Finite Group Databasebeta

← #34191349

Subgroups — #34191349

Order 20224.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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#23764372C212yesmaximal
2#23764219C211yesmaximal, characteristic
3#23761281C211yesmaximal, characteristic
4#23761281C212yesmaximal
5C79×#2845C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579084—22no—
8C2×#14579072C2212yes—
9C2×#14579072C2211yescharacteristic
10#19098325—22no—
11#19098325—22no—
12#398×C79—22no—
13#398×C79—22no—
14#398×C79—22no—
15#382×C79C2212yes—
16#382×C79C2211yescharacteristic
17C22×Dic158—21no—
18C22×C79⋊C8—21no—
19C22×C79⋊C8—22no—
20#14579161—22no—
21#14579161—21no—
22#14579161Q812yes—
23C2×C4⋊Dic79—22no—
24#14579084—44no—
25#14579072—22no—
26#14579072—22no—
27#14579072—22no—
28C2×C4⋊C4×C79—22no—
29C2×C4⋊C4×C79—22no—
30C2×C4⋊C4×C79—22no—
31C2×C4⋊C4×C79D412yes—
32C2×C4⋊C4×C79D412yes—
33C2×C4⋊C4×C79D412yes—
34C2×C42×C79C2311yescharacteristic
35#95×C79—22no—
36#95×C79—22no—
37#95×C79—22no—
38#95×C79—22no—
39C2×Dic158—22no—
40C2×Dic158—42no—
41C2×Dic158—42no—
42C2×Dic158—41no—
43C22×Dic79—41no—
44C2×C79⋊C8—22no—
45C2×C79⋊C8—24no—
46C2×C79⋊C8—41no—
47C2×C79⋊C8—42no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—41no—
50C4⋊Dic79—24no—
51C4⋊Dic79—42no—
52C22×C4×C79—22no—
53C22×C4×C79—22no—
54C22×C4×C79—22no—
55C22×C4×C79C2×D411yescharacteristic
56C22×C4×C79C2×D412yes—
57C4⋊C4×C79—42no—
58C4⋊C4×C79—42no—
59C4⋊C4×C79—42no—
60C4⋊C4×C79—82no—
61C4⋊C4×C79—82no—
62C4⋊C4×C79—82no—
63C42×C79C2×Q811yescharacteristic
64C42×C79#4112yes—
65C42×C79#4111yescharacteristic
66C23×C79#12011yescharacteristic
67C2×C4×C79—21no—
68C2×C4×C79—21no—
69C2×C4×C79—22no—
70C2×C4×C79—22no—
71C2×C4×C79—42no—
72C2×C4×C79—42no—
73C2×C4×C79—42no—
74C2×C4×C79—42no—
75C2×C4×C79—42no—
76C2×C4×C79—42no—
77C2×C4×C79—42no—
78C2×C4×C79—42no—
79C2×C4×C79—42no—
80C2×C4×C79C22⋊Q812yes—
81C2×C4×C79C22⋊Q811yescharacteristic
82C2×C4×C79#12412yes—
83C2×C4×C79#12411yescharacteristic
84C2×Dic79—42no—
85C2×Dic79—81no—
86C2×Dic79—81no—
87Dic158—44no—
88Dic158—82no—
89C79⋊C8—42no—
90C79⋊C8—44no—
91C22×C79#39311yescharacteristic
92C22×C79#45311yescharacteristic
93C22×C79#45411yescharacteristic
94C22×C79#45512yes—
95C22×C79#45612yes—
96C4×C79—21no—
97C4×C79—21no—
98C4×C79—22no—
99C4×C79—22no—
100C4×C79—82no—
101C4×C79—82no—
102C4×C79—82no—
103Dic79—82no—
104#10361—791nomaximal, Sylow
105C2×C79#166112yes—
106C2×C79#166211yescharacteristic
107C2×C79#166311yescharacteristic
108C2×C79#202711yescharacteristic
109C2×C79#202812yes—
110#2845D7911yescharacteristic
111#1993—792no—
112#1960—791no—
113#1295—791no—
114#1295—792no—
115C79#1036111yescharacteristic, complemented, Sylow
116C2×C4⋊Q8—791no—
117C2×#103—1582no—
118C2×#97—791no—
119C2×#97—792no—
120#398—22no—
121#398—22no—
122#398—22no—
123#382C2×D7911yescharacteristic
124#382C2×D7912yes—
125#340—1582no—
126#340—1582no—
127C22×Q8—1581no—
128C22×C8—1582no—
129C22×C8—1581no—
130C4⋊Q8—792no—
131C4⋊Q8—1581no—
132C4⋊Q8—1582no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—1582no—
137C2×C4⋊C4C22⋊D7912yes—
138C2×C4⋊C4C22⋊D7912yes—
139C2×C4⋊C4C22⋊D7912yes—
140C2×C42C22×D7911yescharacteristic
141#103—3164no—
142#97—1582no—
143#97—1582no—
144#97—1582no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149C2×Q8—3162no—
150C2×Q8—3162no—
151C2×Q8—3161no—
152C2×Q8—1582no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—3161no—
157C22×C4C2×C22⋊D7912yes—
158C22×C4C2×C22⋊D7911yescharacteristic
159C2×C8—3161no—
160C2×C8—3162no—
161C2×C8—3162no—
162C2×C8—3161no—
163C2×C8—1582no—
164C2×C8—1584no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—3162no—
172C4⋊C4—1584no—
173C42Q8×D7911yescharacteristic
174C42#1192962512yes—
175C42#1192962511yescharacteristic
176C23#1457913011yescharacteristic
177Q8—3164no—
178Q8—6322no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—3162no—
193C2×C4—6321no—
194C2×C4—6321no—
195C2×C4#1457913712yes—
196C2×C4#1457913711yescharacteristic
197C2×C4#1457915012yes—
198C2×C4#1457915011yescharacteristic
199C8—3162no—
200C8—3164no—
201C22#1909847411yescharacteristic
202C22#1909873111yescharacteristic
203C22#1909873611yescharacteristic
204C22#1909874112yes—
205C22#1909874612yes—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—82no—
212C4—82no—
213C4—6322no—
214C2#2376271712yes—
215C2#2376272311yescharacteristic
216C2#2376272811yescharacteristic
217C2#2376455311yescharacteristic
218C2#2376455812yes—
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2845C2
socle, prime radicalC23×C79#120
hypercenter#2845D79
hypocenterC79#10361
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2845

Complements

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

NH ≅ G/Ntype
115. C79#10361semidirect

Maximal subgroups

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

subgroupindex
#237643722
#237642192
#237612812
C79×#28452
#1036179