584no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22#1909843611yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—41no—
267C4—6322no—
268C4#1909873111yescharacteristic
269C4#1909874111yescharacteristic
270C2—41no—
271C2—81no—
272C2—81no—
273C2#2376361011yescharacteristic
274C2#2376364012yes—
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2980C2
socleC22×C79#385
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#2980D79
hypocenterC79#8918
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #23762168C2semidirect
4. #23761292C2semidirect
13. #19098303C4semidirect
14. #19098303C4semidirect
15. #19098303C4semidirect
143. C79#8918semidirect

Maximal subgroups

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

subgroupindex
#237642332
#237621682
#237619922
#237612922
C79×#29802
#891879