Finite Group Databasebeta

← #34185873

Subgroups — #34185873

Order 20224.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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#23773782C211yesmaximal, characteristic
2#23762138C211yesmaximal, characteristic
3#23762084C212yesmaximal
4#23761291C212yesmaximal
5C79×#2986C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7#19098933C414yes—
8#19098915—22no—
9#19098587C2212yes—
10#19098571—22no—
11C4×#11929608C2211yescharacteristic
12#19098310—22no—
13#19098310—22no—
14#19098309—22no—
15#533×C79C2211yescharacteristic
16#446×C79—22no—
17C8×Q8×C79—22no—
18#433×C79—22no—
19#431×C79C2212yes—
20C22×Dic158—21no—
21C2×#11929608—21no—
22C2×#11929608—22no—
23#14579161—22no—
24#14579161—21no—
25#14579161C2×C412yes—
26C2×C4⋊Dic79—22no—
27C4⋊(C79⋊C8)—22no—
28C4⋊(C79⋊C8)—22no—
29#14579084—28no—
30#14579072C2×C412yes—
31C4×C79⋊C8—21no—
32C4⋊Q8×C79D412yes—
33#125×C79C2×C412yes—
34C22⋊Q8×C79—22no—
35C4×Q8×C79—22no—
36#117×C79D412yes—
37C2×C42×C79C2311yescharacteristic
38C4⋊C8×C79—24no—
39C4⋊C8×C79D412yes—
40C4⋊C8×C79D412yes—
41C22⋊C8×C79—22no—
42#97×C79—22no—
43C4×C8×C79—22no—
44C4×C8×C79—22no—
45C2×Dic158—22no—
46C2×Dic158—44no—
47C2×Dic158—41no—
48C22×Dic79—41no—
49#11929608—42no—
50#11929608—82no—
51C2×C79⋊C8—22no—
52C2×C79⋊C8—42no—
53C4⋊Dic79—24no—
54C4⋊Dic79—42no—
55C2×Q8×C79—22no—
56C22×C4×C79C2×D411yescharacteristic
57C22×C4×C79#4112yes—
58C2×C8×C79—22no—
59C2×C8×C79—22no—
60C4⋊C4×C79—24no—
61C4⋊C4×C79—22no—
62C4⋊C4×C79C22⋊C414yes—
63C22⋊C4×C79—22no—
64C42×C79—22no—
65C42×C79C22×C411yescharacteristic
66C42×C79C2×D411yescharacteristic
67C42×C79C2×D412yes—
68C23×C79#12311yescharacteristic
69Q8×C79—42no—
70Q8×C79—42no—
71C2×C4×C79—22no—
72C2×C4×C79—22no—
73C2×C4×C79—22no—
74C2×C4×C79—24no—
75C2×C4×C79C2×C22⋊C411yescharacteristic
76C2×C4×C79C4×D412yes—
77C2×C4×C79#12011yescharacteristic
78C2×C4×C79#12112yes—
79C2×Dic79—42no—
80C2×Dic79—82no—
81Dic158—44no—
82Dic158—82no—
83C8×C79—42no—
84C8×C79—42no—
85C79⋊C8—42no—
86C79⋊C8—82no—
87C22×C79—21no—
88C22×C79—22no—
89C22×C79#38611yescharacteristic
90C4×C79—21no—
91C4×C79—22no—
92C4×C79—44no—
93C4×C79#45412yes—
94C4×C79#47112yes—
95C4×C79#47312yes—
96Dic79—82no—
97#9436—791nomaximal, Sylow
98C2×C79—41no—
99C2×C79#188811yescharacteristic
100C2×C79#190812yes—
101#3139—791no—
102#2986D7911yescharacteristic
103#1561—791no—
104#1549—792no—
105#1299—792no—
106C79#943611yescharacteristic, complemented, Sylow
107#533C2×D7911yescharacteristic
108C2×C4⋊Q8—791no—
109#491—794no—
110#487—1582no—
111#446—22no—
112C8×Q8—22no—
113#433—22no—
114#431C2×D7912yes—
115#423—792no—
116#419—1582no—
117C4×M16—791no—
118#332—1582no—
119#332—1582no—
120#332—1582no—
121C22×Q8—1581no—
122C2×M16—1582no—
123C2×M16—1581no—
124C4⋊Q8—792no—
125C4⋊Q8—1582no—
126C4⋊Q8—1581no—
127C4⋊Q8D31612yes—
128#125C4×D7912yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D7912yes—
132C2×C4⋊C4—1582no—
133C2×C42C22×D7911yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—1582no—
136C4⋊C8—1582no—
137C4⋊C8C22⋊D7912yes—
138C4⋊C8C22⋊D7912yes—
139#103—1588no—
140C22⋊C8—22no—
141#97—22no—
142#97—792no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—1581no—
146C2×Q8—22no—
147C2×Q8—3164no—
148C2×Q8—3161no—
149C2×Q8—1582no—
150C22×C4—3161no—
151C22×C4C2×C22⋊D7911yescharacteristic
152C22×C4#1192962412yes—
153M16—3162no—
154M16—6322no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—3162no—
158C2×C8—1582no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—3162no—
162C4⋊C4—1584no—
163C4⋊C4(C2×D79)⋊C414yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D7911yescharacteristic
167C42C2×C22⋊D7912yes—
168C42C2×D31611yescharacteristic
169C23#1457914411yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—3164no—
173Q8—6322no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—3162no—
179C2×C4—6322no—
180C2×C4C2×(C2×D79)⋊C411yescharacteristic
181C2×C4C4×C22⋊D7912yes—
182C2×C4#1457913011yescharacteristic
183C2×C4#1457913512yes—
184C8—42no—
185C8—42no—
186C8—3162no—
187C8—6322no—
188C22—21no—
189C22—22no—
190C22#1909844211yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—6322no—
195C4#1909873612yes—
196C4#1909882612yes—
197C4#1909883812yes—
198C2—41no—
199C2#2376380311yescharacteristic
200C2#2376391512yes—
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2986C2
socleC22×C79#386
prime radicalC23×C79#123
cosocleC2×C42×C79C23
hypercenter#2986D79
hypocenterC79#9436
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
106. C79#9436semidirect

Maximal subgroups

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

subgroupindex
#237737822
#237621382
#237620842
#237612912
C79×#29862
#943679