Finite Group Databasebeta

← #34191343

Subgroups — #34191343

Order 20224.

400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 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#23771672C211yesmaximal, characteristic
2#23764373C212yesmaximal
3#23764219C211yesmaximal, characteristic
4#23761282C212yesmaximal
5C79×#1295C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579084—21no—
8C2×#14579084—21no—
9C2×#14579072C2211yescharacteristic
10#19098505—22no—
11#19098502—21no—
12#19098484—22no—
13#19098483—21no—
14#19098429C2212yes—
15#19098326—22no—
16#19098326—22no—
17C2×#97×C79C2212yes—
18#382×C79C2211yescharacteristic
19C22×Dic158—21no—
20C22×Dic158—21no—
21C22×C79⋊C8—21no—
22#14579161—22no—
23#14579161—21no—
24#14579161Q812yescomplemented
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79D412yes—
28C2×C4⋊Dic79D412yes—
29C2×Dic79⋊C4—22no—
30#14579084—42no—
31#14579084—42no—
32#14579072—22no—
33C79⋊#95—21no—
34C79⋊#95—22no—
35C79⋊#95—21no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C22×C8×C79—22no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C42×C79C2311yescharacteristic
42#97×C79—22no—
43#97×C79—22no—
44#95×C79—22no—
45C2×Dic158—22no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C2×Dic158—41no—
50C2×Dic158—41no—
51C2×Dic158—81no—
52C2×Dic158—82no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C22×Dic79—41no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—41no—
58C2×C79⋊C8—41no—
59C4⋊Dic79—24no—
60C4⋊Dic79—42no—
61C4⋊Dic79—42no—
62C4⋊Dic79—42no—
63C4⋊Dic79—81no—
64Dic79⋊C4—82no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79C2×D412yes—
68C22×C4×C79C2×D411yescharacteristic
69C2×C8×C79—24no—
70C2×C8×C79—42no—
71C2×C8×C79—42no—
72C4⋊C4×C79—41no—
73C4⋊C4×C79—41no—
74C42×C79C2×Q811yescharacteristic
75C42×C79#4111yescharacteristic
76C42×C79#4112yes—
77C23×C79#12011yescharacteristic
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—22no—
81C2×C4×C79—22no—
82C2×C4×C79—41no—
83C2×C4×C79—41no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79C22⋊Q812yes—
87C2×C4×C79C22⋊Q811yescharacteristic
88C2×C4×C79#12412yes—
89C2×C4×C79#12411yescharacteristic
90C2×Dic79—42no—
91C2×Dic79—42no—
92C2×Dic79—42no—
93C2×Dic79—42no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C2×Dic79—42no—
97C2×Dic79—81no—
98C2×Dic79—81no—
99Dic158—42no—
100Dic158—42no—
101Dic158—81no—
102Dic158—81no—
103Dic158—84no—
104C8×C79—44no—
105C79⋊C8—42no—
106C22×C79#39311yescharacteristic
107C22×C79#45311yescharacteristic
108C22×C79#45411yescharacteristic
109C22×C79#45512yes—
110C22×C79#45612yes—
111C4×C79—21no—
112C4×C79—21no—
113C4×C79—22no—
114C4×C79—22no—
115C4×C79—81no—
116C4×C79—81no—
117Dic79—82no—
118Dic79—82no—
119Dic79—82no—
120#10359—791nomaximal, Sylow
121C2×C79#165911yescharacteristic
122C2×C79#166011yescharacteristic
123C2×C79#166412yes—
124C2×C79#202711yescharacteristic
125C2×C79#202812yes—
126#2844—791no—
127#1993—792no—
128#1960—791no—
129#1295—792no—
130#1295D7911yescharacteristic
131C79#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—791no—
133C2×#103—1581no—
134C2×#103—1581no—
135C2×#97—791no—
136C2×#97C2×D7912yes—
137#400—1582no—
138#400—1581no—
139#395—1582no—
140#395—1581no—
141#382—792no—
142#382C2×D7911yescharacteristic
143#340—1582no—
144#340—1582no—
145C22×Q8—1581no—
146C22×Q8—1581no—
147C22×C8—22no—
148C22×C8—1581no—
149C4⋊Q8—792no—
150C4⋊Q8—1581no—
151C4⋊Q8—1582no—
152C2×C4⋊C4—792no—
153C2×C4⋊C4—792no—
154C2×C4⋊C4—1582no—
155C2×C4⋊C4—1582no—
156C2×C4⋊C4—1581no—
157C2×C4⋊C4D31611yescharacteristic
158C2×C4⋊C4D31611yescharacteristic
159C2×C42C22×D7911yescharacteristic
160#103—3162no—
161#103—3162no—
162#97—22no—
163#97—22no—
164#97—1582no—
165#95—22no—
166#95—1582no—
167#95—1581no—
168#95—1581no—
169#95—1582no—
170#95—1582no—
171C2×Q8—3161no—
172C2×Q8—3161no—
173C2×Q8—3161no—
174C2×Q8—3161no—
175C2×Q8—3161no—
176C2×Q8—6321no—
177C2×Q8—6322no—
178C2×Q8—1582no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—3161no—
182C22×C4—1582no—
183C22×C4—1582no—
184C22×C4D4×D7912yes—
185C22×C4C2×D31611yescharacteristic
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—3161no—
190C2×C8—3161no—
191C2×C8—1582no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—3162no—
195C4⋊C4—3162no—
196C4⋊C4—3162no—
197C4⋊C4—6322no—
198C4⋊C4—6321no—
199C4⋊C4—1584no—
200C42Q8×D7911yescharacteristic
201C42#1192962311yescharacteristic
202C42#1192962412yes—
203C23#1457913111yescharacteristic
204Q8—3162no—
205Q8—3162no—
206Q8—6321no—
207Q8—6321no—
208Q8—6324no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—3162no—
218C2×C4—3162no—
219C2×C4—3162no—
220C2×C4—3162no—
221C2×C4—3162no—
222C2×C4—3162no—
223C2×C4—3162no—
224C2×C4—6321no—
225C2×C4—6321no—
226C2×C4#1457913912yes—
227C2×C4#1457914111yescharacteristic
228C2×C4#1457914712yes—
229C2×C4#1457914911yescharacteristic
230C8—44no—
231C8—3162no—
232C22#1909847611yescharacteristic
233C22#1909873311yescharacteristic
234C22#1909873811yescharacteristic
235C22#1909874212yes—
236C22#1909874712yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—81no—
243C4—6322no—
244C4—6322no—
245C4—6322no—
246C2#2376270711yescharacteristic
247C2#2376271211yescharacteristic
248C2#2376273512yes—
249C2#2376455711yescharacteristic
250C2#2376456012yes—
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1295C2
socle, prime radicalC23×C79#120
hypercenter#1295D79
hypocenterC79#10359
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
24. #14579161Q8semidirect
131. C79#10359semidirect

Maximal subgroups

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

subgroupindex
#237716722
#237643732
#237642192
#237612822
C79×#12952
#1035979