Finite Group Databasebeta

← #34197772

Subgroups — #34197772

Order 20224.

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

normal subgroups only

This group has 334 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#23772106C211yesmaximal, characteristic
2#23765213C212yesmaximal, complemented
3#23761338C211yesmaximal, characteristic, complemented
4#23761335C212yesmaximal, complemented
5C79×#1321C211yesmaximal, characteristic, complemented
6C2×#14579168C411yescharacteristic
7C22×#11929607C411yescharacteristic
8#19099070C411yescharacteristic
9#19098638—22no—
10#19098631—22no—
11#19098589C411yescharacteristic
12#19098526C2211yescharacteristic
13#19098521C2211yescharacteristic
14#19098383—22no—
15#19098345C2212yes—
16#19098340—22no—
17#408×C79C2211yescharacteristic
18#349×C79C2212yes—
19C22×C4×D79C2×C411yescharacteristic
20#14579168—42no—
21C2×#11929608—21no—
22C2×#11929608C2×C411yescharacteristic
23C2×#11929607—21no—
24C2×#11929607—21no—
25C2×#11929607—42no—
26C22×C79⋊C8—21no—
27C22×C79⋊C8C2×C411yescharacteristic
28#14579142—21no—
29#14579137—21no—
30#14579132—21no—
31#14579129—21no—
32C4×C22⋊D79—22no—
33#14579120C2×C411yescharacteristic
34#14579101—22no—
35Dic79⋊C8—22no—
36(C2×D79)⋊C8—22no—
37#14579071D412yes—
38C8×Dic79D412yes—
39C2×#41×C79C2×C411yescharacteristic
40C2×M16×C79C2311yescharacteristic
41C22×C8×C79C2×C411yescharacteristic
42M32×C79—22no—
43C22⋊C8×C79—22no—
44C23×D79M1611yescharacteristic
45C2×C22⋊D79—22no—
46C22×Dic79M1611yescharacteristic
47C2×C4×D79—21no—
48C2×C4×D79—21no—
49C2×C4×D79—41no—
50C2×C4×D79—41no—
51#11929608—41no—
52#11929608—42no—
53#11929607—42no—
54#11929607—42no—
55C2×C79⋊C8—22no—
56C2×C79⋊C8—21no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8C22⋊C411yescharacteristic
59C2×C79⋊C8C22⋊C411yescharacteristic
60C4⋊Dic79—21no—
61Dic79⋊C4—22no—
62C22⋊Dic79—22no—
63C22⋊Dic79—21no—
64(C2×D79)⋊C4—22no—
65C4×Dic79C22⋊C412yes—
66C79⋊C16—42no—
67#41×C79—41no—
68#41×C79—41no—
69C2×Q8×C79M1611yescharacteristic
70C2×D4×C79—21no—
71C2×D4×C79M1611yescharacteristic
72C22×C4×C79—21no—
73C22×C4×C79C22×C411yescharacteristic
74M16×C79—41no—
75C2×C8×C79—21no—
76C2×C8×C79—41no—
77C2×C8×C79C2×D412yes—
78C16×C79—42no—
79C23×C79—21no—
80C23×C79C2×M1611yescharacteristic
81C22×D79—21no—
82C22×D79—41no—
83C22×D79—81no—
84Q8×C79—41no—
85D4×C79—41no—
86D4×C79—42no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—41no—
91C2×C4×C79—41no—
92C2×C4×C79C2×C22⋊C411yescharacteristic
93C2×C4×C79C2×M1611yescharacteristic
94C22⋊D79—82no—
95C2×Dic79—22no—
96C2×Dic79—21no—
97C2×Dic79—41no—
98C4×D79—42no—
99C4×D79—41no—
100C4×D79—41no—
101C8×C79—22no—
102C8×C79—41no—
103C79⋊C8—21no—
104C79⋊C8—41no—
105C79⋊C8#10012yes—
106C22×C79—21no—
107C22×C79—41no—
108C22×C79—42no—
109C22×C79—41no—
110C22×C79#40711yescharacteristic
111C2×D79—41no—
112C2×D79—41no—
113C2×D79—41no—
114C2×D79—82no—
115C4×C79—21no—
116C4×C79—41no—
117C4×C79#41011yescharacteristic
118C4×C79C2×#10011yescharacteristic
119Dic79—42no—
120Dic79—41no—
121#11538—791nomaximal, Sylow
122C2×C79—21no—
123C2×C79—41no—
124C2×C79—81no—
125C2×C79#146611yescharacteristic
126D79—81no—
127#2930—791no—
128#2166—792no—
129#1321—791no—
130#1321D7911yescharacteristic, complemented
131#1320—792no—
132C79#1153811yescharacteristic, complemented, Sylow
133C2×#131—791no—
134C22×M16—791no—
135#525—791no—
136#436—1582no—
137#435—1582no—
138#424—791no—
139#408C2×D7911yescharacteristic
140#407—791no—
141#405—791no—
142#364—1582no—
143#349—792no—
144#349C2×D7912yes—
145#347—1582no—
146C2×#41C4×D7911yescharacteristic
147C23×C4—791no—
148#131—3162no—
149C2×M16—3162no—
150C2×M16—791no—
151C2×M16—1581no—
152C2×M16—1581no—
153C2×M16—1581no—
154C2×M16C22×D7911yescharacteristic
155C22×C8—791no—
156C22×C8—1581no—
157C22×C8C4×D7911yescharacteristic
158#123—1581no—
159C22⋊Q8—1581no—
160#121—1581no—
161#120—1581no—
162C4×D4—1582no—
163#117—791no—
164M32—22no—
165M32—1582no—
166C4⋊C8—1582no—
167C22⋊C8—22no—
168C22⋊C8—1582no—
169#97—792no—
170C4×C8—792no—
171C24—791no—
172#41—41no—
173#41—41no—
174C2×Q8#1192960711yescharacteristic
175C2×D4—21no—
176C2×D4—1582no—
177C2×D4#1192960711yescharacteristic
178C22×C4—21no—
179C22×C4—3161no—
180C22×C4—3161no—
181C22×C4—791no—
182C22×C4—1581no—
183C22×C4—1581no—
184C22×C4C2×C4×D7911yescharacteristic
185M16—41no—
186M16—3162no—
187M16—3162no—
188M16—3162no—
189M16—3161no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—3162no—
193C2×C8—791no—
194C2×C8—791no—
195C2×C8—1581no—
196C2×C8—1582no—
197C2×C8D4×D7912yes—
198C4⋊C4—1582no—
199C4⋊C4—1581no—
200C22⋊C4—1582no—
201C22⋊C4—1582no—
202C22⋊C4—1581no—
203C42—792no—
204C16—42no—
205C16—3162no—
206C23—21no—
207C23—3161no—
208C23—6321no—
209C23—1581no—
210C23M16×D7911yescharacteristic
211Q8—41no—
212D4—41no—
213D4—42no—
214D4—6322no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—3162no—
221C2×C4—3161no—
222C2×C4—3161no—
223C2×C4—3161no—
224C2×C4—1582no—
225C2×C4—1581no—
226C2×C4C22⋊C4×D7911yescharacteristic
227C2×C4C2×#1192960711yescharacteristic
228C8—22no—
229C8—41no—
230C8—3161no—
231C8—792no—
232C8—1581no—
233C22—21no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—3161no—
238C22—3161no—
239C22—3161no—
240C22—6322no—
241C22#1909852511yescharacteristic
242C4—21no—
243C4—41no—
244C4—3162no—
245C4—3161no—
246C4#1909853511yescharacteristic
247C4#100×D7911yescharacteristic
248C2—21no—
249C2—41no—
250C2—81no—
251C2—6321no—
252C2#2376173911yescharacteristic
subgroupG/H
centerC4#19098535
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#1321C2
socleC2×C79#1466
prime radical#19099070C4
cosocleC2×M16×C79C23
hypercenter#1321D79
hypocenterC79#11538
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #23765213C2semidirect
3. #23761338C2semidirect
4. #23761335C2semidirect
5. C79×#1321C2semidirect
130. #1321D79semidirect
132. C79#11538semidirect

Maximal subgroups

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

subgroupindex
#237721062
#237652132
#237613382
#237613352
C79×#13212
#1153879