Finite Group Databasebeta

← #34205699

Subgroups — #34205699

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098980C211yesmaximal, characteristic, complemented
2#23765136C211yesmaximal, characteristic, complemented
3C2×#19098370C211yesmaximal, characteristic, complemented
4#23763454C211yesmaximal, characteristic, complemented
5#23761513C211yesmaximal, characteristic, complemented
6#23761497C211yesmaximal, characteristic, complemented
7C79×#3067C211yesmaximal, characteristic
8#19098980—22no—
9#19098980—22no—
10C2×#14579101—21no—
11C2×#14579101C2211yescharacteristic
12C22×C79⋊C16—21no—
13C22×C79⋊C16C2211yescharacteristic
14C2×#14579097C2211yescharacteristic
15#19098601C2211yescharacteristic
16#19098580—21no—
17#19098373—22no—
18#19098370—21no—
19#19098370—21no—
20#19098335—21no—
21#575×C79—21no—
22C2×#135×C79C2211yescharacteristic
23C2×#131×C79C2211yescharacteristic
24#473×C79C412yescomplemented
25#472×C79C412yescomplemented
26#471×C79—22no—
27C2×#108×C79C2211yescharacteristic
28#413×C79—21no—
29C2×#11929608—21no—
30#14579120—21no—
31#14579101—24no—
32#14579101—21no—
33#14579101D412yes—
34C2×C79⋊C16—24no—
35C2×C79⋊C16—21no—
36C2×C79⋊C16—21no—
37C2×C79⋊C16—21no—
38C2×C79⋊C16D411yescharacteristic
39C2×C79⋊C16D411yescharacteristic
40#14579097—21no—
41#14579097—21no—
42#14579095—21no—
43#14579093—21no—
44#14579088—42no—
45C2×#41×C79—21no—
46C2×#41×C79D411yescharacteristic
47#137×C79—22no—
48#137×C79—41no—
49AGL(1,ℤ/8)×C79—22no—
50AGL(1,ℤ/8)×C79—41no—
51#135×C79—21no—
52#135×C79—21no—
53#135×C79—21no—
54#135×C79—21no—
55#135×C79—42no—
56C2×Q16×C79C2×C411yescharacteristic
57C2×SD16×C79—21no—
58C2×D8×C79C2×C411yescharacteristic
59#131×C79—21no—
60#131×C79—21no—
61#131×C79C2×C412yes—
62#131×C79D412yes—
63C2×M16×C79—21no—
64C2×M16×C79—21no—
65C2×M16×C79D411yescharacteristic
66C22×C8×C79—21no—
67C22×C8×C79C2311yescharacteristic
68#108×C79—21no—
69#108×C79C2×C412yes—
70#101×C79—22no—
71#100×C79—22no—
72#11929608—41no—
73#11929608—41no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—41no—
76C22⋊Dic79—41no—
77C4×Dic79—41no—
78C79⋊C16—21no—
79C79⋊C16—22no—
80C79⋊C16—21no—
81#41×C79—21no—
82#41×C79—21no—
83#41×C79—41no—
84#41×C79—41no—
85#41×C79—41no—
86#41×C79—41no—
87#41×C79C22⋊C412yes—
88#41×C79D812yes—
89C2×Q8×C79—21no—
90C2×Q8×C79SD1611yescharacteristic
91C2×D4×C79—21no—
92C2×D4×C79—21no—
93C2×D4×C79—41no—
94C2×D4×C79SD1611yescharacteristic
95C22×C4×C79—21no—
96C22×C4×C79—41no—
97C22×C4×C79C2×D411yescharacteristic
98Q16×C79—21no—
99Q16×C79—21no—
100Q16×C79—42no—
101SD16×C79—41no—
102SD16×C79—41no—
103SD16×C79—42no—
104SD16×C79—42no—
105D8×C79—21no—
106D8×C79—21no—
107D8×C79—42no—
108M16×C79—22no—
109M16×C79—22no—
110M16×C79—22no—
111M16×C79—21no—
112M16×C79—41no—
113M16×C79C22⋊C412yes—
114C2×C8×C79—22no—
115C2×C8×C79—22no—
116C2×C8×C79—21no—
117C2×C8×C79—41no—
118C2×C8×C79C22×C411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79C2×D411yescharacteristic
122C2×C8×C79#4111yescharacteristic
123C2×C8×C79#4111yescharacteristic
124C23×C79—21no—
125C23×C79—41no—
126C23×C79#13511yescharacteristic
127Q8×C79—21no—
128Q8×C79—41no—
129Q8×C79—41no—
130Q8×C79#10212yes—
131D4×C79—22no—
132D4×C79—22no—
133D4×C79—21no—
134D4×C79—41no—
135D4×C79—41no—
136D4×C79—81no—
137D4×C79—81no—
138D4×C79#10212yes—
139C2×C4×C79—21no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—41no—
143C2×C4×C79—81no—
144C2×C4×C79—81no—
145C2×C4×C79C2×C22⋊C411yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79#12111yescharacteristic
148C2×C4×C79C2×D811yescharacteristic
149C2×C4×C79C2×SD1611yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×Dic79—81no—
152C8×C79—22no—
153C8×C79—42no—
154C8×C79C4×D411yescharacteristic
155C8×C79C4×D411yescharacteristic
156C8×C79#12011yescharacteristic
157C8×C79#12311yescharacteristic
158C79⋊C8—81no—
159C22×C79—22no—
160C22×C79—41no—
161C22×C79—42no—
162C22×C79—81no—
163C22×C79—161no—
164C22×C79#41611yescharacteristic
165C22×C79#46511yescharacteristic
166C22×C79#46611yescharacteristic
167C4×C79—22no—
168C4×C79—81no—
169C4×C79#38611yescharacteristic
170C4×C79C2×#10211yescharacteristic
171C4×C79#46511yescharacteristic
172C4×C79#46611yescharacteristic
173Dic79—161no—
174#13071—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79—21no—
178C2×C79—42no—
179C2×C79—161no—
180C2×C79#189311yescharacteristic
181C2×#504—791no—
182#3067D7911yescharacteristic
183#2145—791no—
184C2×#359—791no—
185#1814—791no—
186#1388—791no—
187#1382—791no—
188C79#1307111yescharacteristic, complemented, Sylow
189#575—21no—
190C2×#135C2×D7911yescharacteristic
191C2×#131C2×D7911yescharacteristic
192#504—1582no—
193#504—1582no—
194C2×M32—791no—
195C2×M32—1581no—
196C22×C16—791no—
197C22×C16—1581no—
198#473Dic7912yescomplemented
199#472Dic7912yescomplemented
200#471—22no—
201C2×#108—791no—
202C2×#108C2×D7911yescharacteristic
203#427—791no—
204#421—1581no—
205#413—21no—
206#360—1582no—
207#359—1581no—
208#359—1581no—
209#344—1581no—
210C2×#41—21no—
211C2×#41C22⋊D7911yescharacteristic
212#137—22no—
213#137—41no—
214AGL(1,ℤ/8)—22no—
215AGL(1,ℤ/8)—41no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135—42no—
221C2×Q16C2×Dic7911yescharacteristic
222C2×SD16—21no—
223C2×D8C2×Dic7911yescharacteristic
224#131—21no—
225#131—21no—
226#131C2×Dic7912yes—
227#131C22⋊D7912yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1581no—
231C2×M16C22⋊D7911yescharacteristic
232C22×C8—21no—
233C22×C8C22×D7911yescharacteristic
234#117—1581no—
235M32—792no—
236M32—1584no—
237M32—1581no—
238C2×C16—791no—
239C2×C16—791no—
240C2×C16—1584no—
241C2×C16—1581no—
242C2×C16—1581no—
243C2×C16—1581no—
244#108—21no—
245#108—1581no—
246#108—1581no—
247#108C2×Dic7912yes—
248#107—1581no—
249#106—1581no—
250#104—3162no—
251#101—22no—
252#100—22no—
253#41—21no—
254#41—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41C22⋊Dic7912yes—
260#41#1192960912yes—
261C2×Q8—21no—
262C2×Q8#1192961211yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#1192961211yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×C22⋊D7911yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—42no—
273SD16—42no—
274SD16—42no—
275SD16—41no—
276SD16—41no—
277D8—21no—
278D8—21no—
279D8—42no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—41no—
285M16—3161no—
286M16—3161no—
287M16C22⋊Dic7912yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—3161no—
293C2×C8C22×Dic7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8D4×D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8#1192962311yescharacteristic
298C2×C8#1192962311yescharacteristic
299C4⋊C4—3161no—
300C22⋊C4—3161no—
301C42—3161no—
302C16—1581no—
303C16—1581no—
304C16—1582no—
305C23—21no—
306C23—41no—
307C23#1457918111yescharacteristic
308Q8—21no—
309Q8—41no—
310Q8—41no—
311Q8#1457908212yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4#1457908212yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—6321no—
327C2×C4C2×C22⋊Dic7911yescharacteristic
328C2×C4#1457913311yescharacteristic
329C2×C4#1457913311yescharacteristic
330C2×C4C2×#1192960911yescharacteristic
331C2×C4C2×#1192961211yescharacteristic
332C2×C4#1457918111yescharacteristic
333C8—22no—
334C8—42no—
335C8—6321no—
336C8D4×Dic7911yescharacteristic
337C8D4×Dic7911yescharacteristic
338C8#1457912911yescharacteristic
339C8#1457914211yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22#1909855911yescharacteristic
346C22#1909879011yescharacteristic
347C22#1909879711yescharacteristic
348C4—22no—
349C4—81no—
350C4—12641no—
351C4#1909844311yescharacteristic
352C4C2×#1457908211yescharacteristic
353C4#1909879011yescharacteristic
354C4#1909879711yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2#2376382911yescharacteristic
subgroupG/H
centerC4#19098797
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3067C2
socleC2×C79#1893
cosocleC22×C8×C79C23
hypercenter#3067D79
hypocenterC79#13071
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. C2×#19098980C2semidirect
2. #23765136C2semidirect
3. C2×#19098370C2semidirect
4. #23763454C2semidirect
5. #23761513C2semidirect
6. #23761497C2semidirect
24. #473×C79C4semidirect
25. #472×C79C4semidirect
188. C79#13071semidirect
198. #473Dic79semidirect
199. #472Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989802
#237651362
C2×#190983702
#237634542
#237615132
#237614972
C79×#30672
#1307179