Finite Group Databasebeta

← #34297751

Subgroups — #34297751

Order 20224.

1302 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098616C211yesmaximal, characteristic
2#23765807C211yesmaximal, characteristic
3#23764362C214yesmaximal, complemented
4#23764353C214yesmaximal, complemented
5#23763486C214yesmaximal
6C22×C4⋊Q8×C79C211yesmaximal, characteristic
7C2×#14579120—21no—
8#19098616—24no—
9C2×C4⋊(C79⋊C8)C2212yes—
10C2×C22⋊(C79⋊C8)C2212yes—
11C2×C4×C79⋊C8C2212yes—
12#19098433C2212yes—
13#19098429—21no—
14#19098423—21no—
15#19098423C2212yes—
16C4×C4⋊Dic79—21no—
17C4×C4⋊Dic79C2212yes—
18#19098325C22116yes—
19C23×Q8×C79—21no—
20C2×C4⋊Q8×C79—21no—
21C2×C4⋊Q8×C79—21no—
22C2×C4⋊Q8×C79—24no—
23C2×C4⋊Q8×C79C418yescomplemented
24C2×C4⋊Q8×C79C2212yes—
25C2×C4⋊Q8×C79C2212yes—
26C22×C4⋊C4×C79—21no—
27C22×C4⋊C4×C79C2212yes—
28C22×C42×C79C2211yescharacteristic
29C22×C79⋊C8C2314yes—
30#14579120—44no—
31C2×C4⋊Dic79—22no—
32C2×C4⋊Dic79—22no—
33C2×C4⋊Dic79C2314yes—
34C2×C22⋊Dic79—22no—
35C2×C4×Dic79—22no—
36C4⋊(C79⋊C8)—24no—
37C22⋊(C79⋊C8)—24no—
38C4×C79⋊C8—24no—
39C79⋊#95—22no—
40C79⋊#95—22no—
41C22×Q8×C79—28no—
42C22×Q8×C79—22no—
43C22×Q8×C79—22no—
44C22×Q8×C79—44no—
45C22×Q8×C79—44no—
46C23×C4×C79—22no—
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C23×C4×C79C2311yescharacteristic
50C4⋊Q8×C79—28no—
51C4⋊Q8×C79—28no—
52C4⋊Q8×C79—24no—
53C4⋊Q8×C79—24no—
54C4⋊Q8×C79C2×C418yes—
55C4⋊Q8×C79C2×C418yes—
56C2×C4⋊C4×C79—28no—
57C2×C4⋊C4×C79—22no—
58C2×C4⋊C4×C79—22no—
59C2×C4⋊C4×C79—22no—
60C2×C4⋊C4×C79—24no—
61C2×C4⋊C4×C79C2×C418yes—
62C2×C4⋊C4×C79C2314yes—
63C2×C42×C79C2×C414yes—
64C2×C42×C79D411yescharacteristic
65C2×C42×C79D411yescharacteristic
66C2×C42×C79D414yes—
67C2×C42×C79C2311yescharacteristic
68C2×C42×C79C2311yescharacteristic
69C22×Dic79—24no—
70C2×C79⋊C8—24no—
71C2×C79⋊C8—28no—
72C4⋊Dic79—28no—
73C4⋊Dic79—44no—
74C4⋊Dic79—44no—
75C22⋊Dic79—44no—
76C4×Dic79—44no—
77C24×C79C2×D411yescharacteristic
78C2×Q8×C79—28no—
79C2×Q8×C79—28no—
80C2×Q8×C79—432no—
81C2×Q8×C79—44no—
82C2×Q8×C79—44no—
83C2×Q8×C79—44no—
84C2×Q8×C79—44no—
85C22×C4×C79—24no—
86C22×C4×C79—24no—
87C22×C4×C79—28no—
88C22×C4×C79—24no—
89C22×C4×C79—44no—
90C22×C4×C79—44no—
91C22×C4×C79C22⋊C414yes—
92C22×C4×C79C22⋊C414yes—
93C22×C4×C79D814yes—
94C22×C4×C79SD1614yes—
95C22×C4×C79C22×C412yes—
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D412yes—
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2411yescharacteristic
103C4⋊C4×C79—28no—
104C4⋊C4×C79—28no—
105C4⋊C4×C79—28no—
106C4⋊C4×C79—24no—
107C4⋊C4×C79C22×C418yes—
108C42×C79C22⋊C414yes—
109C42×C79C22⋊C414yes—
110C42×C79C22×C412yes—
111C42×C79C22×C412yes—
112C42×C79C2×D412yes—
113C42×C79C2×D412yes—
114C23×C79—22no—
115C23×C79—22no—
116C23×C79C2×C22⋊C412yes—
117C23×C79C2×D812yes—
118C23×C79C2×SD1612yes—
119C23×C79C22×D411yescharacteristic
120Q8×C79—416no—
121Q8×C79—416no—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—24no—
125C2×C4×C79—24no—
126C2×C4×C79—28no—
127C2×C4×C79—28no—
128C2×C4×C79—44no—
129C2×C4×C79—44no—
130C2×C4×C79—416no—
131C2×C4×C79#102116yes—
132C2×C4×C79C2×C22⋊C412yes—
133C2×C4×C79C2×C22⋊C412yes—
134C2×C4×C79C2×C22⋊C412yes—
135C2×C4×C79C2×C22⋊C412yes—
136C2×C4×C79C2×C22⋊C412yes—
137C2×C4×C79C2×D812yes—
138C2×C4×C79C2×D812yes—
139C2×C4×C79C2×SD1612yes—
140C2×C4×C79C2×SD1612yes—
141C2×C4×C79#13714yes—
142C2×C4×C79C23×C411yescharacteristic
143C2×C4×C79C22×D411yescharacteristic
144C2×C4×C792−1+414yes—
145C2×Dic79—44no—
146C2×Dic79—44no—
147C2×Dic79—44no—
148C79⋊C8—44no—
149C22×C79—22no—
150C22×C79—22no—
151C22×C79—24no—
152C22×C79—24no—
153C22×C79C2×#10214yes—
154C22×C79C22×C22⋊C411yescharacteristic
155C22×C79#53612yes—
156C22×C79C22×D811yescharacteristic
157C22×C79C22×SD1611yescharacteristic
158C22×C79C2×#13712yes—
159C4×C79—22no—
160C4×C79—22no—
161C4×C79—48no—
162C4×C79C2×#10214yes—
163C4×C79C2×#10214yes—
164C4×C79#41914yes—
165C4×C79#51912yes—
166C4×C79#53612yes—
167Dic79—84no—
168#21833—791nomaximal, Sylow
169C2×C79—24no—
170C2×C79#231011yescharacteristic
171C2×C79C22×#10211yescharacteristic
172C2×C79C2×#41911yescharacteristic
173C2×C79#322612yes—
174C2×C79#322912yes—
175C22×C4⋊Q8D7911yescharacteristic
176C2×#431—791no—
177#2285—791no—
178#1990—794no—
179#1988—794no—
180#1823—794no—
181C79#2183311yescharacteristic, complemented, Sylow
182C23×Q8—21no—
183C2×C4⋊Q8—21no—
184C2×C4⋊Q8—21no—
185C2×C4⋊Q8—24no—
186C2×C4⋊Q8Dic7918yescomplemented
187C2×C4⋊Q8C2×D7912yes—
188C2×C4⋊Q8C2×D7912yes—
189C2×#117—1581no—
190C22×C4⋊C4—21no—
191C22×C4⋊C4C2×D7912yes—
192C22×C42C2×D7911yescharacteristic
193#431—1584no—
194C2×C4⋊C8—792no—
195C2×C22⋊C8—792no—
196C2×C4×C8—792no—
197#384—792no—
198#382—1581no—
199#380—792no—
200#380—1581no—
201C4×C4⋊C4—792no—
202C4×C4⋊C4—1581no—
203#340—7916no—
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4C22⋊D7911yescharacteristic
211C23×C4C22⋊D7911yescharacteristic
212C23×C4C22×D7911yescharacteristic
213C22×C8—794no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8C2×Dic7918yes—
219C4⋊Q8C2×Dic7918yes—
220#117—3164no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—794no—
227C2×C4⋊C4—1582no—
228C2×C4⋊C4—1582no—
229C2×C4⋊C4C2×Dic7918yes—
230C2×C4⋊C4C22×D7914yes—
231C2×C22⋊C4—1582no—
232C2×C42—1582no—
233C2×C42C2×Dic7914yes—
234C2×C42C22⋊D7911yescharacteristic
235C2×C42C22⋊D7911yescharacteristic
236C2×C42C22⋊D7914yes—
237C2×C42C22×D7911yescharacteristic
238C2×C42C22×D7911yescharacteristic
239C4⋊C8—1584no—
240C22⋊C8—1584no—
241C4×C8—1584no—
242#95—1582no—
243#95—1582no—
244C24C2×C22⋊D7911yescharacteristic
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—432no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—1584no—
259C22×C4C22⋊Dic7914yes—
260C22×C4C22⋊Dic7914yes—
261C22×C4#1192960914yes—
262C22×C4#1192961214yes—
263C22×C4C22×Dic7912yes—
264C22×C4C2×C22⋊D7911yescharacteristic
265C22×C4C2×C22⋊D7911yescharacteristic
266C22×C4C2×C22⋊D7912yes—
267C22×C4C2×C22⋊D7911yescharacteristic
268C22×C4C2×C22⋊D7911yescharacteristic
269C22×C4C2×C22⋊D7911yescharacteristic
270C22×C4C23×D7911yescharacteristic
271C2×C8—1588no—
272C2×C8—1584no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—3164no—
278C4⋊C4—3164no—
279C4⋊C4—1588no—
280C4⋊C4C22×Dic7918yes—
281C22⋊C4—3164no—
282C42—3164no—
283C42C22⋊Dic7914yes—
284C42C22⋊Dic7914yes—
285C42C22×Dic7912yes—
286C42C22×Dic7912yes—
287C42C2×C22⋊D7912yes—
288C42C2×C22⋊D7912yes—
289C23—22no—
290C23—22no—
291C23C2×C22⋊Dic7912yes—
292C23C2×#1192960912yes—
293C23C2×#1192961212yes—
294C23C22×C22⋊D7911yescharacteristic
295Q8—416no—
296Q8—416no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—3164no—
307C2×C4—3164no—
308C2×C4—3164no—
309C2×C4#14579082116yes—
310C2×C4C2×C22⋊Dic7912yes—
311C2×C4C2×C22⋊Dic7912yes—
312C2×C4C2×C22⋊Dic7912yes—
313C2×C4C2×C22⋊Dic7912yes—
314C2×C4C2×C22⋊Dic7912yes—
315C2×C4C2×#1192960912yes—
316C2×C4C2×#1192960912yes—
317C2×C4C2×#1192961212yes—
318C2×C4C2×#1192961212yes—
319C2×C4#1457919214yes—
320C2×C4C23×Dic7911yescharacteristic
321C2×C4C22×C22⋊D7911yescharacteristic
322C2×C4#1457920914yes—
323C8—3164no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—22no—
328C22C2×#1457908214yes—
329C22C22×C22⋊Dic7911yescharacteristic
330C22#1909913712yes—
331C22C22×#1192960911yescharacteristic
332C22C22×#1192961211yescharacteristic
333C22C2×#1457919212yes—
334C4—22no—
335C4—22no—
336C4—48no—
337C4—6324no—
338C4C2×#1457908214yes—
339C4C2×#1457908214yes—
340C4#1909857214yes—
341C4#1909904112yes—
342C4#1909913712yes—
343C2—24no—
344C2#2376598611yescharacteristic
345C2C22×#1457908211yescharacteristic
346C2C2×#1909857211yescharacteristic
347C2#2377452112yes—
348C2#2377454412yes—
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC22×C4⋊Q8×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenterC22×C4⋊Q8D79
hypocenterC79#21833
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #23764362C2semidirect
4. #23764353C2semidirect
23. C2×C4⋊Q8×C79C4semidirect
181. C79#21833semidirect
186. C2×C4⋊Q8Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986162
#237658072
#237643622
#237643532
#237634862
C22×C4⋊Q8×C792
#2183379