Finite Group Databasebeta

← #34201491

Subgroups — #34201491

Order 20224.

344 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 67 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#23774175C211yesmaximal, characteristic
2#23765602C212yesmaximal, complemented
3#23761411C212yesmaximal, complemented
4#23761324C211yesmaximal, characteristic
5C79×#2076C211yesmaximal, characteristic
6C2×#14579140—21no—
7#19098899—22no—
8#19098804—21no—
9#19098804C2212yes—
10#19098721—22no—
11#19098388—22no—
12#19098366—22no—
13C22⋊(C79⋊C16)C2212yes—
14C2×#107×C79C2211yescharacteristic
15C2×C22⋊C8×C79C2211yescharacteristic
16#380×C79—21no—
17#340×C79—21no—
18#340×C79C2211yescharacteristic
19C22×Dic158—21no—
20C2×Dic316—21no—
21C2×Dic316D412yes—
22#14579140—22no—
23#14579140—22no—
24#14579140—41no—
25#14579140—41no—
26C2×C4⋊Dic79—21no—
27C2×Dic79⋊C4—41no—
28C2×C22⋊Dic79—41no—
29C2×C79⋊C16—22no—
30#14579095—21no—
31#14579095D412yes—
32#14579086—22no—
33#14579086—22no—
34C23×C4×C79D411yescharacteristic
35C22×C8×C79D411yescharacteristic
36C22×C8×C79C2311yescharacteristic
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C22⋊C4×C79—41no—
40#107×C79—22no—
41C22⋊C8×C79—22no—
42#95×C79—41no—
43C2×Dic158—22no—
44C2×Dic158—41no—
45C2×Dic158—81no—
46C2×Dic158—81no—
47C22×Dic79—41no—
48C22×Dic79—41no—
49Dic316—42no—
50Dic316—81no—
51C4⋊Dic79—22no—
52C4⋊Dic79—41no—
53Dic79⋊C4—42no—
54Dic79⋊C4—42no—
55C22⋊Dic79—42no—
56C22⋊Dic79—42no—
57C79⋊C16—42no—
58C24×C79D811yescharacteristic
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—41no—
62C22×C4×C79—41no—
63C22×C4×C79—41no—
64C22×C4×C79—41no—
65C22×C4×C79D811yescharacteristic
66C22×C4×C79C2×D411yescharacteristic
67C2×C8×C79—21no—
68C2×C8×C79—41no—
69C2×C8×C79—41no—
70C2×C8×C79C2×D412yes—
71C4⋊C4×C79—41no—
72C4⋊C4×C79—41no—
73C4⋊C4×C79—41no—
74C4⋊C4×C79—41no—
75C22⋊C4×C79—82no—
76C23×C79—21no—
77C23×C79—21no—
78C23×C79—41no—
79C23×C79—41no—
80C23×C79Q3212yes—
81C23×C79C2×D811yescharacteristic
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—41no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—81no—
89C2×C4×C79—81no—
90C2×C4×C79—81no—
91C2×C4×C79—81no—
92C2×C4×C79—81no—
93C2×C4×C79—81no—
94C2×C4×C79—81no—
95C2×C4×C79#12011yescharacteristic
96C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—82no—
100C2×Dic79—82no—
101Dic158—42no—
102Dic158—42no—
103Dic158—82no—
104C8×C79—22no—
105C8×C79—41no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79—22no—
109C22×C79—42no—
110C22×C79—41no—
111C22×C79—41no—
112C22×C79—81no—
113C22×C79#44711yescharacteristic
114C22×C79C2×Q3211yescharacteristic
115C22×C79#51011yescharacteristic
116C4×C79—21no—
117C4×C79—41no—
118C4×C79—161no—
119C4×C79—161no—
120C4×C79#45511yescharacteristic
121C4×C79#45811yescharacteristic
122Dic79—82no—
123Dic79—82no—
124#12282—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79—42no—
128C2×C79#160711yescharacteristic
129C2×C79#218811yescharacteristic
130C2×C79#219011yescharacteristic
131#3189—791no—
132#2236—792no—
133#2076D7911yescharacteristic
134#1348—792no—
135#1314—791no—
136C79#1228211yescharacteristic, complemented, Sylow
137C2×C22⋊Q8—1581no—
138#484—1582no—
139#467—792no—
140#467—1581no—
141#451—1582no—
142C2×#107C2×D7911yescharacteristic
143C2×C22⋊C8C2×D7911yescharacteristic
144#380—21no—
145#366—1582no—
146#358—1582no—
147C22⋊C16—792no—
148#340—21no—
149#340C2×D7911yescharacteristic
150C22×Q8—1581no—
151C23×C4C22⋊D7911yescharacteristic
152C2×Q16—792no—
153C2×Q16—1581no—
154C22×C8C22⋊D7911yescharacteristic
155C22×C8C22×D7911yescharacteristic
156C22⋊Q8—3161no—
157C22⋊Q8—3161no—
158C22⋊Q8—1582no—
159C22⋊Q8—1582no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—3161no—
163C2×C4⋊C4—1581no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—3161no—
166C2×C16—1582no—
167#107—22no—
168#107—792no—
169#107—1581no—
170#103—1582no—
171#103—1582no—
172C22⋊C8—22no—
173#95—41no—
174C24#1192960911yescharacteristic
175C2×Q8—3161no—
176C2×Q8—6321no—
177C2×Q8—6321no—
178C2×Q8—1582no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—3161no—
186C22×C4—3161no—
187C22×C4#1192960911yescharacteristic
188C22×C4C2×C22⋊D7911yescharacteristic
189Q16—3162no—
190Q16—6321no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8D4×D7912yes—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—3162no—
200C4⋊C4—3161no—
201C4⋊C4—3162no—
202C4⋊C4—1582no—
203C22⋊C4—82no—
204C22⋊C4—3162no—
205C22⋊C4—3162no—
206C16—3162no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1457910712yes—
212C23C2×#1192960911yescharacteristic
213Q8—3162no—
214Q8—3162no—
215Q8—6322no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—3162no—
230C2×C4—3162no—
231C2×C4—6322no—
232C2×C4—6322no—
233C2×C4#1457912911yescharacteristic
234C2×C4#1457918611yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1909869611yescharacteristic
245C22C2×#1457910711yescharacteristic
246C22#1909900311yescharacteristic
247C4—21no—
248C4—41no—
249C4—161no—
250C4—161no—
251C4—6322no—
252C4—6322no—
253C4#1909874011yescharacteristic
254C4#1909875211yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2376238511yescharacteristic
259C2#2376533311yescharacteristic
260C2#2376534511yescharacteristic
subgroupG/H
centerC22#19098696
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#2076C2
socleC22×C79#447
prime radicalC24×C79D8
hypercenter#2076D79
hypocenterC79#12282
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

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

NH ≅ G/Ntype
2. #23765602C2semidirect
3. #23761411C2semidirect
136. C79#12282semidirect

Maximal subgroups

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

subgroupindex
#237741752
#237656022
#237614112
#237613242
C79×#20762
#1228279