Finite Group Databasebeta

← #34256279

Subgroups — #34256279

Order 20224.

1016 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772420C211yesmaximal, characteristic, complemented
2#23771828C211yesmaximal, characteristic, complemented
3#23762011C213yesmaximal, complemented
4#23762007C213yesmaximal, complemented
5#23761992C213yesmaximal, complemented
6#23761988C213yesmaximal, complemented
7C79×#3517C211yesmaximal, characteristic
8C2×#14579168—21no—
9Q8×C79⋊C8—21no—
10#19098641—23no—
11#19098620C2213yes—
12#19098519—23no—
13C4×#11929608C2213yes—
14C4×#11929608C2213yes—
15C2×#14579072C2213yes—
16#19098305C2211yescharacteristic
17#19098305C2213yes—
18#19098303C2213yes—
19#19098303C2213yes—
20#19098303C2216yes—
21#557×C79C411yescharacteristic
22#553×C79—23no—
23#550×C79—21no—
24#550×C79C413yes—
25#550×C79C411yescharacteristic
26#547×C79C413yes—
27#540×C79—23no—
28#532×C79C2213yes—
29#529×C79C2213yes—
30#520×C79—21no—
31C4×#41×C79C2211yescharacteristic
32#14579168—28no—
33C2×#11929608—23no—
34C2×#11929608—23no—
35C22×C79⋊C8—23no—
36C4⋊(C79⋊C8)—23no—
37C22⋊(C79⋊C8)—23no—
38#14579072—23no—
39#14579072C2311yescharacteristic
40#14579072C2313yes—
41#14579072C2311yescharacteristic
42#14579072C2313yes—
43C4×C79⋊C8—23no—
44C4×C79⋊C8—23no—
45C2×#41×C79—21no—
46C2×#41×C79C2×C416yes—
47C2×#41×C79D411yescharacteristic
48C4⋊Q8×C79C2×C413yes—
49#127×C79—23no—
50#127×C79C2×C413yes—
51#126×C79—23no—
52#125×C79—23no—
53#125×C79C2311yescharacteristic
54#124×C79C2313yes—
55#123×C79—23no—
56#123×C79—26no—
57C22⋊Q8×C79C2×C413yes—
58#121×C79—23no—
59#121×C79—23no—
60#121×C79—23no—
61#121×C79—23no—
62#121×C79—26no—
63#121×C79C2×C413yes—
64C4×Q8×C79—21no—
65C4×Q8×C79C2×C413yes—
66C4×Q8×C79C2×C411yescharacteristic
67C4×Q8×C79D411yescharacteristic
68C4×D4×C79—23no—
69C4×D4×C79—23no—
70C4×D4×C79—23no—
71C4×D4×C79—23no—
72C4×D4×C79C2×C413yes—
73C4×D4×C79C2×C413yes—
74C4×D4×C79D413yes—
75#117×C79—23no—
76#117×C79D413yes—
77C2×C4⋊C4×C79—23no—
78C2×C42×C79C2313yes—
79#11929608—212no—
80#11929608—46no—
81C2×C79⋊C8—212no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—23no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—23no—
86#41×C79—24no—
87#41×C79—44no—
88#41×C79—86no—
89C2×Q8×C79C22⋊C411yescharacteristic
90C2×Q8×C79C22×C413yes—
91C2×D4×C79—23no—
92C2×D4×C79—26no—
93C2×D4×C79—23no—
94C2×D4×C79—23no—
95C2×D4×C79C22⋊C413yes—
96C2×D4×C79C22×C413yes—
97C22×C4×C79—23no—
98C22×C4×C79—26no—
99C22×C4×C79C22×C413yes—
100C22×C4×C79C2×D413yes—
101C22×C4×C79C2×D413yes—
102C4⋊C4×C79—23no—
103C4⋊C4×C79—23no—
104C4⋊C4×C79—23no—
105C4⋊C4×C79—23no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79C22⋊C413yes—
109C4⋊C4×C79C22⋊C413yes—
110C4⋊C4×C79C22×C411yescharacteristic
111C4⋊C4×C79C22×C411yescharacteristic
112C22⋊C4×C79—23no—
113C22⋊C4×C79—23no—
114C22⋊C4×C79—23no—
115C22⋊C4×C79—23no—
116C22⋊C4×C79—23no—
117C22⋊C4×C79C22⋊C416yes—
118C42×C79—23no—
119C42×C79C22×C413yes—
120C42×C79C2×D413yes—
121C42×C79C2×D413yes—
122C42×C79C2411yescharacteristic
123C23×C79—23no—
124C23×C79C2×C22⋊C413yes—
125Q8×C79—43no—
126Q8×C79#10414yes—
127D4×C79—26no—
128D4×C79—43no—
129D4×C79—46no—
130D4×C79—83no—
131D4×C79—83no—
132D4×C79—86no—
133C2×C4×C79—26no—
134C2×C4×C79—21no—
135C2×C4×C79—23no—
136C2×C4×C79—21no—
137C2×C4×C79—43no—
138C2×C4×C79—43no—
139C2×C4×C79—46no—
140C2×C4×C79—86no—
141C2×C4×C79C2×C22⋊C413yes—
142C2×C4×C79C2×C22⋊C413yes—
143C2×C4×C79C2×C22⋊C413yes—
144C2×C4×C79C23×C411yescharacteristic
145C2×C4×C79C22×D411yescharacteristic
146C2×C4×C79C22×D411yescharacteristic
147C79⋊C8—22no—
148C79⋊C8—26no—
149C79⋊C8—22no—
150C79⋊C8—43no—
151C22×C79—23no—
152C22×C79—23no—
153C22×C79—23no—
154C22×C79—43no—
155C22×C79—43no—
156C22×C79—43no—
157C22×C79C22×C22⋊C411yescharacteristic
158C4×C79—21no—
159C4×C79—23no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—23no—
163C4×C79—43no—
164C4×C79—41no—
165C4×C79—41no—
166C4×C79C2×#10416yes—
167#17655—791nomaximal, Sylow
168C2×C79—43no—
169C2×C79—83no—
170C2×C79#288811yescharacteristic
171C2×C79C22×#10411yescharacteristic
172C2×C79#290111yescharacteristic
173#3517D7911yescharacteristic
174#2971—791no—
175#2875—791no—
176#1530—793no—
177#1529—793no—
178#1525—793no—
179#1524—793no—
180C79#1765511yescharacteristic, complemented, Sylow
181C2×#131—1581no—
182#557Dic7911yescharacteristic
183#553—23no—
184#550—21no—
185#550Dic7913yes—
186#550Dic7911yescharacteristic
187#547Dic7913yes—
188#540—23no—
189#532C2×D7913yes—
190#529C2×D7913yes—
191#520—21no—
192C4×#41C2×D7911yescharacteristic
193C8×Q8—1581no—
194#436—1583no—
195#432—793no—
196#405—1583no—
197C4×M16—793no—
198C4×M16—793no—
199C2×#97—793no—
200#330—791no—
201#330—793no—
202#329—793no—
203#329—793no—
204#329—796no—
205C2×#41—21no—
206C2×#41C2×Dic7916yes—
207C2×#41C22⋊D7911yescharacteristic
208#131—1588no—
209C2×M16—1583no—
210C2×M16—1583no—
211C22×C8—1583no—
212C4⋊Q8C2×Dic7913yes—
213#127—23no—
214#127C2×Dic7913yes—
215#126—23no—
216#125—23no—
217#125C22×D7911yescharacteristic
218#124C22×D7913yes—
219#123—23no—
220#123—26no—
221C22⋊Q8C2×Dic7913yes—
222#121—23no—
223#121—23no—
224#121—23no—
225#121—23no—
226#121—26no—
227#121C2×Dic7913yes—
228C4×Q8—21no—
229C4×Q8C2×Dic7913yes—
230C4×Q8C2×Dic7911yescharacteristic
231C4×Q8C22⋊D7911yescharacteristic
232C4×D4—23no—
233C4×D4—23no—
234C4×D4—23no—
235C4×D4—23no—
236C4×D4C2×Dic7913yes—
237C4×D4C2×Dic7913yes—
238C4×D4C22⋊D7913yes—
239#117—23no—
240#117C22⋊D7913yes—
241C2×C4⋊C4—23no—
242C2×C42C22×D7913yes—
243C4⋊C8—1583no—
244C22⋊C8—1583no—
245#97—791no—
246#97—793no—
247#97—791no—
248#97—793no—
249#97—1583no—
250C4×C8—1583no—
251C4×C8—1583no—
252#41—24no—
253#41—44no—
254#41—86no—
255C2×Q8C22⋊Dic7911yescharacteristic
256C2×Q8C22×Dic7913yes—
257C2×D4—23no—
258C2×D4—26no—
259C2×D4—23no—
260C2×D4—23no—
261C2×D4C22⋊Dic7913yes—
262C2×D4C22×Dic7913yes—
263C22×C4—23no—
264C22×C4—26no—
265C22×C4C22×Dic7913yes—
266C22×C4C2×C22⋊D7913yes—
267C22×C4C2×C22⋊D7913yes—
268M16—3166no—
269M16—15812no—
270C2×C8—1583no—
271C2×C8—15812no—
272C2×C8—1581no—
273C2×C8—1583no—
274C2×C8—1581no—
275C4⋊C4—23no—
276C4⋊C4—23no—
277C4⋊C4—23no—
278C4⋊C4—23no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C22⋊Dic7913yes—
282C4⋊C4C22⋊Dic7913yes—
283C4⋊C4C22×Dic7911yescharacteristic
284C4⋊C4C22×Dic7911yescharacteristic
285C22⋊C4—23no—
286C22⋊C4—23no—
287C22⋊C4—23no—
288C22⋊C4—23no—
289C22⋊C4—23no—
290C22⋊C4C22⋊Dic7916yes—
291C42—23no—
292C42C22×Dic7913yes—
293C42C2×C22⋊D7913yes—
294C42C2×C22⋊D7913yes—
295C42C23×D7911yescharacteristic
296C23—23no—
297C23C2×C22⋊Dic7913yes—
298Q8—43no—
299Q8#1457908814yes—
300D4—26no—
301D4—43no—
302D4—46no—
303D4—83no—
304D4—83no—
305D4—86no—
306C2×C4—26no—
307C2×C4—21no—
308C2×C4—23no—
309C2×C4—21no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—46no—
313C2×C4—86no—
314C2×C4C2×C22⋊Dic7913yes—
315C2×C4C2×C22⋊Dic7913yes—
316C2×C4C2×C22⋊Dic7913yes—
317C2×C4C23×Dic7911yescharacteristic
318C2×C4C22×C22⋊D7911yescharacteristic
319C2×C4C22×C22⋊D7911yescharacteristic
320C8—3163no—
321C8—1582no—
322C8—1586no—
323C8—1582no—
324C22—23no—
325C22—23no—
326C22—23no—
327C22—43no—
328C22—43no—
329C22—43no—
330C22C22×C22⋊Dic7911yescharacteristic
331C4—21no—
332C4—21no—
333C4—23no—
334C4—21no—
335C4—23no—
336C4—41no—
337C4—41no—
338C4—43no—
339C4C2×#1457908816yes—
340C2—43no—
341C2—83no—
342C2#2377187911yescharacteristic
343C2C22×#1457908811yescharacteristic
344C2#2377194711yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
Fitting, prime radicalC79×#3517C2
socleC22×C79C22×C22⋊C4
cosocleC42×C79C24
hypercenter#3517D79
hypocenterC79#17655
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3517

Complements

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

NH ≅ G/Ntype
1. #23772420C2semidirect
2. #23771828C2semidirect
3. #23762011C2semidirect
4. #23762007C2semidirect
5. #23761992C2semidirect
6. #23761988C2semidirect
180. C79#17655semidirect

Maximal subgroups

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

subgroupindex
#237724202
#237718282
#237620112
#237620072
#237619922
#237619882
C79×#35172
#1765579