Finite Group Databasebeta

← #34183464

Subgroups — #34183464

Order 20224.

544 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772118C211yesmaximal, characteristic, complemented
2#23764199C211yesmaximal, characteristic, complemented
3#23762049C211yesmaximal, characteristic, complemented
4#23761825C211yesmaximal, characteristic, complemented
5#23761296C211yesmaximal, characteristic, complemented
6#23761244C211yesmaximal, characteristic, complemented
7C79×#3100C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098635—22no—
10#19098635—22no—
11#19098616C2211yescharacteristic
12#19098587C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14#19098530—21no—
15C2×C22⋊(C79⋊C8)—21no—
16C2×C22⋊(C79⋊C8)—21no—
17C2×#14579072C2211yescharacteristic
18#19098309—21no—
19#19098307—21no—
20D4⋊(C79⋊C8)—22no—
21#532×C79C2211yescharacteristic
22C2×C4×D4×C79C2211yescharacteristic
23#480×C79C412yes—
24#474×C79—21no—
25#459×C79—21no—
26#447×C79C412yes—
27C4×D8×C79—22no—
28#431×C79C2211yescharacteristic
29#14579168—42no—
30#14579168—42no—
31C2×#11929608—22no—
32C2×#11929608—21no—
33C22×C79⋊C8—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8D411yescharacteristic
36C22×C79⋊C8D411yescharacteristic
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)D412yes—
42C22⋊(C79⋊C8)—24no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)—42no—
45C22⋊(C79⋊C8)—42no—
46#14579072—21no—
47#14579072—21no—
48C4×C79⋊C8—21no—
49C2×#41×C79—21no—
50C22×D4×C79C2×C411yescharacteristic
51C23×C4×C79—21no—
52C2×D8×C79—22no—
53C4⋊Q8×C79D411yescharacteristic
54#127×C79D411yescharacteristic
55#124×C79—21no—
56#121×C79—21no—
57#121×C79C2×C412yes—
58C4×D4×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79—21no—
61C4×D4×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79D412yes—
64C2×C4⋊C4×C79C2×C411yescharacteristic
65C2×C22⋊C4×C79—21no—
66C2×C42×C79C2311yescharacteristic
67#107×C79—22no—
68C4⋊C8×C79—21no—
69#102×C79—22no—
70#102×C79—22no—
71C22⋊C8×C79C2×C412yes—
72C4×C8×C79—21no—
73#11929608—44no—
74#11929608—41no—
75#11929608—41no—
76C2×C79⋊C8—21no—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—22no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—44no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8D812yescomplemented
88C24×C79—21no—
89#41×C79—41no—
90#41×C79—41no—
91#41×C79—41no—
92#41×C79—41no—
93C2×Q8×C79—21no—
94C2×D4×C79—22no—
95C2×D4×C79—21no—
96C2×D4×C79—21no—
97C2×D4×C79—21no—
98C2×D4×C79—21no—
99C2×D4×C79—42no—
100C2×D4×C79C22⋊C412yes—
101C22×C4×C79—22no—
102C22×C4×C79—21no—
103C22×C4×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79—41no—
109C22×C4×C79C22×C411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79#4111yescharacteristic
112D8×C79—44no—
113C2×C8×C79—22no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79C22⋊C412yes—
118C22⋊C4×C79—22no—
119C22⋊C4×C79—22no—
120C22⋊C4×C79—41no—
121C22⋊C4×C79—41no—
122C42×C79C2×D411yescharacteristic
123C42×C79C2×D411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79#4111yescharacteristic
126C23×C79—22no—
127C23×C79—22no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79—41no—
134C23×C79C4×D411yescharacteristic
135Q8×C79—41no—
136Q8×C79—41no—
137D4×C79—22no—
138D4×C79—22no—
139D4×C79—42no—
140D4×C79—41no—
141D4×C79—41no—
142D4×C79—42no—
143D4×C79—42no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—81no—
147C2×C4×C79—22no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—81no—
156C2×C4×C79—81no—
157C2×C4×C79—81no—
158C2×C4×C79—81no—
159C2×C4×C79C2×C22⋊C411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79#12011yescharacteristic
162C2×C4×C79#12111yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79#12311yescharacteristic
165C2×C4×C79C2×D811yescharacteristic
166C2×C4×C79#13511yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
169C8×C79—42no—
170C79⋊C8—22no—
171C79⋊C8—42no—
172C79⋊C8—41no—
173C79⋊C8—81no—
174C79⋊C8—81no—
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—42no—
178C22×C79—42no—
179C22×C79—42no—
180C22×C79—42no—
181C22×C79—42no—
182C22×C79—42no—
183C22×C79—81no—
184C22×C79—81no—
185C22×C79—81no—
186C22×C79—81no—
187C22×C79#38611yescharacteristic
188C22×C79C4×D811yescharacteristic
189C22×C79#44211yescharacteristic
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—42no—
195C4×C79—81no—
196C4×C79#44711yescharacteristic
197C4×C79#44911yescharacteristic
198C4×C79#45911yescharacteristic
199C4×C79#46311yescharacteristic
200#9049—791nomaximal, Sylow
201C2×C79—21no—
202C2×C79—21no—
203C2×C79—82no—
204C2×C79—82no—
205C2×C79#187911yescharacteristic
206C2×C79#188111yescharacteristic
207C2×C79#189111yescharacteristic
208#3100D7911yescharacteristic
209#2932—791no—
210#1955—791no—
211#1540—791no—
212#1486—791no—
213#1301—791no—
214#1281—791no—
215C79#904911yescharacteristic, complemented, Sylow
216C2×#131—1581no—
217#532C2×D7911yescharacteristic
218C2×C4×D4C2×D7911yescharacteristic
219#480Dic7912yes—
220#474—21no—
221#459—21no—
222#447Dic7912yes—
223C4×D8—22no—
224#435—1582no—
225#435—1582no—
226#431—791no—
227#431C2×D7911yescharacteristic
228#423—791no—
229C2×C4⋊C8—791no—
230#408—1581no—
231C2×C22⋊C8—1581no—
232C2×C22⋊C8—1581no—
233C2×#97—791no—
234#332—1581no—
235#331—1581no—
236D4⋊C8—1582no—
237C2×#41—21no—
238C22×D4C2×Dic7911yescharacteristic
239C23×C4—21no—
240C2×D8—22no—
241#131—3162no—
242#131—3162no—
243C2×M16—1582no—
244C2×M16—1581no—
245C22×C8—791no—
246C22×C8—791no—
247C22×C8—1581no—
248C22×C8—1582no—
249C4⋊Q8C22⋊D7911yescharacteristic
250#127C22⋊D7911yescharacteristic
251#124—21no—
252#121—21no—
253#121C2×Dic7912yes—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C22⋊D7912yes—
260C2×C4⋊C4C2×Dic7911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C42C22×D7911yescharacteristic
263#107—22no—
264C4⋊C8—21no—
265C4⋊C8—792no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269C4⋊C8—1581no—
270#102—22no—
271#102—22no—
272C22⋊C8—3162no—
273C22⋊C8—3162no—
274C22⋊C8—1581no—
275C22⋊C8—1584no—
276C22⋊C8C2×Dic7912yes—
277#97—1581no—
278#97—1581no—
279C4×C8—21no—
280C4×C8—1581no—
281C24—21no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4C22⋊Dic7912yes—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C22×Dic7911yescharacteristic
303C22×C4D4×D7911yescharacteristic
304C22×C4#1192962311yescharacteristic
305D8—44no—
306M16—3161no—
307M16—3161no—
308M16—3164no—
309C2×C8—22no—
310C2×C8—3164no—
311C2×C8—3161no—
312C2×C8—3161no—
313C2×C8—3161no—
314C2×C8—3161no—
315C2×C8—792no—
316C2×C8—1581no—
317C2×C8—1582no—
318C2×C8—1581no—
319C2×C8—1581no—
320C2×C8—1581no—
321C2×C8—1581no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4C22⋊Dic7912yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42D4×D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C42C2×C22⋊D7911yescharacteristic
333C42#1192962311yescharacteristic
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23D4×Dic7911yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C2×C22⋊Dic7911yescharacteristic
368C2×C4D4×Dic7911yescharacteristic
369C2×C4#1457912911yescharacteristic
370C2×C4#1457913311yescharacteristic
371C2×C4#1457913311yescharacteristic
372C2×C4#1457914211yescharacteristic
373C2×C4D8×D7911yescharacteristic
374C2×C4#1457918011yescharacteristic
375C2×C4#1457918511yescharacteristic
376C2×C4#1457918511yescharacteristic
377C8—42no—
378C8—3162no—
379C8—3161no—
380C8—6321no—
381C8—6321no—
382C8—1582no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#1909844311yescharacteristic
396C22D8×Dic7911yescharacteristic
397C22#1909867511yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4#1909869511yescharacteristic
405C4#1909870511yescharacteristic
406C4#1909875611yescharacteristic
407C4#1909877611yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2—82no—
412C2#2376375011yescharacteristic
413C2#2376376411yescharacteristic
414C2#2376381911yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3100C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3100D79
hypocenterC79#9049
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #23772118C2semidirect
2. #23764199C2semidirect
3. #23762049C2semidirect
4. #23761825C2semidirect
5. #23761296C2semidirect
6. #23761244C2semidirect
87. C2×C79⋊C8D8semidirect
215. C79#9049semidirect

Maximal subgroups

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

subgroupindex
#237721182
#237641992
#237620492
#237618252
#237612962
#237612442
C79×#31002
#904979