Finite Group Databasebeta

← #34205692

Subgroups — #34205692

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776503C211yesmaximal, characteristic, complemented
2C2×#19098373C211yesmaximal, characteristic, complemented
3#23765128C211yesmaximal, characteristic, complemented
4#23763459C211yesmaximal, characteristic, complemented
5#23761511C211yesmaximal, characteristic, complemented
6#23761500C211yesmaximal, characteristic, complemented
7C79×#3067C211yesmaximal, characteristic
8#19098980—22no—
9#19098980—22no—
10C2×#14579101—22no—
11C2×#14579101C2211yescharacteristic
12C2×#14579101C2211yescharacteristic
13C2×#14579097C2211yescharacteristic
14C2×#14579093C2211yescharacteristic
15#19098564—21no—
16#19098373—22no—
17#19098370—22no—
18#19098331—21no—
19#575×C79—21no—
20C2×#135×C79C2211yescharacteristic
21C2×#131×C79C2211yescharacteristic
22#473×C79C412yescomplemented
23#472×C79C412yescomplemented
24#471×C79—22no—
25C2×#108×C79C2211yescharacteristic
26#413×C79—21no—
27C2×#11929608—21no—
28C2×C4⋊Dic79—21no—
29#14579101—24no—
30#14579101—21no—
31#14579101—21no—
32#14579101—21no—
33#14579101D412yes—
34C2×C79⋊C16—24no—
35C2×C79⋊C16—21no—
36C2×C79⋊C16D412yes—
37#14579097—21no—
38#14579097—21no—
39#14579093—21no—
40#14579093—21no—
41#14579085—41no—
42#14579082—41no—
43C2×#41×C79—21no—
44C2×#41×C79D411yescharacteristic
45#137×C79—22no—
46#137×C79—41no—
47AGL(1,ℤ/8)×C79—22no—
48AGL(1,ℤ/8)×C79—41no—
49#135×C79—21no—
50#135×C79—21no—
51#135×C79—21no—
52#135×C79—21no—
53#135×C79—42no—
54C2×Q16×C79C2×C411yescharacteristic
55C2×SD16×C79—21no—
56C2×D8×C79C2×C411yescharacteristic
57#131×C79—21no—
58#131×C79—21no—
59#131×C79C2×C412yes—
60#131×C79D412yes—
61C2×M16×C79—21no—
62C2×M16×C79—21no—
63C2×M16×C79D411yescharacteristic
64C22×C8×C79—21no—
65C22×C8×C79C2311yescharacteristic
66#108×C79—21no—
67#108×C79C2×C412yes—
68#101×C79—22no—
69#100×C79—22no—
70C22×Dic79—41no—
71#11929608—41no—
72#11929608—41no—
73C2×C79⋊C8—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C79⋊C16—22no—
77C79⋊C16—22no—
78#41×C79—21no—
79#41×C79—21no—
80#41×C79—41no—
81#41×C79—41no—
82#41×C79—41no—
83#41×C79—41no—
84#41×C79C22⋊C412yes—
85#41×C79SD1612yes—
86C2×Q8×C79—21no—
87C2×Q8×C79D811yescharacteristic
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79—41no—
91C2×D4×C79D811yescharacteristic
92C22×C4×C79—21no—
93C22×C4×C79—41no—
94C22×C4×C79C2×D411yescharacteristic
95Q16×C79—21no—
96Q16×C79—21no—
97Q16×C79—42no—
98SD16×C79—41no—
99SD16×C79—41no—
100SD16×C79—42no—
101SD16×C79—42no—
102D8×C79—21no—
103D8×C79—21no—
104D8×C79—42no—
105M16×C79—22no—
106M16×C79—22no—
107M16×C79—22no—
108M16×C79—21no—
109M16×C79—41no—
110M16×C79C22⋊C412yes—
111C2×C8×C79—22no—
112C2×C8×C79—22no—
113C2×C8×C79—21no—
114C2×C8×C79—41no—
115C2×C8×C79C22×C411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79#4111yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C23×C79—21no—
122C23×C79—41no—
123C23×C79#13511yescharacteristic
124Q8×C79—21no—
125Q8×C79—41no—
126Q8×C79—41no—
127Q8×C79#10212yes—
128D4×C79—22no—
129D4×C79—22no—
130D4×C79—21no—
131D4×C79—41no—
132D4×C79—41no—
133D4×C79—81no—
134D4×C79—81no—
135D4×C79#10212yes—
136C2×C4×C79—21no—
137C2×C4×C79—22no—
138C2×C4×C79—22no—
139C2×C4×C79—41no—
140C2×C4×C79—81no—
141C2×C4×C79—81no—
142C2×C4×C79C2×C22⋊C411yescharacteristic
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79C2×D811yescharacteristic
146C2×C4×C79C2×SD1611yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×Dic79—81no—
149C2×Dic79—161no—
150C8×C79—22no—
151C8×C79—42no—
152C8×C79C4×D411yescharacteristic
153C8×C79C4×D411yescharacteristic
154C8×C79#12011yescharacteristic
155C8×C79#12311yescharacteristic
156C79⋊C8—81no—
157C22×C79—22no—
158C22×C79—41no—
159C22×C79—81no—
160C22×C79—81no—
161C22×C79—161no—
162C22×C79#41611yescharacteristic
163C22×C79#46511yescharacteristic
164C22×C79#46611yescharacteristic
165C4×C79—22no—
166C4×C79—81no—
167C4×C79#38611yescharacteristic
168C4×C79C2×#10211yescharacteristic
169C4×C79#46511yescharacteristic
170C4×C79#46611yescharacteristic
171Dic79—161no—
172#13070—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—42no—
177C2×C79—161no—
178C2×C79#189311yescharacteristic
179#3407—791no—
180#3067D7911yescharacteristic
181C2×#360—791no—
182#2143—791no—
183#1815—791no—
184#1388—791no—
185#1383—791no—
186C79#1307011yescharacteristic, complemented, Sylow
187#575—21no—
188C2×#135C2×D7911yescharacteristic
189C2×#131C2×D7911yescharacteristic
190#504—1582no—
191#504—1582no—
192C2×M32—791no—
193C2×M32—791no—
194C2×M32—1582no—
195#473Dic7912yescomplemented
196#472Dic7912yescomplemented
197#471—22no—
198C2×#108—791no—
199C2×#108C2×D7911yescharacteristic
200C2×#106—791no—
201#417—1581no—
202#413—21no—
203#360—1582no—
204#359—1582no—
205#343—1581no—
206C2×#41—21no—
207C2×#41C22⋊D7911yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic7911yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic7911yescharacteristic
220#131—21no—
221#131—21no—
222#131C2×Dic7912yes—
223#131C22⋊D7912yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—1581no—
227C2×M16C22⋊D7911yescharacteristic
228C22×C8—21no—
229C22×C8C22×D7911yescharacteristic
230C2×C4⋊C4—1581no—
231M32—792no—
232M32—1584no—
233M32—1581no—
234M32—1581no—
235M32—1581no—
236C2×C16—792no—
237C2×C16—1584no—
238C2×C16—1581no—
239#108—21no—
240#108—1581no—
241#108—1581no—
242#108C2×Dic7912yes—
243#106—1581no—
244#106—1581no—
245#103—3161no—
246#102—3161no—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41C22⋊Dic7912yes—
256#41#1192961212yes—
257C2×Q8—21no—
258C2×Q8#1192960911yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1192960911yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—3161no—
266C22×C4C2×C22⋊D7911yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—42no—
270SD16—42no—
271SD16—42no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—41no—
282M16—3161no—
283M16—3161no—
284M16C22⋊Dic7912yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—3161no—
290C2×C8C22×Dic7911yescharacteristic
291C2×C8D4×D7911yescharacteristic
292C2×C8D4×D7911yescharacteristic
293C2×C8C2×C22⋊D7911yescharacteristic
294C2×C8#1192962311yescharacteristic
295C2×C8#1192962311yescharacteristic
296C4⋊C4—3161no—
297C4⋊C4—3161no—
298C16—1582no—
299C16—1582no—
300C23—21no—
301C23—41no—
302C23#1457918111yescharacteristic
303Q8—21no—
304Q8—41no—
305Q8—41no—
306Q8#1457908212yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4#1457908212yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—6321no—
322C2×C4—12641no—
323C2×C4C2×C22⋊Dic7911yescharacteristic
324C2×C4#1457913311yescharacteristic
325C2×C4#1457913311yescharacteristic
326C2×C4C2×#1192960911yescharacteristic
327C2×C4C2×#1192961211yescharacteristic
328C2×C4#1457918111yescharacteristic
329C8—22no—
330C8—42no—
331C8—6321no—
332C8D4×Dic7911yescharacteristic
333C8D4×Dic7911yescharacteristic
334C8#1457912911yescharacteristic
335C8#1457914211yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22#1909855911yescharacteristic
342C22#1909879011yescharacteristic
343C22#1909879711yescharacteristic
344C4—22no—
345C4—81no—
346C4—12641no—
347C4#1909844311yescharacteristic
348C4C2×#1457908211yescharacteristic
349C4#1909879011yescharacteristic
350C4#1909879711yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2376382911yescharacteristic
subgroupG/H
centerC2#23763829
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3067C2
socleC2×C79#1893
cosocleC22×C8×C79C23
hypercenter#3067D79
hypocenterC79#13070
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #23776503C2semidirect
2. C2×#19098373C2semidirect
3. #23765128C2semidirect
4. #23763459C2semidirect
5. #23761511C2semidirect
6. #23761500C2semidirect
22. #473×C79C4semidirect
23. #472×C79C4semidirect
186. C79#13070semidirect
195. #473Dic79semidirect
196. #472Dic79semidirect

Maximal subgroups

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

subgroupindex
#237765032
C2×#190983732
#237651282
#237634592
#237615112
#237615002
C79×#30672
#1307079