Finite Group Databasebeta

← #34291585

Subgroups — #34291585

Order 20224.

1002 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098591C211yesmaximal, characteristic
2C2×#19098590C211yesmaximal, characteristic
3C2×#19098586C211yesmaximal, characteristic
4C2×#19098518C211yesmaximal, characteristic
5#23765882C211yesmaximal, characteristic
6#23764122C214yesmaximal, complemented
7#23764117C214yesmaximal, complemented
8C2×#19098317C211yesmaximal, characteristic
9C79×#1729C211yesmaximal, characteristic
10C22×#11929608C2211yescharacteristic
11C23×C79⋊C8C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13C22×C4×Dic79C2211yescharacteristic
14#19098591—24no—
15#19098590—24no—
16#19098586—24no—
17C2×C4⋊(C79⋊C8)C2214yes—
18C2×Dic79⋊C8C2214yes—
19C2×Dic79⋊C8C2214yes—
20#19098518—24no—
21C2×#14579071C2212yes—
22C2×C8×Dic79C2212yes—
23#19098459C412yes—
24#19098459C412yes—
25#19098435C414yes—
26C4×Dic79⋊C4C2214yes—
27#19098317C2212yes—
28#19098317C2212yes—
29C22×M16×C79C2211yescharacteristic
30C23×C8×C79C2211yescharacteristic
31C2×#117×C79C2211yescharacteristic
32#336×C79C2212yes—
33#336×C79C2212yes—
34C23×Dic79C2×C412yes—
35C2×#11929608—22no—
36C2×#11929608—22no—
37C2×#11929608D414yes—
38C22×C79⋊C8—22no—
39C22×C79⋊C8—22no—
40C22×C79⋊C8—22no—
41C22×C79⋊C8C2312yes—
42C22×C79⋊C8C2312yes—
43#14579120D414yes—
44#14579120Q814yes—
45C2×C4⋊Dic79C2×C412yes—
46C2×Dic79⋊C4C2×C414yes—
47C2×Dic79⋊C4C2×C414yes—
48C2×C22⋊Dic79C2×C412yes—
49C2×C4×Dic79—22no—
50C2×C4×Dic79—22no—
51C2×C4×Dic79C2×C411yescharacteristic
52C2×C4×Dic79C2×C411yescharacteristic
53C2×C4×Dic79C2312yes—
54C2×C4×Dic79C2312yes—
55C4⋊(C79⋊C8)—28no—
56Dic79⋊C8—28no—
57Dic79⋊C8—28no—
58#14579071—24no—
59C8×Dic79—24no—
60C79⋊#95C2×C414yes—
61C79⋊#95C2×C414yes—
62C23×C4×C79C2311yescharacteristic
63C2×M16×C79—22no—
64C2×M16×C79—22no—
65C2×M16×C79—22no—
66C22×C8×C79—22no—
67C22×C8×C79—22no—
68C22×C8×C79Q814yes—
69C22×C8×C79C2312yes—
70C22×C8×C79C2311yescharacteristic
71C22×C8×C79C2311yescharacteristic
72#117×C79—22no—
73#117×C79—22no—
74C2×C4⋊C4×C79C2×C412yes—
75C2×C22⋊C4×C79C2×C412yes—
76C2×C42×C79C2312yes—
77C22×Dic79—24no—
78C22×Dic79—24no—
79C22×Dic79—42no—
80C22×Dic79C22×C412yes—
81C22×Dic79C22×C414yes—
82C22×Dic79C22×C412yes—
83#11929608—28no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—44no—
90C2×C79⋊C8C2×D414yes—
91C4⋊Dic79C4⋊C418yes—
92Dic79⋊C4—28no—
93Dic79⋊C4—44no—
94C22⋊Dic79C4⋊C418yes—
95C4×Dic79—22no—
96C4×Dic79—22no—
97C4×Dic79—22no—
98C4×Dic79—22no—
99C4×Dic79C2×D414yes—
100C4×Dic79C2×Q814yes—
101C24×C79C22×C411yescharacteristic
102C22×C4×C79C22×C414yes—
103C22×C4×C79C22×C411yescharacteristic
104C22×C4×C79C2×D412yes—
105C22×C4×C79C2×D412yes—
106C22×C4×C79C2×Q812yes—
107C22×C4×C79C2×Q812yes—
108C22×C4×C79#4112yes—
109C22×C4×C79#4112yes—
110C22×C4×C79C2411yescharacteristic
111M16×C79—44no—
112C2×C8×C79—28no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C2×C8×C79—22no—
116C2×C8×C79—22no—
117C2×C8×C79—21no—
118C2×C8×C79—21no—
119C2×C8×C79—22no—
120C2×C8×C79C2×Q812yes—
121C2×C8×C79C2×Q812yes—
122C4⋊C4×C79—24no—
123C22⋊C4×C79—24no—
124C42×C79—22no—
125C42×C79—22no—
126C23×C79—22no—
127C23×C79—22no—
128C23×C79C2×C4⋊C412yes—
129C23×C79C4×D412yes—
130C23×C79C4×Q812yes—
131C23×C79C23×C411yescharacteristic
132C2×C4×C79—24no—
133C2×C4×C79—24no—
134C2×C4×C79—24no—
135C2×C4×C79C2×C4⋊C412yes—
136C2×C4×C79C4×D412yes—
137C2×C4×C79C4×Q812yes—
138C2×C4×C79#12114yes—
139C2×C4×C79C22⋊Q814yes—
140C2×C4×C79#12514yes—
141C2×C4×C79C4⋊Q814yes—
142C2×C4×C79C22×D411yescharacteristic
143C2×C4×C79C22×D411yescharacteristic
144C2×C4×C79C22×Q811yescharacteristic
145C2×C4×C79C22×Q811yescharacteristic
146C2×C4×C79C2×#4111yescharacteristic
147C2×C4×C79C2×#4111yescharacteristic
148C2×Dic79—22no—
149C2×Dic79—24no—
150C2×Dic79—24no—
151C2×Dic79—22no—
152C2×Dic79—22no—
153C2×Dic79—22no—
154C2×Dic79—22no—
155C2×Dic79—48no—
156C2×Dic79C2×C4⋊C418yes—
157C2×Dic79#13114yes—
158C8×C79—22no—
159C8×C79—22no—
160C8×C79—42no—
161C79⋊C8—24no—
162C79⋊C8—42no—
163C22×C79—24no—
164C22×C79—24no—
165C22×C79—22no—
166C22×C79—22no—
167C22×C79#38914yes—
168C22×C79C22×C4⋊C411yescharacteristic
169C22×C79C2×C4×D411yescharacteristic
170C22×C79C2×C4×Q811yescharacteristic
171C22×C79C2×#13112yes—
172C22×C79#56812yes—
173C4×C79—44no—
174C4×C79#38914yes—
175C4×C79C2×#12111yescharacteristic
176C4×C79C2×C22⋊Q811yescharacteristic
177C4×C79C2×#12511yescharacteristic
178C4×C79C2×C4⋊Q811yescharacteristic
179Dic79—28no—
180Dic79—24no—
181Dic79—42no—
182#21148—791nomaximal, Sylow
183C2×C79—24no—
184C2×C79C2×#38911yescharacteristic
185C2×C79#290612yes—
186C2×C79#293312yes—
187C2×C79#296011yescharacteristic
188C2×C79#296111yescharacteristic
189C2×#424—791no—
190C2×#424—791no—
191C2×#423—791no—
192C2×#405—791no—
193#2296—791no—
194#1940—794no—
195#1939—794no—
196#1729D7911yescharacteristic
197C2×#336—791no—
198C79#2114811yescharacteristic, complemented, Sylow
199C22×M16—791no—
200C22×M16C2×D7911yescharacteristic
201C23×C8—791no—
202C23×C8C2×D7911yescharacteristic
203C2×#117—791no—
204C2×#117C2×D7911yescharacteristic
205C22×C42—791no—
206#424—1584no—
207#424—1584no—
208#423—1584no—
209C2×C4⋊C8—794no—
210C2×C4⋊C8—794no—
211C2×C4⋊C8—794no—
212#405—1584no—
213C2×#97—792no—
214C2×C4×C8—792no—
215#389—792no—
216#389—792no—
217#385—794no—
218C4×C4⋊C4—794no—
219#336—792no—
220#336—792no—
221#336C2×D7912yes—
222#336C2×D7912yes—
223C23×C4—792no—
224C23×C4C22×D7911yescharacteristic
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—794no—
229C2×M16—1582no—
230C2×M16—1582no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—792no—
234C22×C8—792no—
235C22×C8—1582no—
236C22×C8—1582no—
237C22×C8—1582no—
238C22×C8Dic15814yes—
239C22×C8C22×D7912yes—
240C22×C8C22×D7911yescharacteristic
241C22×C8C22×D7911yescharacteristic
242#117—22no—
243#117—22no—
244#117—794no—
245#117—794no—
246C2×C4⋊C4—794no—
247C2×C4⋊C4—792no—
248C2×C4⋊C4—794no—
249C2×C4⋊C4C4×D7912yes—
250C2×C22⋊C4—792no—
251C2×C22⋊C4C4×D7912yes—
252C2×C42—792no—
253C2×C42—791no—
254C2×C42—791no—
255C2×C42—792no—
256C2×C42—1582no—
257C2×C42—1582no—
258C2×C42C22×D7912yes—
259C4⋊C8—1588no—
260C4⋊C8—1588no—
261C4⋊C8—1588no—
262#97—1584no—
263C4×C8—1584no—
264#95—794no—
265#95—794no—
266C24C2×C4×D7911yescharacteristic
267C22×C4—3162no—
268C22×C4—792no—
269C22×C4—794no—
270C22×C4—792no—
271C22×C4—1584no—
272C22×C4—1584no—
273C22×C4C2×C4×D7914yes—
274C22×C4C2×C4×D7911yescharacteristic
275C22×C4D4×D7912yes—
276C22×C4D4×D7912yes—
277C22×C4Q8×D7912yes—
278C22×C4C2×Dic15812yes—
279C22×C4#1192962312yes—
280C22×C4#1192962412yes—
281C22×C4C23×D7911yescharacteristic
282M16—44no—
283M16—1588no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—28no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—3164no—
293C2×C8—794no—
294C2×C8—1582no—
295C2×C8—1582no—
296C2×C8—1582no—
297C2×C8—1582no—
298C2×C8—1582no—
299C2×C8C2×Dic15812yes—
300C2×C8C2×Dic15812yes—
301C4⋊C4—24no—
302C4⋊C4—3164no—
303C4⋊C4—798no—
304C4⋊C4—1588no—
305C22⋊C4—24no—
306C22⋊C4—798no—
307C42—22no—
308C42—22no—
309C42—794no—
310C42—794no—
311C42—1582no—
312C42—1582no—
313C42—1582no—
314C42—1582no—
315C23—22no—
316C23—22no—
317C23C4⋊C4×D7912yes—
318C23(C22⋊D79)⋊C412yes—
319C23C4×Dic15812yes—
320C23C22×C4×D7911yescharacteristic
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—3168no—
325C2×C4—798no—
326C2×C4—794no—
327C2×C4—1582no—
328C2×C4—1582no—
329C2×C4—1582no—
330C2×C4—1582no—
331C2×C4—1584no—
332C2×C4—1584no—
333C2×C4—1582no—
334C2×C4C4⋊C4×D7912yes—
335C2×C4(C22⋊D79)⋊C412yes—
336C2×C4C4×Dic15812yes—
337C2×C4#1457913414yes—
338C2×C4#1457913814yes—
339C2×C4#1457915114yes—
340C2×C4#1457916014yes—
341C2×C4C2×D4×D7911yescharacteristic
342C2×C4C2×D4×D7911yescharacteristic
343C2×C4C2×Q8×D7911yescharacteristic
344C2×C4C22×Dic15811yescharacteristic
345C2×C4C2×#1192962311yescharacteristic
346C2×C4C2×#1192962411yescharacteristic
347C8—22no—
348C8—22no—
349C8—42no—
350C8—3162no—
351C8—1584no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22#1909846014yes—
357C22C2×C4⋊C4×D7911yescharacteristic
358C22C2×(C22⋊D79)⋊C411yescharacteristic
359C22C2×C4×Dic15811yescharacteristic
360C22#131×D7912yes—
361C22#1909932812yes—
362C4—44no—
363C4—3162no—
364C4—1588no—
365C4—1584no—
366C4#1909846014yes—
367C4C2×#1457913411yescharacteristic
368C4C2×#1457913811yescharacteristic
369C4C2×#1457915111yescharacteristic
370C4C2×#1457916011yescharacteristic
371C2—24no—
372C2C2×#1909846011yescharacteristic
373C2#2377196712yes—
374C2#2377213012yes—
375C2#2377233211yescharacteristic
376C2#2377233811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D79
commutator, socleC23×C79C23×C4
FittingC79×#1729C2
prime radicalC24×C79C22×C4
cosocleC22×C4×C79C24
hypercenter#1729D79
hypocenterC79#21148
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
6. #23764122C2semidirect
7. #23764117C2semidirect
198. C79#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#190985912
C2×#190985902
C2×#190985862
C2×#190985182
#237658822
#237641222
#237641172
C2×#190983172
C79×#17292
#2114879