Finite Group Databasebeta

← #34441317

Subgroups — #34441317

Order 20224.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098981C211yesmaximal, characteristic
2#23771997C211yesmaximal, characteristic
3#23771996C211yesmaximal, characteristic
4#23765188C211yesmaximal, characteristic, complemented
5#23765187C212yesmaximal, complemented
6#23765185C212yesmaximal, complemented
7#23765182C211yesmaximal, characteristic, complemented
8#23765181C212yesmaximal, complemented
9C2×#19098390C211yesmaximal, characteristic, complemented
10#367×D79C212yesmaximal, complemented
11C79×#2159C211yesmaximal, characteristic, complemented
12C2×#14579167C2211yescharacteristic
13#19099044—21no—
14#19099043—21no—
15#19098981C418yescomplemented
16C2×#14579100C2212yes—
17C22×C79⋊C16C2211yescharacteristic
18C2×C16×D79C2212yes—
19#19098606—21no—
20#19098604—22no—
21#19098603—21no—
22#19098602—22no—
23#19098601C2211yescharacteristic
24#19098600C2212yes—
25#19098600C2212yes—
26C2×#14579095C2211yescharacteristic
27#107×D79C2212yes—
28#107×D79C2212yes—
29#19098391C2212yes—
30#19098390C2211yescharacteristic
31#19098390C2212yes—
32#19098390C2212yes—
33#19098390C2211yescharacteristic
34#19098389C2212yes—
35#19098388C2211yescharacteristic
36#19098388C2211yescharacteristic
37C2×M32×C79C2211yescharacteristic
38#427×C79C2211yescharacteristic
39C2×#107×C79C2211yescharacteristic
40#367×C79C2212yes—
41#366×C79C2212yes—
42C2×#11929624D411yescharacteristic
43#14579167C2×C414yes—
44#14579167Q814yes—
45C2×#11929608D411yescharacteristic
46C2×#11929607D412yes—
47C2×C8×D79C2312yes—
48C4×Dic158—41no—
49#14579126—41no—
50C4×D316—41no—
51#14579123—41no—
52C4×C22⋊D79—42no—
53(C22⋊D79)⋊C4—42no—
54#14579120—21no—
55#14579119—22no—
56#14579119—22no—
57C2×C4⋊Dic79—21no—
58C4⋊C4×D79—22no—
59C4⋊C4×D79—22no—
60#14579100C2×C418yescomplemented
61C2×C79⋊C16C2×C414yescomplemented
62C2×C79⋊C16C2311yescharacteristic
63C2×C79⋊C16C2311yescharacteristic
64C16×D79C2×C418yescomplemented
65#14579095—21no—
66#14579095C2311yescharacteristic
67#14579095C2311yescharacteristic
68#14579095C2312yes—
69#14579094—22no—
70#14579094—22no—
71#14579093—21no—
72#14579092—22no—
73#14579092—22no—
74C22×C8×C79C2311yescharacteristic
75#117×C79—21no—
76C2×C4⋊C4×C79—21no—
77M32×C79C2×C414yescomplemented
78C2×C16×C79C2312yes—
79#107×C79—21no—
80#107×C79C2312yes—
81#107×C79C2311yescharacteristic
82#107×C79C2311yescharacteristic
83#106×C79—21no—
84#11929624C4⋊C414yes—
85#11929624Q1614yes—
86C2×Dic158D811yescharacteristic
87C2×D316D811yescharacteristic
88C2×C22⋊D79D812yes—
89C22×Dic79—41no—
90C2×C4×D79—42no—
91C2×C4×D79—42no—
92C2×C4×D79C2×D412yes—
93#11929608C4⋊C414yes—
94#11929607C4⋊C418yes—
95C2×C79⋊C8C2×D412yes—
96C8×D79C22×C414yes—
97C8×D79C2×Q814yes—
98C4⋊Dic79—21no—
99C4⋊Dic79—22no—
100C4⋊Dic79—21no—
101C4⋊Dic79—41no—
102Dic79⋊C4—42no—
103Dic79⋊C4—42no—
104C22⋊Dic79—41no—
105(C2×D79)⋊C4—42no—
106(C2×D79)⋊C4—42no—
107C4×Dic79—42no—
108C4×Dic79—42no—
109C4×Dic79—41no—
110C79⋊C16C22×C412yes—
111C79⋊C16C22×C412yes—
112C22×C4×C79—41no—
113C22×C4×C79C2×D411yescharacteristic
114C2×C8×C79C22×C412yes—
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79C2×Q812yes—
117C2×C8×C79C2411yescharacteristic
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—41no—
122C22⋊C4×C79—41no—
123C42×C79—41no—
124C16×C79C22×C414yes—
125C23×C79C2×D811yescharacteristic
126C22×D79C2×D812yes—
127C2×C4×C79—41no—
128C2×C4×C79—42no—
129C2×C4×C79—41no—
130C2×C4×C79—161no—
131C2×C4×C79C2×C4⋊C412yes—
132C2×C4×C79C2×D811yescharacteristic
133C2×C4×C79C2×Q1612yes—
134C2×C4×C79C22×D411yescharacteristic
135C22⋊D79#10718yes—
136C2×Dic79—41no—
137C2×Dic79—42no—
138C2×Dic79—41no—
139C2×Dic79—161no—
140C2×Dic79C2×D812yes—
141D316#10714yes—
142C4×D79—162no—
143C4×D79—162no—
144C4×D79C2×C4⋊C414yes—
145C4×D79C2×Q1614yes—
146Dic158#10714yes—
147C8×C79C2×C4⋊C412yes—
148C8×C79C23×C411yescharacteristic
149C8×C79C22×Q811yescharacteristic
150C79⋊C8C2×C4⋊C414yes—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79C2×#10712yes—
154C22×C79C22×D811yescharacteristic
155C2×D79—22no—
156C2×D79—22no—
157C2×D79C2×#10714yes—
158C4×C79—81no—
159C4×C79—82no—
160C4×C79—81no—
161C4×C79C2×#10712yes—
162C4×C79C22×C4⋊C411yescharacteristic
163C4×C79C22×Q1611yescharacteristic
164Dic79—81no—
165Dic79—82no—
166Dic79—81no—
167Dic79C2×#10714yes—
168#33411—791nomaximal, Sylow
169C2×C79—22no—
170C2×C79C22×#10711yescharacteristic
171C2×C79#341611yescharacteristic
172C2×C79#341611yescharacteristic
173D79—24no—
174C2×#504—791no—
175#2913—791no—
176#2913—791no—
177#2159—791no—
178#2159—792no—
179#2159—792no—
180#2159D7911yescharacteristic, complemented
181#2158—791no—
182#2158—792no—
183C2×#367—791no—
184C2×#367—792no—
185C79#3341111yescharacteristic, complemented, Sylow
186C2×#131—791no—
187#520—1581no—
188#520—1581no—
189#504—798no—
190C2×M32—792no—
191C2×M32C2×D7911yescharacteristic
192C22×C16—791no—
193C22×C16—792no—
194#428—1582no—
195#428—1582no—
196#428—1581no—
197#428—1581no—
198#427—791no—
199#427—792no—
200#427—792no—
201#427C2×D7911yescharacteristic
202C2×#107—791no—
203C2×#107—792no—
204C2×#107—792no—
205C2×#107C2×D7911yescharacteristic
206#367—792no—
207#367—791no—
208#367—792no—
209#367—792no—
210#367—791no—
211#367C2×D7912yes—
212#366—792no—
213#366—791no—
214#366—791no—
215#366C2×D7912yes—
216C2×#41—791no—
217#131—794no—
218#131—794no—
219C2×M16—791no—
220C2×M16—792no—
221C22×C8—792no—
222C22×C8C22×D7911yescharacteristic
223C4×Q8—3161no—
224C4×Q8—3161no—
225C4×D4—3161no—
226C4×D4—3162no—
227C4×D4—3161no—
228C4×D4—3162no—
229#117—21no—
230#117—1582no—
231#117—1582no—
232#117—1581no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1582no—
235C2×C4⋊C4—1582no—
236C2×C4⋊C4—1581no—
237M32—798no—
238M32C4×D7914yescomplemented
239C2×C16—794no—
240C2×C16—791no—
241C2×C16—791no—
242C2×C16—798no—
243C2×C16C22×D7912yes—
244#107—21no—
245#107—791no—
246#107—792no—
247#107—791no—
248#107—1582no—
249#107—1582no—
250#107—1581no—
251#107C22×D7911yescharacteristic
252#107C22×D7911yescharacteristic
253#107C22×D7912yes—
254#106—21no—
255#106—1582no—
256#106—1582no—
257#106—1581no—
258#41—794no—
259#41—794no—
260C2×Q8—791no—
261C2×D4—792no—
262C2×D4—791no—
263C22×C4—41no—
264C22×C4—3162no—
265C22×C4—3162no—
266C22×C4—3161no—
267C22×C4—792no—
268C22×C4D4×D7911yescharacteristic
269M16—794no—
270M16—798no—
271C2×C8—792no—
272C2×C8—794no—
273C2×C8—794no—
274C2×C8C2×C4×D7912yes—
275C2×C8D4×D7911yescharacteristic
276C2×C8Q8×D7912yes—
277C2×C8C23×D7911yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—3162no—
283C4⋊C4—3162no—
284C4⋊C4—3161no—
285C4⋊C4—1581no—
286C4⋊C4—1582no—
287C4⋊C4—1581no—
288C22⋊C4—41no—
289C22⋊C4—3162no—
290C22⋊C4—3162no—
291C22⋊C4—3161no—
292C42—41no—
293C42—3162no—
294C42—3162no—
295C42—3161no—
296C16—792no—
297C16—792no—
298C16C2×C4×D7914yes—
299C23—792no—
300C23D8×D7911yescharacteristic
301Q8—794no—
302D4—798no—
303D4—794no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—161no—
308C2×C4—3161no—
309C2×C4—3162no—
310C2×C4—3161no—
311C2×C4—792no—
312C2×C4—794no—
313C2×C4—794no—
314C2×C4—12642no—
315C2×C4—12642no—
316C2×C4—12641no—
317C2×C4C4⋊C4×D7912yes—
318C2×C4D8×D7911yescharacteristic
319C2×C4Q16×D7912yes—
320C2×C4C2×D4×D7911yescharacteristic
321C8—794no—
322C8C4⋊C4×D7912yes—
323C8C22×C4×D7911yescharacteristic
324C8C2×Q8×D7911yescharacteristic
325C22—21no—
326C22—21no—
327C22—794no—
328C22—1582no—
329C22—1582no—
330C22#107×D7912yes—
331C22C2×D8×D7911yescharacteristic
332C4—81no—
333C4—82no—
334C4—81no—
335C4—794no—
336C4—6322no—
337C4—6321no—
338C4—6321no—
339C4#107×D7912yes—
340C4C2×C4⋊C4×D7911yescharacteristic
341C4C2×Q16×D7911yescharacteristic
342C2—22no—
343C2—1584no—
344C2C2×#107×D7911yescharacteristic
345C2#2377654511yescharacteristic
346C2#2377654611yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2159C2
socleC22×C79C22×D8
prime radicalC2×#11929624D4
cosocleC2×C8×C79C24
hypercenter#2159D79
hypocenterC79#33411
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #23765188C2semidirect
5. #23765187C2semidirect
6. #23765185C2semidirect
7. #23765182C2semidirect
8. #23765181C2semidirect
9. C2×#19098390C2semidirect
10. #367×D79C2semidirect
11. C79×#2159C2semidirect
15. #19098981C4semidirect
60. #14579100C2×C4semidirect
61. C2×C79⋊C16C2×C4semidirect
64. C16×D79C2×C4semidirect
77. M32×C79C2×C4semidirect
180. #2159D79semidirect
185. C79#33411semidirect
238. M32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989812
#237719972
#237719962
#237651882
#237651872
#237651852
#237651822
#237651812
C2×#190983902
#367×D792
C79×#21592
#3341179