Finite Group Databasebeta

← #34177574

Subgroups — #34177574

Order 20224.

494 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 87 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#23764938C211yesmaximal, characteristic, complemented
2#23764933C212yesmaximal, complemented
3#23761226C211yesmaximal, characteristic, complemented
4#23761221C212yesmaximal, complemented
5C79×#2733C211yesmaximal, characteristic
6C2×#14579094—21no—
7C2×#14579092—22no—
8C2×#14579085—21no—
9C2×#14579082—22no—
10#19098433C2211yescharacteristic
11#19098318C2211yescharacteristic
12#19098318C2212yes—
13C2×#124×C79—21no—
14#394×C79—21no—
15#392×C79—22no—
16#391×C79C2211yescharacteristic
17#390×C79C2212yes—
18#388×C79—22no—
19C22×C79⋊C8—22no—
20C22×C79⋊C8—21no—
21C2×C4⋊Dic79—22no—
22C2×C4⋊Dic79—21no—
23#14579094—42no—
24#14579092—44no—
25#14579085—42no—
26#14579082—44no—
27C22×Q8×C79D411yescharacteristic
28C22×D4×C79—21no—
29C22×D4×C79—21no—
30C22×D4×C79D412yes—
31#124×C79—42no—
32#124×C79—42no—
33C2×C4⋊C4×C79D412yes—
34C2×C4⋊C4×C79D411yescharacteristic
35C2×C22⋊C4×C79—22no—
36C2×C22⋊C4×C79—22no—
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79—22no—
39C2×C42×C79—21no—
40C2×C42×C79C2311yescharacteristic
41#95×C79—21no—
42#95×C79—22no—
43C22×Dic79—41no—
44C2×C79⋊C8—24no—
45C2×C79⋊C8—22no—
46C2×C79⋊C8—42no—
47C2×C79⋊C8—41no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—41no—
50C4⋊Dic79—42no—
51C4⋊Dic79—41no—
52C4⋊Dic79—42no—
53C4⋊Dic79—41no—
54C24×C79—22no—
55C2×Q8×C79—21no—
56C2×Q8×C79—41no—
57C2×Q8×C79—41no—
58C2×Q8×C79SD1612yes—
59C2×D4×C79—22no—
60C2×D4×C79—22no—
61C2×D4×C79—42no—
62C2×D4×C79—42no—
63C2×D4×C79—42no—
64C2×D4×C79—81no—
65C2×D4×C79—81no—
66C2×D4×C79—82no—
67C2×D4×C79—81no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79C2×D411yescharacteristic
72C22×C4×C79C2×D412yes—
73C4⋊C4×C79—42no—
74C4⋊C4×C79—41no—
75C22⋊C4×C79—44no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C22⋊C4×C79—42no—
79C22⋊C4×C79—42no—
80C22⋊C4×C79—82no—
81C42×C79—41no—
82C42×C79—42no—
83C23×C79—42no—
84C23×C79—42no—
85C23×C79—42no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79#12011yescharacteristic
91Q8×C79—42no—
92Q8×C79—81no—
93D4×C79—44no—
94D4×C79—82no—
95D4×C79—84no—
96D4×C79—84no—
97C2×C4×C79—22no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—41no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—41no—
109C2×C4×C79C2×SD1611yescharacteristic
110C2×C4×C79#13512yes—
111C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
112C2×C4×C79#13712yes—
113C2×Dic79—81no—
114C2×Dic79—82no—
115C79⋊C8—42no—
116C79⋊C8—44no—
117C22×C79—82no—
118C22×C79—82no—
119C22×C79—82no—
120C22×C79—82no—
121C22×C79—82no—
122C22×C79—82no—
123C22×C79—82no—
124C22×C79#45011yescharacteristic
125C22×C79#45212yes—
126C22×C79#45311yescharacteristic
127C22×C79#45312yes—
128C22×C79#45811yescharacteristic
129C4×C79—21no—
130C4×C79—21no—
131C4×C79—22no—
132C4×C79—22no—
133C4×C79—42no—
134C4×C79—82no—
135C4×C79—81no—
136Dic79—161no—
137#7830—791nomaximal, Sylow
138C2×C79—162no—
139C2×C79#161311yescharacteristic
140C2×C79#161812yes—
141C2×C79#163112yes—
142C2×C79#163411yescharacteristic
143C2×C79#165511yescharacteristic
144#2733D7911yescharacteristic
145#2095—791no—
146#2094—792no—
147#1272—791no—
148#1270—792no—
149C79#783011yescharacteristic, complemented, Sylow
150C2×#124—21no—
151C2×#107—1581no—
152C2×#106—1582no—
153C2×#103—1581no—
154C2×#102—1582no—
155#394—21no—
156#392—22no—
157#391C2×D7911yescharacteristic
158#390C2×D7912yes—
159#388—22no—
160#384—791no—
161#336—792no—
162#336—791no—
163C22×Q8C22⋊D7911yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22⋊D7912yes—
167C22×C8—1582no—
168C22×C8—1581no—
169#124—42no—
170#124—42no—
171C2×C4⋊C4—1582no—
172C2×C4⋊C4—1581no—
173C2×C4⋊C4C22⋊D7912yes—
174C2×C4⋊C4C22⋊D7911yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—21no—
180C2×C42C22×D7911yescharacteristic
181#107—3162no—
182#106—3164no—
183#103—3162no—
184#102—3164no—
185#95—21no—
186#95—22no—
187C24—22no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8#1192961212yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—82no—
200C2×D4—81no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—3161no—
205C22×C4C2×C22⋊D7911yescharacteristic
206C22×C4C2×C22⋊D7912yes—
207C2×C8—3162no—
208C2×C8—3161no—
209C2×C8—3162no—
210C2×C8—3161no—
211C2×C8—1584no—
212C2×C8—1582no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—3162no—
216C4⋊C4—3161no—
217C4⋊C4—3162no—
218C4⋊C4—3161no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42—41no—
226C42—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1457913011yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—44no—
238D4—82no—
239D4—84no—
240D4—84no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—6322no—
254C2×C4—6321no—
255C2×C4C2×#1192961211yescharacteristic
256C2×C4#1457918112yes—
257C2×C4#1457918711yescharacteristic
258C2×C4#1457919112yes—
259C8—3164no—
260C8—3162no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1909871411yescharacteristic
269C22#1909872512yes—
270C22#1909873112yes—
271C22#1909873111yescharacteristic
272C22#1909875311yescharacteristic
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—82no—
280C4—12641no—
281C2—162no—
282C2#2376242011yescharacteristic
283C2#2376244812yes—
284C2#2376252712yes—
285C2#2376254511yescharacteristic
286C2#2376268911yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#2733C2
socleC23×C79#120
hypercenter#2733D79
hypocenterC79#7830
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #23764938C2semidirect
2. #23764933C2semidirect
3. #23761226C2semidirect
4. #23761221C2semidirect
149. C79#7830semidirect

Maximal subgroups

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

subgroupindex
#237649382
#237649332
#237612262
#237612212
C79×#27332
#783079