Finite Group Databasebeta

← #34197671

Subgroups — #34197671

Order 20224.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#23765803C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761345C212yesmaximal, complemented
4#23761343C212yesmaximal, complemented
5C79×#1464C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C22×C4×Dic79C411yescharacteristic
8#19098620—21no—
9#19098432C411yescharacteristic
10#19098432C2211yescharacteristic
11#19098428C411yescharacteristic
12#19098428C2211yescharacteristic
13#19098421—21no—
14#19098421—21no—
15C4×Dic79⋊C4—21no—
16C4×Dic79⋊C4—21no—
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#432×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C23×Dic79—21no—
24#14579120—41no—
25#14579120—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79—22no—
35C2×C4×Dic79—22no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79C2×C411yescharacteristic
38C2×C4×Dic79C2×C411yescharacteristic
39C4⋊(C79⋊C8)—41no—
40C22⋊(C79⋊C8)—22no—
41#14579072—41no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47#117×C79—41no—
48#117×C79—41no—
49C2×C4⋊C4×C79C2×C411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C4⋊C8×C79—41no—
56C22⋊C8×C79—22no—
57#97×C79—41no—
58C22×Dic79—24no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—42no—
64C22×Dic79—42no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—41no—
67C4⋊Dic79—41no—
68Dic79⋊C4—42no—
69Dic79⋊C4—42no—
70Dic79⋊C4—42no—
71Dic79⋊C4—42no—
72C22⋊Dic79—42no—
73C4×Dic79—24no—
74C4×Dic79—21no—
75C4×Dic79—21no—
76C4×Dic79—41no—
77C4×Dic79—41no—
78C4×Dic79M1612yes—
79C4×Dic79M1612yes—
80C24×C79C22⋊C411yescharacteristic
81C22×C4×C79—22no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22×C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C2×C8×C79—42no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C22⋊C4×C79—41no—
97C42×C79—21no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79#10112yes—
106C23×C79C2×C22⋊C411yescharacteristic
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
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—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79C2×M1611yescharacteristic
126C2×C4×C79C2×M1611yescharacteristic
127C2×Dic79—22no—
128C2×Dic79—22no—
129C2×Dic79—22no—
130C2×Dic79—22no—
131C2×Dic79—48no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C8×C79—82no—
141C79⋊C8—82no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—22no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79#40711yescharacteristic
150C22×C79C2×#9911yescharacteristic
151C22×C79C2×#10111yescharacteristic
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160Dic79—24no—
161Dic79—42no—
162Dic79—81no—
163Dic79—81no—
164#11513—791nomaximal, Sylow
165C2×C79—21no—
166C2×C79—21no—
167C2×C79—42no—
168C2×C79#146311yescharacteristic
169C2×C79#212611yescharacteristic
170C2×C79#213211yescharacteristic
171#2285—791no—
172#1464—791no—
173#1464D7911yescharacteristic
174#1325—792no—
175#1324—792no—
176C79#1151311yescharacteristic, complemented, Sylow
177C2×#117—791no—
178C2×#117C2×D7911yescharacteristic
179C22×C42—791no—
180#432—21no—
181#432—1581no—
182#384—791no—
183#384—791no—
184#382—791no—
185#382—791no—
186#380—1581no—
187#380—1581no—
188C4×C4⋊C4—1581no—
189C4×C4⋊C4—1581no—
190#324—792no—
191#324C2×D7912yes—
192#323—21no—
193#323—1581no—
194C23×C4—1581no—
195C23×C4C4×D7911yescharacteristic
196#117—41no—
197#117—41no—
198#117—3161no—
199#117—3161no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4C4×D7911yescharacteristic
207C2×C4⋊C4C22×D7911yescharacteristic
208C2×C22⋊C4—1581no—
209C2×C22⋊C4D31611yescharacteristic
210C2×C22⋊C4C22⋊D7911yescharacteristic
211C2×C42—791no—
212C2×C42—791no—
213C2×C42—1581no—
214C2×C42—1582no—
215C2×C42—1582no—
216C2×C42—1581no—
217C2×C42D31611yescharacteristic
218C2×C42C22⋊D7911yescharacteristic
219C4⋊C8—41no—
220C4⋊C8—3161no—
221C22⋊C8—22no—
222C22⋊C8—1582no—
223#97—41no—
224#97—3161no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228#95—1581no—
229C24(C2×D79)⋊C411yescharacteristic
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—3162no—
236C22×C4—3162no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4—1584no—
240C22×C4—1581no—
241C22×C4—1581no—
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4(C2×D79)⋊C411yescharacteristic
245C22×C4C2×C4×D7911yescharacteristic
246C22×C4C2×C22⋊D7911yescharacteristic
247C22×C4C2×D31611yescharacteristic
248C2×C8—42no—
249C2×C8—3162no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—3161no—
253C4⋊C4—3161no—
254C4⋊C4—3162no—
255C4⋊C4—3162no—
256C4⋊C4—3162no—
257C4⋊C4—3162no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—3162no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—3161no—
266C42—3161no—
267C42—792no—
268C42—792no—
269C42—1584no—
270C42—1581no—
271C42—1581no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1457907912yes—
277C23C2×(C2×D79)⋊C411yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3168no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—1582no—
304C2×C4—1582no—
305C2×C4—1582no—
306C2×C4—1582no—
307C2×C4#1457907511yescharacteristic
308C2×C4#1457907511yescharacteristic
309C2×C4M16×D7911yescharacteristic
310C2×C4M16×D7911yescharacteristic
311C8—82no—
312C8—6322no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22#1909852611yescharacteristic
321C22C2×#1457907511yescharacteristic
322C22C2×#1457907911yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3162no—
332C4—6321no—
333C4—6321no—
334C4—1584no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2376172911yescharacteristic
339C2#2376506311yescharacteristic
340C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11513
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
4. #23761343C2semidirect
176. C79#11513semidirect

Maximal subgroups

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

subgroupindex
#237658032
#237617342
#237613452
#237613432
C79×#14642
#1151379