Finite Group Databasebeta

← #34187564

Subgroups — #34187564

Order 20224.

506 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098942C211yesmaximal, characteristic
2#23764352C211yesmaximal, characteristic
3#23763269C211yesmaximal, characteristic
4C2×#19098313C211yesmaximal, characteristic
5C2×#19098309C211yesmaximal, characteristic
6C2×#19098308C211yesmaximal, characteristic
7C79×#1988C211yesmaximal, characteristic
8C22×#11929614—21no—
9C22×#11929614—21no—
10C2×#14579161C2211yescharacteristic
11#19098942—22no—
12#19098942C414yescomplemented
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C2×#14579072—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098325—22no—
18#19098318—21no—
19#19098313—22no—
20#19098309—22no—
21#19098308—22no—
22C2×C4⋊Q8×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24C4×C4⋊C4×C79C2211yescharacteristic
25#340×C79—22no—
26C22×Dic158—21no—
27C2×#11929614—24no—
28C2×#11929614—24no—
29C2×#11929614—42no—
30C2×#11929614—42no—
31C2×#11929614—42no—
32C2×#11929614—42no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8D411yescharacteristic
36C22×C79⋊C8D411yescharacteristic
37#14579161—21no—
38#14579161—21no—
39#14579161—21no—
40#14579161C2×C412yes—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)D412yes—
47#14579072—42no—
48C4×C79⋊C8C2×C412yes—
49C4×C79⋊C8D412yes—
50C22×Q8×C79—21no—
51C22×C8×C79—21no—
52C4⋊Q8×C79—21no—
53C4⋊Q8×C79—21no—
54C4⋊Q8×C79—21no—
55C4⋊Q8×C79C2×C412yes—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79D412yes—
59C2×C42×C79—21no—
60C2×C42×C79C2311yescharacteristic
61C4⋊C8×C79—21no—
62C4⋊C8×C79—21no—
63#95×C79—21no—
64C2×Dic158—22no—
65C2×Dic158—41no—
66C2×Dic158—41no—
67C2×Dic158—41no—
68C2×Dic158—41no—
69C2×Dic158—41no—
70C22×Dic79—41no—
71#11929614—48no—
72#11929614—48no—
73C2×C79⋊C8—22no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—21no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C2×C79⋊C8—42no—
80C2×C79⋊C8C22⋊C412yes—
81C2×C79⋊C8C22⋊C412yes—
82C2×C79⋊C8D812yes—
83C2×C79⋊C8SD1612yes—
84C4⋊Dic79—22no—
85C4⋊Dic79—22no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88C2×Q8×C79—22no—
89C2×Q8×C79—41no—
90C2×Q8×C79—41no—
91C2×Q8×C79—41no—
92C2×Q8×C79—41no—
93C2×Q8×C79—41no—
94C22×C4×C79—22no—
95C22×C4×C79—41no—
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79#4111yescharacteristic
99C2×C8×C79—41no—
100C2×C8×C79—42no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—22no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79—41no—
107C42×C79—41no—
108C42×C79—41no—
109C42×C79C22×C411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79C2×D411yescharacteristic
112C42×C79#4111yescharacteristic
113C23×C79#12111yescharacteristic
114Q8×C79—42no—
115Q8×C79—42no—
116Q8×C79—81no—
117Q8×C79—81no—
118C2×C4×C79—22no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—81no—
124C2×C4×C79—81no—
125C2×C4×C79C2×C22⋊C411yescharacteristic
126C2×C4×C79C4×D411yescharacteristic
127C2×C4×C79C4×D411yescharacteristic
128C2×C4×C79#12111yescharacteristic
129C2×C4×C79#12411yescharacteristic
130C2×C4×C79#12711yescharacteristic
131C2×C4×C79C2×D811yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C79#13712yes—
136C2×C4×C79#13712yes—
137C2×Dic79—42no—
138C2×Dic79—81no—
139C2×Dic79—81no—
140Dic158—42no—
141Dic158—42no—
142Dic158—81no—
143Dic158—81no—
144C8×C79—81no—
145C79⋊C8—22no—
146C79⋊C8—42no—
147C79⋊C8—81no—
148C79⋊C8#10214yes—
149C22×C79#39011yescharacteristic
150C22×C79#45911yescharacteristic
151C22×C79#46011yescharacteristic
152C22×C79#47012yes—
153C22×C79#47312yes—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—82no—
157C4×C79—82no—
158C4×C79C2×#10211yescharacteristic
159C4×C79#41811yescharacteristic
160C4×C79#44112yes—
161C4×C79#45911yescharacteristic
162C4×C79#46011yescharacteristic
163C4×C79#49711yescharacteristic
164C4×C79#49711yescharacteristic
165Dic79—82no—
166#9742—791nomaximal, Sylow
167C2×C79#168912yes—
168C2×C79#169012yes—
169C2×C79#197111yescharacteristic
170C2×C79#197911yescharacteristic
171C2×C79#198211yescharacteristic
172C2×#494—791no—
173#1988—791no—
174#1988D7911yescharacteristic
175#1770—791no—
176C2×#334—791no—
177C2×#332—791no—
178C2×#332—791no—
179C79#974211yescharacteristic, complemented, Sylow
180C22×Q16—1581no—
181C22×Q16—1581no—
182C2×C4⋊Q8—791no—
183C2×C4⋊Q8C2×D7911yescharacteristic
184#494—794no—
185#494—1582no—
186C2×C4⋊C8—791no—
187C2×C4⋊C8—791no—
188C2×C4⋊C8C2×D7911yescharacteristic
189C2×#97—1581no—
190C2×C4×C8—791no—
191C4×C4⋊C4C2×D7911yescharacteristic
192#340—22no—
193#340—1582no—
194#336—1581no—
195#334—1582no—
196#332—1582no—
197#332—1582no—
198C22×Q8—21no—
199C22×Q8—1581no—
200C2×Q16—3162no—
201C2×Q16—3162no—
202C2×Q16—3162no—
203C2×Q16—3162no—
204C2×Q16—1584no—
205C2×Q16—1584no—
206C22×C8—21no—
207C22×C8—791no—
208C22×C8—791no—
209C22×C8—1581no—
210C22×C8—1581no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—792no—
215C4⋊Q8—1581no—
216C4⋊Q8—1581no—
217C4⋊Q8—1581no—
218C4⋊Q8C4×D7912yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4C22⋊D7912yes—
224C2×C42—21no—
225C2×C42C22×D7911yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—792no—
229C4⋊C8—1581no—
230C4⋊C8—1581no—
231C4⋊C8—1581no—
232#97—3162no—
233C4×C8—792no—
234C4×C8—792no—
235#95—21no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—3161no—
243C2×Q8—3161no—
244C2×Q8—3161no—
245C2×Q8—3161no—
246C2×Q8—3161no—
247C2×Q8—1582no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—3161no—
251C22×C4D4×D7911yescharacteristic
252C22×C4C2×C22⋊D7911yescharacteristic
253C22×C4#1192962311yescharacteristic
254Q16—3168no—
255Q16—3168no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—3161no—
259C2×C8—3161no—
260C2×C8—3161no—
261C2×C8—3162no—
262C2×C8—792no—
263C2×C8—792no—
264C2×C8—792no—
265C2×C8—792no—
266C2×C8—1582no—
267C2×C8—1581no—
268C2×C8—1581no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—3161no—
276C4⋊C4—3161no—
277C4⋊C4—1582no—
278C4⋊C4—1582no—
279C42—41no—
280C42—41no—
281C42C2×C4×D7911yescharacteristic
282C42D4×D7911yescharacteristic
283C42D4×D7911yescharacteristic
284C42#1192962411yescharacteristic
285C23#1457913311yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—81no—
289Q8—81no—
290Q8—3162no—
291Q8—3162no—
292Q8—6321no—
293Q8—6321no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—3162no—
302C2×C4—6321no—
303C2×C4—6321no—
304C2×C4C22⋊C4×D7911yescharacteristic
305C2×C4(C22⋊D79)⋊C411yescharacteristic
306C2×C4C4×C22⋊D7911yescharacteristic
307C2×C4#1457913411yescharacteristic
308C2×C4#1457914711yescharacteristic
309C2×C4#1457915711yescharacteristic
310C2×C4D8×D7911yescharacteristic
311C2×C4SD16×D7911yescharacteristic
312C2×C4#1457918511yescharacteristic
313C2×C4#1457918811yescharacteristic
314C2×C4#1457919112yes—
315C2×C4#1457919212yes—
316C8—81no—
317C8—3162no—
318C8—794no—
319C8—6321no—
320C8—1582no—
321C22#1909846311yescharacteristic
322C22#1909875611yescharacteristic
323C22#1909876111yescharacteristic
324C22#1909881912yes—
325C22#1909883612yes—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—82no—
330C4—6322no—
331C4#102×D7911yescharacteristic
332C4#1909856611yescharacteristic
333C4#1909866812yes—
334C4#1909875711yescharacteristic
335C4#1909876211yescharacteristic
336C4#1909895511yescharacteristic
337C4#1909895811yescharacteristic
338C2#2376288712yes—
339C2#2376289212yes—
340C2#2376427011yescharacteristic
341C2#2376431411yescharacteristic
342C2#2376432511yescharacteristic
subgroupG/H
centerC23#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1988C2
socle, prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#1988D79
hypocenterC79#9742
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
12. #19098942C4semidirect
179. C79#9742semidirect

Maximal subgroups

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

subgroupindex
C2×#190989422
#237643522
#237632692
C2×#190983132
C2×#190983092
C2×#190983082
C79×#19882
#974279