Finite Group Databasebeta

← #34197503

Subgroups — #34197503

Order 20224.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#23772097C211yesmaximal, characteristic
2C22⋊C16×D79C212yesmaximal, complemented
3#23761341C212yesmaximal, complemented
4#23761326C211yesmaximal, characteristic, complemented
5C79×#1315C211yesmaximal, characteristic, complemented
6C2×#14579168C411yescharacteristic
7C22×C8×D79C2211yescharacteristic
8#19099070C411yescharacteristic
9C2×C16×D79—22no—
10#19098637—22no—
11(C22⋊D79)⋊C8—22no—
12#19098589C2211yescharacteristic
13#19098526C411yescharacteristic
14#19098521C411yescharacteristic
15C22⋊(C79⋊C16)C2212yes—
16(C2×D79)⋊C16—22no—
17#408×C79C2211yescharacteristic
18C22⋊C16×C79C2212yes—
19C22×C4×D79C2×C411yescharacteristic
20#14579168—42no—
21C2×#11929608—21no—
22C2×#11929608C2×C411yescharacteristic
23C22×C79⋊C8—21no—
24C22×C79⋊C8C2×C411yescharacteristic
25C2×C8×D79—42no—
26C2×C8×D79D412yes—
27C2×C8×D79D412yes—
28#14579142—21no—
29#14579137C812yes—
30#14579132C812yes—
31#14579129—21no—
32C4×C22⋊D79—22no—
33#14579120C2×C411yescharacteristic
34C2×C79⋊C16—22no—
35C16×D79—44no—
36Dic79⋊C8—22no—
37(C2×D79)⋊C8—22no—
38#14579071—21no—
39C8×Dic79—21no—
40C2×#41×C79C2×C411yescharacteristic
41C2×M16×C79C2×C411yescharacteristic
42C22×C8×C79C2311yescharacteristic
43C2×C16×C79—22no—
44C22⋊C8×C79—22no—
45C23×D79M1611yescharacteristic
46C2×C22⋊D79—22no—
47C22×Dic79M1611yescharacteristic
48C2×C4×D79—41no—
49C2×C4×D79—41no—
50C2×C4×D79C22⋊C412yes—
51C2×C4×D79C2×C812yes—
52#11929608—41no—
53#11929608—42no—
54C2×C79⋊C8—22no—
55C2×C79⋊C8—21no—
56C2×C79⋊C8—42no—
57C2×C79⋊C8C22⋊C411yescharacteristic
58C2×C79⋊C8C22⋊C411yescharacteristic
59C8×D79—24no—
60C8×D79—42no—
61C4⋊Dic79C2×C812yes—
62Dic79⋊C4—22no—
63C22⋊Dic79—22no—
64C22⋊Dic79—21no—
65(C2×D79)⋊C4—22no—
66C4×Dic79—21no—
67C79⋊C16—42no—
68#41×C79—41no—
69#41×C79—41no—
70C2×Q8×C79C2×C811yescharacteristic
71C2×D4×C79—21no—
72C2×D4×C79C2×C811yescharacteristic
73C22×C4×C79—21no—
74C22×C4×C79C22×C411yescharacteristic
75M16×C79—41no—
76C2×C8×C79—21no—
77C2×C8×C79—41no—
78C2×C8×C79C2×D412yes—
79C16×C79—42no—
80C23×C79—21no—
81C23×C79C2×M1611yescharacteristic
82C22×D79—24no—
83C22×D79—41no—
84C22×D79C22⋊C812yes—
85Q8×C79—41no—
86D4×C79—41no—
87D4×C79—42no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79C2×C22⋊C411yescharacteristic
94C2×C4×C79C22×C811yescharacteristic
95C22⋊D79—44no—
96C2×Dic79—22no—
97C2×Dic79—41no—
98C2×Dic79C22⋊C812yes—
99C4×D79—22no—
100C4×D79—22no—
101C4×D79—42no—
102C8×C79—22no—
103C8×C79—41no—
104C79⋊C8—21no—
105C79⋊C8—41no—
106C79⋊C8#9912yes—
107C22×C79—22no—
108C22×C79—21no—
109C22×C79—42no—
110C22×C79—41no—
111C22×C79C2×C22⋊C811yescharacteristic
112C2×D79—22no—
113C2×D79—22no—
114C2×D79—44no—
115C2×D79#35014yes—
116C4×C79—21no—
117C4×C79—41no—
118C4×C79C2×#9911yescharacteristic
119C4×C79#41311yescharacteristic
120Dic79—22no—
121Dic79—42no—
122#11468—791nomaximal, Sylow
123C2×C79—41no—
124C2×C79—81no—
125C2×C79#145911yescharacteristic
126C2×C79C2×#35012yes—
127D79—24no—
128#2929—791no—
129C2×C22⋊C16—792no—
130#1323—792no—
131#1315—791no—
132#1315D7911yescharacteristic, complemented
133C79#1146811yescharacteristic, complemented, Sylow
134C2×#131—791no—
135C23×C8—791no—
136#525—791no—
137C22×C16—1582no—
138#435—1582no—
139C8×D4—1582no—
140#424—791no—
141#408C2×D7911yescharacteristic
142#407—791no—
143#405—791no—
144C22⋊C16—792no—
145C22⋊C16—1582no—
146C22⋊C16C2×D7912yes—
147C2×#41C4×D7911yescharacteristic
148C23×C4—791no—
149#131—3162no—
150C2×M16—791no—
151C2×M16—1581no—
152C2×M16C4×D7911yescharacteristic
153C22×C8—3162no—
154C22×C8—792no—
155C22×C8—791no—
156C22×C8—792no—
157C22×C8—1581no—
158C22×C8C22×D7911yescharacteristic
159#123—1581no—
160C22⋊Q8—792no—
161#121—792no—
162#120—1581no—
163C4×D4—1582no—
164#117—791no—
165C2×C16—22no—
166C2×C16—3164no—
167C2×C16—1582no—
168C4⋊C8—1582no—
169C22⋊C8—22no—
170C22⋊C8—1582no—
171#97—1581no—
172C4×C8—1581no—
173C24—791no—
174#41—41no—
175#41—41no—
176C2×Q8C8×D7911yescharacteristic
177C2×D4—21no—
178C2×D4—1582no—
179C2×D4C8×D7911yescharacteristic
180C22×C4—21no—
181C22×C4—3161no—
182C22×C4—3161no—
183C22×C4—791no—
184C22×C4—792no—
185C22×C4—792no—
186C22×C4C2×C4×D7911yescharacteristic
187M16—41no—
188M16—3162no—
189M16—3161no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—3162no—
193C2×C8—3162no—
194C2×C8—791no—
195C2×C8—791no—
196C2×C8—1584no—
197C2×C8—1581no—
198C2×C8—1582no—
199C2×C8D4×D7912yes—
200C4⋊C4—792no—
201C4⋊C4—1582no—
202C22⋊C4—1582no—
203C22⋊C4—1582no—
204C22⋊C4—1581no—
205C42—1581no—
206C16—42no—
207C16—3162no—
208C23—21no—
209C23—3161no—
210C23—792no—
211C23—1584no—
212C23M16×D7911yescharacteristic
213Q8—41no—
214D4—41no—
215D4—42no—
216D4—3164no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3162no—
223C2×C4—3161no—
224C2×C4—792no—
225C2×C4—1582no—
226C2×C4—1582no—
227C2×C4—1582no—
228C2×C4C22⋊C4×D7911yescharacteristic
229C2×C4C2×C8×D7911yescharacteristic
230C8—22no—
231C8—41no—
232C8—3161no—
233C8—792no—
234C8—1581no—
235C22—22no—
236C22—21no—
237C22—42no—
238C22—41no—
239C22—3164no—
240C22—794no—
241C22—1582no—
242C22—1582no—
243C22C22⋊C8×D7911yescharacteristic
244C4—21no—
245C4—41no—
246C4—3162no—
247C4—1582no—
248C4#99×D7911yescharacteristic
249C4#1909854411yescharacteristic
250C2—41no—
251C2—81no—
252C2—1584no—
253C2#2376171511yescharacteristic
254C2#350×D7912yes—
subgroupG/H
centerC2×C4C22⋊C4×D79
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC79×#1315C2
socleC22×C79C2×C22⋊C8
prime radical#19099070C4
cosocleC22×C8×C79C23
hypercenter#1315D79
hypocenterC79#11468
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D79C2semidirect
3. #23761341C2semidirect
4. #23761326C2semidirect
5. C79×#1315C2semidirect
132. #1315D79semidirect
133. C79#11468semidirect

Maximal subgroups

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

subgroupindex
#237720972
C22⋊C16×D792
#237613412
#237613262
C79×#13152
#1146879