Finite Group Databasebeta

← #34201470

Subgroups — #34201470

Order 20224.

396 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765578C212yesmaximal, complemented
2#23764823C211yesmaximal, characteristic, complemented
3#23761409C212yesmaximal, complemented
4#23761324C211yesmaximal, characteristic, complemented
5C79×#3186C211yesmaximal, characteristic
6C2×#14579095C2211yescharacteristic, complemented
7#19098423—21no—
8#19098388—22no—
9#19098364—22no—
10C22⋊(C79⋊C16)C2212yescomplemented
11#19098326—21no—
12#19098326C2211yescharacteristic, complemented
13C2×#121×C79—21no—
14#480×C79—22no—
15#466×C79—21no—
16#466×C79C2212yes—
17#447×C79—22no—
18C2×C22⋊C8×C79C2211yescharacteristic
19C2×C4⋊Dic79—21no—
20C2×C4⋊Dic79—21no—
21C2×C22⋊Dic79—41no—
22C2×C79⋊C16—22no—
23#14579095—22no—
24C79⋊#95—41no—
25C22×D4×C79—21no—
26C22×D4×C79—41no—
27C23×C4×C79D411yescharacteristic
28C2×D8×C79—21no—
29C2×D8×C79D412yes—
30C22×C8×C79D411yescharacteristic
31C22×C8×C79C2311yescharacteristic
32#121×C79—22no—
33#121×C79—22no—
34#121×C79—41no—
35#121×C79—41no—
36C2×C4⋊C4×C79—21no—
37C2×C22⋊C4×C79—41no—
38#107×C79—21no—
39#107×C79D412yes—
40#102×C79—22no—
41#102×C79—22no—
42C22⋊C8×C79—22no—
43C22×Dic79—41no—
44C22×Dic79—41no—
45C4⋊Dic79—41no—
46C4⋊Dic79—41no—
47C4⋊Dic79—41no—
48C4⋊Dic79—41no—
49C22⋊Dic79—82no—
50C79⋊C16—42no—
51C24×C79—41no—
52C24×C79D811yescharacteristic
53C2×D4×C79—22no—
54C2×D4×C79—41no—
55C2×D4×C79—42no—
56C2×D4×C79—42no—
57C2×D4×C79—81no—
58C2×D4×C79—81no—
59C2×D4×C79—82no—
60C2×D4×C79—82no—
61C22×C4×C79—21no—
62C22×C4×C79—21no—
63C22×C4×C79—41no—
64C22×C4×C79—41no—
65C22×C4×C79—41no—
66C22×C4×C79D811yescharacteristic
67C22×C4×C79C2×D411yescharacteristic
68D8×C79—42no—
69D8×C79—81no—
70C2×C8×C79—21no—
71C2×C8×C79—41no—
72C2×C8×C79—41no—
73C2×C8×C79C2×D412yes—
74C4⋊C4×C79—22no—
75C4⋊C4×C79—41no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C23×C79—21no—
79C23×C79—21no—
80C23×C79—41no—
81C23×C79—41no—
82C23×C79—42no—
83C23×C79—82no—
84C23×C79—82no—
85C23×C79—82no—
86C23×C79D1612yes—
87C23×C79C2×D811yescharacteristic
88D4×C79—42no—
89D4×C79—42no—
90D4×C79—82no—
91D4×C79—84no—
92D4×C79—82no—
93D4×C79—82no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—41no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—42no—
101C2×C4×C79—81no—
102C2×C4×C79—82no—
103C2×C4×C79#12011yescharacteristic
104C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
105C2×Dic79—81no—
106C2×Dic79—81no—
107C2×Dic79—81no—
108C2×Dic79—81no—
109C2×Dic79—81no—
110C2×Dic79—81no—
111C8×C79—22no—
112C8×C79—41no—
113C22×C79—21no—
114C22×C79—21no—
115C22×C79—22no—
116C22×C79—42no—
117C22×C79—41no—
118C22×C79—41no—
119C22×C79—81no—
120C22×C79—82no—
121C22×C79—82no—
122C22×C79—82no—
123C22×C79—164no—
124C22×C79#44711yescharacteristic
125C22×C79C2×D1611yescharacteristic
126C22×C79#50911yescharacteristic
127C4×C79—21no—
128C4×C79—41no—
129C4×C79—82no—
130C4×C79#45511yescharacteristic
131C4×C79#45811yescharacteristic
132Dic79—161no—
133Dic79—161no—
134#12278—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—42no—
138C2×C79—162no—
139C2×C79#160711yescharacteristic
140C2×C79#218411yescharacteristic
141C2×C79#219111yescharacteristic
142#3186D7911yescharacteristic
143#2232—792no—
144#2075—791no—
145#1347—792no—
146#1314—791no—
147C79#1227811yescharacteristic, complemented, Sylow
148C2×#121—21no—
149#480—22no—
150#466—21no—
151#466C2×D7912yes—
152#447—22no—
153C2×#107—791no—
154C2×C22⋊C8C2×D7911yescharacteristic
155#380—1581no—
156#366—1582no—
157#357—1582no—
158C22⋊C16—792no—
159#340—791no—
160#340—1581no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4C22⋊D7911yescharacteristic
164C2×D8—21no—
165C2×D8C22⋊D7912yes—
166C22×C8C22⋊D7911yescharacteristic
167C22×C8C22×D7911yescharacteristic
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1581no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—3161no—
177C2×C16—1582no—
178#107—21no—
179#107—1582no—
180#107C22⋊D7912yes—
181#102—22no—
182#102—22no—
183C22⋊C8—22no—
184#95—3161no—
185C24—41no—
186C24#1192960911yescharacteristic
187C2×D4—22no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—82no—
194C2×D4—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—3161no—
201C22×C4—3161no—
202C22×C4#1192960911yescharacteristic
203C22×C4C2×C22⋊D7911yescharacteristic
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×C22⋊D7912yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—3161no—
213C4⋊C4—3161no—
214C4⋊C4—3161no—
215C4⋊C4—3161no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—6322no—
219C16—3162no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23#1457910212yes—
229C23C2×#1192960911yescharacteristic
230D4—42no—
231D4—42no—
232D4—82no—
233D4—84no—
234D4—82no—
235D4—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—6321no—
246C2×C4—6321no—
247C2×C4—6321no—
248C2×C4—6321no—
249C2×C4—6321no—
250C2×C4—6321no—
251C2×C4#1457913011yescharacteristic
252C2×C4#1457918711yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#1909869711yescharacteristic
267C22C2×#1457910211yescharacteristic
268C22#1909899911yescharacteristic
269C4—21no—
270C4—41no—
271C4—82no—
272C4—12641no—
273C4—12641no—
274C4#1909874111yescharacteristic
275C4#1909875311yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—162no—
280C2#2376238611yescharacteristic
281C2#2376531211yescharacteristic
282C2#2376535111yescharacteristic
subgroupG/H
centerC22#19098697
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3186C2
socleC22×C79#447
hypercenter#3186D79
hypocenterC79#12278
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #23765578C2semidirect
2. #23764823C2semidirect
3. #23761409C2semidirect
4. #23761324C2semidirect
6. C2×#14579095C22semidirect
10. C22⋊(C79⋊C16)C22semidirect
12. #19098326C22semidirect
147. C79#12278semidirect

Maximal subgroups

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

subgroupindex
#237655782
#237648232
#237614092
#237613242
C79×#31862
#1227879