Finite Group Databasebeta

← #34175694

Subgroups — #34175694

Order 20224.

228 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 43 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#23763609C211yesmaximal, characteristic
2#23761221C211yesmaximal, characteristic
3C79×#1318C211yesmaximal, characteristic
4#19099038C411yescharacteristic
5#19098603—21no—
6C2×#14579078C411yescharacteristic
7#19098323—21no—
8#19098318—21no—
9#390×C79C2211yescharacteristic
10#324×C79—21no—
11C2×#11929608—21no—
12C22×C79⋊C8—41no—
13D4×Dic79—41no—
14D4×Dic79—41no—
15#14579120—21no—
16C2×C22⋊Dic79—41no—
17#14579094—41no—
18#14579092—41no—
19#14579078—21no—
20#14579078C812yescomplemented
21C22×D4×C79C2×C411yescharacteristic
22C2×C4⋊C4×C79D411yescharacteristic
23C2×C22⋊C4×C79—41no—
24C2×C42×C79D411yescharacteristic
25C22⋊C8×C79—41no—
26C22×Dic79—41no—
27C22×Dic79—41no—
28#11929608—22no—
29#11929608—41no—
30C2×C79⋊C8—41no—
31C2×C79⋊C8—42no—
32C2×C79⋊C8—81no—
33C2×C79⋊C8—81no—
34C4⋊Dic79—21no—
35C4⋊Dic79—21no—
36C22⋊Dic79—42no—
37C22⋊Dic79—41no—
38C4×Dic79—41no—
39C24×C79—21no—
40C2×D4×C79—21no—
41C2×D4×C79—41no—
42C2×D4×C79—41no—
43C2×D4×C79C2×C811yescharacteristic
44C2×D4×C79M1611yescharacteristic
45C22×C4×C79—21no—
46C22×C4×C79—21no—
47C22×C4×C79C22⋊C411yescharacteristic
48C2×C8×C79—81no—
49C4⋊C4×C79—41no—
50C22⋊C4×C79—81no—
51C22⋊C4×C79—81no—
52C42×C79—41no—
53C23×C79—21no—
54C23×C79—21no—
55C23×C79—41no—
56C23×C79—41no—
57C23×C79—81no—
58C23×C79—81no—
59C23×C79#9911yescharacteristic
60D4×C79—41no—
61D4×C79—41no—
62D4×C79—42no—
63C2×C4×C79—21no—
64C2×C4×C79—21no—
65C2×C4×C79—21no—
66C2×C4×C79—21no—
67C2×C4×C79—41no—
68C2×C4×C79—41no—
69C2×C4×C79—41no—
70C2×C4×C79—81no—
71C2×C4×C79C22⋊C811yescharacteristic
72C2×C4×C79#10111yescharacteristic
73C2×Dic79—41no—
74C2×Dic79—41no—
75C2×Dic79—161no—
76C2×Dic79—161no—
77C8×C79—161no—
78C79⋊C8—42no—
79C79⋊C8—82no—
80C22×C79—21no—
81C22×C79—21no—
82C22×C79—41no—
83C22×C79—41no—
84C22×C79—81no—
85C22×C79—81no—
86C22×C79—81no—
87C22×C79—81no—
88C22×C79—161no—
89C22×C79#32411yescharacteristic
90C22×C79AΣL(1,𝔽16)11yescharacteristic
91C22×C79#35211yescharacteristic
92C4×C79—21no—
93C4×C79—21no—
94C4×C79—41no—
95C4×C79—41no—
96C4×C79—81no—
97Dic79—81no—
98Dic79—81no—
99#7375—791nomaximal, Sylow
100C2×C79—21no—
101C2×C79—21no—
102C2×C79—81no—
103C2×C79—81no—
104C2×C79#132611yescharacteristic
105C2×C79#140211yescharacteristic
106C2×C79#140311yescharacteristic
107#1856—791no—
108#1318D7911yescharacteristic
109#1270—791no—
110C79#737511yescharacteristic, complemented, Sylow
111#518—791no—
112#428—1581no—
113C2×#100—791no—
114#390C2×D7911yescharacteristic
115#339—1581no—
116#336—1581no—
117#324—21no—
118C22×D4C4×D7911yescharacteristic
119C2×M16—1581no—
120C22×C8—3161no—
121C4×D4—3161no—
122C4×D4—3161no—
123#117—1581no—
124C2×C4⋊C4C22⋊D7911yescharacteristic
125C2×C22⋊C4—41no—
126C2×C22⋊C4—3161no—
127C2×C42D31611yescharacteristic
128#107—3161no—
129#106—3161no—
130#100—792no—
131#100—1581no—
132C22⋊C8—41no—
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C8×D7911yescharacteristic
138C2×D4#1192960711yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—3161no—
142C22×C4—3161no—
143C22×C4(C2×D79)⋊C411yescharacteristic
144M16—3161no—
145M16—1582no—
146C2×C8—81no—
147C2×C8—3161no—
148C2×C8—3162no—
149C2×C8—6321no—
150C2×C8—6321no—
151C4⋊C4—41no—
152C4⋊C4—1581no—
153C4⋊C4—1581no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C22⋊C4—3162no—
157C22⋊C4—3161no—
158C42—41no—
159C42—3161no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23#1457907511yescharacteristic
167D4—41no—
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—3161no—
179C2×C4—3161no—
180C2×C4—12641no—
181C2×C4—12641no—
182C2×C4(C2×D79)⋊C811yescharacteristic
183C2×C4#1457907911yescharacteristic
184C8—161no—
185C8—3162no—
186C8—6322no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22#1909828811yescharacteristic
197C22#1909834911yescharacteristic
198C22#1909835411yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—6321no—
205C4—6321no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2#2376134711yescharacteristic
211C2#2376154411yescharacteristic
212C2#2376154711yescharacteristic
subgroupG/H
centerC22#19098349
commutatorC2×D4×C79C2×C8
Frattini#390C2×D79
FittingC79×#1318C2
socleC22×C79AΣL(1,𝔽16)
prime radicalC22×D4×C79C2×C4
cosocle#390×C79C22
hypercenter#1318D79
hypocenterC79#7375
LCSG ▷ C2×D4×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1318

Complements

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

NH ≅ G/Ntype
20. #14579078C8semidirect
110. C79#7375semidirect

Maximal subgroups

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

subgroupindex
#237636092
#237612212
C79×#13182
#737579