Finite Group Databasebeta

← #34198469

Subgroups — #34198469

Order 20224.

536 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#23766636C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic, complemented
3#23761405C212yesmaximal, complemented
4#23761402C212yesmaximal, complemented
5C79×#1742C211yesmaximal, characteristic, complemented
6C2×C4×D316C411yescharacteristic
7C2×C4×C22⋊D79—21no—
8#19098616—21no—
9#19098493C2211yescharacteristic
10#19098488C411yescharacteristic
11#19098485—21no—
12#19098480C2211yescharacteristic
13#19098479—21no—
14#19098472C411yescharacteristic
15C4×C4⋊Dic79C411yescharacteristic
16C4×(C2×D79)⋊C4—21no—
17#19098289C2212yes—
18#19098287—21no—
19C22×M16×C79—21no—
20C2×#117×C79C2211yescharacteristic
21#343×C79—21no—
22#343×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C22×C4×D79—21no—
26C4×D316—41no—
27C4×D316—41no—
28C4×C22⋊D79—42no—
29C4×C22⋊D79—42no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×C4×Dic79—21no—
39C4⋊(C79⋊C8)—41no—
40C22⋊(C79⋊C8)—22no—
41C4×C79⋊C8—41no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95C2×C411yescharacteristic
45C79⋊#95C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—24no—
48C2×M16×C79—22no—
49C2×M16×C79—41no—
50C2×M16×C79—42no—
51C22×C8×C79—41no—
52#117×C79—41no—
53#117×C79—41no—
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79C2311yescharacteristic
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79C2×C411yescharacteristic
59C2×C42×C79D411yescharacteristic
60C23×D79—21no—
61C2×D316—41no—
62C2×D316—42no—
63C2×C22⋊D79—42no—
64C2×C22⋊D79—42no—
65C2×C22⋊D79—42no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C2×C4×D79—41no—
70C2×C4×D79—41no—
71C2×C4×D79—42no—
72C2×C4×D79—42no—
73C2×C79⋊C8—42no—
74C4⋊Dic79—41no—
75C4⋊Dic79—42no—
76Dic79⋊C4—41no—
77Dic79⋊C4—41no—
78C22⋊Dic79—42no—
79(C2×D79)⋊C4—42no—
80(C2×D79)⋊C4—42no—
81(C2×D79)⋊C4—42no—
82(C2×D79)⋊C4—41no—
83(C2×D79)⋊C4—41no—
84C4×Dic79—42no—
85C24×C79C22⋊C411yescharacteristic
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—22no—
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79D812yes—
93C22×C4×C79SD1612yes—
94C22×C4×C79C22×C411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97M16×C79—24no—
98M16×C79—44no—
99M16×C79—42no—
100C2×C8×C79—42no—
101C2×C8×C79—44no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79—41no—
106C42×C79—21no—
107C42×C79—21no—
108C42×C79—21no—
109C42×C79—21no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79#10212yes—
115C23×C79C2×C22⋊C411yescharacteristic
116C22×D79—22no—
117C22×D79—41no—
118C22×D79—41no—
119C22×D79—41no—
120C22×D79—41no—
121C22×D79—41no—
122C22×D79—41no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—22no—
126C2×C4×C79—22no—
127C2×C4×C79—22no—
128C2×C4×C79—22no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79#9911yescharacteristic
138C2×C4×C79#9911yescharacteristic
139C2×C4×C79#10212yes—
140C2×C4×C79#10412yes—
141C2×C4×C79C2×D811yescharacteristic
142C2×C4×C79C2×SD1611yescharacteristic
143C22⋊D79—48no—
144C2×Dic79—22no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153D316—42no—
154D316—42no—
155C4×D79—82no—
156C4×D79—82no—
157C8×C79—44no—
158C79⋊C8—82no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—22no—
162C22×C79—42no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—81no—
166C22×C79C2×#9911yescharacteristic
167C22×C79C2×#10211yescharacteristic
168C22×C79C2×#10411yescharacteristic
169C2×D79—42no—
170C2×D79—42no—
171C2×D79—42no—
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C2×D79—81no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—41no—
179C4×C79—41no—
180C4×C79—41no—
181C4×C79—41no—
182C4×C79#32712yes—
183C4×C79#32712yes—
184Dic79—42no—
185Dic79—81no—
186Dic79—81no—
187#11697—791nomaximal, Sylow
188C2×C79—21no—
189C2×C79—21no—
190C2×C79—42no—
191C2×C79C2×#32711yescharacteristic
192C2×C79#212011yescharacteristic
193C2×C79#213211yescharacteristic
194D79—82no—
195#2392—791no—
196#1742D7911yescharacteristic, complemented
197#1458—791no—
198#1346—792no—
199#1345—792no—
200C79#1169711yescharacteristic, complemented, Sylow
201C22×M16—21no—
202C2×C4×D4—791no—
203C2×C4×D4—1581no—
204C2×#117C2×D7911yescharacteristic
205#431—1581no—
206#398—791no—
207#397—791no—
208#396—1581no—
209#394—791no—
210#394—1581no—
211#392—791no—
212C4×C4⋊C4—791no—
213C4×C22⋊C4—1581no—
214#343—21no—
215#343C2×D7912yes—
216#324—792no—
217#323—1581no—
218C22×D4—791no—
219C22×D4—1581no—
220C23×C4—1581no—
221C23×C4C22⋊D7911yescharacteristic
222C2×M16—24no—
223C2×M16—22no—
224C2×M16—41no—
225C2×M16—42no—
226C22×C8—41no—
227C4×D4—3161no—
228C4×D4—3161no—
229C4×D4—3162no—
230C4×D4—3162no—
231#117—41no—
232#117—41no—
233C2×C4⋊C4—791no—
234C2×C4⋊C4—1581no—
235C2×C4⋊C4—1581no—
236C2×C4⋊C4C22⋊D7911yescharacteristic
237C2×C4⋊C4C22×D7911yescharacteristic
238C2×C22⋊C4—1581no—
239C2×C22⋊C4—1581no—
240C2×C22⋊C4—1581no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4—1581no—
243C2×C22⋊C4C4×D7911yescharacteristic
244C2×C22⋊C4D31611yescharacteristic
245C2×C42—1581no—
246C2×C42C4×D7911yescharacteristic
247C2×C42D31611yescharacteristic
248C4⋊C8—3161no—
249C22⋊C8—1582no—
250C4×C8—3161no—
251#95—791no—
252#95—791no—
253#95—1581no—
254#95—1581no—
255C24—1581no—
256C24(C2×D79)⋊C411yescharacteristic
257C2×D4—3161no—
258C2×D4—3162no—
259C2×D4—3162no—
260C2×D4—3162no—
261C2×D4—3162no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—3161no—
266C22×C4—3161no—
267C22×C4—3162no—
268C22×C4—3162no—
269C22×C4—1581no—
270C22×C4—1581no—
271C22×C4—1581no—
272C22×C4(C2×D79)⋊C411yescharacteristic
273C22×C4(C2×D79)⋊C411yescharacteristic
274C22×C4(C2×D79)⋊C411yescharacteristic
275C22×C4#1192960912yes—
276C22×C4#1192961112yes—
277C22×C4C2×C4×D7911yescharacteristic
278C22×C4C2×C22⋊D7911yescharacteristic
279C22×C4C2×D31611yescharacteristic
280M16—24no—
281M16—44no—
282M16—42no—
283C2×C8—42no—
284C2×C8—44no—
285C2×C8—3162no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—3161no—
289C4⋊C4—3162no—
290C4⋊C4—3161no—
291C4⋊C4—3161no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—3162no—
295C22⋊C4—3162no—
296C22⋊C4—3162no—
297C22⋊C4—3162no—
298C22⋊C4—3161no—
299C22⋊C4—3161no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—3162no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—3161no—
310C23—3161no—
311C23—3161no—
312C23—3161no—
313C23—3161no—
314C23—3161no—
315C23—1582no—
316C23#1457908112yes—
317C23C2×(C2×D79)⋊C411yescharacteristic
318D4—3162no—
319D4—3162no—
320D4—3168no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—3161no—
336C2×C4—3161no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3161no—
343C2×C4—6322no—
344C2×C4—6322no—
345C2×C4—1582no—
346C2×C4#1457907511yescharacteristic
347C2×C4#1457907511yescharacteristic
348C2×C4#1457908112yes—
349C2×C4#1457908912yes—
350C2×C4C2×#1192960911yescharacteristic
351C2×C4C2×#1192961111yescharacteristic
352C8—44no—
353C8—6322no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—3162no—
362C22—3162no—
363C22—3162no—
364C22—6321no—
365C22—6321no—
366C22—6321no—
367C22—6321no—
368C22C2×#1457907511yescharacteristic
369C22C2×#1457908111yescharacteristic
370C22C2×#1457908911yescharacteristic
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3162no—
378C4—6321no—
379C4—6321no—
380C4#1909829612yes—
381C4#1909829612yes—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—6322no—
386C2C2×#1909829611yescharacteristic
387C2#2376504211yescharacteristic
388C2#2376508311yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1742C2
socleC22×C79C2×#99
prime radical#19098472C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1742D79
hypocenterC79#11697
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. #23761714C2semidirect
3. #23761405C2semidirect
4. #23761402C2semidirect
5. C79×#1742C2semidirect
196. #1742D79semidirect
200. C79#11697semidirect

Maximal subgroups

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

subgroupindex
#237666362
#237617142
#237614052
#237614022
C79×#17422
#1169779