Finite Group Databasebeta

← #34205100

Subgroups — #34205100

Order 20224.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#23766581C211yesmaximal, characteristic
2#23763794C211yesmaximal, characteristic
3#23763617C212yesmaximal
4#23761526C212yesmaximal
5C2×#356×C79C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098605—22no—
8#19098572—21no—
9C2×#14579080C2211yescharacteristic
10#19098470—21no—
11#19098470C2212yes—
12#19098449—22no—
13#19098449—22no—
14#19098431—21no—
15#19098324—22no—
16#19098323—21no—
17C2×C4⋊Q8×C79C2211yescharacteristic
18C2×#101×C79C2212yes—
19#356×C79C414yescomplemented
20C2×#11929608—21no—
21Q8×Dic79—42no—
22#14579120—21no—
23#14579120—22no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79D412yes—
26C2×C4×Dic79—21no—
27C2×C4×Dic79D412yes—
28#14579095—42no—
29#14579093—42no—
30#14579085—42no—
31#14579080—22no—
32C79⋊#95—21no—
33C79⋊#95—22no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C22×Q8×C79D411yescharacteristic
39C22×Q8×C79C2311yescharacteristic
40C2×M16×C79—22no—
41C4⋊Q8×C79—22no—
42C4⋊Q8×C79C2×C412yes—
43C4⋊Q8×C79Q812yes—
44C2×C4⋊C4×C79—22no—
45C2×C42×C79D411yescharacteristic
46#101×C79—22no—
47#101×C79C2×C414yes—
48C22×Dic79—22no—
49C22×Dic79—22no—
50#11929608—41no—
51#11929608—41no—
52C2×C79⋊C8—41no—
53C4⋊Dic79—22no—
54C4⋊Dic79—22no—
55C4⋊Dic79—22no—
56C4⋊Dic79—42no—
57C4⋊Dic79—81no—
58C22⋊Dic79—41no—
59C4×Dic79—42no—
60C4×Dic79—42no—
61C4×Dic79—41no—
62C4×Dic79—81no—
63C2×Q8×C79—21no—
64C2×Q8×C79—41no—
65C2×Q8×C79—41no—
66C2×Q8×C79—41no—
67C2×Q8×C79—41no—
68C2×Q8×C79C4⋊C412yes—
69C2×Q8×C79C22×C411yescharacteristic
70C2×Q8×C79C2×Q811yescharacteristic
71C2×Q8×C79#4112yes—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79C2×D412yes—
76M16×C79—24no—
77M16×C79—42no—
78C2×C8×C79—42no—
79C4⋊C4×C79—24no—
80C4⋊C4×C79—24no—
81C42×C79—21no—
82C42×C79C4⋊C412yes—
83C23×C79#12011yescharacteristic
84Q8×C79—41no—
85Q8×C79—41no—
86Q8×C79—42no—
87Q8×C79—42no—
88Q8×C79—81no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—22no—
92C2×C4×C79—21no—
93C2×C4×C79—22no—
94C2×C4×C79—22no—
95C2×C4×C79—42no—
96C2×C4×C79—41no—
97C2×C4×C79—41no—
98C2×C4×C79C2×C4⋊C411yescharacteristic
99C2×C4×C79C4×D412yes—
100C2×C4×C79C22⋊Q812yes—
101C2×C4×C79#12311yescharacteristic
102C2×Dic79—44no—
103C2×Dic79—44no—
104C2×Dic79—42no—
105C2×Dic79—42no—
106C2×Dic79—42no—
107C8×C79—44no—
108C79⋊C8—81no—
109C22×C79—22no—
110C22×C79#38511yescharacteristic
111C22×C79B(4,𝔽2)11yescharacteristic
112C22×C79#45811yescharacteristic
113C4×C79—21no—
114C4×C79—22no—
115C4×C79—22no—
116C4×C79—21no—
117C4×C79—42no—
118C4×C79—42no—
119Dic79—82no—
120Dic79—82no—
121Dic79—82no—
122#12971—791nomaximal, Sylow
123C2×C79—21no—
124C2×C79—21no—
125C2×C79#186711yescharacteristic
126C2×C79#220312yes—
127#2381—791no—
128C2×#356D7911yescharacteristic
129#1886—791no—
130#1857—792no—
131#1394—792no—
132C79#1297111yescharacteristic, complemented, Sylow
133C2×C4⋊Q8C2×D7911yescharacteristic
134#519—791no—
135#428—1582no—
136#419—1581no—
137C2×#101—791no—
138C2×#101C2×D7912yes—
139#391—792no—
140#391—1581no—
141#387—1582no—
142#387—1582no—
143#383—1581no—
144#356Dic7914yescomplemented
145#339—1582no—
146#339—1581no—
147C22×Q8D31611yescharacteristic
148C22×Q8C22×D7911yescharacteristic
149C2×M16—22no—
150C2×M16—1581no—
151C4⋊Q8—22no—
152C4⋊Q8Dic15812yes—
153C4⋊Q8C2×Dic7912yes—
154C4×Q8—3162no—
155#117—1582no—
156#117—1581no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—1581no—
160C2×C42—792no—
161C2×C42—1581no—
162C2×C42D31611yescharacteristic
163#107—3162no—
164#106—3162no—
165#103—3162no—
166#101—22no—
167#101—1582no—
168#101C2×Dic7914yes—
169#95—1581no—
170#95—1582no—
171#95—1581no—
172#95—1582no—
173#95—1582no—
174#95—1582no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic7912yes—
181C2×Q8C22×Dic7911yescharacteristic
182C2×Q8C2×Dic15811yescharacteristic
183C2×Q8#1192962312yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—1582no—
187C22×C4—1582no—
188C22×C4D4×D7912yes—
189C22×C4C2×D31611yescharacteristic
190M16—24no—
191M16—42no—
192M16—3161no—
193M16—3161no—
194C2×C8—42no—
195C2×C8—3161no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—3162no—
199C4⋊C4—6321no—
200C4⋊C4—1582no—
201C4⋊C4—1582no—
202C4⋊C4—1582no—
203C22⋊C4—3161no—
204C42—21no—
205C42—3162no—
206C42—3162no—
207C42—3161no—
208C42—6321no—
209C42C4⋊Dic7912yes—
210C23#1457913111yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—42no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—3164no—
226C2×C4—3164no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4C2×C4⋊Dic7911yescharacteristic
231C2×C4D4×Dic7912yes—
232C2×C4#1457913812yes—
233C2×C4#1457914511yescharacteristic
234C8—44no—
235C8—6321no—
236C22—22no—
237C22#1909843811yescharacteristic
238C22#1909875111yescharacteristic
239C22#1909875411yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—6322no—
247C4—6322no—
248C4—6322no—
249C2—21no—
250C2—21no—
251C2#2376368311yescharacteristic
252C2#2376542212yes—
subgroupG/H
centerC22#19098751
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#356×C79C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
cosocleC22×Q8×C79C23
hypercenterC2×#356D79
hypocenterC79#12971
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. #356×C79C4semidirect
132. C79#12971semidirect
144. #356Dic79semidirect

Maximal subgroups

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

subgroupindex
#237665812
#237637942
#237636172
#237615262
C2×#356×C792
#1297179