Finite Group Databasebeta

← #34199200

Subgroups — #34199200

Order 20224.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#23768780C211yesmaximal, characteristic
2C2×#19098289C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761405C212yesmaximal, complemented
5C79×#1821C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7C2×C22⋊(C79⋊C8)—21no—
8#19098493C411yescharacteristic
9#19098493C2211yescharacteristic
10#19098483C411yescharacteristic
11#19098483C2211yescharacteristic
12#19098458—21no—
13#19098458—21no—
14#19098436—21no—
15#19098436—21no—
16C4×C22⋊Dic79C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C22×C79⋊C8—41no—
25#14579140—41no—
26#14579140—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48C2×C4⋊C4×C79—41no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C2×Dic158—41no—
57C2×Dic158—81no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—82no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68Dic79⋊C4—81no—
69Dic79⋊C4—81no—
70C22⋊Dic79—24no—
71C22⋊Dic79—41no—
72C22⋊Dic79—41no—
73C4×Dic79—81no—
74C24×C79C22⋊C411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—22no—
82C22×C4×C79—41no—
83C22×C4×C79—81no—
84C22×C4×C79—81no—
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22×C411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91M16×C79—42no—
92M16×C79—42no—
93C2×C8×C79—42no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C23×C79—22no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79#9911yescharacteristic
103C23×C79#9911yescharacteristic
104C23×C79C2×C22⋊C411yescharacteristic
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79—81no—
124C2×C4×C79#13512yes—
125C2×Dic79—22no—
126C2×Dic79—22no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137Dic158—82no—
138C8×C79—82no—
139C79⋊C8—82no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79#35612yes—
150C22×C79C2×#9911yescharacteristic
151C22×C79#41611yescharacteristic
152C22×C79#42111yescharacteristic
153C4×C79—22no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159Dic79—42no—
160Dic79—42no—
161Dic79—81no—
162Dic79—81no—
163#11854—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79—22no—
167C2×C79—41no—
168C2×C79#151611yescharacteristic
169C2×C79#212111yescharacteristic
170C2×C79C2×#35611yescharacteristic
171#2583—791no—
172#1821D7911yescharacteristic
173C2×#324—791no—
174#1354—792no—
175#1346—792no—
176C79#1185411yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—791no—
178C22×C4⋊C4C2×D7911yescharacteristic
179#407—21no—
180C2×C22⋊C8—1581no—
181#398—791no—
182#398—791no—
183#395—791no—
184#395—791no—
185#389—1581no—
186#389—1581no—
187#385—1581no—
188#385—1581no—
189C4×C22⋊C4—791no—
190#343—21no—
191#343C2×D7912yes—
192#324—792no—
193#324—1581no—
194C22×Q8—791no—
195C23×C4C4×D7911yescharacteristic
196C23×C4D31611yescharacteristic
197C23×C4C22⋊D7911yescharacteristic
198C2×M16—22no—
199C22×C8—3161no—
200C22⋊Q8—3161no—
201C22⋊Q8—3161no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—791no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C4×D7911yescharacteristic
211C2×C4⋊C4D31611yescharacteristic
212C2×C4⋊C4C22⋊D7911yescharacteristic
213C2×C4⋊C4C22×D7911yescharacteristic
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C42—1581no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—3161no—
220C22⋊C8—1582no—
221#95—791no—
222#95—791no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226#95—1581no—
227C24(C2×D79)⋊C411yescharacteristic
228C2×Q8—3161no—
229C2×Q8—6321no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—1581no—
241C22×C4—1581no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4(C2×D79)⋊C411yescharacteristic
245C22×C4(C2×D79)⋊C411yescharacteristic
246C22×C4(C2×D79)⋊C411yescharacteristic
247C22×C4C2×C4×D7911yescharacteristic
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×D31611yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—42no—
253C2×C8—3162no—
254C2×C8—6322no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3161no—
260C4⋊C4—3162no—
261C4⋊C4—3162no—
262C4⋊C4—3162no—
263C4⋊C4—6321no—
264C4⋊C4—6321no—
265C22⋊C4—3161no—
266C22⋊C4—3161no—
267C22⋊C4—1584no—
268C42—6321no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1457907511yescharacteristic
274C23#1457907511yescharacteristic
275C23C2×(C2×D79)⋊C411yescharacteristic
276Q8—6322no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—1582no—
307C2×C4—1582no—
308C2×C4#1457918112yes—
309C8—82no—
310C8—6322no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1909836112yes—
321C22C2×#1457907511yescharacteristic
322C22#1909855711yescharacteristic
323C22#1909858111yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—3162no—
331C4—3162no—
332C4—6321no—
333C4—6321no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2376195411yescharacteristic
339C2#2376504711yescharacteristic
340C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1821C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1821D79
hypocenterC79#11854
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761405C2semidirect
176. C79#11854semidirect

Maximal subgroups

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

subgroupindex
#237687802
C2×#190982892
#237614292
#237614052
C79×#18212
#1185479