Finite Group Databasebeta

← #34205187

Subgroups — #34205187

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098980C211yesmaximal, characteristic, complemented
2#23765028C211yesmaximal, characteristic, complemented
3C2×#19098347C211yesmaximal, characteristic, complemented
4#23765021C211yesmaximal, characteristic, complemented
5#23761489C211yesmaximal, characteristic, complemented
6#23761486C211yesmaximal, characteristic, complemented
7C79×#2955C211yesmaximal, characteristic
8#19098980—22no—
9#19098980—22no—
10C2×#14579101—21no—
11C2×#14579101C2211yescharacteristic
12C2×#14579101C2211yescharacteristic
13C2×#14579101C2211yescharacteristic
14C22×C79⋊C16—21no—
15C22×C79⋊C16C2211yescharacteristic
16#19098347—21no—
17#19098347—21no—
18#19098347—22no—
19#19098345—22no—
20#575×C79C411yescharacteristic
21C2×#131×C79C2211yescharacteristic
22C2×#131×C79C2211yescharacteristic
23#444×C79—22no—
24#443×C79—22no—
25#430×C79C411yescharacteristic
26#421×C79C411yescharacteristic
27#421×C79C411yescharacteristic
28#405×C79C2211yescharacteristic
29#14579101—24no—
30#14579101—21no—
31#14579101—21no—
32#14579101—21no—
33#14579101—21no—
34#14579101—21no—
35#14579101D412yes—
36C2×C79⋊C16—24no—
37C2×C79⋊C16—21no—
38C2×C79⋊C16—21no—
39C2×C79⋊C16—21no—
40C2×C79⋊C16—21no—
41C2×C79⋊C16—21no—
42C2×C79⋊C16D411yescharacteristic
43C2×C79⋊C16D411yescharacteristic
44C2×#41×C79C2×C411yescharacteristic
45C2×#41×C79C2×C411yescharacteristic
46#137×C79—21no—
47#137×C79C812yes—
48AGL(1,ℤ/8)×C79—21no—
49AGL(1,ℤ/8)×C79C812yes—
50#135×C79—22no—
51#131×C79—21no—
52#131×C79—21no—
53#131×C79—21no—
54#131×C79—21no—
55#131×C79—21no—
56#131×C79—21no—
57#131×C79D412yes—
58#131×C79D412yes—
59C2×M16×C79—21no—
60C2×M16×C79—21no—
61C2×M16×C79C2×C411yescharacteristic
62C2×M16×C79C2×C411yescharacteristic
63C2×M16×C79C2×C411yescharacteristic
64C22×C8×C79—21no—
65C22×C8×C79—21no—
66C22×C8×C79C2311yescharacteristic
67#117×C79C2×C411yescharacteristic
68#108×C79—22no—
69#104×C79—22no—
70#104×C79—22no—
71#97×C79—21no—
72C4×C8×C79—21no—
73C79⋊C16—21no—
74C79⋊C16—22no—
75C79⋊C16—21no—
76C79⋊C16—41no—
77C79⋊C16—41no—
78#41×C79—21no—
79#41×C79—21no—
80#41×C79—21no—
81#41×C79—21no—
82#41×C79—21no—
83#41×C79—21no—
84#41×C79C22⋊C412yes—
85#41×C79C2×C812yes—
86C2×Q8×C79C2×C811yescharacteristic
87C2×Q8×C79M1611yescharacteristic
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79C2×C811yescharacteristic
91C2×D4×C79M1611yescharacteristic
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79C22×C411yescharacteristic
95Q16×C79—22no—
96SD16×C79—22no—
97SD16×C79—22no—
98D8×C79—22no—
99M16×C79—22no—
100M16×C79—22no—
101M16×C79—21no—
102M16×C79—21no—
103M16×C79—21no—
104M16×C79—21no—
105M16×C79—41no—
106M16×C79—41no—
107M16×C79C22⋊C412yes—
108M16×C79C2×C812yes—
109C2×C8×C79—22no—
110C2×C8×C79—22no—
111C2×C8×C79—21no—
112C2×C8×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C4⋊C4×C79—21no—
123C22⋊C4×C79—21no—
124C42×C79—21no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79#13111yescharacteristic
128Q8×C79—21no—
129Q8×C79—21no—
130Q8×C79—21no—
131Q8×C79C22⋊C812yes—
132D4×C79—22no—
133D4×C79—22no—
134D4×C79—21no—
135D4×C79—21no—
136D4×C79—21no—
137D4×C79—41no—
138D4×C79—41no—
139D4×C79C22⋊C812yes—
140C2×C4×C79—21no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—21no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79C2×C22⋊C411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79C4×D411yescharacteristic
150C2×C4×C79C22×C811yescharacteristic
151C2×C4×C79C2×M1611yescharacteristic
152C2×C4×C79#13111yescharacteristic
153C8×C79—22no—
154C8×C79—22no—
155C8×C79—41no—
156C8×C79#12011yescharacteristic
157C8×C79#12111yescharacteristic
158C8×C79#12111yescharacteristic
159C8×C79#12311yescharacteristic
160C22×C79—22no—
161C22×C79—41no—
162C22×C79—42no—
163C22×C79—41no—
164C22×C79—81no—
165C22×C79#40811yescharacteristic
166C22×C79C8×D411yescharacteristic
167C22×C79#43511yescharacteristic
168C4×C79—22no—
169C4×C79—41no—
170C4×C79—81no—
171C4×C79#38611yescharacteristic
172C4×C79C2×C22⋊C811yescharacteristic
173C4×C79C8×D411yescharacteristic
174C4×C79#43511yescharacteristic
175#12987—791nomaximal, Sylow
176C2×C79—21no—
177C2×C79—21no—
178C2×C79—21no—
179C2×C79—42no—
180C2×C79—81no—
181C2×C79#186911yescharacteristic
182C2×#504—791no—
183#2955D7911yescharacteristic
184#2117—791no—
185C2×#350—791no—
186#2115—791no—
187#1379—791no—
188#1378—791no—
189C79#1298711yescharacteristic, complemented, Sylow
190#575Dic7911yescharacteristic
191C2×#131C2×D7911yescharacteristic
192C2×#131C2×D7911yescharacteristic
193#504—1582no—
194#504—1582no—
195C2×M32—791no—
196C2×M32—791no—
197C2×M32—791no—
198C2×M32—1581no—
199C22×C16—791no—
200C22×C16—1581no—
201#444—22no—
202#443—22no—
203#430Dic7911yescharacteristic
204#421Dic7911yescharacteristic
205#421Dic7911yescharacteristic
206#405C2×D7911yescharacteristic
207#350—1581no—
208#350—1581no—
209#350—1582no—
210#349—1582no—
211C2×#41C2×Dic7911yescharacteristic
212C2×#41C2×Dic7911yescharacteristic
213#137—21no—
214#137C79⋊C812yes—
215AGL(1,ℤ/8)—21no—
216AGL(1,ℤ/8)C79⋊C812yes—
217#135—22no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131—21no—
224#131C22⋊D7912yes—
225#131C22⋊D7912yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16C2×Dic7911yescharacteristic
229C2×M16C2×Dic7911yescharacteristic
230C2×M16C2×Dic7911yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×D7911yescharacteristic
234#117C2×Dic7911yescharacteristic
235M32—792no—
236M32—1581no—
237M32—1581no—
238M32—1581no—
239M32—1581no—
240M32—1584no—
241M32—1581no—
242C2×C16—791no—
243C2×C16—791no—
244C2×C16—1581no—
245C2×C16—1581no—
246C2×C16—1584no—
247C2×C16—1581no—
248C2×C16—1581no—
249C2×C16—1581no—
250#108—22no—
251#104—22no—
252#104—22no—
253#97—21no—
254C4×C8—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260#41—21no—
261#41C22⋊Dic7912yes—
262#41C2×C79⋊C812yes—
263C2×Q8C2×C79⋊C811yescharacteristic
264C2×Q8#1192960811yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×C79⋊C811yescharacteristic
268C2×D4#1192960811yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×Dic7911yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16C22⋊Dic7912yes—
285M16C2×C79⋊C812yes—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8D4×D7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8#1192962311yescharacteristic
298C2×C8#1192962311yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—3161no—
303C16—3161no—
304C16—1581no—
305C16—1581no—
306C16—1582no—
307C23—21no—
308C23—21no—
309C23#1457916811yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C22⋊(C79⋊C8)12yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C22⋊(C79⋊C8)12yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊Dic7911yescharacteristic
330C2×C4D4×Dic7911yescharacteristic
331C2×C4D4×Dic7911yescharacteristic
332C2×C4C22×C79⋊C811yescharacteristic
333C2×C4C2×#1192960811yescharacteristic
334C2×C4#1457916811yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8#1457912911yescharacteristic
339C8#1457913311yescharacteristic
340C8#1457913311yescharacteristic
341C8#1457914211yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22#1909853011yescharacteristic
348C22D4×C79⋊C811yescharacteristic
349C22#1909863511yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4#1909844311yescharacteristic
354C4C2×C22⋊(C79⋊C8)11yescharacteristic
355C4D4×C79⋊C811yescharacteristic
356C4#1909863511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2376369311yescharacteristic
subgroupG/H
centerC8#14579133
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC79×#2955C2
socleC2×C79#1869
prime radical#575×C79C4
cosocleC22×C8×C79C23
hypercenter#2955D79
hypocenterC79#12987
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C2×#19098980C2semidirect
2. #23765028C2semidirect
3. C2×#19098347C2semidirect
4. #23765021C2semidirect
5. #23761489C2semidirect
6. #23761486C2semidirect
189. C79#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#190989802
#237650282
C2×#190983472
#237650212
#237614892
#237614862
C79×#29552
#1298779