Finite Group Databasebeta

← #34182548

Subgroups — #34182548

Order 20224.

388 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763571C211yesmaximal, characteristic, complemented
2#23762188C211yesmaximal, characteristic, complemented
3#23761839C211yesmaximal, characteristic
4#23761782C211yesmaximal, characteristic
5#23761289C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#2991C211yesmaximal, characteristic
8#19098616C2211yescharacteristic
9C2×#14579096—22no—
10#19098587C2211yescharacteristic
11C4×#11929608C2211yescharacteristic
12C2×C4×C79⋊C8C2211yescharacteristic
13#19098315—21no—
14#19098313—21no—
15Q8⋊(C79⋊C8)—21no—
16Q8⋊(C79⋊C8)C411yescharacteristic
17Q8⋊(C79⋊C8)C411yescharacteristic
18D4⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)C411yescharacteristic
20D4⋊(C79⋊C8)C411yescharacteristic
21C4×#41×C79C2211yescharacteristic
22C8×Q8×C79—21no—
23#436×C79—22no—
24#436×C79—21no—
25#431×C79—21no—
26#431×C79C2211yescharacteristic
27#423×C79—21no—
28#423×C79C2211yescharacteristic
29C2×#11929608—21no—
30C2×#11929608—21no—
31C22×C79⋊C8—21no—
32#14579096—44no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)C2×C411yescharacteristic
36C4⋊(C79⋊C8)C2×C411yescharacteristic
37C22⋊(C79⋊C8)—21no—
38#14579072—21no—
39C4×C79⋊C8—21no—
40C4×C79⋊C8—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8C2×C411yescharacteristic
43C4×C79⋊C8C2×C411yescharacteristic
44C2×#41×C79Q811yescharacteristic
45C2×M16×C79—22no—
46C2×M16×C79D412yes—
47C4×Q8×C79C2×C411yescharacteristic
48C4×Q8×C79D411yescharacteristic
49C4×D4×C79—21no—
50C4×D4×C79—21no—
51C4×D4×C79C2×C411yescharacteristic
52C4×D4×C79D411yescharacteristic
53#117×C79—21no—
54#117×C79Q811yescharacteristic
55C2×C42×C79—21no—
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—22no—
58C4⋊C8×C79—21no—
59C4⋊C8×C79—21no—
60C4⋊C8×C79D411yescharacteristic
61C4⋊C8×C79D411yescharacteristic
62C22⋊C8×C79—22no—
63C22⋊C8×C79—21no—
64C4×C8×C79—22no—
65C4×C8×C79—21no—
66#11929608—22no—
67#11929608—41no—
68#11929608—42no—
69C2×C79⋊C8—22no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8—41no—
77#41×C79—22no—
78#41×C79—41no—
79C2×Q8×C79C4⋊C411yescharacteristic
80C2×D4×C79—21no—
81C2×D4×C79C4⋊C411yescharacteristic
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×Q811yescharacteristic
87C22×C4×C79#4111yescharacteristic
88M16×C79—42no—
89M16×C79—82no—
90C2×C8×C79—21no—
91C2×C8×C79—22no—
92C2×C8×C79—21no—
93C4⋊C4×C79—21no—
94C4⋊C4×C79—21no—
95C4⋊C4×C79C4⋊C411yescharacteristic
96C4⋊C4×C79C4⋊C411yescharacteristic
97C22⋊C4×C79—21no—
98C22⋊C4×C79—21no—
99C22⋊C4×C79—21no—
100C42×C79—21no—
101C42×C79—21no—
102C42×C79—21no—
103C42×C79C22×C411yescharacteristic
104C42×C79C2×D411yescharacteristic
105C42×C79C2×D411yescharacteristic
106C42×C79#4111yescharacteristic
107C23×C79—21no—
108C23×C79C22⋊Q811yescharacteristic
109Q8×C79—21no—
110Q8×C79#10812yescomplemented
111D4×C79—21no—
112D4×C79—21no—
113D4×C79—42no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
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—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79C2×C4⋊C411yescharacteristic
129C2×C4×C79C4×D411yescharacteristic
130C2×C4×C79C4×D411yescharacteristic
131C2×C4×C79#12011yescharacteristic
132C2×C4×C79C22⋊Q811yescharacteristic
133C2×C4×C79#12311yescharacteristic
134C8×C79—42no—
135C8×C79—41no—
136C8×C79—41no—
137C79⋊C8—22no—
138C79⋊C8—22no—
139C79⋊C8—42no—
140C79⋊C8—41no—
141C79⋊C8—41no—
142C79⋊C8—81no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79#38511yescharacteristic
150C4×C79—21no—
151C4×C79—22no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79C2×#10811yescharacteristic
157C4×C79#43011yescharacteristic
158C4×C79#44311yescharacteristic
159C4×C79#44411yescharacteristic
160C4×C79#45311yescharacteristic
161C4×C79#45611yescharacteristic
162#8899—791nomaximal, Sylow
163C2×C79—41no—
164C2×C79—81no—
165C2×C79#185511yescharacteristic
166C2×C79#185911yescharacteristic
167C2×C79#186611yescharacteristic
168#2991D7911yescharacteristic
169#1848—791no—
170#1574—791no—
171#1489—791no—
172#1476—791no—
173#1299—791no—
174#1278—791no—
175C79#889911yescharacteristic, complemented, Sylow
176C4×#41C2×D7911yescharacteristic
177C8×Q8—21no—
178#436—22no—
179#436—21no—
180#431—21no—
181#431—791no—
182#431C2×D7911yescharacteristic
183C2×#108—1582no—
184#423—21no—
185#423—791no—
186#423C2×D7911yescharacteristic
187C4×M16—791no—
188C2×C4×C8—791no—
189#335—1581no—
190#334—1581no—
191Q8⋊C8—791no—
192Q8⋊C8—791no—
193Q8⋊C8—1581no—
194D4⋊C8—791no—
195D4⋊C8—791no—
196D4⋊C8—1581no—
197C2×#41Dic15811yescharacteristic
198C2×M16—22no—
199C2×M16—1581no—
200C2×M16—1581no—
201C2×M16C22⋊D7912yes—
202C22×C8—1581no—
203C4×Q8C4×D7911yescharacteristic
204C4×Q8C22⋊D7911yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C4×D7911yescharacteristic
208C4×D4C22⋊D7911yescharacteristic
209#117—21no—
210#117Dic15811yescharacteristic
211C2×C42—21no—
212C2×C42C22×D7911yescharacteristic
213#108—3164no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—791no—
218C4⋊C8—791no—
219C4⋊C8—1581no—
220C4⋊C8—1581no—
221C4⋊C8C22⋊D7911yescharacteristic
222C4⋊C8C22⋊D7911yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—21no—
225C22⋊C8—1581no—
226#97—1581no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—791no—
230C4×C8—791no—
231C4×C8—1581no—
232C4×C8—1581no—
233C4×C8—1581no—
234#41—22no—
235#41—41no—
236C2×Q8Dic79⋊C411yescharacteristic
237C2×D4—21no—
238C2×D4Dic79⋊C411yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C2×C22⋊D7911yescharacteristic
243C22×C4C2×Dic15811yescharacteristic
244C22×C4#1192962411yescharacteristic
245M16—42no—
246M16—82no—
247M16—3161no—
248M16—3162no—
249M16—1582no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—21no—
253C2×C8—3161no—
254C2×C8—3161no—
255C2×C8—1581no—
256C2×C8—1581no—
257C2×C8—1582no—
258C2×C8—1582no—
259C2×C8—1581no—
260C2×C8—1581no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4Dic79⋊C411yescharacteristic
264C4⋊C4Dic79⋊C411yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42C2×C4×D7911yescharacteristic
272C42C2×C22⋊D7911yescharacteristic
273C42C2×C22⋊D7911yescharacteristic
274C42#1192962411yescharacteristic
275C23—21no—
276C23#1457914011yescharacteristic
277Q8—21no—
278Q8#1457909612yescomplemented
279D4—21no—
280D4—21no—
281D4—42no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4C2×Dic79⋊C411yescharacteristic
297C2×C4C4×C22⋊D7911yescharacteristic
298C2×C4C4×C22⋊D7911yescharacteristic
299C2×C4#1457913011yescharacteristic
300C2×C4#1457914011yescharacteristic
301C2×C4#1457914411yescharacteristic
302C8—42no—
303C8—41no—
304C8—41no—
305C8—3162no—
306C8—3161no—
307C8—3161no—
308C8—6321no—
309C8—1582no—
310C8—1582no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1909843611yescharacteristic
318C4—21no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4C2×#1457909611yescharacteristic
325C4#1909861111yescharacteristic
326C4#1909867811yescharacteristic
327C4#1909868311yescharacteristic
328C4#1909873111yescharacteristic
329C4#1909874611yescharacteristic
330C2—41no—
331C2—81no—
332C2#2376360511yescharacteristic
333C2#2376362611yescharacteristic
334C2#2376367511yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2991C2
socleC22×C79#385
prime radicalC2×#41×C79Q8
cosocleC2×C42×C79C23
hypercenter#2991D79
hypocenterC79#8899
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
1. #23763571C2semidirect
2. #23762188C2semidirect
5. #23761289C2semidirect
6. #23761236C2semidirect
110. Q8×C79#108semidirect
175. C79#8899semidirect
278. Q8#14579096semidirect

Maximal subgroups

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

subgroupindex
#237635712
#237621882
#237618392
#237617822
#237612892
#237612362
C79×#29912
#889979