Finite Group Databasebeta

← #34183792

Subgroups — #34183792

Order 20224.

602 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 115 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#23772441C211yesmaximal, characteristic, complemented
2#23766122C211yesmaximal, characteristic
3#23761790C212yesmaximal, complemented
4#23761228C212yesmaximal, complemented
5C79×#1460C211yesmaximal, characteristic, complemented
6C2×#14579157C411yescharacteristic
7C2×C4×D316C2211yescharacteristic
8#19098641—24no—
9#19098616C2212yes—
10#19098527—22no—
11#19098466C411yescharacteristic
12#19098464C411yescharacteristic
13#19098442—22no—
14#19098439—22no—
15#19098428C411yescharacteristic
16#19098409C2211yescharacteristic
17#19098290—24no—
18C2×C4×D4×C79C2211yescharacteristic
19#431×C79C2212yes—
20#323×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—22no—
23C22×C4×D79—21no—
24C2×#11929608—24no—
25#14579157—22no—
26#14579157—22no—
27C4×D316—22no—
28C4×D316—22no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C22⋊Dic79—22no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—22no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35C4⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—24no—
37C22⋊(C79⋊C8)—22no—
38C4×C79⋊C8D414yescomplemented
39C79⋊#95—22no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C22×D4×C79C2×C411yescharacteristic
43C23×C4×C79D412yes—
44C4×D4×C79—22no—
45C4×D4×C79—22no—
46C2×C4⋊C4×C79C2×C411yescharacteristic
47C2×C22⋊C4×C79D412yes—
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C4×C8×C79—22no—
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—21no—
55C2×D316—22no—
56C2×D316M1612yescomplemented
57C2×C22⋊D79—28no—
58C2×C22⋊D79—82no—
59C2×C22⋊D79—82no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C4×D79—42no—
64C2×C4×D79—82no—
65#11929608—84no—
66C2×C79⋊C8—24no—
67C4⋊Dic79—22no—
68C22⋊Dic79—82no—
69(C2×D79)⋊C4—41no—
70(C2×D79)⋊C4—41no—
71(C2×D79)⋊C4—82no—
72C4×Dic79—21no—
73C4×Dic79—22no—
74C4×Dic79M1612yescomplemented
75C24×C79C22⋊C412yes—
76C2×D4×C79—22no—
77C2×D4×C79—21no—
78C2×D4×C79—21no—
79C2×D4×C79—22no—
80C22×C4×C79—22no—
81C22×C4×C79—22no—
82C22×C4×C79—82no—
83C22×C4×C79C22⋊C412yes—
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C2×C8×C79—42no—
88C4⋊C4×C79—22no—
89C22⋊C4×C79—24no—
90C42×C79C2×D412yes—
91C42×C79#4112yes—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—22no—
95C23×C79—42no—
96C23×C79—42no—
97C23×C79C2×C22⋊C411yescharacteristic
98C22×D79—22no—
99C22×D79—42no—
100C22×D79—42no—
101C22×D79—41no—
102C22×D79—41no—
103D4×C79—42no—
104D4×C79—42no—
105C2×C4×C79—22no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—44no—
109C2×C4×C79—42no—
110C2×C4×C79—82no—
111C2×C4×C79C4×D412yes—
112C2×C4×C79#12011yescharacteristic
113C2×C4×C79#12112yes—
114C2×C4×C79#12311yescharacteristic
115C2×C4×C79C2×M1611yescharacteristic
116C2×C4×C79C2×M1611yescharacteristic
117C2×C4×C79#13112yes—
118C22⋊D79—88no—
119C2×Dic79—22no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—42no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127D316—24no—
128D316—42no—
129C4×D79—84no—
130C8×C79—82no—
131C79⋊C8—44no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—44no—
135C22×C79—42no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—82no—
139C22×C79#38611yescharacteristic
140C22×C79#40711yescharacteristic
141C22×C79#40811yescharacteristic
142C2×D79—44no—
143C2×D79—42no—
144C2×D79—82no—
145C2×D79—81no—
146C2×D79—81no—
147C4×C79—22no—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—42no—
151C4×C79#43512yes—
152C4×C79#43612yes—
153Dic79—42no—
154Dic79—81no—
155Dic79—81no—
156#9105—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—82no—
160C2×C79#187011yescharacteristic
161C2×C79#191011yescharacteristic
162C2×C79#191111yescharacteristic
163D79—82no—
164#2975—791no—
165#2328—791no—
166#1478—792no—
167#1460D7911yescharacteristic, complemented
168#1274—792no—
169C79#910511yescharacteristic, complemented, Sylow
170C2×#127—791no—
171C2×C4×D4—791no—
172C2×C4×D4C2×D7911yescharacteristic
173#436—1584no—
174#431—792no—
175#431C2×D7912yes—
176#407—1582no—
177#390—791no—
178#390—791no—
179#386—1582no—
180#386—1582no—
181#382—791no—
182#376—791no—
183D4⋊C8—1584no—
184#323—22no—
185C22×D4—791no—
186C22×D4—1582no—
187C22×D4C4×D7911yescharacteristic
188C23×C4—1581no—
189C23×C4D31612yes—
190C2×M16—1584no—
191#127—1582no—
192#127—1582no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—1582no—
196C4×D4—1582no—
197C2×C4⋊C4—791no—
198C2×C4⋊C4C4×D7911yescharacteristic
199C2×C22⋊C4—1581no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4—1582no—
202C2×C22⋊C4C22⋊D7912yes—
203C2×C42—791no—
204C2×C42—791no—
205C2×C42C22×D7911yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—1582no—
208C22⋊C8—22no—
209C22⋊C8—1584no—
210C22⋊C8—1582no—
211C4×C8—22no—
212C4×C8—794no—
213#95—1582no—
214#95—1581no—
215#95—1581no—
216C24—1581no—
217C24(C2×D79)⋊C412yes—
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—792no—
223C2×D4—6322no—
224C2×D4—6322no—
225C2×D4—1588no—
226C2×D4—1582no—
227C2×D4—1581no—
228C2×D4—1582no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—3162no—
233C22×C4—6322no—
234C22×C4—1581no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4(C2×D79)⋊C412yes—
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4C2×D31611yescharacteristic
241M16—6324no—
242C2×C8—42no—
243C2×C8—1584no—
244C4⋊C4—22no—
245C4⋊C4—1582no—
246C22⋊C4—24no—
247C22⋊C4—3161no—
248C22⋊C4—3161no—
249C22⋊C4—6322no—
250C22⋊C4—6322no—
251C42—792no—
252C42—1582no—
253C42—1581no—
254C42D4×D7912yes—
255C42#1192962312yes—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—3162no—
262C23—3162no—
263C23—3161no—
264C23—3161no—
265C23—1582no—
266C23C2×(C2×D79)⋊C411yescharacteristic
267D4—42no—
268D4—42no—
269D4—3162no—
270D4—6328no—
271D4—1584no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3162no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—6324no—
286C2×C4—1582no—
287C2×C4(C22⋊D79)⋊C412yes—
288C2×C4#1457913111yescharacteristic
289C2×C4#1457913312yes—
290C2×C4#1457914511yescharacteristic
291C2×C4M16×D7911yescharacteristic
292C2×C4M16×D7911yescharacteristic
293C2×C4#1457916712yes—
294C8—82no—
295C8—3164no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—3164no—
304C22—3162no—
305C22—6322no—
306C22—6321no—
307C22—6321no—
308C22#1909844011yescharacteristic
309C22#1909852611yescharacteristic
310C22#1909852911yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—3162no—
316C4—6321no—
317C4—6321no—
318C4#1909863712yes—
319C4#1909863812yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—6322no—
324C2#2376369511yescharacteristic
325C2#2376392511yescharacteristic
326C2#2376393011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1460C2
socleC22×C79#386
prime radicalC2×#14579157C4
cosocleC2×C42×C79C23
hypercenter#1460D79
hypocenterC79#9105
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
1. #23772441C2semidirect
3. #23761790C2semidirect
4. #23761228C2semidirect
5. C79×#1460C2semidirect
38. C4×C79⋊C8D4semidirect
56. C2×D316M16semidirect
74. C4×Dic79M16semidirect
167. #1460D79semidirect
169. C79#9105semidirect

Maximal subgroups

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

subgroupindex
#237724412
#237661222
#237617902
#237612282
C79×#14602
#910579