Finite Group Databasebeta

← #34205063

Subgroups — #34205063

Order 20224.

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

normal subgroups only

This group has 672 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
1C2×#19098356C211yesmaximal, characteristic
2#23763799C211yesmaximal, characteristic
3#23763680C212yesmaximal, complemented
4#23761518C212yesmaximal, complemented
5C79×#2366C211yesmaximal, characteristic
6#19099038C2211yescharacteristic
7#19098568—21no—
8C2×#14579078C2211yescharacteristic
9C2×#14579076C2212yes—
10#19098436—22no—
11#19098356C414yescomplemented
12#19098329—22no—
13#19098323—21no—
14C2×#127×C79C2211yescharacteristic
15C2×C4×D4×C79—22no—
16#390×C79—22no—
17#386×C79—21no—
18#386×C79C2212yes—
19C4×C4⋊C4×C79—21no—
20C2×#11929608—21no—
21D4×Dic79—41no—
22D4×Dic79—41no—
23#14579120—21no—
24C2×Dic79⋊C4—42no—
25C2×C22⋊Dic79—22no—
26C2×C22⋊Dic79—22no—
27#14579082—41no—
28#14579082—41no—
29#14579078—22no—
30#14579076—22no—
31#14579076C2×C414yescomplemented
32C79⋊#95—42no—
33C22×D4×C79—22no—
34C22×D4×C79D411yescharacteristic
35C22×D4×C79C2311yescharacteristic
36C23×C4×C79—21no—
37C23×C4×C79D412yes—
38#127×C79—22no—
39#127×C79C2×C412yes—
40#127×C79Q812yes—
41C4×D4×C79—48no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79D412yes—
48C2×C42×C79—22no—
49C2×C42×C79D411yescharacteristic
50#95×C79—22no—
51C22×Dic79—22no—
52C22×Dic79—22no—
53C22×Dic79—42no—
54#11929608—41no—
55#11929608—41no—
56C2×C79⋊C8—41no—
57C4⋊Dic79—21no—
58C4⋊Dic79—21no—
59Dic79⋊C4—84no—
60C22⋊Dic79—24no—
61C22⋊Dic79—42no—
62C22⋊Dic79—42no—
63C22⋊Dic79—41no—
64C4×Dic79—41no—
65C24×C79—21no—
66C24×C79C2×D412yes—
67C2×D4×C79—24no—
68C2×D4×C79—24no—
69C2×D4×C79—21no—
70C2×D4×C79—41no—
71C2×D4×C79—41no—
72C2×D4×C79—44no—
73C2×D4×C79—44no—
74C2×D4×C79—41no—
75C2×D4×C79—41no—
76C2×D4×C79C4⋊C412yescomplemented
77C2×D4×C79C22×C411yescharacteristic
78C2×D4×C79C2×Q811yescharacteristic
79C2×D4×C79#4112yes—
80C22×C4×C79—21no—
81C22×C4×C79—22no—
82C22×C4×C79—22no—
83C22×C4×C79—44no—
84C22×C4×C79—44no—
85C22×C4×C79—42no—
86C22×C4×C79—81no—
87C22×C4×C79C2×D411yescharacteristic
88C4⋊C4×C79—24no—
89C4⋊C4×C79—42no—
90C22⋊C4×C79—44no—
91C22⋊C4×C79—44no—
92C22⋊C4×C79—42no—
93C22⋊C4×C79—81no—
94C42×C79—21no—
95C42×C79—44no—
96C42×C79C4⋊C412yescomplemented
97C23×C79—22no—
98C23×C79—22no—
99C23×C79—22no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—42no—
103C23×C79—42no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79C4×D412yes—
107C23×C79#12011yescharacteristic
108C23×C79C22⋊Q812yes—
109D4×C79—41no—
110D4×C79—41no—
111D4×C79—42no—
112D4×C79—42no—
113D4×C79—48no—
114D4×C79—48no—
115D4×C79—81no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—24no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79—44no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—82no—
129C2×C4×C79—84no—
130C2×C4×C79C2×C4⋊C411yescharacteristic
131C2×C4×C79#12311yescharacteristic
132C2×Dic79—44no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—82no—
136C2×Dic79—82no—
137C2×Dic79—82no—
138C2×Dic79—82no—
139C79⋊C8—81no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—22no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—42no—
146C22×C79—44no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79—82no—
151C22×C79—82no—
152C22×C79—81no—
153C22×C79—81no—
154C22×C79#38511yescharacteristic
155C22×C79B(4,𝔽2)11yescharacteristic
156C22×C79#45811yescharacteristic
157C4×C79—24no—
158C4×C79—21no—
159C4×C79—21no—
160C4×C79—44no—
161C4×C79—82no—
162Dic79—84no—
163Dic79—81no—
164Dic79—81no—
165#12964—791nomaximal, Sylow
166C2×C79—21no—
167C2×C79—21no—
168C2×C79—42no—
169C2×C79—42no—
170C2×C79—82no—
171C2×C79#186711yescharacteristic
172C2×C79#219611yescharacteristic
173C2×C79#219811yescharacteristic
174#2366D7911yescharacteristic
175C2×#353—791no—
176#1887—791no—
177#1867—792no—
178#1390—792no—
179C79#1296411yescharacteristic, complemented, Sylow
180C2×#127C2×D7911yescharacteristic
181#518—791no—
182C2×C4×D4—22no—
183#418—1581no—
184C2×#100—791no—
185C2×#99—792no—
186#390—22no—
187#386—21no—
188#386C2×D7912yes—
189#385—1582no—
190C4×C4⋊C4—21no—
191#353—794no—
192#342—1582no—
193#339—1581no—
194C22×D4—22no—
195C22×D4C22⋊D7911yescharacteristic
196C22×D4C22×D7911yescharacteristic
197C23×C4—21no—
198C23×C4C22⋊D7912yes—
199C2×M16—1581no—
200#127—22no—
201#127Dic15812yes—
202#127C4×D7912yes—
203C4×D4—48no—
204C4×D4—3161no—
205C4×D4—3161no—
206#117—1581no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—3162no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1582no—
214C2×C22⋊C4—1582no—
215C2×C22⋊C4C22⋊D7912yes—
216C2×C42—22no—
217C2×C42C22⋊D7911yescharacteristic
218#102—3161no—
219#102—3161no—
220#100—1582no—
221#99—794no—
222#99—1582no—
223#95—22no—
224#95—3162no—
225C24—21no—
226C24C2×C22⋊D7912yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4Dic79⋊C412yescomplemented
237C2×D4C2×C4×D7911yescharacteristic
238C2×D4C2×Dic15811yescharacteristic
239C2×D4#1192962412yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—3162no—
248C22×C4—1582no—
249C22×C4—1582no—
250C22×C4C2×C22⋊D7911yescharacteristic
251M16—3161no—
252M16—3161no—
253C2×C8—3161no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—6324no—
257C4⋊C4—1581no—
258C4⋊C4—1581no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4—3162no—
264C22⋊C4—3162no—
265C22⋊C4—3161no—
266C22⋊C4—1584no—
267C42—21no—
268C42—44no—
269C42—3161no—
270C42Dic79⋊C412yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23C4×C22⋊D7912yes—
281C23#1457913011yescharacteristic
282C23#1457914012yes—
283D4—41no—
284D4—41no—
285D4—42no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—3164no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—6322no—
308C2×C4—6322no—
309C2×C4—6322no—
310C2×C4—6322no—
311C2×C4C2×Dic79⋊C411yescharacteristic
312C2×C4#1457914411yescharacteristic
313C8—6321no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1909843611yescharacteristic
329C22#1909875011yescharacteristic
330C22#1909875311yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—6324no—
337C4—6321no—
338C4—6321no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#2376368111yescharacteristic
345C2#2376538411yescharacteristic
346C2#2376539611yescharacteristic
subgroupG/H
centerC22#19098750
commutatorC2×D4×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#2366C2
socleC22×C79B(4,𝔽2)
prime radicalC2×#127×C79C22
cosocleC22×D4×C79C23
hypercenter#2366D79
hypocenterC79#12964
LCSG ▷ C2×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2366

Complements

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

NH ≅ G/Ntype
3. #23763680C2semidirect
4. #23761518C2semidirect
11. #19098356C4semidirect
31. #14579076C2×C4semidirect
76. C2×D4×C79C4⋊C4semidirect
96. C42×C79C4⋊C4semidirect
179. C79#12964semidirect
236. C2×D4Dic79⋊C4semidirect
270. C42Dic79⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#190983562
#237637992
#237636802
#237615182
C79×#23662
#1296479