Finite Group Databasebeta

← #34177334

Subgroups — #34177334

Order 20224.

374 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 85 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#23770352C211yesmaximal, characteristic
2#23764959C211yesmaximal, characteristic
3#23764873C211yesmaximal, characteristic
4#23764871C211yesmaximal, characteristic
5#23761222C211yesmaximal, characteristic
6#23761222C211yesmaximal, characteristic
7C79×#1271C211yesmaximal, characteristic
8C2×#14579151—21no—
9C2×#14579095—21no—
10C2×#14579094—21no—
11#19098495—21no—
12#19098493—21no—
13#19098493—21no—
14#19098462—21no—
15#19098461C2211yescharacteristic
16#19098461C2211yescharacteristic
17#19098448—21no—
18#19098433C2211yescharacteristic
19#19098326—21no—
20#19098326—21no—
21#19098326—21no—
22#19098325—21no—
23#19098318C2211yescharacteristic
24#389×C79C2211yescharacteristic
25#336×C79C2211yescharacteristic
26#336×C79C2211yescharacteristic
27C22×C79⋊C8—21no—
28#14579151—42no—
29#14579151—42no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4⋊Dic79D411yescharacteristic
37C2×C4⋊Dic79D411yescharacteristic
38C2×C4⋊Dic79D411yescharacteristic
39C2×Dic79⋊C4—21no—
40C2×Dic79⋊C4—21no—
41C2×Dic79⋊C4—21no—
42C2×Dic79⋊C4—21no—
43C2×C4×Dic79—21no—
44#14579095—42no—
45#14579094—42no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C22×C8×C79—21no—
52C22×C8×C79—21no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C42×C79C2311yescharacteristic
57#95×C79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—41no—
63C2×C79⋊C8—22no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8—41no—
66C4⋊Dic79—21no—
67C4⋊Dic79—21no—
68C4⋊Dic79—21no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79—42no—
72C4⋊Dic79—42no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79—41no—
77C4⋊Dic79—81no—
78C4⋊Dic79Q1612yes—
79Dic79⋊C4—42no—
80Dic79⋊C4—42no—
81Dic79⋊C4—41no—
82Dic79⋊C4—41no—
83Dic79⋊C4—41no—
84Dic79⋊C4—41no—
85C4×Dic79—42no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C2×C8×C79—22no—
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C4⋊C4×C79—21no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79D812yes—
102C42×C79—41no—
103C23×C79#12011yescharacteristic
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79C2×D811yescharacteristic
117C2×C4×C79C2×Q1611yescharacteristic
118C2×C4×C79#13511yescharacteristic
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C79#13711yescharacteristic
122C2×Dic79—22no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—81no—
135C2×Dic79—81no—
136C2×Dic79—81no—
137C8×C79—42no—
138C8×C79—81no—
139C79⋊C8—42no—
140C22×C79#44711yescharacteristic
141C22×C79#44911yescharacteristic
142C22×C79#45111yescharacteristic
143C22×C79#45411yescharacteristic
144C22×C79#45611yescharacteristic
145C22×C79#45611yescharacteristic
146C22×C79#45811yescharacteristic
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—42no—
154C4×C79—81no—
155Dic79—42no—
156Dic79—81no—
157Dic79—81no—
158Dic79—81no—
159Dic79—161no—
160#7792—791nomaximal, Sylow
161C2×C79#160911yescharacteristic
162C2×C79#161011yescharacteristic
163C2×C79#161711yescharacteristic
164C2×C79#162511yescharacteristic
165C2×C79#164811yescharacteristic
166C2×C79#164911yescharacteristic
167C2×C79#166311yescharacteristic
168#2726—791no—
169#2099—791no—
170#2083—791no—
171#2083—791no—
172#1271—791no—
173#1271—791no—
174#1271D7911yescharacteristic
175C79#779211yescharacteristic, complemented, Sylow
176C2×#125—1581no—
177C2×#107—1581no—
178C2×#107—1581no—
179#398—1581no—
180#398—1581no—
181#398—1581no—
182#389—791no—
183#389—791no—
184#389—1581no—
185#389C2×D7911yescharacteristic
186#387—1581no—
187#384—791no—
188#340—1581no—
189#340—1581no—
190#340—1581no—
191#340—1581no—
192#336—791no—
193#336C2×D7911yescharacteristic
194#336C2×D7911yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—1581no—
198#125—3162no—
199#125—3162no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—791no—
203C2×C4⋊C4—791no—
204C2×C4⋊C4—791no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1581no—
213C2×C4⋊C4—1581no—
214C2×C4⋊C4D31611yescharacteristic
215C2×C4⋊C4D31611yescharacteristic
216C2×C42—1581no—
217C2×C42C22×D7911yescharacteristic
218#107—3162no—
219#107—3162no—
220#95—21no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—3161no—
229C22×C4—1581no—
230C22×C4—1581no—
231C22×C4—1581no—
232C22×C4—1581no—
233C22×C4D4×D7911yescharacteristic
234C22×C4D4×D7911yescharacteristic
235C22×C4C2×D31611yescharacteristic
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—3161no—
243C2×C8—3161no—
244C2×C8—1582no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—3161no—
250C4⋊C4—3161no—
251C4⋊C4—3162no—
252C4⋊C4—3162no—
253C4⋊C4—3162no—
254C4⋊C4—3162no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—3161no—
258C4⋊C4—3161no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—3161no—
262C4⋊C4—3161no—
263C4⋊C4—792no—
264C4⋊C4—6321no—
265C4⋊C4—1581no—
266C4⋊C4—1581no—
267C4⋊C4—1581no—
268C4⋊C4D63212yes—
269C42—41no—
270C42—3162no—
271C23#1457913111yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—6321no—
296C2×C4—6321no—
297C2×C4—6321no—
298C2×C4—1582no—
299C2×C4C2×D63211yescharacteristic
300C2×C4Q16×D7911yescharacteristic
301C2×C4#1457918211yescharacteristic
302C2×C4#1457918511yescharacteristic
303C2×C4#1457918911yescharacteristic
304C2×C4#1457919011yescharacteristic
305C8—42no—
306C8—81no—
307C8—3162no—
308C22#1909869911yescharacteristic
309C22#1909870911yescharacteristic
310C22#1909872011yescharacteristic
311C22#1909873711yescharacteristic
312C22#1909874711yescharacteristic
313C22#1909874811yescharacteristic
314C22#1909875411yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—81no—
323C4—3162no—
324C4—6321no—
325C4—6321no—
326C4—6321no—
327C4—12641no—
328C2#2376239811yescharacteristic
329C2#2376240511yescharacteristic
330C2#2376244311yescharacteristic
331C2#2376249111yescharacteristic
332C2#2376264311yescharacteristic
333C2#2376265111yescharacteristic
334C2#2376273011yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1271C2
socle, prime radicalC23×C79#120
hypercenter#1271D79
hypocenterC79#7792
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
175. C79#7792semidirect

Maximal subgroups

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

subgroupindex
#237703522
#237649592
#237648732
#237648712
#237612222
C79×#12712
#779279