Finite Group Databasebeta

← #34201082

Subgroups — #34201082

Order 20224.

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

normal subgroups only

This group has 370 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#23764567C211yesmaximal, characteristic
2#23762424C211yesmaximal, characteristic
3#23762356C211yesmaximal, characteristic
4#23761424C211yesmaximal, characteristic
5#23761421C211yesmaximal, characteristic
6#23761330C211yesmaximal, characteristic
7C79×#2841C211yesmaximal, characteristic
8#19099189C2211yescharacteristic
9#19098898—21no—
10#19098719—21no—
11#19098426—21no—
12#19098330C2211yescharacteristic
13#19098329—21no—
14#19098299—21no—
15#19098299—21no—
16#19098287C2211yescharacteristic
17#19098287C2211yescharacteristic
18#400×C79—21no—
19#399×C79C2211yescharacteristic
20#398×C79—21no—
21#398×C79—21no—
22#396×C79—21no—
23#395×C79—21no—
24#393×C79C2211yescharacteristic
25#381×C79C2211yescharacteristic
26#380×C79—21no—
27C2×#11929614—41no—
28#14579140—21no—
29#14579140—21no—
30#14579138—41no—
31#14579138—41no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34#14579094—41no—
35#14579085—41no—
36#14579084—41no—
37C22⋊(C79⋊C8)—21no—
38C22⋊(C79⋊C8)—21no—
39C79⋊#95—41no—
40C79⋊#95—41no—
41C22×Q8×C79D411yescharacteristic
42C23×C4×C79D411yescharacteristic
43C2×C4⋊C4×C79—21no—
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×C79D411yescharacteristic
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79C2311yescharacteristic
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79D411yescharacteristic
61#95×C79D411yescharacteristic
62C2×Dic158—21no—
63C2×Dic158—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—41no—
67#11929614—82no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74C22⋊Dic79—41no—
75C22⋊Dic79—41no—
76C22⋊Dic79—41no—
77C22⋊Dic79—41no—
78C24×C79C2×D411yescharacteristic
79C2×Q8×C79—41no—
80C2×Q8×C79—81no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—41no—
89C22×C4×C79—41no—
90C22×C4×C79Q1612yescomplemented
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C4⋊C4×C79—41no—
94C4⋊C4×C79—81no—
95C4⋊C4×C79—81no—
96C4⋊C4×C79—81no—
97C4⋊C4×C79—81no—
98C4⋊C4×C79—81no—
99C4⋊C4×C79—81no—
100C22⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—81no—
103C22⋊C4×C79—81no—
104C23×C79—21no—
105C23×C79—21no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79#12011yescharacteristic
109C23×C79C2×Q1611yescharacteristic
110C23×C79#13711yescharacteristic
111Q8×C79—42no—
112Q8×C79—81no—
113C2×C4×C79—22no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—42no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
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—81no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—81no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142C2×Dic79—81no—
143C2×Dic79—81no—
144Dic158—81no—
145Dic158—81no—
146C79⋊C8—81no—
147C79⋊C8—81no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—81no—
157C22×C79#45111yescharacteristic
158C22×C79#45511yescharacteristic
159C22×C79B(4,𝔽2)11yescharacteristic
160C4×C79—42no—
161C4×C79—41no—
162C4×C79—41no—
163C4×C79—81no—
164C4×C79—81no—
165C4×C79—81no—
166C4×C79—81no—
167C4×C79—81no—
168Dic79—81no—
169Dic79—81no—
170Dic79—161no—
171#12219—791nomaximal, Sylow
172C2×C79—21no—
173C2×C79—21no—
174C2×C79—41no—
175C2×C79—41no—
176C2×C79#160211yescharacteristic
177C2×C79#220011yescharacteristic
178C2×C79#220111yescharacteristic
179#2841D7911yescharacteristic
180#2030—791no—
181#1614—791no—
182#1602—791no—
183#1352—791no—
184#1351—791no—
185#1317—791no—
186C79#1221911yescharacteristic, complemented, Sylow
187#543—791no—
188#484—1581no—
189#451—1581no—
190#400—21no—
191#399C2×D7911yescharacteristic
192#398—21no—
193#398—21no—
194#396—21no—
195#395—21no—
196#393C2×D7911yescharacteristic
197#381—1581no—
198#381C2×D7911yescharacteristic
199#380—21no—
200#342—791no—
201#342—1581no—
202#328—1581no—
203#328—1581no—
204#323—791no—
205#323—791no—
206C22×Q8C22⋊D7911yescharacteristic
207C23×C4C22⋊D7911yescharacteristic
208C2×Q16—3161no—
209C22⋊Q8—3161no—
210C22⋊Q8—3161no—
211C22⋊Q8—1581no—
212C22⋊Q8—1581no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D7911yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—1581no—
223C2×C22⋊C4—1581no—
224C2×C22⋊C4C22⋊D7911yescharacteristic
225C2×C22⋊C4C22×D7911yescharacteristic
226#107—3161no—
227#103—3161no—
228#103—3161no—
229C22⋊C8—1581no—
230C22⋊C8—1581no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—3161no—
238#95—3161no—
239#95C22⋊D7911yescharacteristic
240#95C22⋊D7911yescharacteristic
241C24C2×C22⋊D7911yescharacteristic
242C2×Q8—41no—
243C2×Q8—81no—
244C2×Q8—1581no—
245C2×Q8—1581no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—3161no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4#1192961412yescomplemented
259C22×C4C2×C22⋊D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261Q16—6322no—
262C2×C8—3161no—
263C2×C8—3161no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—3161no—
272C4⋊C4—3161no—
273C4⋊C4—1581no—
274C4⋊C4—1581no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—3161no—
280C22⋊C4—3161no—
281C22⋊C4—3161no—
282C22⋊C4—3161no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23#1457913011yescharacteristic
288C23C2×#1192961411yescharacteristic
289C23#1457919211yescharacteristic
290Q8—42no—
291Q8—81no—
292Q8—6321no—
293Q8—6321no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
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—81no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—6321no—
321C2×C4—6321no—
322C2×C4—6321no—
323C2×C4—6321no—
324C2×C4—6321no—
325C8—6321no—
326C8—6321no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22#1909871911yescharacteristic
337C22#1909874111yescharacteristic
338C22#1909875011yescharacteristic
339C4—41no—
340C4—41no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—6321no—
348C4—6321no—
349C4—12641no—
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2#2376235711yescharacteristic
355C2#2376540711yescharacteristic
356C2#2376541311yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2841C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2841D79
hypocenterC79#12219
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2841

Complements

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

NH ≅ G/Ntype
90. C22×C4×C79Q16semidirect
186. C79#12219semidirect
258. C22×C4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645672
#237624242
#237623562
#237614242
#237614212
#237613302
C79×#28412
#1221979