Finite Group Databasebeta

← #34204266

Subgroups — #34204266

Order 20224.

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

normal subgroups only

This group has 500 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#124×Dic79C211yesmaximal, characteristic
2C2×#19098360C211yesmaximal, characteristic
3#23763394C212yesmaximal, complemented
4#23761526C212yesmaximal, complemented
5C2×#355×C79C211yesmaximal, characteristic
6C2×Q8×Dic79C2211yescharacteristic
7C2×D4×Dic79C411yescharacteristic
8#19098586—22no—
9C2×#14579080C2212yes—
10#19098470C2211yescharacteristic
11#19098466C411yescharacteristic
12#19098456—22no—
13#19098429C411yescharacteristic
14C22⋊C4×Dic79—22no—
15C42×Dic79C411yescharacteristic
16#19098360—22no—
17#19098331—22no—
18C2×#124×C79C2211yescharacteristic
19C2×#101×C79C2212yes—
20#355×C79—22no—
21C23×Dic79—21no—
22C2×#11929608—22no—
23Q8×Dic79—41no—
24Q8×Dic79—41no—
25D4×Dic79—41no—
26D4×Dic79—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79D412yes—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79C2×C411yescharacteristic
35C2×C4×Dic79C2×C411yescharacteristic
36C2×C4×Dic79D412yes—
37Dic79⋊C8—42no—
38Dic79⋊C8—42no—
39#14579080—22no—
40#14579080—22no—
41C79⋊#95—22no—
42C79⋊#95—22no—
43C22×Q8×C79C2311yescharacteristic
44C22×D4×C79C2×C411yescharacteristic
45C2×M16×C79—22no—
46#124×C79—22no—
47#124×C79D412yes—
48#124×C79Q812yes—
49C2×C22⋊C4×C79—22no—
50C2×C42×C79C2×C411yescharacteristic
51#101×C79—22no—
52#101×C79—22no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55C22×Dic79—44no—
56C22×Dic79—41no—
57C22×Dic79—41no—
58C22×Dic79C22⋊C411yescharacteristic
59C22×Dic79C22⋊C412yes—
60C22×Dic79C22⋊C411yescharacteristic
61#11929608—42no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—42no—
66C4⋊Dic79—41no—
67C22⋊Dic79—44no—
68C22⋊Dic79—42no—
69C4×Dic79—22no—
70C4×Dic79—22no—
71C4×Dic79—21no—
72C4×Dic79—21no—
73C4×Dic79—41no—
74C4×Dic79—42no—
75C4×Dic79—42no—
76C24×C79—21no—
77C2×Q8×C79—41no—
78C2×Q8×C79—41no—
79C2×Q8×C79C2×D411yescharacteristic
80C2×Q8×C79C2×Q811yescharacteristic
81C2×Q8×C79#4112yes—
82C2×D4×C79—21no—
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79C4⋊C412yes—
86C22×C4×C79—21no—
87C22×C4×C79C22×C411yescharacteristic
88C22×C4×C79C2×D412yes—
89M16×C79—42no—
90M16×C79—42no—
91C2×C8×C79—42no—
92C22⋊C4×C79—24no—
93C22⋊C4×C79—24no—
94C42×C79—21no—
95C42×C79C4⋊C412yes—
96C23×C79—22no—
97C23×C79—42no—
98C23×C79—42no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79C2×C22⋊C411yescharacteristic
102Q8×C79—41no—
103Q8×C79—41no—
104Q8×C79—42no—
105D4×C79—42no—
106D4×C79—42no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79C2×C4⋊C411yescharacteristic
115C2×C4×C79#11711yescharacteristic
116C2×C4×C79#12112yes—
117C2×C4×C79C22⋊Q812yes—
118C2×Dic79—22no—
119C2×Dic79—22no—
120C2×Dic79—21no—
121C2×Dic79—22no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—41no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130C2×Dic79#9911yescharacteristic
131C2×Dic79#9911yescharacteristic
132C8×C79—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—44no—
137C22×C79—42no—
138C22×C79—82no—
139C22×C79—81no—
140C22×C79—81no—
141C22×C79#38011yescharacteristic
142C22×C79C2×#9911yescharacteristic
143C22×C79#41011yescharacteristic
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—22no—
147C4×C79—22no—
148C4×C79—42no—
149Dic79—21no—
150Dic79—42no—
151Dic79—41no—
152Dic79—42no—
153Dic79—81no—
154Dic79#35512yescomplemented
155#12801—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79—82no—
159C2×C79#179911yescharacteristic
160C2×C79C2×#35511yescharacteristic
161C2×C79#213211yescharacteristic
162C4×#124—791no—
163C2×#355—791no—
164C2×#355D7911yescharacteristic
165#1801—792no—
166#1394—792no—
167C79#1280111yescharacteristic, complemented, Sylow
168C2×#124C2×D7911yescharacteristic
169C2×C4×Q8—791no—
170C2×C4×D4—791no—
171#423—1582no—
172C2×#101—792no—
173C2×#101C2×D7912yes—
174#391—791no—
175#390—791no—
176#388—1582no—
177#382—791no—
178C4×C22⋊C4—1582no—
179C43—791no—
180#355—22no—
181#355—1582no—
182#343—1582no—
183C22×Q8C22×D7911yescharacteristic
184C22×D4C4×D7911yescharacteristic
185C23×C4—1581no—
186C2×M16—22no—
187C2×M16—1582no—
188#124—22no—
189#124Dic15812yes—
190#124C22⋊D7912yes—
191C4×Q8—3161no—
192C4×Q8—3161no—
193C4×D4—3161no—
194C4×D4—3161no—
195C2×C4⋊C4—791no—
196C2×C4⋊C4—792no—
197C2×C4⋊C4—791no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1581no—
200C2×C22⋊C4—1582no—
201C2×C42—791no—
202C2×C42—791no—
203C2×C42—792no—
204C2×C42—1581no—
205C2×C42—1581no—
206C2×C42C4×D7911yescharacteristic
207C4⋊C8—3162no—
208C4⋊C8—3162no—
209#101—22no—
210#101—22no—
211#101—1582no—
212#101—1582no—
213#95—1582no—
214#95—1582no—
215C24—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8C2×C22⋊D7911yescharacteristic
219C2×Q8C2×Dic15811yescharacteristic
220C2×Q8#1192962312yes—
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4Dic79⋊C412yes—
225C22×C4—21no—
226C22×C4—3161no—
227C22×C4—3161no—
228C22×C4—3164no—
229C22×C4—792no—
230C22×C4—791no—
231C22×C4—791no—
232C22×C4—1581no—
233C22×C4—1581no—
234C22×C4C2×C4×D7911yescharacteristic
235C22×C4D4×D7912yes—
236M16—42no—
237M16—42no—
238M16—3162no—
239M16—3162no—
240C2×C8—42no—
241C2×C8—3162no—
242C4⋊C4—3161no—
243C4⋊C4—3161no—
244C4⋊C4—3162no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—3164no—
248C22⋊C4—3162no—
249C42—21no—
250C42—3161no—
251C42—3162no—
252C42—3162no—
253C42—1582no—
254C42—1582no—
255C42—1581no—
256C42—1581no—
257C42Dic79⋊C412yes—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23C22⋊C4×D7911yescharacteristic
264Q8—41no—
265Q8—41no—
266Q8—42no—
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3162no—
280C2×C4—3162no—
281C2×C4—3161no—
282C2×C4—791no—
283C2×C4—791no—
284C2×C4—6322no—
285C2×C4—6322no—
286C2×C4—1582no—
287C2×C4—1581no—
288C2×C4—1582no—
289C2×C4—1582no—
290C2×C4C2×Dic79⋊C411yescharacteristic
291C2×C4#1457911711yescharacteristic
292C2×C4#1457913312yes—
293C2×C4#1457913812yes—
294C8—82no—
295C8—6322no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22#1909842111yescharacteristic
304C22#99×D7911yescharacteristic
305C22#1909853511yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—42no—
311C4—3162no—
312C4—3161no—
313C4—3162no—
314C4—792no—
315C4—6321no—
316C4—1581no—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#2376338611yescharacteristic
321C2#355×D7911yescharacteristic
322C2#2376508211yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#355×C79C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC22×Q8×C79C23
hypercenterC2×#355D79
hypocenterC79#12801
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #23763394C2semidirect
4. #23761526C2semidirect
154. Dic79#355semidirect
167. C79#12801semidirect

Maximal subgroups

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

subgroupindex
#124×Dic792
C2×#190983602
#237633942
#237615262
C2×#355×C792
#1280179