Finite Group Databasebeta

← #34204542

Subgroups — #34204542

Order 20224.

614 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 99 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#23765992C211yesmaximal, characteristic
2#23763328C211yesmaximal, characteristic
3#23761526C214yesmaximal, complemented
4C79×#1785C211yesmaximal, characteristic
5C2×#14579120C411yescharacteristic
6C22×C4⋊Dic79C411yescharacteristic
7C2×#14579080C2212yes—
8C2×#14579078—21no—
9#19098527—21no—
10#19098470C2212yes—
11#19098466C412yes—
12#19098443—21no—
13#19098443—21no—
14#19098423—21no—
15#19098423—21no—
16C22×#41×C79C2211yescharacteristic
17C2×#101×C79C2212yes—
18C2×#100×C79—21no—
19#407×C79—21no—
20C23×Dic79—21no—
21C2×#11929608—22no—
22#14579120—42no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—42no—
26C2×C4⋊Dic79C2×C412yes—
27C2×C22⋊Dic79—21no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C4×Dic79C2×C412yes—
31#14579080—24no—
32#14579078—42no—
33C22⋊(C79⋊C8)—42no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C2×#41×C79—24no—
37C2×#41×C79—42no—
38C2×#41×C79—42no—
39C22×Q8×C79C2311yescharacteristic
40C22×D4×C79C2×C411yescharacteristic
41C22×D4×C79D411yescharacteristic
42C22×D4×C79D411yescharacteristic
43C23×C4×C79C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C2×M16×C79—22no—
47#101×C79—24no—
48#100×C79—42no—
49C22⋊C8×C79—42no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—82no—
54#11929608—42no—
55#11929608—42no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—42no—
59C4⋊Dic79—42no—
60C22⋊Dic79—42no—
61C22⋊Dic79—82no—
62C22⋊Dic79—82no—
63C4×Dic79—42no—
64C24×C79C22⋊C411yescharacteristic
65C24×C79C22⋊C411yescharacteristic
66C24×C79C22⋊C411yescharacteristic
67#41×C79—416no—
68C2×Q8×C79—41no—
69C2×Q8×C79—41no—
70C2×Q8×C79#4114yes—
71C2×D4×C79—22no—
72C2×D4×C79—22no—
73C2×D4×C79—22no—
74C2×D4×C79—22no—
75C2×D4×C79—22no—
76C2×D4×C79—41no—
77C2×D4×C79—41no—
78C2×D4×C79—41no—
79C2×D4×C79—41no—
80C22×C4×C79—22no—
81C22×C4×C79—22no—
82C22×C4×C79—22no—
83C22×C4×C79—22no—
84C22×C4×C79—22no—
85C22×C4×C79—22no—
86C22×C4×C79—22no—
87C22×C4×C79—41no—
88C22×C4×C79—41no—
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—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79—41no—
109C23×C79—41no—
110C23×C79#9912yes—
111C23×C79C2×C22⋊C411yescharacteristic
112Q8×C79—44no—
113D4×C79—44no—
114D4×C79—44no—
115D4×C79—44no—
116C2×C4×C79—28no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—44no—
126C2×C4×C79—44no—
127C2×C4×C79#9912yes—
128C2×C4×C79#11712yes—
129C2×C4×C79#12312yes—
130C2×C4×C79#12312yes—
131C2×Dic79—42no—
132C2×Dic79—42no—
133C2×Dic79—42no—
134C2×Dic79—42no—
135C2×Dic79—42no—
136C2×Dic79—42no—
137C2×Dic79—84no—
138C8×C79—82no—
139C79⋊C8—82no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—22no—
143C22×C79—42no—
144C22×C79—42no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79—41no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79—41no—
155C22×C79—81no—
156C22×C79—81no—
157C22×C79#38111yescharacteristic
158C22×C79C2×#9911yescharacteristic
159C22×C79C2×#9911yescharacteristic
160C4×C79—22no—
161C4×C79—22no—
162C4×C79—22no—
163C4×C79—22no—
164Dic79—82no—
165Dic79—82no—
166#12863—791nomaximal, Sylow
167C2×C79—21no—
168C2×C79—21no—
169C2×C79—42no—
170C2×C79—81no—
171C2×C79—81no—
172C2×C79#181911yescharacteristic
173C2×C79#213411yescharacteristic
174C2×C79#213511yescharacteristic
175#2311—791no—
176#1785—791no—
177#1785D7911yescharacteristic
178#1394—794no—
179C79#1286311yescharacteristic, complemented, Sylow
180C22×#41C2×D7911yescharacteristic
181C2×#117—791no—
182C22×C4⋊C4—791no—
183C2×#101—792no—
184C2×#101C2×D7912yes—
185C2×#100—21no—
186C2×#100—1581no—
187#407—21no—
188#407—1581no—
189#391—792no—
190#390—792no—
191#386—1581no—
192#386—1581no—
193#380—1581no—
194#380—1581no—
195C2×#41—24no—
196C2×#41—42no—
197C2×#41—42no—
198C22×Q8C22×D7911yescharacteristic
199C22×D4C4×D7911yescharacteristic
200C22×D4D31611yescharacteristic
201C22×D4C22⋊D7911yescharacteristic
202C23×C4—1581no—
203C23×C4C4×D7911yescharacteristic
204C23×C4D31611yescharacteristic
205C23×C4C22⋊D7911yescharacteristic
206C2×M16—22no—
207C2×M16—1582no—
208#117—3162no—
209C2×C4⋊C4—3162no—
210C2×C4⋊C4—792no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1581no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4—1582no—
215C2×C22⋊C4—1582no—
216C2×C42—792no—
217#101—24no—
218#101—1584no—
219#100—42no—
220#100—3162no—
221C22⋊C8—42no—
222C22⋊C8—3162no—
223#95—1582no—
224#95—1582no—
225C24(C2×D79)⋊C411yescharacteristic
226C24(C2×D79)⋊C411yescharacteristic
227C24(C2×D79)⋊C411yescharacteristic
228#41—416no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8#1192962314yes—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—3162no—
251C22×C4—6322no—
252C22×C4—1582no—
253C22×C4—1582no—
254C22×C4(C2×D79)⋊C411yescharacteristic
255C22×C4C2×C4×D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4C2×D31611yescharacteristic
258M16—42no—
259M16—42no—
260M16—3162no—
261M16—3162no—
262C2×C8—42no—
263C2×C8—3162no—
264C4⋊C4—3162no—
265C4⋊C4—3162no—
266C4⋊C4—3162no—
267C22⋊C4—3162no—
268C22⋊C4—6322no—
269C22⋊C4—6322no—
270C42—3162no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1457907512yes—
286C23C2×(C2×D79)⋊C411yescharacteristic
287Q8—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—3162no—
303C2×C4—3162no—
304C2×C4—3162no—
305C2×C4—3162no—
306C2×C4—3162no—
307C2×C4—3162no—
308C2×C4—6324no—
309C2×C4#1457907512yes—
310C2×C4#1457911712yes—
311C2×C4#1457914212yes—
312C2×C4#1457914512yes—
313C8—82no—
314C8—6322no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22#1909842411yescharacteristic
333C22C2×#1457907511yescharacteristic
334C22C2×#1457907511yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—6322no—
340C4—6322no—
341C2—21no—
342C2—21no—
343C2—42no—
344C2—81no—
345C2—81no—
346C2#2376347211yescharacteristic
347C2#2376509111yescharacteristic
348C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#1785C2
socleC22×C79C2×#99
prime radicalC22×#41×C79C22
cosocleC22×Q8×C79C23
hypercenter#1785D79
hypocenterC79#12863
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #23761526C2semidirect
179. C79#12863semidirect

Maximal subgroups

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

subgroupindex
#237659922
#237633282
#237615262
C79×#17852
#1286379