Finite Group Databasebeta

← #34201280

Subgroups — #34201280

Order 20224.

386 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764745C211yesmaximal, characteristic, complemented
2#23762407C211yesmaximal, characteristic, complemented
3#23762339C211yesmaximal, characteristic
4#23761406C211yesmaximal, characteristic, complemented
5#23761403C211yesmaximal, characteristic
6#23761346C211yesmaximal, characteristic, complemented
7C79×#2823C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098893—21no—
10#19098714—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18C2×#126×C79—21no—
19#398×C79—21no—
20#398×C79C2211yescharacteristic
21#396×C79—21no—
22#396×C79—21no—
23#396×C79—21no—
24#394×C79—21no—
25#394×C79C2211yescharacteristic
26#384×C79C2211yescharacteristic
27C2×#11929612—41no—
28C2×#11929608—21no—
29#14579160—41no—
30#14579152—41no—
31#14579140—21no—
32#14579140—21no—
33#14579120—21no—
34#14579094—41no—
35#14579092—41no—
36#14579092—41no—
37#14579084—41no—
38#14579082—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×D4×C79D411yescharacteristic
42#126×C79—42no—
43#126×C79—42no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79C2311yescharacteristic
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C22⋊C4×C79—21no—
58C2×C22⋊C4×C79—21no—
59C2×C42×C79D411yescharacteristic
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66#11929612—82no—
67#11929608—22no—
68#11929608—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74C4⋊Dic79—21no—
75C4⋊Dic79—21no—
76Dic79⋊C4—41no—
77Dic79⋊C4—41no—
78C22⋊Dic79—41no—
79C4×Dic79—41no—
80C24×C79—21no—
81C2×D4×C79—41no—
82C2×D4×C79—81no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C4⋊C4×C79—41no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—42no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—81no—
99C4⋊C4×C79—81no—
100C4⋊C4×C79—81no—
101C22⋊C4×C79—42no—
102C22⋊C4×C79—42no—
103C22⋊C4×C79—41no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79—81no—
106C22⋊C4×C79—81no—
107C22⋊C4×C79—81no—
108C42×C79—21no—
109C42×C79SD1612yescomplemented
110C23×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79#12011yescharacteristic
118D4×C79—42no—
119D4×C79—81no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—22no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79C2×SD1611yescharacteristic
142C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
143C2×Dic79—41no—
144C2×Dic79—41no—
145Dic158—81no—
146Dic158—81no—
147C79⋊C8—42no—
148C79⋊C8—81no—
149C79⋊C8—81no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—81no—
153C22×C79—81no—
154C22×C79—81no—
155C22×C79—81no—
156C22×C79—81no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79#45011yescharacteristic
160C22×C79#45611yescharacteristic
161C22×C79B(4,𝔽2)11yescharacteristic
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—42no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79—81no—
168C4×C79—81no—
169C4×C79—81no—
170C4×C79—81no—
171Dic79—81no—
172Dic79—81no—
173#12249—791nomaximal, Sylow
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—161no—
177C2×C79#160411yescharacteristic
178C2×C79#220011yescharacteristic
179C2×C79#220311yescharacteristic
180#2823D7911yescharacteristic
181#2060—791no—
182#1611—791no—
183#1599—791no—
184#1346—791no—
185#1345—791no—
186#1325—791no—
187C79#1224911yescharacteristic, complemented, Sylow
188#544—791no—
189C2×#126—21no—
190#483—1581no—
191#450—1581no—
192#428—1581no—
193#398—21no—
194#398C2×D7911yescharacteristic
195#396—21no—
196#396—21no—
197#396—21no—
198#394—21no—
199#394C2×D7911yescharacteristic
200#384C2×D7911yescharacteristic
201#343—791no—
202#339—1581no—
203#328—1581no—
204#328—1581no—
205#324—791no—
206#324—791no—
207C22×D4C22⋊D7911yescharacteristic
208C2×SD16—3161no—
209C2×M16—1581no—
210C4⋊Q8—3161no—
211#126—42no—
212#126—42no—
213#125—3161no—
214C22⋊Q8—1581no—
215C22⋊Q8—1581no—
216#117—1581no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D7911yescharacteristic
224C2×C4⋊C4C22⋊D7911yescharacteristic
225C2×C4⋊C4C22×D7911yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42C22⋊D7911yescharacteristic
233#107—3161no—
234#106—3161no—
235#106—3161no—
236#103—3161no—
237#102—3161no—
238C22⋊C8—1581no—
239C22⋊C8—1581no—
240#95—21no—
241#95—21no—
242#95C22⋊D7911yescharacteristic
243#95C22⋊D7911yescharacteristic
244C24—21no—
245C2×Q8—1581no—
246C2×Q8—1581no—
247C2×D4—41no—
248C2×D4—81no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C2×C22⋊D7911yescharacteristic
255C22×C4C2×C22⋊D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257SD16—6322no—
258M16—3161no—
259M16—1582no—
260C2×C8—3161no—
261C2×C8—3161no—
262C2×C8—3161no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—3161no—
274C4⋊C4—3161no—
275C4⋊C4—1581no—
276C4⋊C4—1581no—
277C4⋊C4—1581no—
278C4⋊C4—1581no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—3161no—
287C42—21no—
288C42—3161no—
289C42#1192961212yescomplemented
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1457913011yescharacteristic
298Q8—6321no—
299Q8—6321no—
300D4—42no—
301D4—81no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4C2×#1192961211yescharacteristic
326C2×C4#1457918711yescharacteristic
327C8—3162no—
328C8—6321no—
329C8—6321no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1909871411yescharacteristic
340C22#1909874611yescharacteristic
341C22#1909875011yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—6321no—
352C4—6321no—
353C2—21no—
354C2—21no—
355C2—161no—
356C2#2376236911yescharacteristic
357C2#2376540711yescharacteristic
358C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2823C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2823D79
hypocenterC79#12249
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2823

Complements

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

NH ≅ G/Ntype
1. #23764745C2semidirect
2. #23762407C2semidirect
4. #23761406C2semidirect
6. #23761346C2semidirect
109. C42×C79SD16semidirect
187. C79#12249semidirect
289. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237624072
#237623392
#237614062
#237614032
#237613462
C79×#28232
#1224979