Finite Group Databasebeta

← #34175758

Subgroups — #34175758

Order 20224.

296 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 39 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#23761336C211yesmaximal, characteristic, complemented
2#23761285C211yesmaximal, characteristic, complemented
3C79×#2010C211yesmaximal, characteristic
4C4×C22⋊Dic79—21no—
5#19098345—21no—
6#575×C79C411yescharacteristic, complemented
7#525×C79—21no—
8#421×C79—21no—
9#407×C79C2211yescharacteristic
10H(3,ℤ/4)×C79C411yescharacteristic, complemented
11C2×C22⋊Dic79—41no—
12C2×C4×Dic79—41no—
13#14579101—41no—
14C79⋊#95—22no—
15C2×#41×C79—21no—
16C23×C4×C79D411yescharacteristic
17#137×C79—41no—
18AGL(1,ℤ/8)×C79—41no—
19#135×C79—41no—
20C2×M16×C79C2×C411yescharacteristic
21C2×M16×C79D411yescharacteristic
22#123×C79—41no—
23C22⋊Q8×C79—41no—
24#121×C79—41no—
25#120×C79—41no—
26C4×D4×C79—42no—
27#117×C79—21no—
28#104×C79—42no—
29C22⋊C8×C79—41no—
30C22×Dic79—42no—
31C22⋊Dic79—44no—
32C4×Dic79—81no—
33C4×Dic79—81no—
34C79⋊C16—81no—
35C24×C79D811yescharacteristic
36#41×C79—41no—
37#41×C79—41no—
38#41×C79—41no—
39#41×C79—41no—
40C2×Q8×C79—21no—
41C2×D4×C79—21no—
42C2×D4×C79—42no—
43C2×D4×C79—41no—
44C22×C4×C79—21no—
45C22×C4×C79—21no—
46C22×C4×C79—41no—
47C22×C4×C79—41no—
48C22×C4×C79—41no—
49C22×C4×C79C22⋊C411yescharacteristic
50C22×C4×C79SD1611yescharacteristic
51Q16×C79—41no—
52SD16×C79—42no—
53D8×C79—41no—
54M16×C79—41no—
55M16×C79—41no—
56C2×C8×C79—21no—
57C2×C8×C79—21no—
58C4⋊C4×C79—41no—
59C4⋊C4×C79—42no—
60C22⋊C4×C79—42no—
61C22⋊C4×C79—42no—
62C22⋊C4×C79—41no—
63C42×C79—22no—
64C23×C79—21no—
65C23×C79—42no—
66C23×C79—41no—
67C23×C79—41no—
68C23×C79#10211yescharacteristic
69Q8×C79—41no—
70Q8×C79—41no—
71D4×C79—41no—
72D4×C79—41no—
73D4×C79—81no—
74D4×C79—84no—
75D4×C79—81no—
76C2×C4×C79—21no—
77C2×C4×C79—21no—
78C2×C4×C79—21no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—42no—
83C2×C4×C79—41no—
84C2×C4×C79—81no—
85C2×C4×C79—81no—
86C2×C4×C79—81no—
87C2×C4×C79#9911yescharacteristic
88C2×C4×C79#10411yescharacteristic
89C2×Dic79—44no—
90C2×Dic79—82no—
91C2×Dic79—82no—
92C8×C79—41no—
93C8×C79—41no—
94C22×C79—22no—
95C22×C79—21no—
96C22×C79—41no—
97C22×C79—41no—
98C22×C79—42no—
99C22×C79—41no—
100C22×C79—81no—
101C22×C79—82no—
102C22×C79—81no—
103C22×C79#32711yescharacteristic
104C4×C79—21no—
105C4×C79—41no—
106C4×C79—82no—
107C4×C79—81no—
108C4×C79AΣL(1,𝔽16)11yescharacteristic
109C4×C79#35311yescharacteristic
110Dic79—84no—
111#7396—791nomaximal, Sylow
112C2×C79—21no—
113C2×C79—42no—
114C2×C79—41no—
115C2×C79—161no—
116C2×C79#134311yescharacteristic
117#2010D7911yescharacteristic
118#1320—791no—
119#1297—791no—
120C79#739611yescharacteristic, complemented, Sylow
121#575Dic7911yescharacteristic, complemented
122#525—21no—
123#421—21no—
124#407C2×D7911yescharacteristic
125C4×C22⋊C4—1581no—
126#349—1581no—
127H(3,ℤ/4)Dic7911yescharacteristic, complemented
128C2×#41—21no—
129C23×C4C22⋊D7911yescharacteristic
130#137—41no—
131AGL(1,ℤ/8)—41no—
132#135—41no—
133C2×M16C2×Dic7911yescharacteristic
134C2×M16C22⋊D7911yescharacteristic
135#123—41no—
136C22⋊Q8—41no—
137#121—41no—
138#120—41no—
139C4×D4—42no—
140#117—21no—
141C2×C22⋊C4—3161no—
142C2×C42—3161no—
143M32—3161no—
144#104—42no—
145C22⋊C8—41no—
146#95—1582no—
147C24#1192960911yescharacteristic
148#41—41no—
149#41—41no—
150#41—41no—
151#41—41no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—42no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—3162no—
162C22×C4C22⋊Dic7911yescharacteristic
163C22×C4#1192961211yescharacteristic
164Q16—41no—
165SD16—42no—
166D8—41no—
167M16—41no—
168M16—41no—
169C2×C8—21no—
170C2×C8—21no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—3164no—
177C42—22no—
178C42—6321no—
179C42—6321no—
180C16—6321no—
181C23—21no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23#1457908211yescharacteristic
186Q8—41no—
187Q8—41no—
188D4—41no—
189D4—41no—
190D4—81no—
191D4—84no—
192D4—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—3164no—
205C2×C4—6322no—
206C2×C4—6322no—
207C2×C4#1457907611yescharacteristic
208C2×C4#1457908811yescharacteristic
209C8—41no—
210C8—41no—
211C22—22no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22—81no—
218C22—82no—
219C22—81no—
220C22#1909829711yescharacteristic
221C4—21no—
222C4—41no—
223C4—82no—
224C4—81no—
225C4—6324no—
226C4#1909835011yescharacteristic
227C4#1909835611yescharacteristic
228C2—21no—
229C2—41no—
230C2—42no—
231C2—161no—
232C2#2376139711yescharacteristic
subgroupG/H
centerC4#19098356
commutatorC2×M16×C79C2×C4
Frattini#407C2×D79
Fitting, prime radicalC79×#2010C2
socleC2×C79#1343
cosocle#407×C79C22
hypercenter#2010D79
hypocenterC79#7396
LCSG ▷ C2×M16×C79 ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #23761336C2semidirect
2. #23761285C2semidirect
6. #575×C79C4semidirect
10. H(3,ℤ/4)×C79C4semidirect
120. C79#7396semidirect
121. #575Dic79semidirect
127. H(3,ℤ/4)Dic79semidirect

Maximal subgroups

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

subgroupindex
#237613362
#237612852
C79×#20102
#739679