Finite Group Databasebeta

← #34177110

Subgroups — #34177110

Order 20224.

424 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 83 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#23771388C211yesmaximal, characteristic
2#23764730C212yesmaximal
3#23761226C211yesmaximal, characteristic
4#23761222C212yesmaximal
5C79×#2016C211yesmaximal, characteristic
6C2×#14579092—22no—
7#19098502—21no—
8#19098495—22no—
9#19098470—21no—
10#19098461C2212yes—
11#19098450—22no—
12#19098325—22no—
13#19098318C2212yes—
14C22×Q16×C79—21no—
15C2×C4⋊Q8×C79C2211yescharacteristic
16C2×#103×C79—21no—
17#391×C79C2211yescharacteristic
18#336×C79C2211yescharacteristic
19C22×C79⋊C8—22no—
20C2×C4⋊Dic79—21no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79D412yes—
23C2×C4⋊Dic79D412yes—
24C2×Dic79⋊C4—22no—
25C2×C4×Dic79—21no—
26#14579092—44no—
27C79⋊#95—21no—
28C79⋊#95—22no—
29C79⋊#95—22no—
30C79⋊#95—22no—
31C22×Q8×C79—21no—
32C22×Q8×C79D411yescharacteristic
33C2×Q16×C79—42no—
34C2×Q16×C79—42no—
35C2×Q16×C79—42no—
36C22×C8×C79—21no—
37C4⋊Q8×C79—42no—
38C2×C4⋊C4×C79—22no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C42×C79C2311yescharacteristic
41#103×C79—42no—
42#95×C79—22no—
43C22×Dic79—22no—
44C22×Dic79—22no—
45C2×C79⋊C8—24no—
46C2×C79⋊C8—42no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—22no—
49C4⋊Dic79—22no—
50C4⋊Dic79—42no—
51C4⋊Dic79—81no—
52C4⋊Dic79—81no—
53Dic79⋊C4—42no—
54Dic79⋊C4—42no—
55C4×Dic79—81no—
56C2×Q8×C79—21no—
57C2×Q8×C79—22no—
58C2×Q8×C79—41no—
59C2×Q8×C79—41no—
60C2×Q8×C79—41no—
61C2×Q8×C79—41no—
62C2×Q8×C79—41no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79SD1612yes—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—41no—
69C22×C4×C79C2×D411yescharacteristic
70C22×C4×C79C2×D412yes—
71Q16×C79—48no—
72C2×C8×C79—22no—
73C2×C8×C79—41no—
74C2×C8×C79—41no—
75C4⋊C4×C79—24no—
76C4⋊C4×C79—42no—
77C4⋊C4×C79—41no—
78C42×C79—41no—
79C23×C79#12011yescharacteristic
80Q8×C79—42no—
81Q8×C79—42no—
82Q8×C79—42no—
83Q8×C79—81no—
84Q8×C79—81no—
85Q8×C79—81no—
86C2×C4×C79—21no—
87C2×C4×C79—22no—
88C2×C4×C79—22no—
89C2×C4×C79—21no—
90C2×C4×C79—22no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—81no—
97C2×C4×C79—81no—
98C2×C4×C79C2×SD1611yescharacteristic
99C2×C4×C79#13512yes—
100C2×C4×C79#13711yescharacteristic
101C2×C4×C79#13712yes—
102C2×Dic79—42no—
103C2×Dic79—42no—
104C2×Dic79—42no—
105C2×Dic79—42no—
106C2×Dic79—42no—
107C2×Dic79—42no—
108C8×C79—42no—
109C79⋊C8—44no—
110C22×C79#44811yescharacteristic
111C22×C79#45212yes—
112C22×C79#45311yescharacteristic
113C22×C79#45612yes—
114C22×C79B(4,𝔽2)11yescharacteristic
115C4×C79—21no—
116C4×C79—21no—
117C4×C79—22no—
118C4×C79—22no—
119C4×C79—42no—
120C4×C79—82no—
121C4×C79—81no—
122Dic79—82no—
123Dic79—82no—
124#7760—791nomaximal, Sylow
125C2×C79#159611yescharacteristic
126C2×C79#160312yes—
127C2×C79#163411yescharacteristic
128C2×C79#165312yes—
129C2×C79#166011yescharacteristic
130#2809—791no—
131#2058—792no—
132#2016D7911yescharacteristic
133#1272—791no—
134#1271—792no—
135C79#776011yescharacteristic, complemented, Sylow
136C22×Q16—21no—
137C2×C4⋊Q8C2×D7911yescharacteristic
138C2×#106—1582no—
139C2×#103—21no—
140#400—1581no—
141#398—1582no—
142#391—1581no—
143#391C2×D7911yescharacteristic
144#389—792no—
145#387—1582no—
146#340—1582no—
147#336—792no—
148#336C2×D7911yescharacteristic
149C22×Q8—21no—
150C22×Q8C22⋊D7911yescharacteristic
151C2×Q16—42no—
152C2×Q16—42no—
153C2×Q16—42no—
154C22×C8—21no—
155C22×C8—1582no—
156C4⋊Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—792no—
160C2×C4⋊C4—1582no—
161C2×C4⋊C4—1581no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4C22⋊D7911yescharacteristic
164C2×C42—1581no—
165C2×C42C22×D7911yescharacteristic
166#106—3164no—
167#103—42no—
168#95—22no—
169#95—1581no—
170#95—1582no—
171#95—1582no—
172#95—1582no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8#1192961212yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—1582no—
187C22×C4—1582no—
188C22×C4D4×D7912yes—
189C22×C4C2×C22⋊D7911yescharacteristic
190Q16—48no—
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—3162no—
195C2×C8—3162no—
196C2×C8—1584no—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—3162no—
201C4⋊C4—3162no—
202C4⋊C4—3162no—
203C4⋊C4—6321no—
204C4⋊C4—6321no—
205C4⋊C4—1582no—
206C4⋊C4—1582no—
207C42—41no—
208C42—6321no—
209C23#1457912911yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4—3162no—
231C2×C4—3162no—
232C2×C4—3162no—
233C2×C4—3162no—
234C2×C4C2×#1192961211yescharacteristic
235C2×C4#1457918412yes—
236C2×C4#1457919111yescharacteristic
237C2×C4#1457919312yes—
238C8—42no—
239C8—3164no—
240C22#1909870111yescharacteristic
241C22#1909872412yes—
242C22#1909873011yescharacteristic
243C22#1909874412yes—
244C22#1909874911yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—81no—
251C4—82no—
252C4—6322no—
253C4—6322no—
254C2#2376232211yescharacteristic
255C2#2376236212yes—
256C2#2376254311yescharacteristic
257C2#2376267512yes—
258C2#2376270911yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2016C2
socle, prime radicalC23×C79#120
hypercenter#2016D79
hypocenterC79#7760
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
135. C79#7760semidirect

Maximal subgroups

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

subgroupindex
#237713882
#237647302
#237612262
#237612222
C79×#20162
#776079