Finite Group Databasebeta

← #34201297

Subgroups — #34201297

Order 20224.

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

normal subgroups only

This group has 452 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#23764753C211yesmaximal, characteristic
2#23762424C211yesmaximal, characteristic
3#23762356C211yesmaximal, characteristic
4#23761424C211yesmaximal, characteristic
5#23761421C211yesmaximal, characteristic
6#23761348C211yesmaximal, characteristic
7C79×#2677C211yesmaximal, characteristic
8#19099189C2211yescharacteristic
9#19098898—21no—
10#19098719—21no—
11#19098436—21no—
12#19098330C2211yescharacteristic
13#19098329—21no—
14#19098299—21no—
15#19098299—21no—
16#19098287C2211yescharacteristic
17#19098287C2211yescharacteristic
18C2×C4⋊Q8×C79—21no—
19C2×C22⋊Q8×C79—21no—
20#399×C79C2211yescharacteristic
21#398×C79—21no—
22#395×C79—21no—
23#393×C79C2211yescharacteristic
24#388×C79—21no—
25#387×C79—21no—
26#385×C79C2211yescharacteristic
27C2×#11929614—41no—
28#14579140—21no—
29#14579140—21no—
30#14579138—41no—
31#14579138—41no—
32C2×Dic79⋊C4—41no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35#14579094—41no—
36#14579085—41no—
37#14579084—41no—
38C22⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—21no—
40C79⋊#95—41no—
41C22×Q8×C79—21no—
42C22×Q8×C79D411yescharacteristic
43C23×C4×C79D411yescharacteristic
44C4⋊Q8×C79—44no—
45C22⋊Q8×C79—44no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C22⋊C4×C79C2311yescharacteristic
57C2×C42×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—41no—
69#11929614—82no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76Dic79⋊C4—82no—
77C22⋊Dic79—22no—
78C22⋊Dic79—41no—
79C22⋊Dic79—41no—
80C22⋊Dic79—41no—
81C24×C79C2×D411yescharacteristic
82C2×Q8×C79—42no—
83C2×Q8×C79—42no—
84C2×Q8×C79—42no—
85C2×Q8×C79—41no—
86C2×Q8×C79—81no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—42no—
94C22×C4×C79—41no—
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C4⋊C4×C79—24no—
98C4⋊C4×C79—42no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—42no—
101C4⋊C4×C79—42no—
102C4⋊C4×C79—42no—
103C4⋊C4×C79—41no—
104C22⋊C4×C79—21no—
105C22⋊C4×C79—41no—
106C22⋊C4×C79—42no—
107C22⋊C4×C79—42no—
108C22⋊C4×C79Q1612yescomplemented
109C42×C79—42no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79#12011yescharacteristic
115C23×C79C2×Q1611yescharacteristic
116C23×C79#13711yescharacteristic
117Q8×C79—42no—
118Q8×C79—81no—
119Q8×C79—84no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
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—42no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—81no—
144C2×Dic79—42no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—81no—
148C2×Dic79—81no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151Dic158—81no—
152Dic158—81no—
153C79⋊C8—81no—
154C79⋊C8—81no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—41no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79—41no—
163C22×C79—81no—
164C22×C79#45111yescharacteristic
165C22×C79#45611yescharacteristic
166C22×C79B(4,𝔽2)11yescharacteristic
167C4×C79—42no—
168C4×C79—42no—
169C4×C79—42no—
170C4×C79—42no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—81no—
174C4×C79—81no—
175Dic79—82no—
176Dic79—81no—
177Dic79—81no—
178#12252—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—41no—
182C2×C79—41no—
183C2×C79#160511yescharacteristic
184C2×C79#219611yescharacteristic
185C2×C79#220111yescharacteristic
186#2677D7911yescharacteristic
187#2062—791no—
188#1614—791no—
189#1602—791no—
190#1352—791no—
191#1351—791no—
192#1326—791no—
193C79#1225211yescharacteristic, complemented, Sylow
194#543—791no—
195C2×C4⋊Q8—21no—
196C2×C22⋊Q8—21no—
197#484—1581no—
198#451—1581no—
199#399C2×D7911yescharacteristic
200#398—21no—
201#395—21no—
202#393C2×D7911yescharacteristic
203#388—21no—
204#387—21no—
205#385—1581no—
206#385C2×D7911yescharacteristic
207#342—791no—
208#342—1581no—
209#328—1581no—
210#328—1581no—
211#323—791no—
212#323—791no—
213C22×Q8—21no—
214C22×Q8C22⋊D7911yescharacteristic
215C23×C4C22⋊D7911yescharacteristic
216C2×Q16—3161no—
217C4⋊Q8—44no—
218C22⋊Q8—44no—
219C22⋊Q8—3161no—
220C22⋊Q8—3161no—
221C22⋊Q8—1581no—
222C22⋊Q8—1581no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—3161no—
230C2×C4⋊C4C22⋊D7911yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—1581no—
234C2×C22⋊C4—1581no—
235C2×C22⋊C4C22⋊D7911yescharacteristic
236C2×C22⋊C4C22×D7911yescharacteristic
237C2×C42—21no—
238#107—3161no—
239#103—3161no—
240#103—3161no—
241C22⋊C8—1581no—
242C22⋊C8—1581no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—3161no—
248#95C22⋊D7911yescharacteristic
249#95C22⋊D7911yescharacteristic
250C24C2×C22⋊D7911yescharacteristic
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—81no—
256C2×Q8—1581no—
257C2×Q8—1581no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—3161no—
267C22×C4—1581no—
268C22×C4—1581no—
269C22×C4C2×C22⋊D7911yescharacteristic
270C22×C4C2×C22⋊D7911yescharacteristic
271Q16—6322no—
272C2×C8—3161no—
273C2×C8—3161no—
274C4⋊C4—24no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—3161no—
282C4⋊C4—3161no—
283C4⋊C4—6322no—
284C4⋊C4—1581no—
285C4⋊C4—1581no—
286C22⋊C4—21no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—3161no—
291C22⋊C4—3161no—
292C22⋊C4—3161no—
293C22⋊C4—1582no—
294C22⋊C4#1192961412yescomplemented
295C42—42no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23#1457913011yescharacteristic
301C23C2×#1192961411yescharacteristic
302C23#1457919211yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—84no—
306Q8—6321no—
307Q8—6321no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
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—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—3162no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4—6321no—
336C2×C4—6321no—
337C2×C4—6321no—
338C2×C4—6321no—
339C8—6321no—
340C8—6321no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22#1909871911yescharacteristic
351C22#1909874611yescharacteristic
352C22#1909875011yescharacteristic
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—6322no—
362C4—6321no—
363C4—6321no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#2376237411yescharacteristic
369C2#2376538411yescharacteristic
370C2#2376541311yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2677C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2677D79
hypocenterC79#12252
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2677

Complements

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

NH ≅ G/Ntype
108. C22⋊C4×C79Q16semidirect
193. C79#12252semidirect
294. C22⋊C4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237647532
#237624242
#237623562
#237614242
#237614212
#237613482
C79×#26772
#1225279