Finite Group Databasebeta

← #34199426

Subgroups — #34199426

Order 20224.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#23768576C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761426C212yesmaximal, complemented
5C79×#1800C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8#19098502C2211yescharacteristic
9#19098499C411yescharacteristic
10#19098483C2211yescharacteristic
11#19098475C411yescharacteristic
12#19098447—21no—
13#19098443—21no—
14#19098436—21no—
15#19098421—21no—
16C2×C79⋊#95C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×C4×D4×C79C2211yescharacteristic
20#423×C79—21no—
21C2×#100×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C23×Dic79—21no—
25C22×C79⋊C8—41no—
26#14579140—41no—
27#14579140—41no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—41no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×D4×C79D411yescharacteristic
45C23×C4×C79C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48C4×D4×C79—41no—
49C4×D4×C79—41no—
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C4⋊C8×C79—41no—
55C4⋊C8×C79—41no—
56#100×C79—42no—
57C2×Dic158—41no—
58C2×Dic158—81no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—42no—
64C22×Dic79—42no—
65C22×Dic79—81no—
66C2×C79⋊C8—42no—
67C2×C79⋊C8—82no—
68C4⋊Dic79—41no—
69Dic79⋊C4—42no—
70Dic79⋊C4—41no—
71Dic79⋊C4—41no—
72Dic79⋊C4—81no—
73C22⋊Dic79—42no—
74C22⋊Dic79—42no—
75C22⋊Dic79—81no—
76C24×C79C22⋊C411yescharacteristic
77C24×C79C22⋊C411yescharacteristic
78C2×D4×C79—41no—
79C2×D4×C79—41no—
80C2×D4×C79SD1612yes—
81C2×D4×C79Q1612yes—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—81no—
87C22×C4×C79—81no—
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22×C411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93M16×C79—42no—
94M16×C79—42no—
95C2×C8×C79—42no—
96C4⋊C4×C79—41no—
97C22⋊C4×C79—41no—
98C22⋊C4×C79—41no—
99C42×C79—21no—
100C42×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79—41no—
109C23×C79#10312yes—
110C23×C79#10312yes—
111C23×C79C2×C22⋊C411yescharacteristic
112D4×C79—42no—
113D4×C79—42no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—41no—
120C2×C4×C79—42no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—81no—
125C2×C4×C79—81no—
126C2×C4×C79—81no—
127C2×C4×C79#9911yescharacteristic
128C2×C4×C79#9911yescharacteristic
129C2×C4×C79C2×SD1611yescharacteristic
130C2×C4×C79C2×Q1611yescharacteristic
131C2×Dic79—22no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—84no—
144Dic158—82no—
145C8×C79—82no—
146C79⋊C8—82no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—22no—
150C22×C79—22no—
151C22×C79—42no—
152C22×C79—42no—
153C22×C79—42no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79—81no—
159C22×C79C2×#9911yescharacteristic
160C22×C79C2×#10311yescharacteristic
161C22×C79#42111yescharacteristic
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—41no—
167C4×C79—41no—
168C4×C79—41no—
169Dic79—42no—
170Dic79—81no—
171Dic79—81no—
172Dic79—81no—
173#11906—791nomaximal, Sylow
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—42no—
177C2×C79—42no—
178C2×C79#152011yescharacteristic
179C2×C79#212111yescharacteristic
180C2×C79#213211yescharacteristic
181#2564—791no—
182#1800D7911yescharacteristic
183#1459—791no—
184#1354—792no—
185#1353—792no—
186C79#1190611yescharacteristic, complemented, Sylow
187C2×C22⋊Q8—791no—
188C2×C4×D4C2×D7911yescharacteristic
189#423—21no—
190C2×#100—21no—
191C2×C22⋊C8—1581no—
192#400—791no—
193#399—791no—
194#395—791no—
195#393—791no—
196#387—1581no—
197#386—1581no—
198#385—1581no—
199#380—1581no—
200C2×#95—791no—
201#343C2×D7912yes—
202#324—792no—
203#323—1581no—
204C22×Q8—791no—
205C22×D4C22⋊D7911yescharacteristic
206C23×C4—1581no—
207C23×C4C4×D7911yescharacteristic
208C23×C4D31611yescharacteristic
209C2×M16—22no—
210C22×C8—3161no—
211C22⋊Q8—3161no—
212C22⋊Q8—3161no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—791no—
216C2×C4⋊C4—1581no—
217C2×C4⋊C4—1581no—
218C2×C4⋊C4—1581no—
219C2×C4⋊C4C22×D7911yescharacteristic
220C2×C22⋊C4—1581no—
221C2×C22⋊C4—1581no—
222C2×C22⋊C4—1581no—
223C2×C22⋊C4C4×D7911yescharacteristic
224C2×C22⋊C4D31611yescharacteristic
225C2×C42C22⋊D7911yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—41no—
228#100—42no—
229C22⋊C8—3161no—
230C22⋊C8—1582no—
231#95—791no—
232#95—791no—
233#95—1581no—
234#95—1581no—
235#95—1581no—
236#95—1581no—
237#95—1581no—
238C24(C2×D79)⋊C411yescharacteristic
239C24(C2×D79)⋊C411yescharacteristic
240C2×Q8—3161no—
241C2×Q8—6321no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1192961212yes—
245C2×D4#1192961412yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—3162no—
253C22×C4—3162no—
254C22×C4—6321no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4—1581no—
259C22×C4(C2×D79)⋊C411yescharacteristic
260C22×C4(C2×D79)⋊C411yescharacteristic
261C22×C4C2×C4×D7911yescharacteristic
262C22×C4C2×C22⋊D7911yescharacteristic
263C22×C4C2×D31611yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C2×C8—3162no—
268C2×C8—6322no—
269C4⋊C4—41no—
270C4⋊C4—3161no—
271C4⋊C4—3162no—
272C4⋊C4—3161no—
273C4⋊C4—3161no—
274C4⋊C4—6321no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—3162no—
278C22⋊C4—3162no—
279C22⋊C4—6321no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1457908412yes—
291C23#1457908412yes—
292C23C2×(C2×D79)⋊C411yescharacteristic
293Q8—6322no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—3161no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—6324no—
321C2×C4—1582no—
322C2×C4#1457907511yescharacteristic
323C2×C4#1457907511yescharacteristic
324C2×C4C2×#1192961211yescharacteristic
325C2×C4C2×#1192961411yescharacteristic
326C8—82no—
327C8—6322no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×#1457907511yescharacteristic
341C22C2×#1457908411yescharacteristic
342C22#1909858111yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3162no—
351C4—6321no—
352C4—6321no—
353C4—6321no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2376197111yescharacteristic
359C2#2376504711yescharacteristic
360C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1800C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenter#1800D79
hypocenterC79#11906
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761426C2semidirect
186. C79#11906semidirect

Maximal subgroups

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

subgroupindex
#237685762
#237617182
#237614292
#237614262
C79×#18002
#1190679