Finite Group Databasebeta

← #34176005

Subgroups — #34176005

Order 20224.

266 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765085C211yesmaximal, characteristic
2#23762035C211yesmaximal, characteristic
3C79×#1884C211yesmaximal, characteristic
4#19098620—21no—
5#19098547—21no—
6#19098362C412yescomplemented
7#19098360—21no—
8#19098303—21no—
9#19098303C412yescomplemented
10#568×C79C411yescharacteristic
11#532×C79C2211yescharacteristic
12#421×C79—21no—
13#410×C79C411yescharacteristic
14#339×C79—21no—
15C2×#11929608—41no—
16C4⋊(C79⋊C8)—41no—
17#14579080—22no—
18#14579078—41no—
19C22⋊(C79⋊C8)—41no—
20#14579072—22no—
21C2×#41×C79C2×C411yescharacteristic
22C2×#41×C79D411yescharacteristic
23#131×C79—42no—
24C2×M16×C79—21no—
25C2×M16×C79—41no—
26C4⋊Q8×C79C2×C411yescharacteristic
27#127×C79Q811yescharacteristic
28#124×C79C2×C411yescharacteristic
29#124×C79D411yescharacteristic
30#121×C79—41no—
31C4×D4×C79—41no—
32#117×C79—21no—
33C2×C42×C79D411yescharacteristic
34#104×C79—42no—
35#99×C79—41no—
36#11929608—42no—
37#11929608—81no—
38C2×C79⋊C8—42no—
39C2×C79⋊C8—81no—
40#41×C79—41no—
41#41×C79—41no—
42#41×C79—41no—
43#41×C79—41no—
44C2×Q8×C79C4211yescharacteristic
45C2×Q8×C79C22⋊C411yescharacteristic
46C2×D4×C79—21no—
47C2×D4×C79—21no—
48C2×D4×C79—41no—
49C2×D4×C79C4⋊C411yescharacteristic
50C2×D4×C79C4⋊C411yescharacteristic
51C22×C4×C79—21no—
52C22×C4×C79—21no—
53C22×C4×C79—21no—
54C22×C4×C79C22⋊C411yescharacteristic
55M16×C79—22no—
56M16×C79—42no—
57M16×C79—41no—
58C2×C8×C79—41no—
59C2×C8×C79—42no—
60C4⋊C4×C79—41no—
61C4⋊C4×C79—41no—
62C22⋊C4×C79—41no—
63C22⋊C4×C79—41no—
64C22⋊C4×C79—41no—
65C42×C79—21no—
66C42×C79—41no—
67C42×C79C22⋊C411yescharacteristic
68C42×C79C4⋊C411yescharacteristic
69C23×C79—21no—
70C23×C79—21no—
71C23×C79#9911yescharacteristic
72Q8×C79—41no—
73Q8×C79—41no—
74D4×C79—41no—
75D4×C79—41no—
76D4×C79—42no—
77D4×C79—81no—
78D4×C79—82no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—41no—
86C2×C4×C79—41no—
87C2×C4×C79—41no—
88C2×C4×C79—41no—
89C2×C4×C79—81no—
90C2×C4×C79—81no—
91C2×C4×C79#9511yescharacteristic
92C2×C4×C79#9911yescharacteristic
93C8×C79—42no—
94C79⋊C8—82no—
95C79⋊C8—82no—
96C22×C79—21no—
97C22×C79—41no—
98C22×C79—41no—
99C22×C79—41no—
100C22×C79—81no—
101C22×C79—81no—
102C22×C79#34211yescharacteristic
103C4×C79—21no—
104C4×C79—41no—
105C4×C79—41no—
106C4×C79—41no—
107C4×C79—41no—
108C4×C79—161no—
109C4×C79AΣL(1,𝔽16)11yescharacteristic
110C4×C79#35211yescharacteristic
111#7484—791nomaximal, Sylow
112C2×C79—21no—
113C2×C79—41no—
114C2×C79—81no—
115C2×C79—81no—
116C2×C79#139011yescharacteristic
117#2132—791no—
118#1884D7911yescharacteristic
119#1536—791no—
120C79#748411yescharacteristic, complemented, Sylow
121#568Dic7911yescharacteristic
122#532C2×D7911yescharacteristic
123#432—1581no—
124#421—21no—
125#413—1581no—
126#410Dic7911yescharacteristic
127#356—792no—
128#355—1581no—
129#339—21no—
130#329—792no—
131#329—1581no—
132C2×#41C2×Dic7911yescharacteristic
133C2×#41C22⋊D7911yescharacteristic
134#131—42no—
135C2×M16—21no—
136C2×M16—41no—
137C2×M16—3161no—
138C4⋊Q8C4×D7911yescharacteristic
139#127Dic15811yescharacteristic
140#124C4×D7911yescharacteristic
141#124D31611yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145C2×C42C22⋊D7911yescharacteristic
146C4⋊C8—3161no—
147#104—42no—
148#101—1582no—
149#100—3161no—
150#99—41no—
151C22⋊C8—3161no—
152#97—1582no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C4×Dic7911yescharacteristic
158C2×Q8(C2×D79)⋊C411yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4Dic79⋊C411yescharacteristic
163C2×D4C4⋊Dic7911yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22⋊Dic7911yescharacteristic
168M16—22no—
169M16—42no—
170M16—41no—
171M16—3162no—
172M16—6321no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—3162no—
176C2×C8—6321no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42(C2×D79)⋊C411yescharacteristic
185C42Dic79⋊C411yescharacteristic
186C23—21no—
187C23—21no—
188C23#1457907611yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—81no—
195D4—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4C79⋊#9511yescharacteristic
209C2×C4#1457907611yescharacteristic
210C8—42no—
211C8—6322no—
212C8—6322no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1909833011yescharacteristic
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4#1909835011yescharacteristic
227C4#1909835311yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2376151811yescharacteristic
subgroupG/H
centerC2#23761518
commutatorC2×Q8×C79C42
Frattini#532C2×D79
FittingC79×#1884C2
socleC2×C79#1390
prime radical, cosocle#532×C79C22
hypercenter#1884D79
hypocenterC79#7484
LCSG ▷ C2×Q8×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
6. #19098362C4semidirect
9. #19098303C4semidirect
120. C79#7484semidirect

Maximal subgroups

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

subgroupindex
#237650852
#237620352
C79×#18842
#748479