Finite Group Databasebeta

← #34182601

Subgroups — #34182601

Order 20224.

376 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772351C211yesmaximal, characteristic
2#23764173C211yesmaximal, characteristic
3#23761828C211yesmaximal, characteristic
4C2×#19098295C211yesmaximal, characteristic
5#23761295C211yesmaximal, characteristic
6#23761237C211yesmaximal, characteristic
7C79×#1567C211yesmaximal, characteristic
8C2×C4×Dic158C2211yescharacteristic
9#19098692—22no—
10#19098690—21no—
11#19098687—21no—
12#19098616C2211yescharacteristic
13C2×#14579096—21no—
14#19098586—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C4×#11929608C2211yescharacteristic
17M16×Dic79—21no—
18#19098327—21no—
19#19098295C411yescharacteristic
20#19098295C411yescharacteristic
21#19098295C411yescharacteristic
22#19098295C411yescharacteristic
23#19098293—21no—
24#19098293—21no—
25#423×C79C2211yescharacteristic
26#423×C79C2211yescharacteristic
27C2×C4×C8×C79C2211yescharacteristic
28#335×C79—22no—
29C22×Dic158Q811yescharacteristic
30C2×#11929608D411yescharacteristic
31C2×#11929608D411yescharacteristic
32C22×C79⋊C8—21no—
33C4×Dic158—21no—
34C4×Dic158—21no—
35C4×Dic158C2×C411yescharacteristic
36C4×Dic158C2×C411yescharacteristic
37C4×Dic158D411yescharacteristic
38C4×Dic158D411yescharacteristic
39C2×C4⋊Dic79Q811yescharacteristic
40C2×Dic79⋊C4—21no—
41C2×C4×Dic79—21no—
42#14579096—42no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)C2×C411yescharacteristic
46C4⋊(C79⋊C8)C2×C411yescharacteristic
47Dic79⋊C8—22no—
48Dic79⋊C8—22no—
49C22⋊(C79⋊C8)—21no—
50#14579072—21no—
51#14579071—22no—
52C4×C79⋊C8—21no—
53C4×C79⋊C8—21no—
54C8×Dic79—22no—
55C2×M16×C79—21no—
56C2×M16×C79—21no—
57C22×C8×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C4⋊C8×C79—21no—
62C4⋊C8×C79D412yes—
63C4×C8×C79—21no—
64C4×C8×C79C2×C411yescharacteristic
65C4×C8×C79C2×C411yescharacteristic
66C2×Dic158—22no—
67C2×Dic158—21no—
68C2×Dic158—41no—
69C2×Dic158C4⋊C411yescharacteristic
70C2×Dic158C4⋊C411yescharacteristic
71C22×Dic79—21no—
72C22×Dic79—21no—
73#11929608—21no—
74#11929608—21no—
75#11929608—21no—
76#11929608Q1612yes—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C4⋊Dic79—21no—
85C4⋊Dic79C4⋊C411yescharacteristic
86C4⋊Dic79C4⋊C411yescharacteristic
87Dic79⋊C4—21no—
88Dic79⋊C4—21no—
89Dic79⋊C4—21no—
90Dic79⋊C4—21no—
91C4×Dic79—21no—
92C4×Dic79—21no—
93C4×Dic79—21no—
94C4×Dic79—21no—
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×Q811yescharacteristic
97C22×C4×C79#4111yescharacteristic
98M16×C79—41no—
99M16×C79—41no—
100M16×C79—41no—
101M16×C79—41no—
102C2×C8×C79—21no—
103C2×C8×C79—21no—
104C2×C8×C79—22no—
105C2×C8×C79—42no—
106C2×C8×C79—41no—
107C42×C79C22×C411yescharacteristic
108C42×C79C2×D411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79#4111yescharacteristic
111C23×C79C22⋊Q811yescharacteristic
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—41no—
115C2×C4×C79C2×C4⋊C411yescharacteristic
116C2×C4×C79C4×D411yescharacteristic
117C2×C4×C79C4×D411yescharacteristic
118C2×C4×C79#12011yescharacteristic
119C2×C4×C79C22⋊Q811yescharacteristic
120C2×C4×C79#12311yescharacteristic
121C2×C4×C79C2×Q1611yescharacteristic
122C2×C4×C79#13511yescharacteristic
123C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C79#13711yescharacteristic
125C2×Dic79—21no—
126C2×Dic79—21no—
127C2×Dic79—21no—
128C2×Dic79—21no—
129C2×Dic79—42no—
130C2×Dic79—81no—
131Dic158—21no—
132Dic158—21no—
133Dic158—21no—
134Dic158—42no—
135Dic158#10812yescomplemented
136C8×C79—22no—
137C8×C79—41no—
138C8×C79—42no—
139C8×C79—81no—
140C79⋊C8—22no—
141C79⋊C8—41no—
142C79⋊C8—41no—
143C79⋊C8—41no—
144C79⋊C8—81no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79#38511yescharacteristic
148C22×C79#47711yescharacteristic
149C22×C79#47911yescharacteristic
150C4×C79—21no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79C2×#10811yescharacteristic
154C4×C79#43011yescharacteristic
155C4×C79C4×Q1611yescharacteristic
156C4×C79#44011yescharacteristic
157C4×C79#44911yescharacteristic
158C4×C79#45111yescharacteristic
159Dic79—22no—
160Dic79—41no—
161Dic79—41no—
162Dic79—41no—
163#8907—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79#185511yescharacteristic
167C2×C79#186111yescharacteristic
168C2×C79#186311yescharacteristic
169Q8×M16—791no—
170#1950—791no—
171#1567D7911yescharacteristic
172#1487—791no—
173C2×Q8⋊C8—791no—
174#1301—791no—
175#1278—791no—
176C79#890711yescharacteristic, complemented, Sylow
177C2×C4×Q8—791no—
178#446—1581no—
179#446—1582no—
180C8×Q8—1581no—
181#431—791no—
182C2×#108—1581no—
183#423—1581no—
184#423C2×D7911yescharacteristic
185#423C2×D7911yescharacteristic
186C2×C4⋊C8—791no—
187C4×M16—791no—
188C4×M16—1581no—
189C2×C4×C8C2×D7911yescharacteristic
190#341—1581no—
191#335—22no—
192Q8⋊C8—791no—
193Q8⋊C8—791no—
194Q8⋊C8—791no—
195Q8⋊C8—791no—
196Q8⋊C8—1581no—
197Q8⋊C8—1581no—
198C22×Q8—791no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—791no—
202C2×M16—791no—
203C22×C8—21no—
204C22×C8—1581no—
205C4×Q8—791no—
206C4×Q8—791no—
207C4×Q8—791no—
208C4×Q8—791no—
209C4×Q8—1581no—
210C4×Q8—1581no—
211C2×C4⋊C4—791no—
212C2×C4⋊C4—1581no—
213C2×C42—1581no—
214C2×C42C22×D7911yescharacteristic
215#108—3162no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—791no—
220C4⋊C8—791no—
221C4⋊C8—1582no—
222C4⋊C8—1582no—
223C4⋊C8—1581no—
224C4⋊C8—1581no—
225C4⋊C8C22⋊D7912yes—
226C22⋊C8—1581no—
227#97—1582no—
228#97—1581no—
229C4×C8—21no—
230C4×C8—1582no—
231C4×C8—1581no—
232C4×C8—1581no—
233C4×C8C4×D7911yescharacteristic
234C4×C8C4×D7911yescharacteristic
235C2×Q8—3161no—
236C2×Q8—791no—
237C2×Q8—791no—
238C2×Q8—1582no—
239C2×Q8—1581no—
240C22×C4—1581no—
241C22×C4—1581no—
242C22×C4D4×D7911yescharacteristic
243C22×C4Q8×D7911yescharacteristic
244C22×C4#1192962411yescharacteristic
245M16—41no—
246M16—41no—
247M16—41no—
248M16—41no—
249M16—792no—
250M16—1581no—
251M16—1581no—
252M16—1581no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—3161no—
259C2×C8—3161no—
260C2×C8—3161no—
261C2×C8—1581no—
262C2×C8—1581no—
263C2×C8—1581no—
264C2×C8—1581no—
265C4⋊C4—791no—
266C4⋊C4—791no—
267C4⋊C4—1581no—
268C4⋊C4—1581no—
269C4⋊C4—1581no—
270C4⋊C4—1581no—
271C4⋊C4—1581no—
272C42—1581no—
273C42—1581no—
274C42—1581no—
275C42—1581no—
276C42C2×C4×D7911yescharacteristic
277C42D4×D7911yescharacteristic
278C42C2×C22⋊D7911yescharacteristic
279C42#1192962511yescharacteristic
280C23#1457913911yescharacteristic
281Q8—3162no—
282Q8—792no—
283Q8—1581no—
284Q8—1581no—
285Q8—1581no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—3162no—
290C2×C4—6321no—
291C2×C4—1581no—
292C2×C4—1581no—
293C2×C4—1581no—
294C2×C4—1581no—
295C2×C4C4⋊C4×D7911yescharacteristic
296C2×C4C4×C22⋊D7911yescharacteristic
297C2×C4#1457912311yescharacteristic
298C2×C4#1457912911yescharacteristic
299C2×C4#1457913711yescharacteristic
300C2×C4#1457914611yescharacteristic
301C2×C4Q16×D7911yescharacteristic
302C2×C4#1457918011yescharacteristic
303C2×C4#1457918511yescharacteristic
304C2×C4#1457919311yescharacteristic
305C8—22no—
306C8—41no—
307C8—42no—
308C8—81no—
309C8—3161no—
310C8—3161no—
311C8—3161no—
312C8—6321no—
313C8—1582no—
314C22—21no—
315C22—21no—
316C22#1909843411yescharacteristic
317C22#1909885711yescharacteristic
318C22#1909886911yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—3161no—
323C4—3161no—
324C4—3161no—
325C4—1582no—
326C4#108×D7911yescharacteristic
327C4#1909861011yescharacteristic
328C4#1909865711yescharacteristic
329C4#1909866211yescharacteristic
330C4#1909870511yescharacteristic
331C4#1909871711yescharacteristic
332C2—21no—
333C2—21no—
334C2#2376360311yescharacteristic
335C2#2376363911yescharacteristic
336C2#2376365111yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1567C2
socleC22×C79#385
prime radicalC23×C79C22⋊Q8
cosocleC2×C42×C79C23
hypercenter#1567D79
hypocenterC79#8907
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
135. Dic158#108semidirect
176. C79#8907semidirect

Maximal subgroups

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

subgroupindex
#237723512
#237641732
#237618282
C2×#190982952
#237612952
#237612372
C79×#15672
#890779