Finite Group Databasebeta

← #34183123

Subgroups — #34183123

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11929608C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic
3C4×#14579085C211yesmaximal, characteristic
4#23761855C211yesmaximal, characteristic
5#23761302C211yesmaximal, characteristic
6#23761236C211yesmaximal, characteristic
7C79×#3104C211yesmaximal, characteristic
8#19098694—22no—
9#19098694—21no—
10Q8×C79⋊C8—21no—
11#19098587—21no—
12#19098587C2211yescharacteristic
13C2×#14579085—21no—
14C4×#11929608—21no—
15C4×#11929608C2211yescharacteristic
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098322—21no—
20#19098322—21no—
21Q8⋊(C79⋊C8)—22no—
22#533×C79C2211yescharacteristic
23C2×C4×Q8×C79C2211yescharacteristic
24#484×C79—21no—
25#477×C79C411yescharacteristic, complemented
26#477×C79C411yescharacteristic, complemented
27#462×C79C411yescharacteristic, complemented
28#462×C79C411yescharacteristic, complemented
29#451×C79—21no—
30C4×Q16×C79—21no—
31C4×Q16×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—21no—
34C2×#11929608D411yescharacteristic
35C2×#11929608D411yescharacteristic
36C22×C79⋊C8—21no—
37C2×C4⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C4⋊(C79⋊C8)—22no—
40C4⋊(C79⋊C8)—22no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D412yes—
43#14579085—24no—
44#14579072—22no—
45#14579072—21no—
46C4×C79⋊C8—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C79⋊#95—21no—
51C22×Q8×C79D411yescharacteristic
52C2×Q16×C79—21no—
53C2×Q16×C79—21no—
54C4⋊Q8×C79C2×C411yescharacteristic
55C4⋊Q8×C79C2×C411yescharacteristic
56#125×C79—21no—
57C22⋊Q8×C79—21no—
58C22⋊Q8×C79—21no—
59C4×Q8×C79—21no—
60C4×Q8×C79—21no—
61C4×Q8×C79—21no—
62C4×Q8×C79—21no—
63C4×Q8×C79C2×C411yescharacteristic
64C4×Q8×C79C2×C411yescharacteristic
65C4×Q8×C79D411yescharacteristic
66C4×Q8×C79D411yescharacteristic
67#117×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#107×C79—21no—
73#107×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#103×C79—21no—
77#103×C79—21no—
78#103×C79—21no—
79#103×C79—21no—
80C22⋊C8×C79—21no—
81C4×C8×C79—21no—
82C22×Dic79—41no—
83#11929608—21no—
84#11929608—21no—
85#11929608—21no—
86#11929608—41no—
87#11929608—41no—
88#11929608Q1612yescomplemented
89C2×C79⋊C8—22no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—22no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C4⋊Dic79—22no—
96C4⋊Dic79—41no—
97C4×Dic79—41no—
98C4×Dic79—41no—
99C2×Q8×C79—21no—
100C2×Q8×C79—22no—
101C2×Q8×C79—21no—
102C2×Q8×C79—21no—
103C2×Q8×C79—41no—
104C2×Q8×C79C22⋊C411yescharacteristic
105C2×Q8×C79C22⋊C411yescharacteristic
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111Q16×C79—42no—
112Q16×C79—42no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79C22⋊C411yescharacteristic
127C4⋊C4×C79C22⋊C411yescharacteristic
128C22⋊C4×C79—41no—
129C42×C79—21no—
130C42×C79—21no—
131C42×C79—21no—
132C42×C79—21no—
133C42×C79—41no—
134C42×C79C22×C411yescharacteristic
135C42×C79C2×D411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79#4111yescharacteristic
138C23×C79#12111yescharacteristic
139Q8×C79—21no—
140Q8×C79—21no—
141Q8×C79—21no—
142Q8×C79—42no—
143Q8×C79—42no—
144Q8×C79—41no—
145Q8×C79—41no—
146Q8×C79#10412yes—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—41no—
154C2×C4×C79—42no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—81no—
158C2×C4×C79C2×C22⋊C411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79#12011yescharacteristic
162C2×C4×C79#12111yescharacteristic
163C2×C4×C79#12311yescharacteristic
164C2×C4×C79C2×Q1611yescharacteristic
165C2×C4×C79#13511yescharacteristic
166C2×C4×C79#13711yescharacteristic
167C2×C4×C79#13711yescharacteristic
168C2×Dic79—42no—
169C2×Dic79—81no—
170C2×Dic79—81no—
171C8×C79—41no—
172C8×C79—41no—
173C79⋊C8—22no—
174C79⋊C8—42no—
175C79⋊C8—41no—
176C79⋊C8—42no—
177C22×C79—21no—
178C22×C79—21no—
179C22×C79#38611yescharacteristic
180C22×C79#46211yescharacteristic
181C22×C79#46411yescharacteristic
182C4×C79—21no—
183C4×C79—22no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—41no—
187C4×C79—41no—
188C4×C79—41no—
189C4×C79—81no—
190C4×C79—81no—
191C4×C79C2×#10411yescharacteristic
192C4×C79#42111yescharacteristic
193C4×C79C4×Q1611yescharacteristic
194C4×C79#44111yescharacteristic
195C4×C79#45111yescharacteristic
196C4×C79#45211yescharacteristic
197Dic79—82no—
198#8990—791nomaximal, Sylow
199C2×C79—21no—
200C2×C79—21no—
201C2×C79#187111yescharacteristic
202C2×C79#187811yescharacteristic
203C2×C79#188111yescharacteristic
204#3104D7911yescharacteristic
205Q8×M16—791no—
206#1986—791no—
207C4×#103—791no—
208#1493—791no—
209#1303—791no—
210#1278—791no—
211C79#899011yescharacteristic, complemented, Sylow
212#533C2×D7911yescharacteristic
213C2×C4×Q8C2×D7911yescharacteristic
214#484—21no—
215#477Dic7911yescharacteristic, complemented
216#477Dic7911yescharacteristic, complemented
217#462Dic7911yescharacteristic, complemented
218#462Dic7911yescharacteristic, complemented
219#451—21no—
220#446—1582no—
221#446—1581no—
222C8×Q8—1581no—
223C4×Q16—21no—
224C4×Q16—21no—
225#431C2×D7911yescharacteristic
226#423—791no—
227#423—1581no—
228C2×#103—1581no—
229C4×M16—791no—
230C4×M16—1581no—
231C2×C4×C8—791no—
232C4×C4⋊C4—791no—
233#340—1581no—
234#339—1581no—
235#339—1581no—
236Q8⋊C8—1582no—
237C22×Q8C22⋊D7911yescharacteristic
238C2×Q16—21no—
239C2×Q16—21no—
240C2×M16—791no—
241C2×M16—791no—
242C2×M16—1581no—
243C22×C8—1581no—
244C4⋊Q8C2×Dic7911yescharacteristic
245C4⋊Q8C2×Dic7911yescharacteristic
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C2×Dic7911yescharacteristic
254C4×Q8C2×Dic7911yescharacteristic
255C4×Q8C22⋊D7911yescharacteristic
256C4×Q8C22⋊D7911yescharacteristic
257#117—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—1581no—
260C2×C4⋊C4C22⋊D7911yescharacteristic
261C2×C42—21no—
262C2×C42—1581no—
263C2×C42C22×D7911yescharacteristic
264#107—21no—
265#107—21no—
266C4⋊C8—792no—
267C4⋊C8—1582no—
268C4⋊C8—1582no—
269C4⋊C8—1581no—
270C4⋊C8C2×Dic7911yescharacteristic
271C4⋊C8C2×Dic7911yescharacteristic
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#103—1584no—
277C22⋊C8—21no—
278#97—1582no—
279#97—1581no—
280C4×C8—21no—
281C4×C8—1582no—
282C4×C8—1581no—
283C4×C8—1581no—
284C4×C8—1581no—
285#95—1581no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8C22⋊Dic7911yescharacteristic
292C2×Q8C22⋊Dic7911yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3161no—
296C22×C4D4×D7911yescharacteristic
297C22×C4C2×C22⋊D7911yescharacteristic
298C22×C4#1192962311yescharacteristic
299Q16—42no—
300Q16—42no—
301M16—3161no—
302M16—3161no—
303M16—792no—
304M16—1581no—
305M16—1581no—
306M16—1581no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—3161no—
310C2×C8—3161no—
311C2×C8—1582no—
312C2×C8—1582no—
313C2×C8—1581no—
314C2×C8—1581no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—3161no—
327C4⋊C4—1582no—
328C4⋊C4C22⋊Dic7911yescharacteristic
329C4⋊C4C22⋊Dic7911yescharacteristic
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—41no—
336C42—3161no—
337C42—3161no—
338C42C22×Dic7911yescharacteristic
339C42D4×D7911yescharacteristic
340C42C2×C22⋊D7911yescharacteristic
341C42#1192962311yescharacteristic
342C23#1457913311yescharacteristic
343Q8—21no—
344Q8—21no—
345Q8—21no—
346Q8—42no—
347Q8—42no—
348Q8—41no—
349Q8—41no—
350Q8#1457908812yes—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—3162no—
363C2×C4—6321no—
364C2×C4—6321no—
365C2×C4C2×C22⋊Dic7911yescharacteristic
366C2×C4D4×Dic7911yescharacteristic
367C2×C4D4×Dic7911yescharacteristic
368C2×C4#1457912911yescharacteristic
369C2×C4#1457913311yescharacteristic
370C2×C4#1457914211yescharacteristic
371C2×C4Q16×D7911yescharacteristic
372C2×C4#1457918411yescharacteristic
373C2×C4#1457919311yescharacteristic
374C2×C4#1457919311yescharacteristic
375C8—41no—
376C8—41no—
377C8—3162no—
378C8—3162no—
379C8—3161no—
380C8—1582no—
381C22—21no—
382C22—21no—
383C22#1909844311yescharacteristic
384C22#1909877111yescharacteristic
385C22#1909878711yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—6322no—
396C4C2×#1457908811yescharacteristic
397C4#1909858011yescharacteristic
398C4Q16×Dic7911yescharacteristic
399C4#1909867011yescharacteristic
400C4#1909871711yescharacteristic
401C4#1909872411yescharacteristic
402C2—21no—
403C2—21no—
404C2#2376370311yescharacteristic
405C2#2376374311yescharacteristic
406C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3104C2
socleC22×C79#386
prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#3104D79
hypocenterC79#8990
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
25. #477×C79C4semidirect
26. #477×C79C4semidirect
27. #462×C79C4semidirect
28. #462×C79C4semidirect
88. #11929608Q16semidirect
211. C79#8990semidirect
215. #477Dic79semidirect
216. #477Dic79semidirect
217. #462Dic79semidirect
218. #462Dic79semidirect

Maximal subgroups

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

subgroupindex
Q8×#119296082
#237643452
C4×#145790852
#237618552
#237613022
#237612362
C79×#31042
#899079