Finite Group Databasebeta

← #34177552

Subgroups — #34177552

Order 20224.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#23769772C211yesmaximal, characteristic
2#23764964C212yesmaximal, complemented
3#23764933C211yesmaximal, characteristic
4#23761224C212yesmaximal, complemented
5C79×#1270C211yesmaximal, characteristic
6C2×#14579138—22no—
7C2×#14579092—21no—
8C2×#14579082—21no—
9#19098486—22no—
10#19098483—21no—
11#19098461C2212yes—
12#19098443—21no—
13#19098433C2211yescharacteristic
14#19098326—22no—
15#19098326—22no—
16#19098318C2211yescharacteristic
17#390×C79C2211yescharacteristic
18#336×C79C2212yes—
19C22×Dic158—21no—
20C23×Dic79—21no—
21C22×C79⋊C8—21no—
22#14579138—44no—
23#14579138—44no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79D412yes—
27C2×C4⋊Dic79D412yes—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—21no—
32#14579092—42no—
33#14579082—42no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C22×D4×C79D411yescharacteristic
37C22×C8×C79—22no—
38C2×C4⋊C4×C79D411yescharacteristic
39C2×C22⋊C4×C79—22no—
40C2×C42×C79C2311yescharacteristic
41C2×Dic158—44no—
42C2×Dic158—81no—
43C22×Dic79—22no—
44C22×Dic79—22no—
45C22×Dic79—44no—
46C22×Dic79—41no—
47C22×Dic79—81no—
48C2×C79⋊C8—22no—
49C2×C79⋊C8—41no—
50C2×C79⋊C8—41no—
51C4⋊Dic79—22no—
52C4⋊Dic79—42no—
53C4⋊Dic79—41no—
54C4⋊Dic79—41no—
55C4⋊Dic79—44no—
56C4⋊Dic79SD1614yes—
57Dic79⋊C4—44no—
58Dic79⋊C4—42no—
59Dic79⋊C4—42no—
60C22⋊Dic79—44no—
61C22⋊Dic79—81no—
62C24×C79—21no—
63C2×D4×C79—21no—
64C2×D4×C79—41no—
65C2×D4×C79—41no—
66C2×D4×C79Q1612yes—
67C22×C4×C79—21no—
68C22×C4×C79C2×D412yes—
69C22×C4×C79C2×D411yescharacteristic
70C2×C8×C79—42no—
71C2×C8×C79—42no—
72C2×C8×C79—42no—
73C4⋊C4×C79—41no—
74C22⋊C4×C79—42no—
75C22⋊C4×C79—42no—
76C42×C79—41no—
77C23×C79—22no—
78C23×C79—42no—
79C23×C79—41no—
80C23×C79—42no—
81C23×C79—41no—
82C23×C79#12011yescharacteristic
83D4×C79—42no—
84D4×C79—81no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—21no—
89C2×C4×C79—42no—
90C2×C4×C79—41no—
91C2×C4×C79C2×SD1612yes—
92C2×C4×C79C2×Q1611yescharacteristic
93C2×C4×C79AGL(1,ℤ/8)12yes—
94C2×C4×C79#13711yescharacteristic
95C2×Dic79—24no—
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—42no—
101C2×Dic79—84no—
102C2×Dic79—81no—
103C2×Dic79—82no—
104Dic158—84no—
105C8×C79—82no—
106C79⋊C8—42no—
107C22×C79—42no—
108C22×C79—44no—
109C22×C79—82no—
110C22×C79—81no—
111C22×C79—81no—
112C22×C79#45012yes—
113C22×C79#45111yescharacteristic
114C22×C79#45311yescharacteristic
115C22×C79#45412yes—
116C22×C79#45811yescharacteristic
117C4×C79—21no—
118C4×C79—21no—
119C4×C79—22no—
120C4×C79—22no—
121C4×C79—81no—
122Dic79—44no—
123Dic79—82no—
124Dic79—161no—
125#7826—791nomaximal, Sylow
126C2×C79—82no—
127C2×C79#161411yescharacteristic
128C2×C79#161612yes—
129C2×C79#162911yescharacteristic
130C2×C79#163712yes—
131C2×C79#165711yescharacteristic
132#2683—791no—
133#2100—792no—
134#2094—791no—
135#1271—792no—
136#1270D7911yescharacteristic
137C79#782611yescharacteristic, complemented, Sylow
138C2×C22⋊Q8—1582no—
139C2×#106—1581no—
140C2×#102—1581no—
141#396—1582no—
142#395—1581no—
143#390C2×D7911yescharacteristic
144#389—792no—
145#386—1581no—
146#384—791no—
147#340—1582no—
148#340—1582no—
149#336—791no—
150#336C2×D7912yes—
151C22×Q8—1581no—
152C22×D4D31611yescharacteristic
153C23×C4—1581no—
154C22×C8—22no—
155C22×C8—1581no—
156C22⋊Q8—3164no—
157C22⋊Q8—3164no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—792no—
160C2×C4⋊C4—1582no—
161C2×C4⋊C4—1582no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4—1582no—
164C2×C4⋊C4D31611yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—1582no—
167C2×C22⋊C4—1581no—
168C2×C42C22×D7911yescharacteristic
169#106—3162no—
170#102—3162no—
171#95—1581no—
172#95—1582no—
173C24—21no—
174C2×Q8—3164no—
175C2×Q8—6321no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Dic31612yes—
180C22×C4—21no—
181C22×C4—3164no—
182C22×C4—3161no—
183C22×C4—6321no—
184C22×C4—1582no—
185C22×C4—1582no—
186C22×C4D4×D7912yes—
187C22×C4C2×D31611yescharacteristic
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—3161no—
192C2×C8—3161no—
193C2×C8—1582no—
194C4⋊C4—41no—
195C4⋊C4—3162no—
196C4⋊C4—3161no—
197C4⋊C4—3161no—
198C4⋊C4—3164no—
199C4⋊C4—3164no—
200C4⋊C4—3162no—
201C4⋊C4—3162no—
202C4⋊C4—794no—
203C4⋊C4—1582no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—3164no—
207C22⋊C4—6321no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1457913111yescharacteristic
215Q8—6324no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—3162no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—6324no—
230C2×C4—6321no—
231C2×C4—6322no—
232C2×C4—1584no—
233C2×C4SD16×D7912yes—
234C2×C4C2×Dic31611yescharacteristic
235C2×C4#1457918512yes—
236C2×C4#1457919411yescharacteristic
237C8—82no—
238C8—3162no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1909871512yes—
245C22#1909872111yescharacteristic
246C22#1909873311yescharacteristic
247C22#1909873712yes—
248C22#1909875411yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—3164no—
255C4—6322no—
256C4—12641no—
257C2—82no—
258C2#2376242711yescharacteristic
259C2#2376243812yes—
260C2#2376251711yescharacteristic
261C2#2376256812yes—
262C2#2376269911yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1270C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#1270D79
hypocenterC79#7826
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #23764964C2semidirect
4. #23761224C2semidirect
137. C79#7826semidirect

Maximal subgroups

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

subgroupindex
#237697722
#237649642
#237649332
#237612242
C79×#12702
#782679