Finite Group Databasebeta

← #34205337

Subgroups — #34205337

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×#19098645C211yesmaximal, characteristic
3C2×#19098365C211yesmaximal, characteristic
4#23764262C211yesmaximal, characteristic, complemented
5#23761491C211yesmaximal, characteristic, complemented
6#23761477C211yesmaximal, characteristic, complemented
7C2×#358×C79C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C2×#14579123—21no—
10C2×#14579104—22no—
11C2×#14579104—22no—
12C2×#14579104C414yescomplemented
13C22×C79⋊C16C2211yescharacteristic
14#19098645—22no—
15#19098645—22no—
16C2×#14579095C2211yescharacteristic
17C2×#14579085—21no—
18C2×#14579081—21no—
19C2×C8×Dic79C2211yescharacteristic
20#19098470—21no—
21#19098365—22no—
22C22×Q16×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#358×C79—22no—
26C22×D316—21no—
27C22×C4×D79—41no—
28C2×D632—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632C2×C412yes—
32C2×D632D412yes—
33C2×D632D412yes—
34C22×C79⋊C8D411yescharacteristic
35#14579123—24no—
36#14579123—42no—
37C2×C4⋊Dic79—21no—
38C2×(C2×D79)⋊C4—41no—
39C2×C4×Dic79D411yescharacteristic
40#14579104—28no—
41C2×C79⋊C16—21no—
42C2×C79⋊C16—21no—
43C2×C79⋊C16C2×C412yescomplemented
44#14579095—21no—
45#14579095—21no—
46#14579085—42no—
47#14579081—24no—
48C8×Dic79—21no—
49C8×Dic79—21no—
50C79⋊#95—41no—
51C79⋊#95—41no—
52C22×Q8×C79—21no—
53C2×Q16×C79—21no—
54C2×Q16×C79—21no—
55C2×Q16×C79—21no—
56C2×Q16×C79—21no—
57C2×Q16×C79—21no—
58C2×Q16×C79C2×C412yescomplemented
59C22×C8×C79C2311yescharacteristic
60C2×C4⋊C4×C79—21no—
61C2×C16×C79—21no—
62C2×C16×C79—21no—
63C2×C16×C79—21no—
64#107×C79—21no—
65#107×C79D412yes—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—22no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C22×Dic79—41no—
74C22×Dic79D811yescharacteristic
75C22×Dic79D811yescharacteristic
76C2×C4×D79—44no—
77C2×C4×D79—82no—
78C2×C4×D79—82no—
79D632—22no—
80D632—22no—
81D632—22no—
82D632C22⋊C414yes—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88(C2×D79)⋊C4—44no—
89C4×Dic79—21no—
90C4×Dic79—21no—
91C79⋊C16—22no—
92C2×Q8×C79—22no—
93C2×Q8×C79—41no—
94C2×Q8×C79—41no—
95C2×Q8×C79—41no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C22×C4×C79—41no—
99C22×C4×C79—41no—
100C22×C4×C79C2×D411yescharacteristic
101Q16×C79—22no—
102Q16×C79—22no—
103Q16×C79—41no—
104Q16×C79—41no—
105C2×C8×C79C22×C411yescharacteristic
106C2×C8×C79C2×D411yescharacteristic
107C2×C8×C79C2×D411yescharacteristic
108C2×C8×C79C2×D411yescharacteristic
109C2×C8×C79#4111yescharacteristic
110C2×C8×C79#4111yescharacteristic
111C4⋊C4×C79—22no—
112C4⋊C4×C79—41no—
113C16×C79—41no—
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—42no—
127C2×C4×C79—42no—
128C2×C4×C79—81no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79#12011yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#13511yescharacteristic
136C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
138C2×Dic79—21no—
139C2×Dic79—21no—
140C2×Dic79—21no—
141C2×Dic79—21no—
142C2×Dic79—21no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145C2×Dic79—81no—
146C2×Dic79SD3212yescomplemented
147D316—24no—
148D316—42no—
149D316—42no—
150D316—42no—
151C4×D79—88no—
152C8×C79C2×C22⋊C411yescharacteristic
153C8×C79C4×D411yescharacteristic
154C8×C79#12111yescharacteristic
155C8×C79#12311yescharacteristic
156C79⋊C8—41no—
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—82no—
172C4×C79—82no—
173C4×C79#38611yescharacteristic
174C4×C79#44211yescharacteristic
175C4×C79#44911yescharacteristic
176C4×C79#46311yescharacteristic
177Dic79—22no—
178Dic79—41no—
179Dic79—161no—
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×#358D7911yescharacteristic, complemented
192C2×#357—791no—
193#1969—791no—
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×C16C2×D7911yescharacteristic
205C4×D8—1582no—
206C4×D8—1582no—
207C2×#107—791no—
208C2×#107C2×D7911yescharacteristic
209C2×#103—1581no—
210C2×#102—1581no—
211C2×C4×C8—791no—
212#391—1581no—
213#358—22no—
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×C8—791no—
231C22×C8C22×D7911yescharacteristic
232C4×D4—3162no—
233C4×D4—1584no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1581no—
236C2×C22⋊C4—3161no—
237C2×C42—791no—
238SD32—1588no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—792no—
243C2×C16—1581no—
244C2×C16—1581no—
245#107—21no—
246#107—1581no—
247#107—1581no—
248#107C22⋊D7912yes—
249#103—3162no—
250#102—1584no—
251C4×C8—1581no—
252C4×C8—1581no—
253#95—3161no—
254#95—3161no—
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—791no—
273C22×C4—791no—
274C22×C4—6322no—
275C22×C4—6322no—
276C22×C4D4×D7911yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281D8—794no—
282D8—1582no—
283D8—1582no—
284D8—1582no—
285C2×C8—1581no—
286C2×C8—1581no—
287C2×C8—1581no—
288C2×C8C2×C4×D7911yescharacteristic
289C2×C8D4×D7911yescharacteristic
290C2×C8D4×D7911yescharacteristic
291C2×C8C2×C22⋊D7911yescharacteristic
292C2×C8#1192962311yescharacteristic
293C2×C8#1192962411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—3161no—
297C4⋊C4—3161no—
298C22⋊C4—3164no—
299C42—1581no—
300C42—1581no—
301C16—41no—
302C16—1582no—
303C23—3162no—
304C23—3162no—
305C23—3162no—
306C23—6321no—
307C23—6321no—
308C23—6321no—
309C23—6321no—
310C23D8×D7911yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—3162no—
316D4—3162no—
317D4—3162no—
318D4—1584no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—792no—
326C2×C4—6328no—
327C2×C4—6321no—
328C2×C4—6321no—
329C2×C4—6321no—
330C2×C4—1581no—
331C2×C4—1581no—
332C2×C4—1581no—
333C2×C4—1581no—
334C2×C4—1581no—
335C2×C4(C22⋊D79)⋊C411yescharacteristic
336C2×C4#1457912911yescharacteristic
337C2×C4#1457913411yescharacteristic
338C2×C4#1457918211yescharacteristic
339C2×C4#1457918511yescharacteristic
340C2×C4#1457918811yescharacteristic
341C8—3161no—
342C8C22⋊C4×D7911yescharacteristic
343C8C4×C22⋊D7911yescharacteristic
344C8#1457913211yescharacteristic
345C8#1457914311yescharacteristic
346C22—3164no—
347C22—6322no—
348C22—6322no—
349C22—6322no—
350C22—6322no—
351C22—6322no—
352C22—6322no—
353C22#1909864311yescharacteristic
354C22#1909869511yescharacteristic
355C22#1909875711yescharacteristic
356C22SD32×D7911yescharacteristic
357C22#1909899611yescharacteristic
358C22#1909900011yescharacteristic
359C22#1909900511yescharacteristic
360C4—82no—
361C4—82no—
362C4—3161no—
363C4—1582no—
364C4—12641no—
365C4#1909843911yescharacteristic
366C4#1909867211yescharacteristic
367C4#1909870711yescharacteristic
368C4#1909877711yescharacteristic
369C2—6324no—
370C2#2376374511yescharacteristic
371C2#2376524911yescharacteristic
372C2#2376526711yescharacteristic
373C2#2376532711yescharacteristic
374C2#2376533911yescharacteristic
375C2#2376544411yescharacteristic
376C2#2376545811yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#358×C79C2
socleC23×C79C2×D8
prime radicalC22×D632C22
cosocleC22×C8×C79C23
hypercenterC2×#358D79
hypocenterC79#13014
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
4. #23764262C2semidirect
5. #23761491C2semidirect
6. #23761477C2semidirect
7. C2×#358×C79C2semidirect
12. C2×#14579104C4semidirect
43. C2×C79⋊C16C2×C4semidirect
58. C2×Q16×C79C2×C4semidirect
146. C2×Dic79SD32semidirect
191. C2×#358D79semidirect
196. C79#13014semidirect
223. C2×Q16C4×D79semidirect

Maximal subgroups

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

subgroupindex
C22×#145791042
C2×#190986452
C2×#190983652
#237642622
#237614912
#237614772
C2×#358×C792
#1301479