Finite Group Databasebeta

← #34317394

Subgroups — #34317394

Order 20224.

766 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771990C211yesmaximal, characteristic
2#23771989C211yesmaximal, characteristic
3C2×#19098599C211yesmaximal, characteristic
4C2×#19098591C211yesmaximal, characteristic
5#23764914C214yesmaximal, complemented
6#23764909C214yesmaximal, complemented
7#23763138C211yesmaximal, characteristic
8C2×#19098327C211yesmaximal, characteristic
9C79×#1742C211yesmaximal, characteristic
10C22×#11929608C2211yescharacteristic
11C23×C79⋊C8C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13C2×#14579120C2211yescharacteristic
14#19098606—22no—
15#19098605—22no—
16#19098605C2214yes—
17#19098603C2214yes—
18#19098601—22no—
19#19098599—22no—
20#19098599—22no—
21#19098599—22no—
22C2×#14579094C2212yes—
23C2×#14579092C2212yes—
24#19098591—22no—
25#19098591C2212yes—
26#19098591C2212yes—
27C2×Dic79⋊C8—22no—
28#19098331C2214yes—
29#19098327C2212yes—
30#19098327C2212yes—
31C22×M16×C79C2211yescharacteristic
32C22×M16×C79C2211yescharacteristic
33C2×#117×C79C2211yescharacteristic
34#343×C79C2214yes—
35C2×#11929608—21no—
36C2×#11929608—22no—
37C2×#11929608Q814yes—
38C2×#11929608C2312yes—
39C22×C79⋊C8—21no—
40C22×C79⋊C8—22no—
41C22×C79⋊C8—22no—
42C22×C79⋊C8D412yes—
43C22×C79⋊C8D412yes—
44C22×C79⋊C8C2311yescharacteristic
45C22×C79⋊C8C2311yescharacteristic
46#14579120—21no—
47#14579120—21no—
48#14579120—21no—
49#14579120—21no—
50#14579120—21no—
51#14579120—21no—
52#14579120D412yes—
53#14579120C2312yes—
54C2×C4⋊Dic79D412yes—
55C2×C4⋊Dic79C2312yes—
56C2×C22⋊Dic79—21no—
57C2×C22⋊Dic79—21no—
58C2×C4×Dic79—21no—
59C2×C4×Dic79—21no—
60#14579095—24no—
61#14579094—24no—
62#14579094—24no—
63#14579093—24no—
64#14579092—24no—
65#14579092—24no—
66Dic79⋊C8—24no—
67Dic79⋊C8—24no—
68C23×C4×C79C2311yescharacteristic
69C2×M16×C79—21no—
70C2×M16×C79—22no—
71C2×M16×C79—21no—
72C2×M16×C79—22no—
73C2×M16×C79—22no—
74C2×M16×C79Q814yes—
75C2×M16×C79C2312yes—
76C2×M16×C79C2312yes—
77C22×C8×C79—21no—
78C22×C8×C79—21no—
79#117×C79—21no—
80#117×C79—21no—
81#117×C79—21no—
82#117×C79—21no—
83C2×C4⋊C4×C79C2312yes—
84C2×C22⋊C4×C79—21no—
85C2×C42×C79—21no—
86C22×Dic79—22no—
87C22×Dic79—22no—
88#11929608—24no—
89#11929608—22no—
90#11929608C2×Q814yes—
91C2×C79⋊C8—24no—
92C2×C79⋊C8—24no—
93C2×C79⋊C8—24no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—21no—
98C2×C79⋊C8—21no—
99C2×C79⋊C8C2×D412yes—
100C2×C79⋊C8C2×D412yes—
101C4⋊Dic79—22no—
102C4⋊Dic79—22no—
103C4⋊Dic79—22no—
104C4⋊Dic79C2×D414yes—
105C22⋊Dic79—22no—
106C22⋊Dic79—22no—
107C22⋊Dic79—41no—
108C22⋊Dic79—41no—
109C4×Dic79—22no—
110C4×Dic79—22no—
111C4×Dic79—41no—
112C4×Dic79—41no—
113C24×C79#4111yescharacteristic
114C22×C4×C79—22no—
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×Q812yes—
120C22×C4×C79C2×Q812yes—
121C22×C4×C79#4111yescharacteristic
122C22×C4×C79#4112yes—
123C22×C4×C79#4112yes—
124C22×C4×C79C2411yescharacteristic
125M16×C79—24no—
126M16×C79—22no—
127M16×C79—22no—
128M16×C79—22no—
129M16×C79—42no—
130M16×C79C2×Q814yes—
131C2×C8×C79—24no—
132C2×C8×C79—22no—
133C2×C8×C79—22no—
134C2×C8×C79—42no—
135C4⋊C4×C79—22no—
136C4⋊C4×C79—22no—
137C22⋊C4×C79—41no—
138C22⋊C4×C79—41no—
139C42×C79—41no—
140C42×C79—41no—
141C23×C79—42no—
142C23×C79#12111yescharacteristic
143C23×C79#12111yescharacteristic
144C23×C79#12512yes—
145C23×C79#12512yes—
146C23×C79C2×#4111yescharacteristic
147C2×C4×C79—42no—
148C2×C4×C79—82no—
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79C22⋊Q812yes—
152C2×C4×C79C22⋊Q812yes—
153C2×C4×C79C22⋊Q812yes—
154C2×C4×C79C22⋊Q812yes—
155C2×C4×C79#12312yes—
156C2×C4×C79#12312yes—
157C2×C4×C79#12512yes—
158C2×C4×C79#12512yes—
159C2×C4×C79C4⋊Q812yes—
160C2×C4×C79C4⋊Q812yes—
161C2×C4×C79C22×D411yescharacteristic
162C2×C4×C79C22×D411yescharacteristic
163C2×C4×C79C22×Q811yescharacteristic
164C2×C4×C79C22×Q811yescharacteristic
165C2×C4×C79C2×#4111yescharacteristic
166C2×C4×C79C2×#4111yescharacteristic
167C2×Dic79—24no—
168C2×Dic79—44no—
169C2×Dic79—42no—
170C2×Dic79—82no—
171C8×C79—24no—
172C8×C79—42no—
173C79⋊C8—24no—
174C79⋊C8—22no—
175C79⋊C8—22no—
176C22×C79—22no—
177C22×C79—22no—
178C22×C79—22no—
179C22×C79—42no—
180C22×C79—42no—
181C22×C79#39812yes—
182C22×C79#39812yes—
183C22×C79C2×#12111yescharacteristic
184C22×C79C2×#12511yescharacteristic
185C22×C79C2×#12511yescharacteristic
186C4×C79—82no—
187C4×C79#39812yes—
188C4×C79#39812yes—
189C4×C79C2×C22⋊Q811yescharacteristic
190C4×C79C2×C22⋊Q811yescharacteristic
191C4×C79C2×#12311yescharacteristic
192C4×C79C2×C4⋊Q811yescharacteristic
193Dic79—44no—
194Dic79—82no—
195#23674—791nomaximal, Sylow
196C2×C79—21no—
197C2×C79—21no—
198C2×C79—22no—
199C2×C79—22no—
200C2×C79C2×#39811yescharacteristic
201C2×C79#306712yes—
202#2911—791no—
203#2911—791no—
204C2×#427—791no—
205C2×#424—791no—
206#2090—794no—
207#2089—794no—
208#1742—791no—
209#1742D7911yescharacteristic
210C2×#341—791no—
211C79#2367411yescharacteristic, complemented, Sylow
212C22×M16—791no—
213C22×M16C2×D7911yescharacteristic
214C22×M16C2×D7911yescharacteristic
215C23×C8—791no—
216C2×#117—791no—
217C2×#117—791no—
218C2×#117C2×D7911yescharacteristic
219#428—794no—
220#428—794no—
221#428—1582no—
222#428—1582no—
223#427—1582no—
224#427—1582no—
225#427—1582no—
226#427—1582no—
227C2×#107—792no—
228C2×#106—792no—
229#424—792no—
230#424—792no—
231#424—1582no—
232C2×C4⋊C8—1582no—
233#343—794no—
234#343C2×D7914yes—
235#341—792no—
236#341—792no—
237C23×C4C22×D7911yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—794no—
244C2×M16—792no—
245C2×M16—1581no—
246C2×M16—1582no—
247C2×M16Dic15814yes—
248C2×M16C22×D7912yes—
249C2×M16C22×D7912yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—792no—
253C22×C8—792no—
254C22×C8—791no—
255C22×C8—791no—
256C22×C8—1581no—
257C22×C8—1582no—
258C22×C8—1582no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—792no—
264#117—792no—
265#117—1581no—
266#117—1581no—
267#117—1581no—
268#117—1581no—
269#117—1581no—
270#117—1581no—
271C2×C4⋊C4—792no—
272C2×C4⋊C4—792no—
273C2×C4⋊C4C22×D7912yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1581no—
276C2×C22⋊C4—1581no—
277C2×C42—21no—
278C2×C42—1581no—
279C2×C42—1581no—
280#107—1584no—
281#107—1584no—
282#107—1584no—
283#106—1584no—
284#106—1584no—
285#106—1584no—
286C4⋊C8—1584no—
287C4⋊C8—1584no—
288C24#1192962411yescharacteristic
289C22×C4—22no—
290C22×C4—1582no—
291C22×C4—1582no—
292C22×C4D4×D7911yescharacteristic
293C22×C4D4×D7911yescharacteristic
294C22×C4D4×D7911yescharacteristic
295C22×C4D4×D7911yescharacteristic
296C22×C4Q8×D7912yes—
297C22×C4C2×Dic15812yes—
298C22×C4#1192962312yes—
299C22×C4#1192962412yes—
300C22×C4#1192962411yescharacteristic
301C22×C4C23×D7911yescharacteristic
302M16—22no—
303M16—24no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—794no—
308M16—1582no—
309M16—1584no—
310M16C2×Dic15814yes—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—792no—
316C2×C8—792no—
317C2×C8—1581no—
318C2×C8—1581no—
319C2×C8—1581no—
320C2×C8—1581no—
321C2×C8—1584no—
322C2×C8—1584no—
323C2×C8—1584no—
324C2×C8—1582no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—794no—
328C4⋊C4—1582no—
329C4⋊C4—1582no—
330C4⋊C4—1582no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—3161no—
334C22⋊C4—3161no—
335C22⋊C4—1582no—
336C22⋊C4—1582no—
337C42—41no—
338C42—41no—
339C42—3161no—
340C42—3161no—
341C42—1582no—
342C42—1582no—
343C23—42no—
344C23#1457913411yescharacteristic
345C23#1457913411yescharacteristic
346C23#1457915112yes—
347C23#1457915312yes—
348C23C2×#1192962411yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—3164no—
352C2×C4—3162no—
353C2×C4—6322no—
354C2×C4—1584no—
355C2×C4#1457913411yescharacteristic
356C2×C4#1457913411yescharacteristic
357C2×C4#1457913812yes—
358C2×C4#1457913812yes—
359C2×C4#1457913912yes—
360C2×C4#1457913912yes—
361C2×C4#1457914312yes—
362C2×C4#1457914312yes—
363C2×C4#1457915112yes—
364C2×C4#1457915312yes—
365C2×C4#1457916012yes—
366C2×C4#1457916012yes—
367C2×C4C2×D4×D7911yescharacteristic
368C2×C4C2×D4×D7911yescharacteristic
369C2×C4C2×Q8×D7911yescharacteristic
370C2×C4C22×Dic15811yescharacteristic
371C2×C4C2×#1192962311yescharacteristic
372C2×C4C2×#1192962411yescharacteristic
373C8—24no—
374C8—42no—
375C8—1584no—
376C8—1582no—
377C8—1582no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1909849712yes—
384C22#1909849712yes—
385C22C2×#1457913411yescharacteristic
386C22C2×#1457915111yescharacteristic
387C22C2×#1457915311yescharacteristic
388C4—82no—
389C4—3164no—
390C4—6322no—
391C4#1909849712yes—
392C4#1909849712yes—
393C4C2×#1457913811yescharacteristic
394C4C2×#1457913911yescharacteristic
395C4C2×#1457914311yescharacteristic
396C4C2×#1457916011yescharacteristic
397C2—22no—
398C2—22no—
399C2—21no—
400C2—21no—
401C2C2×#1909849711yescharacteristic
402C2#2377311912yes—
subgroupG/H
centerC2×C4#14579134
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1742C2
socleC22×C79C2×#121
prime radicalC24×C79#41
hypercenter#1742D79
hypocenterC79#23674
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #23764914C2semidirect
6. #23764909C2semidirect
211. C79#23674semidirect

Maximal subgroups

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

subgroupindex
#237719902
#237719892
C2×#190985992
C2×#190985912
#237649142
#237649092
#237631382
C2×#190983272
C79×#17422
#2367479