Finite Group Databasebeta

← #34205181

Subgroups — #34205181

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776502C211yesmaximal, characteristic, complemented
2C2×#19098683C211yesmaximal, characteristic
3#23765029C211yesmaximal, characteristic, complemented
4#23765028C211yesmaximal, characteristic, complemented
5#23761483C211yesmaximal, characteristic, complemented
6#23761479C211yesmaximal, characteristic, complemented
7C79×#2115C211yesmaximal, characteristic, complemented
8C2×#14579181C411yescharacteristic
9C2×#14579167C2211yescharacteristic
10#19098981—22no—
11#19098979—22no—
12C2×#14579101C2211yescharacteristic
13C2×#14579101C2211yescharacteristic
14M32×D79—22no—
15#19098683—22no—
16#19098683—22no—
17C2×#14579096C411yescharacteristic
18C2×#14579088C411yescharacteristic
19C2×#14579087C411yescharacteristic
20C2×#14579071C2211yescharacteristic
21#19098348—22no—
22#19098347—22no—
23C2×#131×C79C2211yescharacteristic
24C2×M32×C79C2211yescharacteristic
25C2×M32×C79C2211yescharacteristic
26#349×C79—22no—
27C2×#11929624C2×C411yescharacteristic
28#14579181—22no—
29#14579181—22no—
30C2×#11929614C811yescharacteristic
31C2×#11929612C811yescharacteristic
32C2×#11929611C811yescharacteristic
33C2×#11929609C811yescharacteristic
34#14579167—21no—
35#14579167—21no—
36#14579167D412yes—
37#14579167D412yes—
38C2×#11929608C2×C411yescharacteristic
39C2×#11929607—21no—
40C22×C79⋊C8C2×C411yescharacteristic
41C2×C8×D79—21no—
42C2×C4×Dic79C2×C411yescharacteristic
43#14579101—21no—
44#14579101—21no—
45#14579101—21no—
46#14579101—21no—
47#14579100—24no—
48C2×C79⋊C16—21no—
49C2×C79⋊C16D412yes—
50C16×D79—24no—
51#14579096—22no—
52#14579088—22no—
53#14579087—22no—
54#14579071—21no—
55#14579071—21no—
56C2×#41×C79C2×C411yescharacteristic
57#131×C79—21no—
58#131×C79—21no—
59#131×C79—21no—
60#131×C79—21no—
61C2×M16×C79—21no—
62C2×M16×C79C2×C411yescharacteristic
63C22×C8×C79—21no—
64C22×C8×C79C2311yescharacteristic
65M32×C79—21no—
66M32×C79—21no—
67M32×C79—21no—
68M32×C79D412yes—
69C2×C16×C79—21no—
70C2×C16×C79—21no—
71#11929624—21no—
72#11929624—21no—
73#11929624C22⋊C412yes—
74#11929624M1612yes—
75C2×Dic158C2×C811yescharacteristic
76C2×D316C2×C811yescharacteristic
77C2×C22⋊D79—21no—
78C22×Dic79—21no—
79C2×C4×D79—21no—
80#11929614—22no—
81#11929612—22no—
82#11929611—22no—
83#11929609—22no—
84#11929608—21no—
85#11929608C22⋊C412yes—
86#11929607—22no—
87#11929607—22no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8C2×C811yescharacteristic
92C2×C79⋊C8C2×C811yescharacteristic
93C8×D79—22no—
94C8×D79—22no—
95C4×Dic79—21no—
96C4×Dic79—21no—
97C79⋊C16—22no—
98C79⋊C16—41no—
99#41×C79—21no—
100#41×C79—21no—
101#41×C79—21no—
102#41×C79—21no—
103C2×Q8×C79C2×C811yescharacteristic
104C2×D4×C79—21no—
105C2×D4×C79C2×C811yescharacteristic
106C22×C4×C79—21no—
107C22×C4×C79C22×C411yescharacteristic
108M16×C79—21no—
109M16×C79—21no—
110M16×C79—41no—
111M16×C79—41no—
112C2×C8×C79—21no—
113C2×C8×C79—41no—
114C2×C8×C79—41no—
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79#4111yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C16×C79—22no—
122C16×C79—41no—
123C23×C79—21no—
124C23×C79#13111yescharacteristic
125C22×D79—21no—
126Q8×C79—21no—
127Q8×C79—21no—
128D4×C79—21no—
129D4×C79—21no—
130D4×C79—41no—
131D4×C79—41no—
132C2×C4×C79—21no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79C2×C22⋊C411yescharacteristic
136C2×C4×C79C4×D411yescharacteristic
137C2×C4×C79C4×D411yescharacteristic
138C2×C4×C79C22×C811yescharacteristic
139C2×C4×C79C2×M1611yescharacteristic
140C2×C4×C79#13111yescharacteristic
141C22⋊D79—22no—
142C22⋊D79—22no—
143C2×Dic79—21no—
144C2×Dic79—41no—
145C2×Dic79—81no—
146D316—21no—
147D316C22⋊C812yes—
148C4×D79—22no—
149C4×D79—22no—
150Dic158—21no—
151Dic158C22⋊C812yes—
152C8×C79—41no—
153C8×C79#12011yescharacteristic
154C8×C79#12111yescharacteristic
155C8×C79#12111yescharacteristic
156C8×C79#12311yescharacteristic
157C79⋊C8—22no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79—81no—
163C22×C79#40811yescharacteristic
164C22×C79C8×D411yescharacteristic
165C22×C79#43511yescharacteristic
166C2×D79—22no—
167C2×D79—81no—
168C4×C79—41no—
169C4×C79#38611yescharacteristic
170C4×C79C2×C22⋊C811yescharacteristic
171C4×C79C8×D411yescharacteristic
172C4×C79#43511yescharacteristic
173Dic79—22no—
174Dic79—81no—
175#12986—791nomaximal, Sylow
176C2×C79—21no—
177C2×C79—21no—
178C2×C79—21no—
179C2×C79—81no—
180C2×C79#186911yescharacteristic
181D79—42no—
182#3407—791no—
183C2×#444—791no—
184#2117—791no—
185#2117—791no—
186#2115D7911yescharacteristic, complemented
187#1377—791no—
188#1375—791no—
189C79#1298611yescharacteristic, complemented, Sylow
190C2×#135—791no—
191C2×#131—791no—
192C2×#131C2×D7911yescharacteristic
193#504—1582no—
194#504—1582no—
195C2×M32—791no—
196C2×M32—791no—
197C2×M32—1582no—
198C2×M32C2×D7911yescharacteristic
199C2×M32C2×D7911yescharacteristic
200#444—1582no—
201#444—1582no—
202C2×#108—791no—
203C2×#104—791no—
204C2×#104—791no—
205C2×#97—791no—
206#350—1582no—
207#350—1582no—
208#349—22no—
209C2×#41—791no—
210C2×#41C4×D7911yescharacteristic
211#135—1582no—
212#135—1582no—
213C2×Q16—791no—
214C2×SD16—791no—
215C2×SD16—791no—
216C2×D8—791no—
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—792no—
222#131—792no—
223#131—1581no—
224#131—1581no—
225C2×M16—21no—
226C2×M16—791no—
227C2×M16—1581no—
228C2×M16C4×D7911yescharacteristic
229C22×C8—21no—
230C22×C8—791no—
231C22×C8—1581no—
232C22×C8C22×D7911yescharacteristic
233C2×C42—791no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—1584no—
238M32—1581no—
239M32—1581no—
240M32—1581no—
241M32—1581no—
242M32C22⋊D7912yes—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—792no—
246C2×C16—1584no—
247C2×C16—1581no—
248#108—1582no—
249#104—1582no—
250#104—1582no—
251#97—1581no—
252#97—1581no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—792no—
258#41—792no—
259#41—1581no—
260#41—1581no—
261C2×Q8—791no—
262C2×Q8C8×D7911yescharacteristic
263C2×D4—21no—
264C2×D4—791no—
265C2×D4—1581no—
266C2×D4C8×D7911yescharacteristic
267C22×C4—21no—
268C22×C4—1581no—
269C22×C4—1581no—
270C22×C4C2×C4×D7911yescharacteristic
271Q16—1582no—
272SD16—1582no—
273SD16—1582no—
274D8—1582no—
275M16—21no—
276M16—21no—
277M16—41no—
278M16—41no—
279M16—792no—
280M16—1581no—
281M16—1582no—
282M16—1582no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—1581no—
289C2×C8—1581no—
290C2×C8—1581no—
291C2×C8—1582no—
292C2×C8—1582no—
293C2×C8D4×D7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8D4×D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8#1192962311yescharacteristic
298C2×C8#1192962411yescharacteristic
299C42—1581no—
300C42—1581no—
301C16—22no—
302C16—41no—
303C16—3161no—
304C16—1582no—
305C23—21no—
306C23—1581no—
307C23#1457916711yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—792no—
311Q8—1581no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—792no—
317D4—1582no—
318D4—1582no—
319D4—1581no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—3161no—
324C2×C4—6321no—
325C2×C4—1581no—
326C2×C4—1582no—
327C2×C4—1582no—
328C2×C4C22⋊C4×D7911yescharacteristic
329C2×C4(C22⋊D79)⋊C411yescharacteristic
330C2×C4C4×C22⋊D7911yescharacteristic
331C2×C4C2×C8×D7911yescharacteristic
332C2×C4M16×D7911yescharacteristic
333C2×C4#1457916911yescharacteristic
334C8—41no—
335C8—1582no—
336C8—1581no—
337C8—1581no—
338C8#1457912911yescharacteristic
339C8#1457913211yescharacteristic
340C8#1457913411yescharacteristic
341C8#1457914311yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—6321no—
346C22—1582no—
347C22#1909852811yescharacteristic
348C22C8×C22⋊D7911yescharacteristic
349C22#1909863711yescharacteristic
350C4—41no—
351C4—6321no—
352C4—1582no—
353C4#1909843911yescharacteristic
354C4C22⋊C8×D7911yescharacteristic
355C4(C22⋊D79)⋊C811yescharacteristic
356C4#1909863211yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—3162no—
362C2#2376368911yescharacteristic
subgroupG/H
centerC4#19098439
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC79×#2115C2
socleC2×C79#1869
prime radicalC2×#14579181C4
cosocleC22×C8×C79C23
hypercenter#2115D79
hypocenterC79#12986
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2115

Complements

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

NH ≅ G/Ntype
1. #23776502C2semidirect
3. #23765029C2semidirect
4. #23765028C2semidirect
5. #23761483C2semidirect
6. #23761479C2semidirect
7. C79×#2115C2semidirect
186. #2115D79semidirect
189. C79#12986semidirect

Maximal subgroups

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

subgroupindex
#237765022
C2×#190986832
#237650292
#237650282
#237614832
#237614792
C79×#21152
#1298679