Finite Group Databasebeta

← #34205115

Subgroups — #34205115

Order 20224.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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#23772471C211yesmaximal, characteristic
2#23763784C211yesmaximal, characteristic
3#23763622C212yesmaximal, complemented
4#23761530C212yesmaximal, complemented
5C79×#2132C211yesmaximal, characteristic
6#19099330C2211yescharacteristic
7#19098635—22no—
8#19098635—22no—
9#19098620—21no—
10#19098612—22no—
11#19098580—21no—
12#19098537C2211yescharacteristic
13#19098530—21no—
14#19098530C2212yes—
15#19098327—22no—
16#19098323—21no—
17#532×C79C2211yescharacteristic
18#413×C79C2212yes—
19#356×C79C412yescomplemented
20#355×C79C412yescomplemented
21#14579168—42no—
22C2×#11929608—21no—
23C2×#11929608—21no—
24C2×#11929608—22no—
25C2×#11929608D412yes—
26C22×C79⋊C8—21no—
27C22×C79⋊C8D412yes—
28#14579120—21no—
29#14579097—44no—
30C4⋊(C79⋊C8)—21no—
31C4⋊(C79⋊C8)—21no—
32#14579088—42no—
33#14579076—22no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37#14579072—22no—
38C2×#41×C79D411yescharacteristic
39C2×#41×C79C2311yescharacteristic
40C2×M16×C79—22no—
41C4⋊Q8×C79C2×C411yescharacteristic
42#127×C79Q811yescharacteristic
43#124×C79C2×C411yescharacteristic
44#124×C79Q811yescharacteristic
45#121×C79—22no—
46C4×D4×C79—22no—
47C2×C42×C79D411yescharacteristic
48#101×C79C2×C414yes—
49#100×C79—22no—
50#11929608—22no—
51#11929608—24no—
52#11929608—42no—
53#11929608—41no—
54#11929608—81no—
55C2×C79⋊C8—22no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8—41no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—81no—
61C4⋊Dic79—41no—
62C22⋊Dic79—41no—
63C4×Dic79—41no—
64#41×C79—41no—
65#41×C79—41no—
66#41×C79—41no—
67#41×C79—41no—
68C2×Q8×C79C4⋊C411yescharacteristic
69C2×Q8×C79C22×C411yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—21no—
72C2×D4×C79C4⋊C411yescharacteristic
73C2×D4×C79C2×Q811yescharacteristic
74C2×D4×C79#4112yes—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D412yes—
79M16×C79—24no—
80M16×C79—42no—
81C2×C8×C79—42no—
82C4⋊C4×C79—22no—
83C22⋊C4×C79—22no—
84C22⋊C4×C79—22no—
85C42×C79—21no—
86C42×C79C4⋊C411yescharacteristic
87C42×C79C4⋊C411yescharacteristic
88C23×C79—21no—
89C23×C79C22⋊Q812yes—
90C23×C79#12311yescharacteristic
91Q8×C79—41no—
92Q8×C79—41no—
93D4×C79—41no—
94D4×C79—42no—
95D4×C79—41no—
96D4×C79—81no—
97D4×C79—82no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—81no—
108C2×C4×C79C2×C4⋊C411yescharacteristic
109C2×C4×C79C4×D412yes—
110C2×C4×C79#12011yescharacteristic
111C2×Dic79—81no—
112C8×C79—44no—
113C79⋊C8—42no—
114C79⋊C8—42no—
115C79⋊C8—42no—
116C22×C79—21no—
117C22×C79—22no—
118C22×C79—41no—
119C22×C79—42no—
120C22×C79—42no—
121C22×C79—41no—
122C22×C79#38511yescharacteristic
123C4×C79—21no—
124C4×C79—22no—
125C4×C79—41no—
126C4×C79—41no—
127C4×C79—41no—
128C4×C79B(4,𝔽2)11yescharacteristic
129C4×C79#45811yescharacteristic
130Dic79—161no—
131#12974—791nomaximal, Sylow
132C2×C79—21no—
133C2×C79—41no—
134C2×C79—42no—
135C2×C79—81no—
136C2×C79#186711yescharacteristic
137#2980—791no—
138#2132D7911yescharacteristic
139#1884—791no—
140#1858—792no—
141#1395—792no—
142C79#1297411yescharacteristic, complemented, Sylow
143#568—791no—
144#532C2×D7911yescharacteristic
145#435—1582no—
146#435—1582no—
147#432—1581no—
148#430—1582no—
149#421—1581no—
150#413C2×D7912yes—
151#410—791no—
152#408—792no—
153#408—1581no—
154#356Dic7912yescomplemented
155#355Dic7912yescomplemented
156#341—1582no—
157#339—1581no—
158C2×#41D31611yescharacteristic
159C2×#41C22×D7911yescharacteristic
160#131—3162no—
161C2×M16—22no—
162C2×M16—792no—
163C2×M16—1582no—
164C2×M16—1581no—
165C2×M16—1581no—
166C22×C8—792no—
167C22×C8—1581no—
168C4⋊Q8C2×Dic7911yescharacteristic
169#127Dic15811yescharacteristic
170#124Dic15811yescharacteristic
171#124C2×Dic7911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1581no—
175C2×C42D31611yescharacteristic
176#108—3164no—
177C4⋊C8—1581no—
178C4⋊C8—1581no—
179#104—3162no—
180#101C2×Dic7914yes—
181#100—22no—
182#99—1582no—
183C22⋊C8—1582no—
184C22⋊C8—1582no—
185C22⋊C8—1582no—
186#97—1582no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊Dic7911yescharacteristic
192C2×Q8C22×Dic7911yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊Dic7911yescharacteristic
196C2×D4C2×Dic15811yescharacteristic
197C2×D4#1192962312yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4D4×D7912yes—
201C22×C4C2×D31611yescharacteristic
202M16—24no—
203M16—42no—
204M16—3161no—
205M16—3162no—
206M16—6321no—
207M16—1582no—
208M16—1584no—
209C2×C8—42no—
210C2×C8—3161no—
211C2×C8—3162no—
212C2×C8—3162no—
213C2×C8—6321no—
214C2×C8—1582no—
215C2×C8—1582no—
216C4⋊C4—22no—
217C4⋊C4—3161no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—3161no—
221C42—21no—
222C42—3161no—
223C42C4⋊Dic7911yescharacteristic
224C42C4⋊Dic7911yescharacteristic
225C23—21no—
226C23#1457913812yes—
227C23#1457914511yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—81no—
234D4—82no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—6321no—
246C2×C4C2×C4⋊Dic7911yescharacteristic
247C2×C4D4×Dic7912yes—
248C2×C4#1457913111yescharacteristic
249C8—44no—
250C8—3162no—
251C8—3162no—
252C8—3162no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1909843811yescharacteristic
260C4—22no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—12641no—
266C4#1909875111yescharacteristic
267C4#1909875411yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2376368311yescharacteristic
subgroupG/H
centerC4#19098751
commutatorC2×Q8×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2132C2
socleC2×C79#1867
prime radical#532×C79C22
cosocleC2×#41×C79C23
hypercenter#2132D79
hypocenterC79#12974
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #23763622C2semidirect
4. #23761530C2semidirect
19. #356×C79C4semidirect
20. #355×C79C4semidirect
142. C79#12974semidirect
154. #356Dic79semidirect
155. #355Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724712
#237637842
#237636222
#237615302
C79×#21322
#1297479