Finite Group Databasebeta

← #34189607

Subgroups — #34189607

Order 20224.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#19098972C212yesmaximal
2#23771468C211yesmaximal, characteristic
3#23764219C211yesmaximal, characteristic
4C2×#19098304C212yesmaximal
5C79×#1295C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579153C2212yes—
8#19098972—24no—
9#19098972—24no—
10C2×#14579095—22no—
11C2×#14579093—22no—
12C2×#14579084—22no—
13C2×#14579072C2211yescharacteristic
14#19098503—22no—
15#19098495—22no—
16#19098493—22no—
17#19098304—24no—
18C2×#97×C79C2212yes—
19#382×C79C2211yescharacteristic
20C22×Dic158—21no—
21C22×C79⋊C8—21no—
22#14579161—22no—
23#14579161—21no—
24#14579161D412yes—
25#14579153—22no—
26#14579153—22no—
27#14579153—22no—
28#14579153—22no—
29C2×C4⋊Dic79—22no—
30C2×C4⋊Dic79—22no—
31C2×C4⋊Dic79D412yes—
32C2×C4⋊Dic79D412yes—
33C2×Dic79⋊C4—22no—
34C2×Dic79⋊C4—22no—
35#14579095—28no—
36#14579093—28no—
37#14579084—44no—
38#14579072—22no—
39C79⋊#95—24no—
40C79⋊#95—22no—
41C22×C8×C79—22no—
42C2×C4⋊C4×C79D412yes—
43C2×C42×C79C2311yescharacteristic
44#97×C79—22no—
45#97×C79D414yes—
46#95×C79—22no—
47C2×Dic158—22no—
48C2×Dic158—42no—
49C2×Dic158—42no—
50C2×Dic158—41no—
51C22×Dic79—22no—
52C22×Dic79—22no—
53C22×Dic79—41no—
54C2×C79⋊C8—22no—
55C2×C79⋊C8—41no—
56C2×C79⋊C8—41no—
57C4⋊Dic79—22no—
58C4⋊Dic79—24no—
59C4⋊Dic79—22no—
60C4⋊Dic79—22no—
61C4⋊Dic79—22no—
62C4⋊Dic79—42no—
63C4⋊Dic79—82no—
64Dic79⋊C4—42no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68C22×C4×C79—22no—
69C22×C4×C79C2×D412yes—
70C22×C4×C79C2×D411yescharacteristic
71C2×C8×C79—28no—
72C2×C8×C79—24no—
73C4⋊C4×C79—42no—
74C42×C79C2×D411yescharacteristic
75C42×C79C2×D412yes—
76C42×C79#4111yescharacteristic
77C23×C79#12011yescharacteristic
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—24no—
81C2×C4×C79—42no—
82C2×C4×C79—44no—
83C2×C4×C79#12012yes—
84C2×C4×C79#12111yescharacteristic
85C2×C4×C79#12112yes—
86C2×C4×C79#12711yescharacteristic
87C2×Dic79—42no—
88C2×Dic79—42no—
89C2×Dic79—42no—
90C2×Dic79—42no—
91C2×Dic79—42no—
92C2×Dic79—44no—
93C2×Dic79—81no—
94C2×Dic79—81no—
95Dic158—44no—
96Dic158—82no—
97C8×C79—28no—
98C79⋊C8—42no—
99C22×C79#39211yescharacteristic
100C22×C79#45311yescharacteristic
101C22×C79#45411yescharacteristic
102C22×C79#45614yes—
103C4×C79—22no—
104C4×C79—22no—
105C4×C79—21no—
106C4×C79—21no—
107C4×C79—82no—
108Dic79—82no—
109Dic79—82no—
110Dic79—82no—
111#10087—791nomaximal, Sylow
112C2×C79#166012yes—
113C2×C79#166312yes—
114C2×C79#200512yes—
115C2×C79#200611yescharacteristic
116C2×#501—792no—
117#2822—791no—
118#1960—791no—
119C2×#330—792no—
120#1295D7911yescharacteristic
121C79#1008711yescharacteristic, complemented, Sylow
122C2×C4⋊Q8—791no—
123C2×#125—792no—
124#501—1584no—
125#501—1584no—
126C2×#107—1582no—
127C2×#106—1582no—
128C2×#103—1582no—
129C2×#97—791no—
130C2×#97C2×D7912yes—
131#400—1582no—
132#398—1582no—
133#398—1582no—
134#382C2×D7911yescharacteristic
135#330—1584no—
136C22×Q8—1581no—
137C22×C8—22no—
138C22×C8—1581no—
139C4⋊Q8—792no—
140C4⋊Q8—1582no—
141C4⋊Q8—1581no—
142#125—1582no—
143#125—1582no—
144#125—1582no—
145#125—1582no—
146C2×C4⋊C4—792no—
147C2×C4⋊C4—792no—
148C2×C4⋊C4—1582no—
149C2×C4⋊C4—1582no—
150C2×C4⋊C4—1582no—
151C2×C4⋊C4—1582no—
152C2×C4⋊C4D31612yes—
153C2×C42C22×D7911yescharacteristic
154#107—1588no—
155#106—1588no—
156#103—3164no—
157#97—22no—
158#97—1582no—
159#97D31614yes—
160#95—22no—
161#95—1584no—
162#95—1582no—
163C2×Q8—3162no—
164C2×Q8—3162no—
165C2×Q8—3161no—
166C2×Q8—1582no—
167C22×C4—22no—
168C22×C4—3161no—
169C22×C4—1582no—
170C22×C4—1582no—
171C22×C4D4×D7912yes—
172C22×C4C2×D31611yescharacteristic
173C2×C8—28no—
174C2×C8—24no—
175C2×C8—3161no—
176C2×C8—3161no—
177C2×C8—1582no—
178C4⋊C4—42no—
179C4⋊C4—3162no—
180C4⋊C4—3162no—
181C4⋊C4—3162no—
182C4⋊C4—3162no—
183C4⋊C4—3162no—
184C4⋊C4—6322no—
185C4⋊C4—1582no—
186C4⋊C4—1584no—
187C4⋊C4—1582no—
188C4⋊C4—1582no—
189C4⋊C4—1582no—
190C42D4×D7911yescharacteristic
191C42C2×D31612yes—
192C42#1192962511yescharacteristic
193C23#1457913111yescharacteristic
194Q8—3164no—
195Q8—6322no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—3162no—
202C2×C4—3162no—
203C2×C4—3162no—
204C2×C4—3162no—
205C2×C4—3162no—
206C2×C4—3164no—
207C2×C4—6321no—
208C2×C4—6321no—
209C2×C4#1457913112yes—
210C2×C4#1457913611yescharacteristic
211C2×C4#1457913612yes—
212C2×C4#1457915811yescharacteristic
213C8—28no—
214C8—3162no—
215C22#1909847311yescharacteristic
216C22#1909873311yescharacteristic
217C22#1909873811yescharacteristic
218C22#1909874714yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—6322no—
225C4—6322no—
226C4—6322no—
227C2#2376271212yes—
228C2#2376272912yes—
229C2#2376443912yes—
230C2#2376444411yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1295C2
socle, prime radicalC23×C79#120
hypercenter#1295D79
hypocenterC79#10087
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
121. C79#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#190989722
#237714682
#237642192
C2×#190983042
C79×#12952
#1008779