Finite Group Databasebeta

← #34262184

Subgroups — #34262184

Order 20224.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099194C211yesmaximal, characteristic, complemented
2#23774157C212yesmaximal
3#23762407C214yesmaximal, complemented
4#23762339C214yesmaximal, complemented
5#23761971C212yesmaximal, complemented
6C2×#19098289C211yesmaximal, characteristic, complemented
7C79×#2618C211yesmaximal, characteristic
8#19099194—24no—
9#19099194—22no—
10#19099194C2214yes—
11C2×#14579160—22no—
12C2×#14579152—22no—
13C2×#14579140—21no—
14C2×#14579140C2212yes—
15C2×#14579120—21no—
16#19098893C2214yes—
17#19098879—22no—
18#19098790—24no—
19#19098714C2214yes—
20#19098701—22no—
21C2×C22⋊(C79⋊C8)C2212yes—
22#19098299C2218yes—
23#19098289C2214yes—
24C2×#123×C79—21no—
25C2×#121×C79C2211yescharacteristic
26C22×C4⋊C4×C79C2211yescharacteristic
27#398×C79C2212yes—
28#396×C79—22no—
29#394×C79C2212yes—
30#386×C79—21no—
31#380×C79C2211yescharacteristic
32C22×Dic158—21no—
33C22×Dic158D412yes—
34C2×#11929612—24no—
35C22×C79⋊C8—22no—
36#14579160—28no—
37#14579160—28no—
38#14579152—28no—
39#14579152—28no—
40#14579140—24no—
41#14579140—22no—
42#14579140—22no—
43#14579140—24no—
44#14579140D414yes—
45#14579140C2314yes—
46#14579120—24no—
47#14579120—24no—
48C2×C4⋊Dic79—21no—
49C2×C4⋊Dic79—21no—
50C2×C4⋊Dic79—21no—
51C2×C4⋊Dic79D412yes—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×C22⋊Dic79—22no—
55C2×C4×Dic79—22no—
56#14579092—24no—
57#14579084—24no—
58#14579082—24no—
59C22⋊(C79⋊C8)—22no—
60C22⋊(C79⋊C8)C2314yes—
61C22×D4×C79—21no—
62C22×D4×C79C2311yescharacteristic
63C23×C4×C79D411yescharacteristic
64C23×C4×C79D411yescharacteristic
65C23×C4×C79C2311yescharacteristic
66#123×C79—44no—
67#121×C79—22no—
68#121×C79—22no—
69C2×C4⋊C4×C79—22no—
70C2×C4⋊C4×C79—22no—
71C2×C4⋊C4×C79—22no—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79D412yes—
74C2×C4⋊C4×C79C2311yescharacteristic
75C2×C4⋊C4×C79C2312yes—
76C2×C22⋊C4×C79—21no—
77C2×C22⋊C4×C79—21no—
78C2×C22⋊C4×C79—22no—
79C2×C22⋊C4×C79—22no—
80#95×C79—22no—
81#95×C79C2312yes—
82C2×Dic158—24no—
83C2×Dic158—22no—
84C2×Dic158—82no—
85C2×Dic158—82no—
86C2×Dic158C2×D414yes—
87C22×Dic79—22no—
88C22×Dic79—22no—
89#11929612—84no—
90C2×C79⋊C8—24no—
91C2×C79⋊C8—82no—
92C4⋊Dic79—24no—
93C4⋊Dic79—24no—
94C4⋊Dic79—24no—
95C4⋊Dic79—22no—
96C4⋊Dic79C2×D414yes—
97Dic79⋊C4—28no—
98Dic79⋊C4—24no—
99Dic79⋊C4—24no—
100C22⋊Dic79—24no—
101C22⋊Dic79—24no—
102C4×Dic79—28no—
103C24×C79—21no—
104C24×C79C2×D411yescharacteristic
105C2×D4×C79—22no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—42no—
109C2×D4×C79—42no—
110C2×D4×C79—81no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—22no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—22no—
118C22×C4×C79—81no—
119C22×C4×C79—81no—
120C22×C4×C79SD1614yescomplemented
121C22×C4×C79C2×D412yes—
122C22×C4×C79C2×D412yes—
123C22×C4×C79C2×D411yescharacteristic
124C22×C4×C79C2×D412yes—
125C22×C4×C79C2×D412yes—
126C22×C4×C79C2411yescharacteristic
127C4⋊C4×C79—24no—
128C4⋊C4×C79—24no—
129C4⋊C4×C79—22no—
130C4⋊C4×C79—44no—
131C4⋊C4×C79—81no—
132C22⋊C4×C79—42no—
133C22⋊C4×C79—42no—
134C22⋊C4×C79—44no—
135C22⋊C4×C79—82no—
136C23×C79—41no—
137C23×C79—41no—
138C23×C79—42no—
139C23×C79—42no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—41no—
143C23×C79#12011yescharacteristic
144C23×C79#12011yescharacteristic
145C23×C79C2×SD1612yes—
146C23×C79#13712yes—
147C23×C79C22×D411yescharacteristic
148D4×C79—44no—
149D4×C79—82no—
150D4×C79—82no—
151C2×C4×C79—22no—
152C2×C4×C79—22no—
153C2×C4×C79—24no—
154C2×C4×C79—24no—
155C2×C4×C79—22no—
156C2×C4×C79—42no—
157C2×C4×C79—42no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—42no—
163C2×C4×C79—42no—
164C2×C4×C79—41no—
165C2×C4×C79—42no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—81no—
169C2×C4×C79#12012yes—
170C2×C4×C79C2×SD1614yes—
171C2×C4×C79C22×D412yes—
172C2×Dic79—24no—
173C2×Dic79—24no—
174C2×Dic79—44no—
175C2×Dic79—44no—
176Dic158—44no—
177Dic158—82no—
178Dic158—84no—
179C79⋊C8—44no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—42no—
187C22×C79—42no—
188C22×C79—82no—
189C22×C79—82no—
190C22×C79—81no—
191C22×C79—81no—
192C22×C79—81no—
193C22×C79#44812yes—
194C22×C79#44812yes—
195C22×C79C2×#12011yescharacteristic
196C22×C79C22×SD1611yescharacteristic
197C22×C79C2×#13711yescharacteristic
198C4×C79—24no—
199C4×C79—22no—
200C4×C79—22no—
201C4×C79—42no—
202C4×C79—82no—
203C4×C79—81no—
204Dic79—44no—
205Dic79—44no—
206#18294—791nomaximal, Sylow
207C2×C79—21no—
208C2×C79—21no—
209C2×C79—22no—
210C2×C79—22no—
211C2×C79—161no—
212C2×C79C2×#44811yescharacteristic
213C2×C79#302011yescharacteristic
214C2×C79#302611yescharacteristic
215C2×#544—791no—
216#3187—792no—
217#2618D7911yescharacteristic
218#1611—794no—
219#1599—794no—
220#1520—792no—
221C2×#324—791no—
222C79#1829411yescharacteristic, complemented, Sylow
223#544—794no—
224#544—1584no—
225#544—1582no—
226C2×C4⋊Q8—1582no—
227C2×#125—1582no—
228C2×#123—21no—
229C2×C22⋊Q8—792no—
230C2×C22⋊Q8—1581no—
231C2×#121C2×D7911yescharacteristic
232C2×#117—1581no—
233C22×C4⋊C4C2×D7911yescharacteristic
234#483—794no—
235#481—1582no—
236#465—1584no—
237#450—794no—
238#448—1582no—
239C2×C22⋊C8—792no—
240#398C2×D7912yes—
241#396—22no—
242#394C2×D7912yes—
243#386—21no—
244#380C2×D7911yescharacteristic
245#328—798no—
246#324—794no—
247C22×Q8—792no—
248C22×Q8—1581no—
249C22×D4—21no—
250C22×D4C22×D7911yescharacteristic
251C23×C4C22⋊D7911yescharacteristic
252C23×C4C22⋊D7911yescharacteristic
253C23×C4C22×D7911yescharacteristic
254C2×SD16—1584no—
255C22×C8—1582no—
256C4⋊Q8—1588no—
257C4⋊Q8—1588no—
258#125—1588no—
259#125—1588no—
260#123—44no—
261C22⋊Q8—794no—
262C22⋊Q8—794no—
263C22⋊Q8—1584no—
264C22⋊Q8—1582no—
265C22⋊Q8—1582no—
266C22⋊Q8—1584no—
267#121—22no—
268#121—22no—
269#117—1584no—
270#117—1584no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—792no—
276C2×C4⋊C4—1581no—
277C2×C4⋊C4—1581no—
278C2×C4⋊C4—1581no—
279C2×C4⋊C4—1582no—
280C2×C4⋊C4—1582no—
281C2×C4⋊C4C22⋊D7912yes—
282C2×C4⋊C4C22×D7911yescharacteristic
283C2×C4⋊C4C22×D7912yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—1582no—
289C2×C42—1582no—
290#106—1584no—
291#103—1584no—
292#102—1584no—
293C22⋊C8—794no—
294C22⋊C8—1582no—
295#95—22no—
296#95C22×D7912yes—
297C24—21no—
298C24C2×C22⋊D7911yescharacteristic
299C2×Q8—794no—
300C2×Q8—6322no—
301C2×Q8—6322no—
302C2×Q8—1584no—
303C2×Q8—1582no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—1582no—
320C22×C4—1582no—
321C22×C4#1192961214yescomplemented
322C22×C4D4×D7912yes—
323C22×C4D4×D7912yes—
324C22×C4C2×C22⋊D7912yes—
325C22×C4C2×C22⋊D7912yes—
326C22×C4C2×C22⋊D7911yescharacteristic
327C22×C4C23×D7911yescharacteristic
328SD16—6324no—
329C2×C8—6322no—
330C2×C8—1584no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—81no—
336C4⋊C4—794no—
337C4⋊C4—1588no—
338C4⋊C4—1584no—
339C4⋊C4—1584no—
340C4⋊C4—1584no—
341C4⋊C4—1584no—
342C4⋊C4—1582no—
343C4⋊C4—1584no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—1584no—
349C22⋊C4—1584no—
350C42—1588no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#1457912911yescharacteristic
359C23#1457912911yescharacteristic
360C23C2×#1192961212yes—
361C23#1457919112yes—
362C23C22×C22⋊D7911yescharacteristic
363Q8—3164no—
364Q8—6322no—
365Q8—6324no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—3164no—
388C2×C4—3164no—
389C2×C4—1584no—
390C2×C4—1584no—
391C2×C4#1457912912yes—
392C2×C4C2×#1192961214yes—
393C2×C4C2×D4×D7912yes—
394C8—3164no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#1909870112yes—
409C22#1909870112yes—
410C22C2×#1457912911yescharacteristic
411C22C22×#1192961211yescharacteristic
412C22C2×#1457919111yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—3164no—
420C4—3164no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#1909870111yescharacteristic
427C2#2377276311yescharacteristic
428C2#2377279611yescharacteristic
subgroupG/H
centerC22C2×#14579129
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2618C2
socleC22×C79C2×#120
prime radicalC2×#121×C79C22
hypercenter#2618D79
hypocenterC79#18294
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#19099194C2semidirect
3. #23762407C2semidirect
4. #23762339C2semidirect
5. #23761971C2semidirect
6. C2×#19098289C2semidirect
120. C22×C4×C79SD16semidirect
222. C79#18294semidirect
321. C22×C4#11929612semidirect

Maximal subgroups

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

subgroupindex
C2×#190991942
#237741572
#237624072
#237623392
#237619712
C2×#190982892
C79×#26182
#1829479