Finite Group Databasebeta

← #34205097

Subgroups — #34205097

Order 20224.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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
2C2×#19098362C211yesmaximal, characteristic
3#23763615C212yesmaximal
4#23761526C212yesmaximal
5C79×#1886C211yesmaximal, characteristic
6#19098606—22no—
7C2×#14579080C2212yes—
8#19098470—21no—
9#19098470C2212yes—
10#19098449—22no—
11#19098449—22no—
12#19098431—21no—
13#19098362C414yescomplemented
14#19098322—22no—
15C2×C4⋊Q8×C79C2211yescharacteristic
16#519×C79C2211yescharacteristic
17#419×C79—21no—
18C2×#101×C79C2211yescharacteristic
19#339×C79—21no—
20C2×#11929608—22no—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79D412yes—
23C2×C4×Dic79—21no—
24C2×C4×Dic79D412yes—
25#14579094—42no—
26#14579092—42no—
27#14579080—22no—
28#14579080C2×C414yes—
29C79⋊#95—21no—
30C79⋊#95—22no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C22×Q8×C79D411yescharacteristic
36C22×Q8×C79C2311yescharacteristic
37C2×M16×C79—21no—
38C4⋊Q8×C79—22no—
39C4⋊Q8×C79C2×C412yes—
40C4⋊Q8×C79Q812yes—
41C4×Q8×C79—42no—
42#117×C79—21no—
43#117×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C42×C79D411yescharacteristic
46#103×C79—42no—
47#101×C79—22no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50#11929608—24no—
51#11929608—42no—
52C2×C79⋊C8—42no—
53C4⋊Dic79—42no—
54C4⋊Dic79—81no—
55C4×Dic79—42no—
56C4×Dic79—81no—
57C2×Q8×C79—21no—
58C2×Q8×C79—41no—
59C2×Q8×C79—41no—
60C2×Q8×C79—41no—
61C2×Q8×C79—41no—
62C2×Q8×C79C4⋊C412yes—
63C2×Q8×C79C22×C411yescharacteristic
64C2×Q8×C79C2×Q811yescharacteristic
65C2×Q8×C79#4112yes—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79C2×D411yescharacteristic
69C22×C4×C79C2×D412yes—
70M16×C79—41no—
71M16×C79—41no—
72C2×C8×C79—41no—
73C4⋊C4×C79—24no—
74C4⋊C4×C79—24no—
75C4⋊C4×C79—22no—
76C4⋊C4×C79—22no—
77C4⋊C4×C79—22no—
78C22⋊C4×C79—41no—
79C42×C79—21no—
80C42×C79—42no—
81C42×C79—41no—
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×C79—42no—
99C2×C4×C79C2×C4⋊C411yescharacteristic
100C2×C4×C79C4×D412yes—
101C2×C4×C79C22⋊Q812yes—
102C2×C4×C79#12311yescharacteristic
103C2×Dic79—44no—
104C2×Dic79—44no—
105C2×Dic79—42no—
106C2×Dic79—42no—
107C8×C79—81no—
108C79⋊C8—44no—
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—
119C4×C79—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×#356—791no—
129#1886D7911yescharacteristic
130#1857—792no—
131#1394—792no—
132C79#1297111yescharacteristic, complemented, Sylow
133C2×C4⋊Q8C2×D7911yescharacteristic
134#519C2×D7911yescharacteristic
135#428—1582no—
136#419—21no—
137C2×#101—792no—
138C2×#101C2×D7911yescharacteristic
139#391—792no—
140#391—1581no—
141#387—1582no—
142#387—1582no—
143#383—1581no—
144#356—794no—
145#339—21no—
146#339—1582no—
147C22×Q8C22⋊D7911yescharacteristic
148C22×Q8C22×D7911yescharacteristic
149C2×M16—21no—
150C2×M16—1582no—
151C4⋊Q8—22no—
152C4⋊Q8Dic15812yes—
153C4⋊Q8C4×D7912yes—
154C4×Q8—42no—
155#117—22no—
156#117—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—1581no—
160C2×C42—792no—
161C2×C42—1581no—
162C2×C42C22⋊D7911yescharacteristic
163#107—3162no—
164#106—3162no—
165#103—42no—
166#101—22no—
167#101—794no—
168#101—1582no—
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×Q8Dic79⋊C412yes—
181C2×Q8C2×C4×D7911yescharacteristic
182C2×Q8C2×Dic15811yescharacteristic
183C2×Q8#1192962312yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—1582no—
187C22×C4—1582no—
188C22×C4D4×D7912yes—
189C22×C4C2×C22⋊D7911yescharacteristic
190M16—41no—
191M16—41no—
192M16—3162no—
193M16—1584no—
194C2×C8—41no—
195C2×C8—3162no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—3162no—
202C4⋊C4—6321no—
203C22⋊C4—41no—
204C42—21no—
205C42—42no—
206C42—41no—
207C42—3162no—
208C42—6321no—
209C42Dic79⋊C412yes—
210C23#1457912911yescharacteristic
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—42no—
226C2×C4—3164no—
227C2×C4—3164no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4C2×Dic79⋊C411yescharacteristic
231C2×C4(C22⋊D79)⋊C412yes—
232C2×C4#1457913812yes—
233C2×C4#1457914211yescharacteristic
234C8—81no—
235C8—3164no—
236C22—22no—
237C22#1909843511yescharacteristic
238C22#1909874911yescharacteristic
239C22#1909875211yescharacteristic
240C4—21no—
241C4—22no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—6322no—
248C4—6322no—
249C2—21no—
250C2—21no—
251C2#2376368011yescharacteristic
252C2#2376541912yes—
subgroupG/H
centerC22#19098749
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#1886C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
cosocleC22×Q8×C79C23
hypercenter#1886D79
hypocenterC79#12971
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
13. #19098362C4semidirect
132. C79#12971semidirect

Maximal subgroups

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

subgroupindex
#237665812
C2×#190983622
#237636152
#237615262
C79×#18862
#1297179