Finite Group Databasebeta

← #34183356

Subgroups — #34183356

Order 20224.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771964C211yesmaximal, characteristic
2#23766668C211yesmaximal, characteristic
3#23761830C212yesmaximal, complemented
4#23761302C212yesmaximal, complemented
5C79×#1458C211yesmaximal, characteristic
6C22×#11929608—21no—
7C2×C4×Dic158C2211yescharacteristic
8#19098590—24no—
9#19098587—21no—
10#19098587C2212yes—
11#19098502C411yescharacteristic
12#19098499C411yescharacteristic
13#19098496—21no—
14#19098486—21no—
15#19098483C411yescharacteristic
16#19098475C411yescharacteristic
17C4×Dic79⋊C4—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19C4×C22⋊Dic79—21no—
20#19098295—24no—
21C2×#117×C79C2211yescharacteristic
22#431×C79C2212yes—
23#324×C79—21no—
24#323×C79—21no—
25C22×Dic158C2×C411yescharacteristic
26C2×#11929608—22no—
27C2×#11929608—22no—
28C2×#11929608—42no—
29C2×#11929608—42no—
30C22×C79⋊C8—22no—
31C4×Dic158—22no—
32C4×Dic158—22no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79C2×C411yescharacteristic
35C2×Dic79⋊C4—21no—
36C2×Dic79⋊C4—21no—
37C2×Dic79⋊C4—21no—
38C2×C22⋊Dic79—21no—
39C2×C22⋊Dic79—21no—
40C2×C4×Dic79—21no—
41C2×C4×Dic79—21no—
42C4⋊(C79⋊C8)—24no—
43C4⋊(C79⋊C8)—22no—
44C4⋊(C79⋊C8)D414yes—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95C2×C411yescharacteristic
50C79⋊#95C2×C411yescharacteristic
51C23×C4×C79D411yescharacteristic
52#117×C79—22no—
53#117×C79—22no—
54C2×C4⋊C4×C79C2×C411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C2×C42×C79C2311yescharacteristic
60C4⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62C4×C8×C79—22no—
63C2×Dic158—22no—
64C2×Dic158—42no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69#11929608—44no—
70#11929608—42no—
71#11929608—42no—
72C2×C79⋊C8—24no—
73C2×C79⋊C8—44no—
74C4⋊Dic79—22no—
75C4⋊Dic79—42no—
76Dic79⋊C4—42no—
77Dic79⋊C4—42no—
78Dic79⋊C4—42no—
79C22⋊Dic79—42no—
80C22⋊Dic79—42no—
81C4×Dic79—41no—
82C4×Dic79—41no—
83C4×Dic79—41no—
84C4×Dic79—41no—
85C24×C79C22⋊C411yescharacteristic
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79SD1612yes—
94C22×C4×C79Q1612yes—
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C2×C8×C79—42no—
99C4⋊C4×C79—22no—
100C4⋊C4×C79—22no—
101C22⋊C4×C79—22no—
102C22⋊C4×C79—22no—
103C42×C79—21no—
104C42×C79—21no—
105C42×C79C2×D412yes—
106C42×C79#4112yes—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79#10312yes—
112C23×C79C2×C22⋊C411yescharacteristic
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79#10312yes—
126C2×C4×C79C4×D412yes—
127C2×C4×C79#12011yescharacteristic
128C2×C4×C79#12112yes—
129C2×C4×C79#12311yescharacteristic
130C2×C4×C79C2×SD1611yescharacteristic
131C2×C4×C79C2×Q1611yescharacteristic
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C79#13711yescharacteristic
134C2×Dic79—22no—
135C2×Dic79—22no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146Dic158—24no—
147Dic158—42no—
148C8×C79—82no—
149C79⋊C8—44no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—22no—
153C22×C79—42no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—81no—
157C22×C79#38611yescharacteristic
158C22×C79C2×#10311yescharacteristic
159C22×C79#41711yescharacteristic
160C4×C79—22no—
161C4×C79—22no—
162C4×C79—22no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79#45011yescharacteristic
166C4×C79#45111yescharacteristic
167C4×C79#48311yescharacteristic
168C4×C79#48411yescharacteristic
169Dic79—42no—
170Dic79—42no—
171Dic79—81no—
172Dic79—81no—
173#9033—791nomaximal, Sylow
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—42no—
177C2×C79#187211yescharacteristic
178C2×C79#188411yescharacteristic
179C2×C79#189111yescharacteristic
180#2905—791no—
181#2396—791no—
182#1487—792no—
183#1458D7911yescharacteristic
184#1303—792no—
185C79#903311yescharacteristic, complemented, Sylow
186C22×M16—1581no—
187C2×C4×Q8—791no—
188C2×#117C2×D7911yescharacteristic
189#431C2×D7912yes—
190#424—1584no—
191#423—792no—
192#423—1581no—
193#400—791no—
194#399—791no—
195#398—1581no—
196#396—1581no—
197#395—791no—
198#393—791no—
199C4×C4⋊C4—791no—
200C4×C4⋊C4—1581no—
201C4×C22⋊C4—1581no—
202Q8⋊C8—1584no—
203#324—21no—
204#323—21no—
205C22×Q8—791no—
206C23×C4D31611yescharacteristic
207C2×M16—3162no—
208C2×M16—3162no—
209C2×M16—1582no—
210C2×M16—1582no—
211C22×C8—1582no—
212C4×Q8—1582no—
213C4×Q8—1582no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—791no—
217C2×C4⋊C4—1581no—
218C2×C4⋊C4—1581no—
219C2×C4⋊C4—1581no—
220C2×C4⋊C4—1581no—
221C2×C4⋊C4C4×D7911yescharacteristic
222C2×C4⋊C4C22⋊D7911yescharacteristic
223C2×C22⋊C4—1581no—
224C2×C22⋊C4—1581no—
225C2×C22⋊C4C4×D7911yescharacteristic
226C2×C22⋊C4C22⋊D7911yescharacteristic
227C2×C42—1581no—
228C2×C42—1581no—
229C2×C42D31611yescharacteristic
230C2×C42C22×D7911yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—794no—
233C4⋊C8—1584no—
234C4⋊C8—1582no—
235C22⋊C8—22no—
236C4×C8—22no—
237#95—791no—
238#95—791no—
239#95—1581no—
240#95—1581no—
241#95—1581no—
242#95—1581no—
243C24(C2×D79)⋊C411yescharacteristic
244C2×Q8—3162no—
245C2×Q8—1582no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1581no—
251C22×C4—1581no—
252C22×C4—1581no—
253C22×C4—1581no—
254C22×C4(C2×D79)⋊C411yescharacteristic
255C22×C4(C2×D79)⋊C411yescharacteristic
256C22×C4(C2×D79)⋊C411yescharacteristic
257C22×C4#1192961312yes—
258C22×C4Dic31612yes—
259C22×C4C2×C4×D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261C22×C4C2×D31611yescharacteristic
262M16—3164no—
263M16—3162no—
264M16—3162no—
265C2×C8—42no—
266C2×C8—3164no—
267C2×C8—1584no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—3162no—
271C4⋊C4—3162no—
272C4⋊C4—3162no—
273C4⋊C4—3162no—
274C4⋊C4—1582no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—3162no—
278C22⋊C4—3162no—
279C42—21no—
280C42—21no—
281C42—3161no—
282C42—3161no—
283C42—3161no—
284C42—3161no—
285C42D4×D7912yes—
286C42#1192962312yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1457908612yes—
292C23C2×(C2×D79)⋊C411yescharacteristic
293Q8—3162no—
294Q8—1584no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—1582no—
318C2×C4—1582no—
319C2×C4#1457908612yes—
320C2×C4(C22⋊D79)⋊C412yes—
321C2×C4#1457913111yescharacteristic
322C2×C4#1457913312yes—
323C2×C4#1457914511yescharacteristic
324C2×C4C2×#1192961311yescharacteristic
325C2×C4C2×Dic31611yescharacteristic
326C2×C4#1457918911yescharacteristic
327C2×C4#1457919411yescharacteristic
328C8—82no—
329C8—3164no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1909844011yescharacteristic
338C22C2×#1457908611yescharacteristic
339C22#1909856511yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—3162no—
346C4—3162no—
347C4—6321no—
348C4—6321no—
349C4#1909871611yescharacteristic
350C4#1909872111yescharacteristic
351C4#1909889411yescharacteristic
352C4#1909889911yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2376370511yescharacteristic
357C2#2376378111yescharacteristic
358C2#2376381611yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1458C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1458D79
hypocenterC79#9033
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #23761830C2semidirect
4. #23761302C2semidirect
185. C79#9033semidirect

Maximal subgroups

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

subgroupindex
#237719642
#237666682
#237618302
#237613022
C79×#14582
#903379