Finite Group Databasebeta

← #34444609

Subgroups — #34444609

Order 20224.

534 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773979C211yesmaximal, characteristic, complemented
2#23765554C212yesmaximal, complemented
3#23765542C211yesmaximal, characteristic, complemented
4#23765537C211yesmaximal, characteristic, complemented
5#23765515C212yesmaximal, complemented
6#23765501C212yesmaximal, complemented
7#23765452C212yesmaximal, complemented
8#23765440C211yesmaximal, characteristic, complemented
9#23765435C211yesmaximal, characteristic, complemented
10#23765156C211yesmaximal, characteristic, complemented
11C79×#3168C211yesmaximal, characteristic, complemented
12#19099113—21no—
13C2×#14579104C2212yes—
14C2×#14579102C2212yes—
15C2×#14579101C2212yes—
16#19098968C2211yescharacteristic
17#19098940C2211yescharacteristic
18#19098910—21no—
19#19098799—21no—
20#19098799C2212yes—
21C4×D632—21no—
22C4×D632C2212yes—
23#19098390C2212yes—
24#19098388C2212yes—
25C4⋊(C79⋊C16)C2211yescharacteristic
26C4⋊(C79⋊C16)C2211yescharacteristic
27C4⋊(C79⋊C16)C2212yes—
28#19098367C2212yes—
29#19098364C2212yes—
30#19098363C2212yes—
31#19098363C2212yes—
32#529×C79—21no—
33#499×C79C2211yescharacteristic
34#495×C79C2211yescharacteristic
35#488×C79—21no—
36#467×C79C2211yescharacteristic
37#466×C79C2211yescharacteristic
38#465×C79—21no—
39C4×Q16×C79C2211yescharacteristic
40C4×SD16×C79—21no—
41C4×D8×C79C2211yescharacteristic
42C2×C4×C8×C79C2211yescharacteristic
43C2×#11929624—41no—
44C2×D632—21no—
45C2×D632C2312yes—
46#14579161—21no—
47#14579158—21no—
48#14579149—41no—
49#14579135—22no—
50#14579135—41no—
51C4×D316—22no—
52C4×D316—41no—
53#14579104—42no—
54#14579102—42no—
55#14579101—42no—
56C2×C79⋊C16C2312yes—
57C2×C79⋊C16C2312yes—
58#14579095—21no—
59#14579095C2312yes—
60#14579083—22no—
61#14579083—22no—
62C2×Q16×C79C2311yescharacteristic
63C2×SD16×C79—21no—
64C2×D8×C79C2311yescharacteristic
65C22×C8×C79D411yescharacteristic
66C22×C8×C79C2311yescharacteristic
67#126×C79—41no—
68#125×C79—21no—
69#124×C79—21no—
70#123×C79—41no—
71C22⋊Q8×C79—21no—
72#121×C79—21no—
73C4×Q8×C79—21no—
74C4×D4×C79—21no—
75C4×D4×C79—41no—
76#117×C79—41no—
77C2×C42×C79D411yescharacteristic
78#107×C79C2311yescharacteristic
79#107×C79C2311yescharacteristic
80#106×C79—21no—
81#103×C79—21no—
82#103×C79—21no—
83#102×C79—21no—
84#102×C79—21no—
85C4×C8×C79D411yescharacteristic
86C4×C8×C79D411yescharacteristic
87C4×C8×C79C2311yescharacteristic
88C4×C8×C79C2311yescharacteristic
89#11929624—84no—
90C2×Dic158—41no—
91C2×D316—22no—
92C2×D316—41no—
93C2×D316—41no—
94C2×C22⋊D79—42no—
95C2×C4×D79—42no—
96D632—22no—
97D632—22no—
98D632—42no—
99C4⋊Dic79—22no—
100C4⋊Dic79—41no—
101(C2×D79)⋊C4—42no—
102(C2×D79)⋊C4—42no—
103C79⋊C16—22no—
104C79⋊C16—22no—
105C2×Q8×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79—41no—
108C22×C4×C79—41no—
109C22×C4×C79D811yescharacteristic
110C22×C4×C79D811yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112Q16×C79—21no—
113Q16×C79—21no—
114SD16×C79—42no—
115D8×C79—21no—
116D8×C79—21no—
117C2×C8×C79—21no—
118C2×C8×C79—21no—
119C2×C8×C79—41no—
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79C2×D411yescharacteristic
122C2×C8×C79C2×D411yescharacteristic
123C2×C8×C79C2411yescharacteristic
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C4⋊C4×C79—41no—
129C4⋊C4×C79—41no—
130C22⋊C4×C79—41no—
131C22⋊C4×C79—41no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C42×C79—41no—
136C42×C79D811yescharacteristic
137C42×C79D811yescharacteristic
138C42×C79C2×D411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C23×C79—41no—
141C23×C79C2×D811yescharacteristic
142C22×D79—42no—
143Q8×C79—41no—
144Q8×C79—41no—
145D4×C79—41no—
146D4×C79—41no—
147D4×C79—82no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—22no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—82no—
156C2×C4×C79C2×D811yescharacteristic
157C2×C4×C79C2×D811yescharacteristic
158C2×C4×C79C2×D811yescharacteristic
159C2×C4×C79C2×D811yescharacteristic
160C2×C4×C79C2×D811yescharacteristic
161C2×C4×C79C22×D411yescharacteristic
162C22⋊D79—84no—
163C2×Dic79—42no—
164D316—42no—
165D316—42no—
166D316—82no—
167D316—82no—
168C4×D79—84no—
169Dic158—82no—
170C8×C79—21no—
171C8×C79—21no—
172C8×C79—21no—
173C8×C79H(5,𝔽2)11yescharacteristic
174C8×C792−1+411yescharacteristic
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—21no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79C22×D811yescharacteristic
182C2×D79—82no—
183C2×D79—82no—
184C2×D79—82no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—81no—
190C4×C79—81no—
191C4×C79—81no—
192C4×C79#50912yes—
193C4×C79#53711yescharacteristic
194C4×C79#57511yescharacteristic
195Dic79—82no—
196#33817—791nomaximal, Sylow
197C2×C79—41no—
198C2×C79—161no—
199C2×C79#324511yescharacteristic
200C2×C79C2×#50911yescharacteristic
201C2×C79#341411yescharacteristic
202D79—162no—
203#3169—791no—
204#3168D7911yescharacteristic, complemented
205#2228—792no—
206#2225—791no—
207#2224—791no—
208#2220—792no—
209#2218—792no—
210#2210—792no—
211#2207—791no—
212#2206—791no—
213#2150—791no—
214C79#3381711yescharacteristic, complemented, Sylow
215#532—1581no—
216#529—21no—
217C2×SD32—792no—
218C2×D16—792no—
219C2×M32—792no—
220#500—791no—
221#499C2×D7911yescharacteristic
222#495C2×D7911yescharacteristic
223#493—791no—
224#488—21no—
225#486—1581no—
226#467C2×D7911yescharacteristic
227#466—792no—
228#466—1581no—
229#466C2×D7911yescharacteristic
230#465—21no—
231C4×Q16C2×D7911yescharacteristic
232C4×SD16—21no—
233C4×D8—792no—
234C4×D8—1581no—
235C4×D8C2×D7911yescharacteristic
236C2×C4×C8C2×D7911yescharacteristic
237#367—792no—
238#366—792no—
239C4⋊C16—791no—
240C4⋊C16—791no—
241C4⋊C16—792no—
242#358—792no—
243#357—792no—
244#357—792no—
245#357—792no—
246C2×#41—3161no—
247C2×Q16C22×D7911yescharacteristic
248C2×SD16—21no—
249C2×D8—792no—
250C2×D8—1581no—
251C2×D8C22×D7911yescharacteristic
252C22×C8C22⋊D7911yescharacteristic
253C22×C8C22×D7911yescharacteristic
254C4⋊Q8—1581no—
255#127—1581no—
256#126—41no—
257#125—21no—
258#124—21no—
259#124—3161no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263#121—3161no—
264#121—1582no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—41no—
268C4×D4—3161no—
269C4×D4—1582no—
270#117—41no—
271C2×C42C22⋊D7911yescharacteristic
272SD32—3162no—
273D16—3162no—
274M32—3162no—
275C2×C16—792no—
276C2×C16—792no—
277#107—792no—
278#107—1581no—
279#107C22×D7911yescharacteristic
280#107C22×D7911yescharacteristic
281#106—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286#102—1582no—
287#102—1582no—
288C4×C8C22⋊D7911yescharacteristic
289C4×C8C22⋊D7911yescharacteristic
290C4×C8C22×D7911yescharacteristic
291C4×C8C22×D7911yescharacteristic
292#41—6324no—
293C2×Q8—21no—
294C2×Q8—3161no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3162no—
298C2×D4—3161no—
299C2×D4—3161no—
300C2×D4—1582no—
301C22×C4—41no—
302C22×C4—3162no—
303C22×C4#1192960911yescharacteristic
304C22×C4#1192960911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306Q16—21no—
307Q16—21no—
308SD16—42no—
309D8—21no—
310D8—21no—
311D8—3162no—
312D8—1582no—
313D8—1582no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8C2×C22⋊D7911yescharacteristic
318C2×C8C2×C22⋊D7911yescharacteristic
319C2×C8C2×C22⋊D7911yescharacteristic
320C2×C8C23×D7911yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—3161no—
328C4⋊C4—1582no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—3162no—
335C22⋊C4—3162no—
336C42—41no—
337C42#1192960911yescharacteristic
338C42#1192960911yescharacteristic
339C42C2×C22⋊D7911yescharacteristic
340C42C2×C22⋊D7911yescharacteristic
341C16—1582no—
342C16—1582no—
343C23—41no—
344C23—3162no—
345C23C2×#1192960911yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—6322no—
349D4—41no—
350D4—41no—
351D4—82no—
352D4—3162no—
353D4—3162no—
354D4—6322no—
355D4—6324no—
356D4—6322no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—3162no—
366C2×C4—6324no—
367C2×C4C2×#1192960911yescharacteristic
368C2×C4C2×#1192960911yescharacteristic
369C2×C4C2×#1192960911yescharacteristic
370C2×C4C2×#1192960911yescharacteristic
371C2×C4C2×#1192960911yescharacteristic
372C2×C4C22×C22⋊D7911yescharacteristic
373C8—21no—
374C8—21no—
375C8—21no—
376C8#1457920711yescharacteristic
377C8#1457920811yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—6322no—
385C22—6322no—
386C22—6322no—
387C22C22×#1192960911yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—6322no—
396C4#1909899812yes—
397C4#1909914111yescharacteristic
398C4#1909936111yescharacteristic
399C2—41no—
400C2—161no—
401C2—12642no—
402C2#2377470711yescharacteristic
403C2C2×#1909899811yescharacteristic
404C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3168C2
socleC22×C79C22×D8
hypercenter#3168D79
hypocenterC79#33817
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #23773979C2semidirect
2. #23765554C2semidirect
3. #23765542C2semidirect
4. #23765537C2semidirect
5. #23765515C2semidirect
6. #23765501C2semidirect
7. #23765452C2semidirect
8. #23765440C2semidirect
9. #23765435C2semidirect
10. #23765156C2semidirect
11. C79×#3168C2semidirect
204. #3168D79semidirect
214. C79#33817semidirect

Maximal subgroups

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

subgroupindex
#237739792
#237655542
#237655422
#237655372
#237655152
#237655012
#237654522
#237654402
#237654352
#237651562
C79×#31682
#3381779