Finite Group Databasebeta

← #34205343

Subgroups — #34205343

Order 20224.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579104C211yesmaximal, characteristic
2C2×C4×D632C211yesmaximal, characteristic
3C2×#19098367C211yesmaximal, characteristic, complemented
4C2×#19098363C211yesmaximal, characteristic
5#23761491C211yesmaximal, characteristic, complemented
6#23761478C211yesmaximal, characteristic, complemented
7C79×#1969C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C2×C4×D316—21no—
10C2×#14579104—22no—
11C2×#14579104—22no—
12C2×#14579104C414yescomplemented
13C22×C79⋊C16C2211yescharacteristic
14C22×C79⋊C16C2211yescharacteristic
15C4×D632—22no—
16C4×D632—22no—
17C2×#14579095C2211yescharacteristic
18C2×#14579083—21no—
19#19098367—22no—
20#19098363—22no—
21C22×Q16×C79C2211yescharacteristic
22C2×#107×C79C2211yescharacteristic
23C2×#103×C79—21no—
24C2×C4×C8×C79C2211yescharacteristic
25#391×C79—21no—
26C22×D316—21no—
27C22×C4×D79—41no—
28C2×D632—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632C2×C412yes—
32C2×D632D412yes—
33C2×D632D412yes—
34C4×D316—24no—
35C4×D316—42no—
36C2×C4⋊Dic79—21no—
37C2×(C2×D79)⋊C4—41no—
38#14579104—28no—
39C2×C79⋊C16—21no—
40C2×C79⋊C16—21no—
41C2×C79⋊C16—21no—
42C2×C79⋊C16—21no—
43C2×C79⋊C16—21no—
44C2×C79⋊C16C2×C412yescomplemented
45#14579095—21no—
46#14579095D412yes—
47#14579083—24no—
48C22×Q8×C79—21no—
49C2×Q16×C79—21no—
50C2×Q16×C79—21no—
51C2×Q16×C79—21no—
52C2×Q16×C79—21no—
53C2×Q16×C79—21no—
54C2×Q16×C79C2×C412yescomplemented
55C22×C8×C79D411yescharacteristic
56C22×C8×C79C2311yescharacteristic
57C2×C4⋊C4×C79—21no—
58C2×C42×C79D411yescharacteristic
59#107×C79—21no—
60#107×C79—21no—
61#103×C79—42no—
62C4×C8×C79—21no—
63C4×C8×C79—21no—
64#95×C79—41no—
65#95×C79—41no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—22no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C22×Dic79—41no—
74C2×C4×D79—44no—
75C2×C4×D79—82no—
76C2×C4×D79—82no—
77D632—22no—
78D632—22no—
79D632—22no—
80D632C22⋊C414yes—
81C4⋊Dic79—22no—
82C4⋊Dic79—41no—
83(C2×D79)⋊C4—44no—
84C79⋊C16—22no—
85C79⋊C16—41no—
86C2×Q8×C79—22no—
87C2×Q8×C79—41no—
88C2×Q8×C79—41no—
89C2×Q8×C79—41no—
90C2×Q8×C79—41no—
91C2×Q8×C79—41no—
92C22×C4×C79—41no—
93C22×C4×C79—41no—
94C22×C4×C79D811yescharacteristic
95C22×C4×C79D811yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97Q16×C79—22no—
98Q16×C79—22no—
99Q16×C79—41no—
100Q16×C79—41no—
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79—21no—
104C2×C8×C79C22×C411yescharacteristic
105C2×C8×C79C2×D411yescharacteristic
106C2×C8×C79C2×D411yescharacteristic
107C2×C8×C79C2×D411yescharacteristic
108C2×C8×C79#4111yescharacteristic
109C2×C8×C79#4111yescharacteristic
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C42×C79—21no—
113C42×C79—21no—
114C23×C79C2×D811yescharacteristic
115C22×D79—42no—
116C22×D79—42no—
117C22×D79—42no—
118C22×D79—81no—
119C22×D79—81no—
120C22×D79—81no—
121C22×D79—81no—
122Q8×C79—42no—
123Q8×C79—42no—
124Q8×C79—81no—
125Q8×C79—81no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—42no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79SD3212yescomplemented
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79#12011yescharacteristic
140C2×C4×C79#12111yescharacteristic
141C2×C4×C79#13511yescharacteristic
142C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×Dic79—42no—
145C2×Dic79—81no—
146C2×Dic79—81no—
147D316—24no—
148D316—42no—
149D316—42no—
150D316—42no—
151C4×D79—88no—
152C8×C79—41no—
153C8×C79C2×C22⋊C411yescharacteristic
154C8×C79C4×D411yescharacteristic
155C8×C79#12111yescharacteristic
156C8×C79#12311yescharacteristic
157C22×C79C4×D811yescharacteristic
158C22×C79#44711yescharacteristic
159C22×C79#45911yescharacteristic
160C22×C79C2×SD3211yescharacteristic
161C22×C79#50811yescharacteristic
162C22×C79#50911yescharacteristic
163C22×C79#51011yescharacteristic
164C2×D79—44no—
165C2×D79—82no—
166C2×D79—82no—
167C2×D79—82no—
168C2×D79—82no—
169C2×D79—82no—
170C2×D79—82no—
171C4×C79—22no—
172C4×C79—41no—
173C4×C79—82no—
174C4×C79—161no—
175C4×C79#38611yescharacteristic
176C4×C79#44211yescharacteristic
177C4×C79#44911yescharacteristic
178C4×C79#46311yescharacteristic
179Dic79—82no—
180#13014—791nomaximal, Sylow
181C2×C79#187911yescharacteristic
182C2×C79C4×SD3211yescharacteristic
183C2×C79#217511yescharacteristic
184C2×C79#218711yescharacteristic
185C2×C79#218911yescharacteristic
186C2×C79#220811yescharacteristic
187C2×C79#221011yescharacteristic
188D79—84no—
189C22×SD32—791no—
190C2×C4×D8—791no—
191C2×#358—791no—
192C2×#357—791no—
193#1969D7911yescharacteristic, complemented
194#1380—791no—
195#1374—791no—
196C79#1301411yescharacteristic, complemented, Sylow
197C22×Q16C2×D7911yescharacteristic
198C22×D8—791no—
199C2×C4×D4—1581no—
200C2×SD32—794no—
201C2×SD32—1582no—
202C2×SD32—1582no—
203C22×C16—791no—
204C22×C16—791no—
205C4×D8—1582no—
206C4×D8—1582no—
207C2×#107—791no—
208C2×#107C2×D7911yescharacteristic
209C2×#103—21no—
210C2×#102—1581no—
211C2×C4×C8C2×D7911yescharacteristic
212#391—21no—
213#358—1582no—
214#357—1582no—
215C22×Q8—21no—
216C22×D4—1581no—
217C23×C4—3161no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16C4×D7912yescomplemented
224C2×D8—792no—
225C2×D8—792no—
226C2×D8—792no—
227C2×D8—1581no—
228C2×D8—1581no—
229C2×D8—1581no—
230C22×C8C22⋊D7911yescharacteristic
231C22×C8C22×D7911yescharacteristic
232C4×D4—3162no—
233C4×D4—1584no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1581no—
236C2×C22⋊C4—3161no—
237C2×C42C22⋊D7911yescharacteristic
238SD32—1588no—
239C2×C16—792no—
240C2×C16—1581no—
241C2×C16—1581no—
242C2×C16—1581no—
243C2×C16—1581no—
244C2×C16—1581no—
245#107—21no—
246#107—21no—
247#107—792no—
248#107—1581no—
249#103—42no—
250#102—1584no—
251C4×C8—21no—
252C4×C8—21no—
253#95—41no—
254#95—41no—
255C24—3161no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—3161no—
263C2×D4—3161no—
264C2×D4—3161no—
265C2×D4—1582no—
266C2×D4—1582no—
267C2×D4—1582no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—3164no—
271C22×C4—3161no—
272C22×C4—6322no—
273C22×C4—6322no—
274C22×C4#1192960911yescharacteristic
275C22×C4#1192960911yescharacteristic
276C22×C4C2×C22⋊D7911yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281D8—794no—
282D8—1582no—
283D8—1582no—
284D8—1582no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C2×C4×D7911yescharacteristic
289C2×C8D4×D7911yescharacteristic
290C2×C8D4×D7911yescharacteristic
291C2×C8D4×D7911yescharacteristic
292C2×C8#1192962311yescharacteristic
293C2×C8#1192962411yescharacteristic
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—3161no—
297C4⋊C4—1582no—
298C22⋊C4—3164no—
299C42—21no—
300C42—21no—
301C16—3161no—
302C16—1582no—
303C23—3162no—
304C23—3162no—
305C23—3162no—
306C23—6321no—
307C23—6321no—
308C23—6321no—
309C23—6321no—
310C23C2×#1192960911yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—3162no—
316D4—3162no—
317D4—3162no—
318D4—1584no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—3162no—
331C2×C4—6328no—
332C2×C4—6321no—
333C2×C4—6321no—
334C2×C4#1457910412yescomplemented
335C2×C4C4×C22⋊D7911yescharacteristic
336C2×C4#1457912911yescharacteristic
337C2×C4#1457913211yescharacteristic
338C2×C4#1457918111yescharacteristic
339C2×C4#1457918611yescharacteristic
340C2×C4#1457918711yescharacteristic
341C8—41no—
342C8C22⋊C4×D7911yescharacteristic
343C8(C22⋊D79)⋊C411yescharacteristic
344C8#1457913411yescharacteristic
345C8#1457914311yescharacteristic
346C22—3164no—
347C22—6322no—
348C22—6322no—
349C22—6322no—
350C22—6322no—
351C22—6322no—
352C22—6322no—
353C22C4×#1192960911yescharacteristic
354C22#1909869611yescharacteristic
355C22#1909875511yescharacteristic
356C22C2×#1457910411yescharacteristic
357C22#1909899311yescharacteristic
358C22#1909899811yescharacteristic
359C22#1909900411yescharacteristic
360C4—22no—
361C4—41no—
362C4—82no—
363C4—161no—
364C4—6322no—
365C4#1909843911yescharacteristic
366C4#1909867311yescharacteristic
367C4#1909870611yescharacteristic
368C4#1909877511yescharacteristic
369C2—6324no—
370C2#2376375111yescharacteristic
371C2C4×#1457910411yescharacteristic
372C2#2376526211yescharacteristic
373C2#2376532811yescharacteristic
374C2#2376533811yescharacteristic
375C2#2376544211yescharacteristic
376C2#2376545211yescharacteristic
subgroupG/H
centerC23C2×#11929609
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1969C2
socleC23×C79C2×D8
prime radicalC22×D632C22
cosocleC22×C8×C79C23
hypercenter#1969D79
hypocenterC79#13014
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
3. C2×#19098367C2semidirect
5. #23761491C2semidirect
6. #23761478C2semidirect
7. C79×#1969C2semidirect
12. C2×#14579104C4semidirect
44. C2×C79⋊C16C2×C4semidirect
54. C2×Q16×C79C2×C4semidirect
137. C2×C4×C79SD32semidirect
193. #1969D79semidirect
196. C79#13014semidirect
223. C2×Q16C4×D79semidirect
334. C2×C4#14579104semidirect

Maximal subgroups

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

subgroupindex
C22×#145791042
C2×C4×D6322
C2×#190983672
C2×#190983632
#237614912
#237614782
C79×#19692
#1301479