Finite Group Databasebeta

← #34263174

Subgroups — #34263174

Order 20224.

870 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 191 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#23777210C211yesmaximal, characteristic
2#23774551C212yesmaximal
3#23762441C218yesmaximal
4#23761898C212yesmaximal
5#23761724C211yesmaximal, characteristic
6C79×#2568C211yesmaximal, characteristic
7Q8×Dic158—22no—
8#19099292—22no—
9#19099289—24no—
10#19099194—21no—
11#19099194C2214yes—
12#19099120—22no—
13#19099120C2212yes—
14#19099041—21no—
15#19098966—22no—
16#19098915—22no—
17#19098898C2218yes—
18#19098855—24no—
19#19098616C2212yes—
20#19098299C2218yes—
21#19098289C2214yes—
22C2×C4⋊Q8×C79C2211yescharacteristic
23C2×C4×Q8×C79C2211yescharacteristic
24#400×C79C2214yes—
25#391×C79—21no—
26#391×C79—22no—
27#382×C79—21no—
28C4×C4⋊C4×C79C2211yescharacteristic
29#14579161—22no—
30#14579161—22no—
31#14579161D412yes—
32#14579161D412yes—
33#14579160—24no—
34#14579160—28no—
35#14579153—24no—
36#14579153—22no—
37#14579152—24no—
38#14579152—28no—
39#14579140—24no—
40#14579140—22no—
41#14579140C2314yes—
42Q8×Dic79—24no—
43Q8×Dic79—24no—
44C4×Dic158—24no—
45C4×Dic158—24no—
46C4×Dic158—22no—
47C4×Dic158D414yes—
48#14579120—22no—
49#14579120—24no—
50#14579094—28no—
51C4⋊(C79⋊C8)—22no—
52#14579084—28no—
53C22⋊(C79⋊C8)C2314yes—
54C4×C79⋊C8—22no—
55C22×Q8×C79—21no—
56C22×Q8×C79D411yescharacteristic
57C4⋊Q8×C79—22no—
58C4⋊Q8×C79—22no—
59C4×Q8×C79—22no—
60C4×Q8×C79—22no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C4⋊C4×C79C2312yes—
65C2×C4⋊C4×C79C2312yes—
66C2×C42×C79—21no—
67C2×C42×C79D412yes—
68C2×C42×C79C2311yescharacteristic
69#95×C79—24no—
70#95×C79—24no—
71#95×C79C2312yes—
72C2×Dic158—24no—
73C2×Dic158—22no—
74C2×Dic158C2×D414yes—
75C2×C79⋊C8—24no—
76C4⋊Dic79—24no—
77C4⋊Dic79—24no—
78C4⋊Dic79—24no—
79C4⋊Dic79—22no—
80C4⋊Dic79—22no—
81C4⋊Dic79—22no—
82C4⋊Dic79C2×D414yes—
83Dic79⋊C4—28no—
84Dic79⋊C4—24no—
85Dic79⋊C4—24no—
86C22⋊Dic79—24no—
87C4×Dic79—28no—
88C4×Dic79—24no—
89C2×Q8×C79—22no—
90C2×Q8×C79—22no—
91C2×Q8×C79—44no—
92C2×Q8×C79—41no—
93C2×Q8×C79—41no—
94C2×Q8×C79Q1614yes—
95C22×C4×C79—22no—
96C22×C4×C79—22no—
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D412yes—
99C22×C4×C79C2×D412yes—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2411yescharacteristic
102C4⋊C4×C79—22no—
103C4⋊C4×C79—24no—
104C4⋊C4×C79—24no—
105C4⋊C4×C79—42no—
106C4⋊C4×C79—42no—
107C42×C79—22no—
108C42×C79—22no—
109C42×C79—42no—
110C42×C79C2×D412yes—
111C42×C79C2×D412yes—
112C23×C79C22×D411yescharacteristic
113Q8×C79—28no—
114Q8×C79—82no—
115Q8×C79—82no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—41no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79—44no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79#12011yescharacteristic
128C2×C4×C79#12012yes—
129C2×C4×C79#12011yescharacteristic
130C2×C4×C79C2×Q1614yes—
131C2×C4×C79C2×Q1612yes—
132C2×C4×C79#13712yes—
133C2×C4×C79C22×D412yes—
134C2×Dic79—24no—
135C2×Dic79—24no—
136Dic158—44no—
137Dic158—44no—
138Dic158—84no—
139C79⋊C8—44no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79C2×#12011yescharacteristic
143C22×C79C22×Q1611yescharacteristic
144C22×C79C2×#13711yescharacteristic
145C4×C79—24no—
146C4×C79—21no—
147C4×C79—22no—
148C4×C79—42no—
149C4×C79—82no—
150C4×C79—82no—
151C4×C79#45114yes—
152C4×C79#45412yes—
153Dic79—44no—
154Dic79—44no—
155#18411—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79C2×#45111yescharacteristic
159C2×C79C2×#45411yescharacteristic
160C2×C79#302811yescharacteristic
161#3502—791no—
162#3230—792no—
163#2568D7911yescharacteristic
164#1617—798no—
165#1503—792no—
166#1461—791no—
167C79#1841111yescharacteristic, complemented, Sylow
168Q82—1582no—
169#557—1584no—
170#557—1582no—
171#544—794no—
172#544—1581no—
173#533—792no—
174#533—1582no—
175C2×C4⋊Q8C2×D7911yescharacteristic
176#519—1581no—
177C2×C4×Q8C2×D7911yescharacteristic
178#500—1582no—
179#487—1582no—
180#484—798no—
181#477—1584no—
182#431—792no—
183#400C2×D7914yes—
184#391—21no—
185#391—22no—
186#382—21no—
187C4×C4⋊C4C2×D7911yescharacteristic
188#328—798no—
189#324—794no—
190C22×Q8—21no—
191C22×Q8C22⋊D7911yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—792no—
195C4⋊Q8—792no—
196C4⋊Q8—1582no—
197C4⋊Q8—1582no—
198C4⋊Q8—1588no—
199C4⋊Q8—1584no—
200#125—1584no—
201#125—1588no—
202#125—1584no—
203#125—1582no—
204C22⋊Q8—794no—
205C22⋊Q8—1582no—
206C22⋊Q8—1584no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—794no—
210C4×Q8—1584no—
211C4×Q8—1584no—
212C4×Q8—1584no—
213C4×Q8—1584no—
214C4×Q8—1582no—
215#117—1582no—
216#117—1584no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D7911yescharacteristic
220C2×C4⋊C4C22×D7912yes—
221C2×C4⋊C4C22×D7912yes—
222C2×C42—21no—
223C2×C42C22⋊D7912yes—
224C2×C42C22×D7911yescharacteristic
225#107—1588no—
226C4⋊C8—1582no—
227#103—1588no—
228C22⋊C8—794no—
229C4×C8—1582no—
230#95—24no—
231#95—24no—
232#95C22×D7912yes—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—794no—
239C2×Q8—1584no—
240C2×Q8—1582no—
241C2×Q8#1192961414yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D7911yescharacteristic
245C22×C4C2×C22⋊D7912yes—
246C22×C4C2×C22⋊D7912yes—
247C22×C4C2×C22⋊D7911yescharacteristic
248C22×C4C23×D7911yescharacteristic
249C2×C8—1584no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—794no—
256C4⋊C4—1588no—
257C4⋊C4—1584no—
258C4⋊C4—1584no—
259C4⋊C4—1584no—
260C4⋊C4—1584no—
261C4⋊C4—1582no—
262C4⋊C4—1582no—
263C4⋊C4—1582no—
264C4⋊C4—1584no—
265C22⋊C4—1584no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—1588no—
270C42—1584no—
271C42D4×D7912yes—
272C42D4×D7912yes—
273C23C22×C22⋊D7911yescharacteristic
274Q8—28no—
275Q8—82no—
276Q8—82no—
277Q8—3164no—
278Q8—3164no—
279Q8—6324no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1584no—
292C2×C4—1584no—
293C2×C4#1457912911yescharacteristic
294C2×C4#1457912912yes—
295C2×C4#1457912911yescharacteristic
296C2×C4C2×#1192961414yes—
297C2×C4C2×#1192961412yes—
298C2×C4#1457919112yes—
299C2×C4C2×D4×D7912yes—
300C8—3164no—
301C22—21no—
302C22—21no—
303C22C2×#1457912911yescharacteristic
304C22C22×#1192961411yescharacteristic
305C22C2×#1457919111yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—3164no—
313C4—3164no—
314C4#1909871814yes—
315C4#1909873512yes—
316C2—21no—
317C2—21no—
318C2C2×#1909871811yescharacteristic
319C2C2×#1909873511yescharacteristic
320C2#2377280511yescharacteristic
subgroupG/H
centerC22C2×#14579129
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2568C2
socleC22×C79C2×#120
prime radicalC23×C79C22×D4
hypercenter#2568D79
hypocenterC79#18411
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
167. C79#18411semidirect

Maximal subgroups

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

subgroupindex
#237772102
#237745512
#237624412
#237618982
#237617242
C79×#25682
#1841179