Finite Group Databasebeta

← #34177102

Subgroups — #34177102

Order 20224.

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

normal subgroups only

This group has 536 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#23764536C212yesmaximal, complemented
2#23764499C211yesmaximal, characteristic, complemented
3#23761226C211yesmaximal, characteristic, complemented
4#23761221C212yesmaximal, complemented
5C79×#2733C211yesmaximal, characteristic
6C22×#11929614—21no—
7C2×#14579161C2211yescharacteristic
8C2×#14579084—21no—
9C2×#14579082—22no—
10#19098325—22no—
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×Dic158—21no—
20C2×#11929614—42no—
21C2×#11929614—42no—
22C2×#11929614—42no—
23C22×C79⋊C8—21no—
24C22×C79⋊C8—22no—
25#14579161—42no—
26C2×C4⋊Dic79—22no—
27#14579084—42no—
28#14579082—44no—
29C22×Q8×C79D411yescharacteristic
30C22×D4×C79—21no—
31C22×D4×C79—21no—
32C22×D4×C79D412yes—
33#124×C79—42no—
34#124×C79—42no—
35C2×C4⋊C4×C79D411yescharacteristic
36C2×C4⋊C4×C79D412yes—
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C42×C79—21no—
42C2×C42×C79C2311yescharacteristic
43#95×C79—21no—
44#95×C79—22no—
45C2×Dic158—22no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C2×Dic158—41no—
50C2×Dic158—41no—
51C22×Dic79—41no—
52#11929614—48no—
53C2×C79⋊C8—22no—
54C2×C79⋊C8—41no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—42no—
57C2×C79⋊C8—41no—
58C2×C79⋊C8—42no—
59C4⋊Dic79—24no—
60C4⋊Dic79—42no—
61C24×C79—22no—
62C2×Q8×C79—21no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79D812yes—
66C2×D4×C79—22no—
67C2×D4×C79—22no—
68C2×D4×C79—42no—
69C2×D4×C79—42no—
70C2×D4×C79—42no—
71C2×D4×C79—81no—
72C2×D4×C79—81no—
73C2×D4×C79—81no—
74C2×D4×C79—82no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—22no—
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D412yes—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—41no—
82C22⋊C4×C79—44no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C22⋊C4×C79—42no—
86C22⋊C4×C79—42no—
87C22⋊C4×C79—82no—
88C42×C79—41no—
89C42×C79—42no—
90C23×C79—42no—
91C23×C79—42no—
92C23×C79—42no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79—42no—
96C23×C79—42no—
97C23×C79#12011yescharacteristic
98Q8×C79—42no—
99Q8×C79—81no—
100D4×C79—44no—
101D4×C79—82no—
102D4×C79—84no—
103D4×C79—84no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79C2×D811yescharacteristic
117C2×C4×C79#13512yes—
118C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C79#13712yes—
120C2×Dic79—42no—
121C2×Dic79—81no—
122C2×Dic79—81no—
123Dic158—42no—
124Dic158—42no—
125Dic158—81no—
126Dic158—81no—
127C79⋊C8—42no—
128C79⋊C8—82no—
129C22×C79—82no—
130C22×C79—82no—
131C22×C79—82no—
132C22×C79—82no—
133C22×C79—82no—
134C22×C79—82no—
135C22×C79—82no—
136C22×C79#44711yescharacteristic
137C22×C79#45212yes—
138C22×C79#45311yescharacteristic
139C22×C79#45512yes—
140C22×C79B(4,𝔽2)11yescharacteristic
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—22no—
144C4×C79—22no—
145C4×C79—42no—
146C4×C79—82no—
147C4×C79—81no—
148Dic79—82no—
149#7758—791nomaximal, Sylow
150C2×C79—162no—
151C2×C79#159511yescharacteristic
152C2×C79#160612yes—
153C2×C79#163411yescharacteristic
154C2×C79#164112yes—
155C2×C79#165911yescharacteristic
156#2733D7911yescharacteristic
157#2023—792no—
158#2016—791no—
159#1272—791no—
160#1270—792no—
161C79#775811yescharacteristic, complemented, Sylow
162C22×Q16—1581no—
163C2×C4⋊Q8—791no—
164C2×#124—21no—
165C2×#103—1581no—
166C2×#102—1582no—
167#394—21no—
168#392—22no—
169#391C2×D7911yescharacteristic
170#390C2×D7912yes—
171#388—22no—
172#340—1582no—
173#336—791no—
174#336—792no—
175C22×Q8—1581no—
176C22×Q8C22⋊D7911yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C22⋊D7912yes—
180C2×Q16—3162no—
181C2×Q16—3162no—
182C2×Q16—3162no—
183C22×C8—1582no—
184C22×C8—1581no—
185C4⋊Q8—3162no—
186#124—42no—
187#124—42no—
188C2×C4⋊C4—1582no—
189C2×C4⋊C4C22⋊D7912yes—
190C2×C4⋊C4C22⋊D7911yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—21no—
196C2×C42C22×D7911yescharacteristic
197#103—3162no—
198#102—3164no—
199#95—21no—
200#95—22no—
201C24—22no—
202C2×Q8—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—3161no—
206C2×Q8—3161no—
207C2×Q8—3161no—
208C2×Q8—3161no—
209C2×Q8—3161no—
210C2×Q8—1582no—
211C2×Q8#1192960912yes—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—81no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—3161no—
225C22×C4C2×C22⋊D7911yescharacteristic
226C22×C4C2×C22⋊D7912yes—
227Q16—3168no—
228C2×C8—3161no—
229C2×C8—3161no—
230C2×C8—3162no—
231C2×C8—3162no—
232C2×C8—3162no—
233C2×C8—1582no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—3162no—
237C4⋊C4—1584no—
238C22⋊C4—44no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C42—41no—
245C42—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23#1457913011yescharacteristic
254Q8—42no—
255Q8—81no—
256Q8—3162no—
257Q8—3162no—
258Q8—6321no—
259Q8—6321no—
260D4—44no—
261D4—82no—
262D4—84no—
263D4—84no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—3162no—
277C2×C4—6321no—
278C2×C4—6321no—
279C2×C4C2×#1192960911yescharacteristic
280C2×C4#1457918112yes—
281C2×C4#1457918711yescharacteristic
282C2×C4#1457919112yes—
283C8—3162no—
284C8—6322no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22#1909869711yescharacteristic
293C22#1909872512yes—
294C22#1909873111yescharacteristic
295C22#1909874112yes—
296C22#1909875011yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—81no—
303C4—82no—
304C4—6322no—
305C2—162no—
306C2#2376231811yescharacteristic
307C2#2376238012yes—
308C2#2376254511yescharacteristic
309C2#2376259312yes—
310C2#2376270611yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#2733C2
socleC23×C79#120
hypercenter#2733D79
hypocenterC79#7758
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. #23764536C2semidirect
2. #23764499C2semidirect
3. #23761226C2semidirect
4. #23761221C2semidirect
161. C79#7758semidirect

Maximal subgroups

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

subgroupindex
#237645362
#237644992
#237612262
#237612212
C79×#27332
#775879