Finite Group Databasebeta

← #34205229

Subgroups — #34205229

Order 20224.

476 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#23772011C211yesmaximal, characteristic, complemented
2#23765128C211yesmaximal, characteristic, complemented
3C2×#19098370C211yesmaximal, characteristic, complemented
4C2×#19098347C211yesmaximal, characteristic, complemented
5#23761513C211yesmaximal, characteristic, complemented
6#23761493C211yesmaximal, characteristic, complemented
7C79×#3067C211yesmaximal, characteristic
8#19099330—21no—
9C2×#14579101C2211yescharacteristic
10C22×C79⋊C16C2211yescharacteristic
11#19098612—22no—
12#19098612—21no—
13C2×#14579097—22no—
14C2×#14579097C2211yescharacteristic
15C2×#14579097C2211yescharacteristic
16#19098370—22no—
17#19098370—22no—
18#19098347—22no—
19#575×C79—21no—
20C2×#135×C79C2211yescharacteristic
21C2×#131×C79C2211yescharacteristic
22#473×C79C412yes—
23#472×C79C412yes—
24#471×C79—22no—
25C2×#108×C79C2211yescharacteristic
26#413×C79—21no—
27#14579168—42no—
28#14579168—42no—
29C2×#11929608—24no—
30C2×#11929608—21no—
31C2×#11929608—21no—
32#14579101—21no—
33#14579101—21no—
34C2×C79⋊C16—21no—
35C2×C79⋊C16—21no—
36C2×C79⋊C16—21no—
37C2×C79⋊C16—21no—
38#14579097—24no—
39#14579097—21no—
40#14579097—21no—
41#14579097D412yes—
42#14579097D412yes—
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—
70#11929608—22no—
71#11929608—22no—
72#11929608—44no—
73#11929608—41no—
74#11929608—41no—
75C2×C79⋊C8—44no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C79⋊C16—41no—
79C79⋊C16—41no—
80#41×C79—21no—
81#41×C79—21no—
82#41×C79—41no—
83#41×C79—41no—
84#41×C79—41no—
85#41×C79—41no—
86#41×C79C22⋊C412yes—
87#41×C79D812yes—
88C2×Q8×C79—21no—
89C2×Q8×C79SD1611yescharacteristic
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—41no—
93C2×D4×C79SD1611yescharacteristic
94C22×C4×C79—21no—
95C22×C4×C79—41no—
96C22×C4×C79C2×D411yescharacteristic
97Q16×C79—21no—
98Q16×C79—21no—
99Q16×C79—42no—
100SD16×C79—41no—
101SD16×C79—41no—
102SD16×C79—42no—
103SD16×C79—42no—
104D8×C79—21no—
105D8×C79—21no—
106D8×C79—42no—
107M16×C79—22no—
108M16×C79—22no—
109M16×C79—22no—
110M16×C79—21no—
111M16×C79—41no—
112M16×C79C22⋊C412yes—
113C2×C8×C79—22no—
114C2×C8×C79—22no—
115C2×C8×C79—21no—
116C2×C8×C79—41no—
117C2×C8×C79C22×C411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C2×C8×C79#4111yescharacteristic
123C23×C79—21no—
124C23×C79—41no—
125C23×C79#13711yescharacteristic
126Q8×C79—21no—
127Q8×C79—41no—
128Q8×C79—41no—
129Q8×C79#10212yes—
130D4×C79—22no—
131D4×C79—22no—
132D4×C79—21no—
133D4×C79—41no—
134D4×C79—41no—
135D4×C79—81no—
136D4×C79—81no—
137D4×C79#10212yes—
138C2×C4×C79—21no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—41no—
142C2×C4×C79—81no—
143C2×C4×C79—81no—
144C2×C4×C79C2×C22⋊C411yescharacteristic
145C2×C4×C79#12011yescharacteristic
146C2×C4×C79#12311yescharacteristic
147C2×C4×C79C2×D811yescharacteristic
148C2×C4×C79C2×SD1611yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150C8×C79—22no—
151C8×C79—42no—
152C8×C79C4×D411yescharacteristic
153C8×C79C4×D411yescharacteristic
154C8×C79#12111yescharacteristic
155C8×C79#12111yescharacteristic
156C79⋊C8—42no—
157C79⋊C8—42no—
158C22×C79—22no—
159C22×C79—41no—
160C22×C79—81no—
161C22×C79—81no—
162C22×C79—161no—
163C22×C79#41711yescharacteristic
164C22×C79#44811yescharacteristic
165C22×C79#48011yescharacteristic
166C4×C79—22no—
167C4×C79—81no—
168C4×C79#38611yescharacteristic
169C4×C79C2×#10211yescharacteristic
170C4×C79#44711yescharacteristic
171C4×C79#48111yescharacteristic
172#12995—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—42no—
177C2×C79—161no—
178C2×C79#187411yescharacteristic
179#3067D7911yescharacteristic
180#2916—791no—
181#2143—791no—
182C2×#359—791no—
183C2×#350—791no—
184#1388—791no—
185#1381—791no—
186C79#1299511yescharacteristic, complemented, Sylow
187#575—21no—
188C2×#135C2×D7911yescharacteristic
189#568—1581no—
190C2×#131C2×D7911yescharacteristic
191C2×M32—791no—
192C22×C16—791no—
193#473Dic7912yes—
194#472Dic7912yes—
195#471—22no—
196#430—1582no—
197#430—1581no—
198C2×#108—791no—
199C2×#108—791no—
200C2×#108—1582no—
201C2×#108C2×D7911yescharacteristic
202#413—21no—
203#359—1582no—
204#359—1582no—
205#350—1582no—
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#131—3162no—
223#131—3162no—
224#131C2×Dic7912yes—
225#131C22⋊D7912yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—1584no—
229C2×M16—1581no—
230C2×M16—1581no—
231C2×M16C22⋊D7911yescharacteristic
232C22×C8—21no—
233C22×C8C22×D7911yescharacteristic
234M32—1581no—
235M32—1581no—
236C2×C16—1581no—
237C2×C16—1581no—
238C2×C16—1581no—
239C2×C16—1581no—
240#108—21no—
241#108—792no—
242#108—792no—
243#108—1584no—
244#108—1581no—
245#108—1581no—
246#108C2×Dic7912yes—
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#1192960912yes—
257C2×Q8—21no—
258C2×Q8#1192961211yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1192961211yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4C2×C22⋊D7911yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—42no—
269SD16—42no—
270SD16—42no—
271SD16—41no—
272SD16—41no—
273D8—21no—
274D8—21no—
275D8—42no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—41no—
281M16—3164no—
282M16—3161no—
283M16—3161no—
284M16—1582no—
285M16—1582no—
286M16C22⋊Dic7912yes—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—3164no—
292C2×C8—3161no—
293C2×C8—3161no—
294C2×C8C22×Dic7911yescharacteristic
295C2×C8D4×D7911yescharacteristic
296C2×C8D4×D7911yescharacteristic
297C2×C8C2×C22⋊D7911yescharacteristic
298C2×C8#1192962311yescharacteristic
299C2×C8#1192962311yescharacteristic
300C16—3161no—
301C16—3161no—
302C23—21no—
303C23—41no—
304C23#1457919111yescharacteristic
305Q8—21no—
306Q8—41no—
307Q8—41no—
308Q8#1457908212yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4#1457908212yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊Dic7911yescharacteristic
324C2×C4#1457912911yescharacteristic
325C2×C4#1457914211yescharacteristic
326C2×C4C2×#1192960911yescharacteristic
327C2×C4C2×#1192961211yescharacteristic
328C2×C4#1457918611yescharacteristic
329C8—22no—
330C8—42no—
331C8—3162no—
332C8—3162no—
333C8D4×Dic7911yescharacteristic
334C8D4×Dic7911yescharacteristic
335C8#1457913311yescharacteristic
336C8#1457913311yescharacteristic
337C22—22no—
338C22—41no—
339C22—81no—
340C22—81no—
341C22—161no—
342C22#1909856411yescharacteristic
343C22#1909870111yescharacteristic
344C22#1909887411yescharacteristic
345C4—22no—
346C4—81no—
347C4#1909844311yescharacteristic
348C4C2×#1457908211yescharacteristic
349C4#1909869611yescharacteristic
350C4#1909887911yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2376371811yescharacteristic
subgroupG/H
centerC4#19098696
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3067C2
socleC2×C79#1874
cosocleC22×C8×C79C23
hypercenter#3067D79
hypocenterC79#12995
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #23772011C2semidirect
2. #23765128C2semidirect
3. C2×#19098370C2semidirect
4. C2×#19098347C2semidirect
5. #23761513C2semidirect
6. #23761493C2semidirect
186. C79#12995semidirect

Maximal subgroups

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

subgroupindex
#237720112
#237651282
C2×#190983702
C2×#190983472
#237615132
#237614932
C79×#30672
#1299579