Finite Group Databasebeta

← #34205331

Subgroups — #34205331

Order 20224.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579104C211yesmaximal, characteristic
2C2×Q16×Dic79C211yesmaximal, characteristic, complemented
3C2×#19098365C211yesmaximal, characteristic
4#23764264C211yesmaximal, characteristic
5#23761491C211yesmaximal, characteristic, complemented
6#23761477C211yesmaximal, characteristic, complemented
7C2×#358×C79C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C2×Q8×Dic79—21no—
10C2×#14579104—22no—
11C2×#14579104—22no—
12C2×#14579104C414yescomplemented
13C22×C79⋊C16C2211yescharacteristic
14Q16×Dic79—22no—
15Q16×Dic79—22no—
16C2×#14579095C2211yescharacteristic
17C2×#14579085—21no—
18C2×#14579081—21no—
19C2×C8×Dic79C2211yescharacteristic
20#19098464—21no—
21#19098365—22no—
22C22×Q16×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#358×C79—22no—
26C22×D316—21no—
27C2×D632—21no—
28C2×D632—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632—21no—
32C2×D632C2×C412yes—
33C22×C79⋊C8D411yescharacteristic
34Q8×Dic79—24no—
35Q8×Dic79—42no—
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79—41no—
38C2×(C2×D79)⋊C4—41no—
39C2×(C2×D79)⋊C4—41no—
40C2×C4×Dic79—41no—
41C2×C4×Dic79D411yescharacteristic
42#14579104—28no—
43C2×C79⋊C16—21no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16C2×C412yes—
46#14579095—21no—
47#14579095D412yes—
48#14579085—24no—
49#14579081—42no—
50C8×Dic79—21no—
51C8×Dic79—21no—
52C22×Q8×C79—21no—
53C2×Q16×C79—21no—
54C2×Q16×C79—21no—
55C2×Q16×C79—21no—
56C2×Q16×C79C2×C412yes—
57C2×Q16×C79D412yes—
58C2×Q16×C79D412yes—
59C22×C8×C79C2311yescharacteristic
60C2×C4⋊C4×C79—21no—
61C2×C16×C79—21no—
62C2×C16×C79—21no—
63C2×C16×C79—21no—
64#107×C79—21no—
65#107×C79—21no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—41no—
69C2×D316—41no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C22×Dic79—41no—
74C22×Dic79D811yescharacteristic
75C22×Dic79D811yescharacteristic
76D632—22no—
77D632—22no—
78D632—41no—
79D632—41no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C4⋊Dic79—22no—
84C4⋊Dic79—44no—
85C4⋊Dic79—41no—
86(C2×D79)⋊C4—82no—
87(C2×D79)⋊C4—82no—
88C4×Dic79—21no—
89C4×Dic79—21no—
90C4×Dic79—44no—
91C79⋊C16—22no—
92C2×Q8×C79—22no—
93C2×Q8×C79—22no—
94C2×Q8×C79—22no—
95C2×Q8×C79—41no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C22×C4×C79—41no—
99C22×C4×C79—41no—
100C22×C4×C79C2×D411yescharacteristic
101Q16×C79—22no—
102Q16×C79—22no—
103Q16×C79—22no—
104Q16×C79C22⋊C414yescomplemented
105C2×C8×C79C22×C411yescharacteristic
106C2×C8×C79C2×D411yescharacteristic
107C2×C8×C79C2×D411yescharacteristic
108C2×C8×C79C2×D411yescharacteristic
109C2×C8×C79#4111yescharacteristic
110C2×C8×C79#4111yescharacteristic
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C16×C79—41no—
114C23×C79C2×D811yescharacteristic
115C22×D79—42no—
116C22×D79—81no—
117C22×D79—81no—
118C22×D79—81no—
119C22×D79—81no—
120C22×D79—81no—
121C22×D79—81no—
122Q8×C79—24no—
123Q8×C79—42no—
124Q8×C79—42no—
125Q8×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79#12011yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#13511yescharacteristic
136C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
138C2×Dic79—21no—
139C2×Dic79—21no—
140C2×Dic79—21no—
141C2×Dic79—21no—
142C2×Dic79—21no—
143C2×Dic79—42no—
144C2×Dic79—81no—
145C2×Dic79—81no—
146C2×Dic79SD3212yescomplemented
147D316—42no—
148D316—42no—
149D316—81no—
150D316—81no—
151C8×C79C2×C22⋊C411yescharacteristic
152C8×C79C4×D411yescharacteristic
153C8×C79#12111yescharacteristic
154C8×C79#12311yescharacteristic
155C79⋊C8—41no—
156C22×C79C4×D811yescharacteristic
157C22×C79#44711yescharacteristic
158C22×C79#45911yescharacteristic
159C22×C79C2×SD3211yescharacteristic
160C22×C79#50811yescharacteristic
161C22×C79#50911yescharacteristic
162C22×C79#51011yescharacteristic
163C2×D79—82no—
164C2×D79—82no—
165C2×D79—82no—
166C2×D79—161no—
167C2×D79—161no—
168C2×D79—161no—
169C2×D79—161no—
170C4×C79—44no—
171C4×C79—161no—
172C4×C79#38611yescharacteristic
173C4×C79#44211yescharacteristic
174C4×C79#44911yescharacteristic
175C4×C79#46311yescharacteristic
176Dic79—22no—
177Dic79—41no—
178Dic79—82no—
179#13013—791nomaximal, Sylow
180C2×C79#187911yescharacteristic
181C2×C79C4×SD3211yescharacteristic
182C2×C79#217511yescharacteristic
183C2×C79#218511yescharacteristic
184C2×C79#218611yescharacteristic
185C2×C79#220711yescharacteristic
186C2×C79#221111yescharacteristic
187D79—162no—
188C22×SD32—791no—
189C2×C4×Q16—791no—
190C2×#358D7911yescharacteristic, complemented
191C2×#357—791no—
192#1970—791no—
193#1380—791no—
194#1374—791no—
195C79#1301311yescharacteristic, complemented, Sylow
196C22×Q16C2×D7911yescharacteristic
197C22×D8—791no—
198C2×C4×Q8—1581no—
199C2×SD32—794no—
200C2×SD32—1582no—
201C2×SD32—1582no—
202C22×C16—791no—
203C22×C16C2×D7911yescharacteristic
204C4×Q16—1582no—
205C4×Q16—1582no—
206C2×#107—791no—
207C2×#107C2×D7911yescharacteristic
208C2×#103—1581no—
209C2×#102—1581no—
210C2×C4×C8—791no—
211#390—1581no—
212#358—22no—
213#357—1582no—
214C22×Q8—21no—
215C22×D4—1581no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D7912yes—
220C2×Q16D31612yes—
221C2×Q16C22⋊D7912yes—
222C2×D8—792no—
223C2×D8—1581no—
224C2×D8—1581no—
225C2×D8—1581no—
226C2×D8—1581no—
227C2×D8—1581no—
228C22×C8—791no—
229C22×C8C22×D7911yescharacteristic
230C4×Q8—3162no—
231C4×Q8—1584no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—3161no—
234C2×C4⋊C4—1581no—
235C2×C22⋊C4—3161no—
236C2×C22⋊C4—3161no—
237C2×C42—3161no—
238C2×C42—791no—
239SD32—1588no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—792no—
244C2×C16—1581no—
245C2×C16—1581no—
246#107—21no—
247#107—21no—
248#107—792no—
249#107—1581no—
250#103—1584no—
251#102—3162no—
252C4×C8—1581no—
253C4×C8—1581no—
254C24—3161no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—3161no—
262C2×D4—3161no—
263C2×D4—3161no—
264C2×D4—3161no—
265C2×D4—3161no—
266C2×D4—1582no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—3161no—
270C22×C4—791no—
271C22×C4—791no—
272C22×C4D4×D7911yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16(C2×D79)⋊C414yescomplemented
277D8—3161no—
278D8—3161no—
279D8—1582no—
280D8—1582no—
281C2×C8—1581no—
282C2×C8—1581no—
283C2×C8—1581no—
284C2×C8C2×C4×D7911yescharacteristic
285C2×C8D4×D7911yescharacteristic
286C2×C8C2×C22⋊D7911yescharacteristic
287C2×C8C2×D31611yescharacteristic
288C2×C8#1192962311yescharacteristic
289C2×C8#1192962311yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—3164no—
293C4⋊C4—3161no—
294C4⋊C4—1582no—
295C22⋊C4—6322no—
296C22⋊C4—6322no—
297C42—3164no—
298C42—1581no—
299C42—1581no—
300C16—41no—
301C16—1582no—
302C23—3162no—
303C23—6321no—
304C23—6321no—
305C23—6321no—
306C23—6321no—
307C23—6321no—
308C23—6321no—
309C23D8×D7911yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—3162no—
315D4—3162no—
316D4—6321no—
317D4—6321no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—3162no—
325C2×C4—792no—
326C2×C4—6321no—
327C2×C4—6321no—
328C2×C4—1581no—
329C2×C4—1581no—
330C2×C4—1581no—
331C2×C4—1581no—
332C2×C4—1581no—
333C2×C4(C22⋊D79)⋊C411yescharacteristic
334C2×C4#1457913111yescharacteristic
335C2×C4#1457913311yescharacteristic
336C2×C4#1457918211yescharacteristic
337C2×C4#1457918511yescharacteristic
338C2×C4#1457918811yescharacteristic
339C8—3161no—
340C8C2×(C2×D79)⋊C411yescharacteristic
341C8(C22⋊D79)⋊C411yescharacteristic
342C8#1457913311yescharacteristic
343C8#1457914511yescharacteristic
344C22—6322no—
345C22—6322no—
346C22—6322no—
347C22—12641no—
348C22—12641no—
349C22—12641no—
350C22—12641no—
351C22#1909864311yescharacteristic
352C22#1909869811yescharacteristic
353C22#1909875611yescharacteristic
354C22SD32×D7911yescharacteristic
355C22#1909899611yescharacteristic
356C22#1909900011yescharacteristic
357C22#1909900511yescharacteristic
358C4—44no—
359C4—161no—
360C4—3161no—
361C4—6322no—
362C4—1582no—
363C4#1909844011yescharacteristic
364C4#1909867211yescharacteristic
365C4#1909870911yescharacteristic
366C4#1909878011yescharacteristic
367C2—12642no—
368C2#2376374611yescharacteristic
369C2#2376524911yescharacteristic
370C2#2376526711yescharacteristic
371C2#2376531811yescharacteristic
372C2#2376532611yescharacteristic
373C2#2376543811yescharacteristic
374C2#2376546411yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#358×C79C2
socleC23×C79C2×D8
prime radicalC22×D632C22
cosocleC22×C8×C79C23
hypercenterC2×#358D79
hypocenterC79#13013
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic79C2semidirect
5. #23761491C2semidirect
6. #23761477C2semidirect
7. C2×#358×C79C2semidirect
12. C2×#14579104C4semidirect
104. Q16×C79C22⋊C4semidirect
146. C2×Dic79SD32semidirect
190. C2×#358D79semidirect
195. C79#13013semidirect
276. Q16(C2×D79)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145791042
C2×Q16×Dic792
C2×#190983652
#237642642
#237614912
#237614772
C2×#358×C792
#1301379