Finite Group Databasebeta

← #34198646

Subgroups — #34198646

Order 20224.

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

normal subgroups only

This group has 402 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#23766691C211yesmaximal, characteristic
2#23761724C211yesmaximal, characteristic
3#23761429C212yesmaximal
4#23761405C212yesmaximal
5C4×#101×C79C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098616—21no—
8#19098494—21no—
9#19098493C411yescharacteristic
10#19098493C2211yescharacteristic
11#19098483C411yescharacteristic
12#19098483C2211yescharacteristic
13#19098482—21no—
14C4×Dic79⋊C4—21no—
15C4×Dic79⋊C4—21no—
16#19098409C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20C2×#101×C79—21no—
21C4×M16×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79—21no—
34C4⋊(C79⋊C8)—41no—
35C22⋊(C79⋊C8)—22no—
36C4×C79⋊C8—41no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×Q8×C79D411yescharacteristic
45C2×M16×C79—22no—
46C4×Q8×C79—41no—
47C4×Q8×C79—41no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54#101×C79—24no—
55#97×C79—41no—
56C4×C8×C79—41no—
57C2×Dic158—41no—
58C2×Dic158—42no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—42no—
70C4×Dic79—24no—
71C4×Dic79—41no—
72C4×Dic79—41no—
73C2×Q8×C79—22no—
74C2×Q8×C79—22no—
75C2×Q8×C79—41no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83M16×C79—24no—
84M16×C79—42no—
85C2×C8×C79—42no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C23×C79C2×C22⋊C411yescharacteristic
96Q8×C79—44no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79#9911yescharacteristic
112C2×C4×C79#9911yescharacteristic
113C2×C4×C79#10412yes—
114C2×C4×C79#13512yes—
115C2×Dic79—22no—
116C2×Dic79—22no—
117C2×Dic79—22no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127Dic158—42no—
128Dic158—42no—
129C8×C79—44no—
130C79⋊C8—82no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79C2×#9911yescharacteristic
134C22×C79#41611yescharacteristic
135C22×C79C2×#10411yescharacteristic
136C4×C79—21no—
137C4×C79—22no—
138C4×C79—22no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—41no—
143C4×C79#35612yes—
144Dic79—42no—
145Dic79—42no—
146Dic79—42no—
147Dic79—81no—
148#11736—791nomaximal, Sylow
149C2×C79—21no—
150C2×C79—21no—
151C2×C79#150111yescharacteristic
152C2×C79#212011yescharacteristic
153C2×C79C2×#35611yescharacteristic
154#2399—791no—
155C4×#101D7911yescharacteristic
156#1461—791no—
157#1354—792no—
158#1346—792no—
159C79#1173611yescharacteristic, complemented, Sylow
160C2×C4×Q8—791no—
161C2×C4×Q8C2×D7911yescharacteristic
162#431—1581no—
163C2×#101—21no—
164C4×M16—21no—
165#398—791no—
166#398—791no—
167#398—1581no—
168#395—791no—
169#395—791no—
170#395—1581no—
171C4×C4⋊C4—1581no—
172C4×C4⋊C4—1581no—
173#376—791no—
174#343C2×D7912yes—
175#324—792no—
176#324—1581no—
177C22×Q8—791no—
178C22×Q8C22⋊D7911yescharacteristic
179C2×M16—22no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—3161no—
183C4×Q8—3161no—
184C2×C4⋊C4—791no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4—1581no—
187C2×C4⋊C4—1581no—
188C2×C4⋊C4—1581no—
189C2×C4⋊C4—1581no—
190C2×C4⋊C4C4×D7911yescharacteristic
191C2×C4⋊C4D31611yescharacteristic
192C2×C4⋊C4C22×D7911yescharacteristic
193C2×C42—1581no—
194C2×C42—1581no—
195C2×C42C4×D7911yescharacteristic
196C2×C42D31611yescharacteristic
197C2×C42C22⋊D7911yescharacteristic
198C4⋊C8—3161no—
199#101—24no—
200C22⋊C8—1582no—
201#97—41no—
202C4×C8—41no—
203C4×C8—3161no—
204#95—791no—
205#95—791no—
206#95—1581no—
207#95—1581no—
208#95—1581no—
209#95—1581no—
210#95—1581no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—3161no—
215C2×Q8—3162no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4—1581no—
220C22×C4(C2×D79)⋊C411yescharacteristic
221C22×C4(C2×D79)⋊C411yescharacteristic
222C22×C4(C2×D79)⋊C411yescharacteristic
223C22×C4(C2×D79)⋊C411yescharacteristic
224C22×C4C2×C4×D7911yescharacteristic
225C22×C4C2×C22⋊D7911yescharacteristic
226C22×C4C2×D31611yescharacteristic
227M16—24no—
228M16—42no—
229C2×C8—42no—
230C2×C8—3162no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—3161no—
235C4⋊C4—3162no—
236C4⋊C4—3162no—
237C4⋊C4—3162no—
238C4⋊C4—3162no—
239C4⋊C4—3162no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—3161no—
247C42—3161no—
248C42—1584no—
249C23C2×(C2×D79)⋊C411yescharacteristic
250Q8—44no—
251Q8—3162no—
252Q8—3162no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—3161no—
268C2×C4—3161no—
269C2×C4—3161no—
270C2×C4—3161no—
271C2×C4—3161no—
272C2×C4—3161no—
273C2×C4—3161no—
274C2×C4—3161no—
275C2×C4—3161no—
276C2×C4—1582no—
277C2×C4—1582no—
278C2×C4—1582no—
279C2×C4#1457907511yescharacteristic
280C2×C4#1457907511yescharacteristic
281C2×C4#1457908912yes—
282C2×C4#1457918112yes—
283C8—44no—
284C8—6322no—
285C22—21no—
286C22—21no—
287C22C2×#1457907511yescharacteristic
288C22#1909855711yescharacteristic
289C22C2×#1457908911yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—3162no—
298C4—3162no—
299C4—3162no—
300C4—6321no—
301C4#1909836112yes—
302C2—21no—
303C2—21no—
304C2#2376188911yescharacteristic
305C2#2376504211yescharacteristic
306C2C2×#1909836111yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#101×C79C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#101D79
hypocenterC79#11736
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
159. C79#11736semidirect

Maximal subgroups

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

subgroupindex
#237666912
#237617242
#237614292
#237614052
C4×#101×C792
#1173679