Finite Group Databasebeta

← #34444603

Subgroups — #34444603

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#23773970C211yesmaximal, characteristic, complemented
2#23765557C212yesmaximal, complemented
3#23765539C211yesmaximal, characteristic, complemented
4#23765534C211yesmaximal, characteristic, complemented
5#23765511C212yesmaximal, complemented
6#23765501C212yesmaximal, complemented
7#23765455C212yesmaximal, complemented
8#23765437C211yesmaximal, characteristic, complemented
9#23765432C211yesmaximal, characteristic, complemented
10#23765156C211yesmaximal, characteristic, complemented
11C79×#3169C211yesmaximal, characteristic, complemented
12#19099097—21no—
13C2×#14579105C2212yes—
14C2×#14579102C2212yes—
15C2×#14579101C2212yes—
16#19098965C2211yescharacteristic
17#19098947C2211yescharacteristic
18#19098918—21no—
19#19098804C2211yescharacteristic
20#19098799C2211yescharacteristic
21#19098792—21no—
22C4×Dic316C2211yescharacteristic
23C4×#11929613—21no—
24C4×D632C2211yescharacteristic
25#19098390C2212yes—
26#19098388C2212yes—
27C4⋊(C79⋊C16)C2211yescharacteristic
28C4⋊(C79⋊C16)C2211yescharacteristic
29C4⋊(C79⋊C16)C2212yes—
30#19098366C2212yes—
31#19098364C2212yes—
32#19098364C2212yes—
33#19098363C2212yes—
34#532×C79—21no—
35#500×C79C2211yescharacteristic
36#493×C79C2211yescharacteristic
37#486×C79—21no—
38#466×C79—21no—
39#466×C79C2212yes—
40C4×D8×C79—21no—
41C4×D8×C79C2212yes—
42C2×C4×C8×C79C2211yescharacteristic
43C2×Dic316C2311yescharacteristic
44C2×#11929613—21no—
45C2×D632C2311yescharacteristic
46#14579155—41no—
47#14579153—21no—
48#14579149—21no—
49#14579144—41no—
50#14579140—21no—
51#14579135—21no—
52C4×Dic158—21no—
53C4×D316—21no—
54C4×C22⋊D79—41no—
55#14579118—41no—
56#14579105—42no—
57#14579102—42no—
58#14579101—42no—
59C2×C79⋊C16C2312yes—
60C2×C79⋊C16C2312yes—
61#14579095C2311yescharacteristic
62#14579095C2311yescharacteristic
63#14579093—21no—
64#14579086—21no—
65#14579086—21no—
66#14579083—21no—
67#14579083—21no—
68C2×#41×C79—41no—
69C2×D8×C79—21no—
70C2×D8×C79C2312yes—
71C22×C8×C79D411yescharacteristic
72C22×C8×C79C2311yescharacteristic
73C4⋊Q8×C79—21no—
74#127×C79—21no—
75#124×C79—41no—
76#121×C79—22no—
77#121×C79—41no—
78C4×D4×C79—22no—
79C4×D4×C79—41no—
80C2×C42×C79D411yescharacteristic
81#107×C79—21no—
82#107×C79C2312yes—
83#102×C79—22no—
84#102×C79—22no—
85C4×C8×C79D411yescharacteristic
86C4×C8×C79D411yescharacteristic
87C4×C8×C79C2311yescharacteristic
88C4×C8×C79C2311yescharacteristic
89C2×Dic158—21no—
90C2×D316—21no—
91C2×C22⋊D79—41no—
92C2×C4×D79—41no—
93Dic316—21no—
94Dic316—21no—
95#11929613—42no—
96D632—21no—
97D632—21no—
98C4⋊Dic79—21no—
99C4⋊Dic79—21no—
100Dic79⋊C4—41no—
101Dic79⋊C4—41no—
102Dic79⋊C4—41no—
103Dic79⋊C4—41no—
104C22⋊Dic79—41no—
105(C2×D79)⋊C4—41no—
106(C2×D79)⋊C4—41no—
107(C2×D79)⋊C4—41no—
108(C2×D79)⋊C4—41no—
109C4×Dic79—41no—
110C79⋊C16—22no—
111C79⋊C16—22no—
112#41×C79—84no—
113C2×Q8×C79—41no—
114C2×D4×C79—22no—
115C2×D4×C79—42no—
116C2×D4×C79—41no—
117C2×D4×C79—41no—
118C22×C4×C79—42no—
119C22×C4×C79D811yescharacteristic
120C22×C4×C79D811yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122D8×C79—22no—
123D8×C79—22no—
124D8×C79—42no—
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—41no—
128C2×C8×C79C2×D411yescharacteristic
129C2×C8×C79C2×D411yescharacteristic
130C2×C8×C79C2×D411yescharacteristic
131C2×C8×C79C2411yescharacteristic
132C4⋊C4×C79—22no—
133C4⋊C4×C79—41no—
134C22⋊C4×C79—42no—
135C22⋊C4×C79—42no—
136C42×C79D811yescharacteristic
137C42×C79D811yescharacteristic
138C42×C79C2×D411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C23×C79—42no—
141C23×C79C2×D811yescharacteristic
142C22×D79—41no—
143Q8×C79—82no—
144D4×C79—42no—
145D4×C79—42no—
146D4×C79—82no—
147D4×C79—84no—
148D4×C79—82no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—22no—
152C2×C4×C79—41no—
153C2×C4×C79—42no—
154C2×C4×C79—84no—
155C2×C4×C79C2×D811yescharacteristic
156C2×C4×C79C2×D811yescharacteristic
157C2×C4×C79C2×D811yescharacteristic
158C2×C4×C79C2×D811yescharacteristic
159C2×C4×C79C2×D811yescharacteristic
160C2×C4×C79C22×D411yescharacteristic
161C22⋊D79—82no—
162C2×Dic79—41no—
163C2×Dic79—41no—
164C2×Dic79—41no—
165D316—41no—
166D316—41no—
167C4×D79—82no—
168Dic158—41no—
169Dic158—41no—
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—82no—
179C22×C79—82no—
180C22×C79—82no—
181C22×C79C22×D811yescharacteristic
182C2×D79—81no—
183C2×D79—81no—
184C2×D79—81no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—82no—
190C4×C79#50912yes—
191C4×C79#53711yescharacteristic
192C4×C79#57511yescharacteristic
193Dic79—81no—
194Dic79—81no—
195Dic79—81no—
196#33817—791nomaximal, Sylow
197C2×C79—41no—
198C2×C79—162no—
199C2×C79#324511yescharacteristic
200C2×C79C2×#50911yescharacteristic
201C2×C79#341411yescharacteristic
202D79—161no—
203#3169D7911yescharacteristic, complemented
204#3168—791no—
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—21no—
216#529—1581no—
217C2×SD32—792no—
218C2×D16—792no—
219C2×M32—792no—
220#500C2×D7911yescharacteristic
221#499—791no—
222#495—791no—
223#493C2×D7911yescharacteristic
224#488—1581no—
225#486—21no—
226#467—791no—
227#466—21no—
228#466—791no—
229#466C2×D7912yes—
230#465—1581no—
231C4×Q16—791no—
232C4×SD16—1581no—
233C4×D8—21no—
234C4×D8—791no—
235C4×D8C2×D7912yes—
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—41no—
247C2×Q16—791no—
248C2×SD16—1581no—
249C2×D8—21no—
250C2×D8—791no—
251C2×D8C22×D7912yes—
252C22×C8C22⋊D7911yescharacteristic
253C22×C8C22×D7911yescharacteristic
254C4⋊Q8—21no—
255#127—21no—
256#126—3161no—
257#125—1581no—
258#124—41no—
259#124—1581no—
260#123—3161no—
261C22⋊Q8—1581no—
262#121—22no—
263#121—41no—
264#121—1581no—
265C4×Q8—1581no—
266C4×D4—22no—
267C4×D4—41no—
268C4×D4—3161no—
269C4×D4—1581no—
270#117—3161no—
271C2×C42C22⋊D7911yescharacteristic
272SD32—3162no—
273D16—3162no—
274M32—3162no—
275C2×C16—792no—
276C2×C16—792no—
277#107—21no—
278#107—791no—
279#107—791no—
280#107C22×D7912yes—
281#106—1581no—
282#103—1581no—
283#103—1581no—
284#102—22no—
285#102—22no—
286#102—1581no—
287#102—1581no—
288C4×C8C22⋊D7911yescharacteristic
289C4×C8C22⋊D7911yescharacteristic
290C4×C8C22×D7911yescharacteristic
291C4×C8C22×D7911yescharacteristic
292#41—84no—
293C2×Q8—41no—
294C2×Q8—1581no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—3161no—
300C2×D4—1581no—
301C22×C4—42no—
302C22×C4—3161no—
303C22×C4#1192960911yescharacteristic
304C22×C4#1192960911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306Q16—1581no—
307Q16—1581no—
308SD16—3162no—
309D8—22no—
310D8—22no—
311D8—42no—
312D8—1581no—
313D8—1581no—
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—22no—
322C4⋊C4—41no—
323C4⋊C4—3161no—
324C4⋊C4—3161no—
325C4⋊C4—3161no—
326C4⋊C4—3161no—
327C4⋊C4—1581no—
328C4⋊C4—1581no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—3161no—
332C22⋊C4—3161no—
333C22⋊C4—3161no—
334C22⋊C4—3161no—
335C22⋊C4—3161no—
336C42—3161no—
337C42#1192960911yescharacteristic
338C42#1192960911yescharacteristic
339C42C2×C22⋊D7911yescharacteristic
340C42C2×C22⋊D7911yescharacteristic
341C16—1582no—
342C16—1582no—
343C23—42no—
344C23—3161no—
345C23C2×#1192960911yescharacteristic
346Q8—82no—
347Q8—3161no—
348Q8—3161no—
349D4—42no—
350D4—42no—
351D4—82no—
352D4—84no—
353D4—82no—
354D4—3161no—
355D4—3161no—
356D4—6322no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—3161no—
364C2×C4—3161no—
365C2×C4—3161no—
366C2×C4—6322no—
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—82no—
382C22—82no—
383C22—82no—
384C22—6321no—
385C22—6321no—
386C22—6321no—
387C22C22×#1192960911yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—6321no—
394C4—6321no—
395C4—6321no—
396C4#1909899912yes—
397C4#1909914111yescharacteristic
398C4#1909936111yescharacteristic
399C2—41no—
400C2—162no—
401C2—12641no—
402C2#2377470711yescharacteristic
403C2C2×#1909899911yescharacteristic
404C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3169C2
socleC22×C79C22×D8
hypercenter#3169D79
hypocenterC79#33817
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #23773970C2semidirect
2. #23765557C2semidirect
3. #23765539C2semidirect
4. #23765534C2semidirect
5. #23765511C2semidirect
6. #23765501C2semidirect
7. #23765455C2semidirect
8. #23765437C2semidirect
9. #23765432C2semidirect
10. #23765156C2semidirect
11. C79×#3169C2semidirect
203. #3169D79semidirect
214. C79#33817semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237655572
#237655392
#237655342
#237655112
#237655012
#237654552
#237654372
#237654322
#237651562
C79×#31692
#3381779