Finite Group Databasebeta

← #34184489

Subgroups — #34184489

Order 20224.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098840C211yesmaximal, characteristic
2C2×#19098640C211yesmaximal, characteristic
3#23764203C211yesmaximal, characteristic, complemented
4#23763401C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C2×#332×C79C211yesmaximal, characteristic, complemented
8C22×#11929607—21no—
9C2×C4×D316C2211yescharacteristic
10#19098840—22no—
11#19098840C414yescomplemented
12#19098640—22no—
13#19098640—22no—
14C2×#14579094—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×#14579085—21no—
17C2×#14579085—21no—
18C2×#14579083—21no—
19C2×#14579081—21no—
20C2×(C2×D79)⋊C8—21no—
21C2×C4×C79⋊C8C2211yescharacteristic
22#19098433C2211yescharacteristic
23#19098326—21no—
24C2×C4⋊Q8×C79C2211yescharacteristic
25C2×C4⋊C8×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27#332×C79—22no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C2×#11929607—24no—
31C2×#11929607—42no—
32C2×#11929607—42no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8D411yescharacteristic
35C22×C79⋊C8D411yescharacteristic
36C4×D316—21no—
37C4×D316—21no—
38C4×D316C2×C412yes—
39C4×D316D412yes—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79D411yescharacteristic
44C2×(C2×D79)⋊C4—21no—
45#14579094—24no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)C2×C412yes—
48#14579085—42no—
49#14579085—42no—
50#14579083—42no—
51#14579081—24no—
52(C2×D79)⋊C8—24no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C22×Q8×C79—21no—
56C22×C8×C79—21no—
57C22×C8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79C2×C412yes—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C42×C79C2311yescharacteristic
65C4⋊C8×C79—21no—
66C4⋊C8×C79—21no—
67C4⋊C8×C79—21no—
68C4⋊C8×C79D412yes—
69C23×D79—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316C22⋊C412yes—
76C22×Dic79—21no—
77C22×Dic79—41no—
78C2×C4×D79—22no—
79C2×C4×D79—22no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84#11929607—48no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8D812yes—
94C4⋊Dic79—21no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99C4⋊Dic79—81no—
100C4⋊Dic79C22⋊C412yes—
101(C2×D79)⋊C4—22no—
102(C2×D79)⋊C4—22no—
103C2×Q8×C79—22no—
104C2×Q8×C79—41no—
105C2×Q8×C79—41no—
106C2×Q8×C79—41no—
107C2×Q8×C79—41no—
108C2×Q8×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79#4111yescharacteristic
113C2×C8×C79—22no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C2×C8×C79—41no—
119C4⋊C4×C79—22no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C42×C79C22×C411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79#12111yescharacteristic
128C22×D79—22no—
129C22×D79—41no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135Q8×C79—42no—
136Q8×C79—42no—
137Q8×C79—81no—
138Q8×C79—81no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—42no—
142C2×C4×C79—81no—
143C2×C4×C79—81no—
144C2×C4×C79C2×C22⋊C411yescharacteristic
145C2×C4×C79C4×D411yescharacteristic
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79#12011yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79#12311yescharacteristic
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79#13511yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C2×C4×C79#13711yescharacteristic
157C2×C4×C79#13711yescharacteristic
158C2×Dic79—22no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—81no—
162C2×Dic79—81no—
163C2×Dic79—81no—
164D316—22no—
165D316—22no—
166D316—41no—
167D316—41no—
168C4×D79—42no—
169C4×D79—42no—
170C4×D79—42no—
171C4×D79—42no—
172C8×C79—42no—
173C8×C79—81no—
174C79⋊C8—22no—
175C79⋊C8—42no—
176C79⋊C8—41no—
177C22×C79#38611yescharacteristic
178C22×C79#45911yescharacteristic
179C22×C79#46311yescharacteristic
180C22×C79#46811yescharacteristic
181C22×C79#47011yescharacteristic
182C22×C79#47111yescharacteristic
183C22×C79#47311yescharacteristic
184C2×D79—42no—
185C2×D79—42no—
186C2×D79—42no—
187C2×D79—81no—
188C2×D79—81no—
189C2×D79—81no—
190C2×D79—81no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—82no—
194C4×C79#41711yescharacteristic
195C4×C79#41911yescharacteristic
196C4×C79C4×D811yescharacteristic
197C4×C79#44211yescharacteristic
198C4×C79#44711yescharacteristic
199C4×C79#44911yescharacteristic
200C4×C79#46811yescharacteristic
201C4×C79#47011yescharacteristic
202Dic79—42no—
203Dic79—161no—
204#9227—791nomaximal, Sylow
205C2×C79#168511yescharacteristic
206C2×C79#168911yescharacteristic
207C2×C79#169911yescharacteristic
208C2×C79#170311yescharacteristic
209C2×C79#187911yescharacteristic
210C2×C79#189411yescharacteristic
211C2×C79#190811yescharacteristic
212D79—82no—
213C2×#474—791no—
214C2×#436—791no—
215#1956—791no—
216#1802—791no—
217C2×#332D7911yescharacteristic, complemented
218#1296—791no—
219#1277—791no—
220C79#922711yescharacteristic, complemented, Sylow
221C22×M16—1581no—
222C2×C4⋊Q8C2×D7911yescharacteristic
223C2×C4×D4—791no—
224#474—794no—
225#474—1582no—
226#436—1582no—
227#436—1582no—
228C2×#107—1581no—
229C2×C4⋊C8—791no—
230C2×C4⋊C8C2×D7911yescharacteristic
231C2×C4⋊C8C2×D7911yescharacteristic
232C2×#103—1581no—
233C2×#103—1581no—
234C2×#102—1581no—
235C2×#102—1581no—
236C2×C22⋊C8—1581no—
237C2×C4×C8—791no—
238#384—791no—
239#340—1581no—
240#332—22no—
241C22×Q8—21no—
242C22×D4—791no—
243C23×C4—1581no—
244C2×M16—3162no—
245C2×M16—3162no—
246C2×M16—1584no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—791no—
250C22×C8—791no—
251C22×C8—1581no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C4×D7912yes—
256C4×D4—792no—
257C4×D4—792no—
258C4×D4—1581no—
259C4×D4—1581no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—791no—
263C2×C4⋊C4—1581no—
264C2×C4⋊C4—1581no—
265C2×C4⋊C4—1581no—
266C2×C22⋊C4—1581no—
267C2×C42C22×D7911yescharacteristic
268#107—1584no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—792no—
273C4⋊C8—1581no—
274C4⋊C8C22⋊D7912yes—
275#103—3162no—
276#103—3162no—
277#102—3162no—
278#102—1584no—
279C22⋊C8—1584no—
280C4×C8—1581no—
281C4×C8—1581no—
282C24—1581no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—792no—
290C2×D4—1581no—
291C2×D4—1581no—
292C2×D4—1581no—
293C2×D4—1581no—
294C2×D4—1581no—
295C22×C4—41no—
296C22×C4—3161no—
297C22×C4—3161no—
298C22×C4—3161no—
299C22×C4—3161no—
300C22×C4—3161no—
301C22×C4—1581no—
302C22×C4—1582no—
303C22×C4—1582no—
304C22×C4D4×D7911yescharacteristic
305C22×C4D4×D7911yescharacteristic
306C22×C4#1192962411yescharacteristic
307M16—3168no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—3161no—
315C2×C8—3161no—
316C2×C8—792no—
317C2×C8—1582no—
318C2×C8—1581no—
319C2×C8—1581no—
320C2×C8—1581no—
321C2×C8—1581no—
322C2×C8—1581no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—3161no—
328C4⋊C4—3161no—
329C4⋊C4—3161no—
330C4⋊C4—3161no—
331C4⋊C4—792no—
332C4⋊C4—6321no—
333C4⋊C4—1581no—
334C22⋊C4—1582no—
335C22⋊C4—1582no—
336C42C2×C4×D7911yescharacteristic
337C42D4×D7911yescharacteristic
338C42C2×C22⋊D7911yescharacteristic
339C42#1192962311yescharacteristic
340C23—3161no—
341C23—3161no—
342C23—3161no—
343C23—3161no—
344C23—3161no—
345C23—3161no—
346C23—1582no—
347C23#1457913411yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—3161no—
353D4—3161no—
354D4—1582no—
355D4—1582no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—3161no—
362C2×C4—3161no—
363C2×C4—3162no—
364C2×C4—3162no—
365C2×C4—3162no—
366C2×C4—3162no—
367C2×C4—6321no—
368C2×C4—6321no—
369C2×C4—6321no—
370C2×C4—1582no—
371C2×C4C22⋊C4×D7911yescharacteristic
372C2×C4(C22⋊D79)⋊C411yescharacteristic
373C2×C4C4×C22⋊D7911yescharacteristic
374C2×C4#1457912911yescharacteristic
375C2×C4#1457913211yescharacteristic
376C2×C4#1457914311yescharacteristic
377C2×C4D8×D7911yescharacteristic
378C2×C4#1457918211yescharacteristic
379C2×C4#1457918511yescharacteristic
380C2×C4#1457918811yescharacteristic
381C2×C4#1457918811yescharacteristic
382C2×C4#1457919011yescharacteristic
383C2×C4#1457919311yescharacteristic
384C2×C4#1457919311yescharacteristic
385C8—42no—
386C8—81no—
387C8—3161no—
388C8—3162no—
389C8—1582no—
390C22—3162no—
391C22—3162no—
392C22—3162no—
393C22—6321no—
394C22—6321no—
395C22—6321no—
396C22—6321no—
397C22#1909843911yescharacteristic
398C22#1909875711yescharacteristic
399C22#1909877711yescharacteristic
400C22#1909881211yescharacteristic
401C22#1909882011yescharacteristic
402C22#1909882911yescharacteristic
403C22#1909883711yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—3162no—
408C4—12641no—
409C4#1909856311yescharacteristic
410C4#1909857011yescharacteristic
411C4#1909864311yescharacteristic
412C4#1909867211yescharacteristic
413C4#1909869511yescharacteristic
414C4#1909870711yescharacteristic
415C4#1909880611yescharacteristic
416C4#1909881811yescharacteristic
417C2—6322no—
418C2#2376286811yescharacteristic
419C2#2376288811yescharacteristic
420C2#2376294611yescharacteristic
421C2#2376297411yescharacteristic
422C2#2376374511yescharacteristic
423C2#2376383011yescharacteristic
424C2#2376391211yescharacteristic
subgroupG/H
centerC23#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#332×C79C2
socleC23×C79#121
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenterC2×#332D79
hypocenterC79#9227
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #23764203C2semidirect
5. #23761284C2semidirect
6. #23761234C2semidirect
7. C2×#332×C79C2semidirect
11. #19098840C4semidirect
217. C2×#332D79semidirect
220. C79#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#190988402
C2×#190986402
#237642032
#237634012
#237612842
#237612342
C2×#332×C792
#922779