Finite Group Databasebeta

← C73×#8309 #32917713

Subgroups — C73×#8309 #32917713

Order 18688.

874 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C22×#126×C73C211yesmaximal, characteristic
2C73×#1817C212yesmaximal, complemented
3C2×#324×C73C212yesmaximal, complemented
4C2×#323×C73C212yesmaximal
5C2×#126×C73—24no—
6C2×#126×C73—24no—
7C2×#126×C73—24no—
8C22×C4⋊C4×C73C411yescharacteristic
9C22×C4⋊C4×C73C411yescharacteristic
10C22×C4⋊C4×C73C2211yescharacteristic
11C22×C22⋊C4×C73C411yescharacteristic
12C22×C22⋊C4×C73C411yescharacteristic
13C22×C22⋊C4×C73C2211yescharacteristic
14C22×C42×C73C2211yescharacteristic
15C2×C22⋊C8×C73C2214yes—
16#336×C73—24no—
17#324×C73—24no—
18#323×C73—24no—
19C25×C73C2×C411yescharacteristic
20C23×C4×C73C2×C411yescharacteristic
21C23×C4×C73C2×C411yescharacteristic
22C23×C4×C73C2×C411yescharacteristic
23C23×C4×C73C2×C411yescharacteristic
24C23×C4×C73C2×C411yescharacteristic
25C23×C4×C73C2311yescharacteristic
26C22×C8×C73—24no—
27#126×C73—232no—
28C2×C4⋊C4×C73—22no—
29C2×C4⋊C4×C73—22no—
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73—22no—
32C2×C4⋊C4×C73—22no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73—22no—
36C2×C4⋊C4×C73D414yes—
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—22no—
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73D414yes—
46C2×C42×C73—22no—
47C2×C42×C73—22no—
48C2×C42×C73D414yes—
49C22⋊C8×C73—24no—
50C22⋊C8×C73—24no—
51C24×C73—21no—
52C24×C73—21no—
53C24×C73—24no—
54C24×C73—42no—
55C24×C73—42no—
56C24×C73C22⋊C412yes—
57C24×C73C22×C411yescharacteristic
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—24no—
69C22×C4×C73—24no—
70C22×C4×C73—42no—
71C22×C4×C73—42no—
72C22×C4×C73—42no—
73C22×C4×C73—42no—
74C22×C4×C73C22⋊C412yes—
75C22×C4×C73C22⋊C412yes—
76C22×C4×C73C22⋊C412yes—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C2×D412yes—
80C22×C4×C73C2×D412yes—
81C22×C4×C73C2×D412yes—
82C2×C8×C73—48no—
83C2×C8×C73—44no—
84C4⋊C4×C73—28no—
85C4⋊C4×C73—28no—
86C4⋊C4×C73—28no—
87C22⋊C4×C73—28no—
88C22⋊C4×C73—28no—
89C22⋊C4×C73—28no—
90C42×C73—28no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73—24no—
95C23×C73—42no—
96C23×C73—42no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—42no—
102C23×C73—42no—
103C23×C73—88no—
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73C2×C22⋊C411yescharacteristic
106C23×C73C2×C22⋊C411yescharacteristic
107C23×C73#13114yes—
108C2×C4×C73—24no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—24no—
115C2×C4×C73—24no—
116C2×C4×C73—24no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—42no—
124C2×C4×C73—42no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73#12014yes—
132C8×C73—84no—
133C22×C73—28no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—42no—
139C22×C73—42no—
140C22×C73—42no—
141C22×C73—84no—
142C22×C73—84no—
143C22×C73—84no—
144C22×C73#37911yescharacteristic
145C22×C73#40812yes—
146C22×C73#40812yes—
147C22×C73#40812yes—
148C22×C73#41012yes—
149C22×C73#41312yes—
150C4×C73—24no—
151C4×C73—42no—
152C4×C73—42no—
153C4×C73—42no—
154C4×C73—42no—
155C4×C73—42no—
156#8309C7311yesmaximal, characteristic, direct product factor, Sylow
157C2×C73—22no—
158C2×C73—22no—
159C2×C73—82no—
160C2×C73#147314yes—
161C2×C73#178011yescharacteristic
162C2×C73#178211yescharacteristic
163C2×C73#178511yescharacteristic
164C22×#126C2×C7311yescharacteristic
165#1817C2×C7312yescomplemented
166C2×#324C2×C7312yescomplemented
167C2×#323C2×C7312yes—
168C73#830911yescharacteristic, direct product factor, Sylow
169C2×#126—24no—
170C2×#126—24no—
171C2×#126—24no—
172C22×C4⋊C4C4×C7311yescharacteristic
173C22×C4⋊C4C4×C7311yescharacteristic
174C22×C4⋊C4C22×C7311yescharacteristic
175C22×C22⋊C4C4×C7311yescharacteristic
176C22×C22⋊C4C4×C7311yescharacteristic
177C22×C22⋊C4C22×C7311yescharacteristic
178C22×C42C22×C7311yescharacteristic
179C2×C22⋊C8C22×C7314yes—
180#336—24no—
181#324—24no—
182#323—24no—
183C25C2×C4×C7311yescharacteristic
184C23×C4C2×C4×C7311yescharacteristic
185C23×C4C2×C4×C7311yescharacteristic
186C23×C4C2×C4×C7311yescharacteristic
187C23×C4C2×C4×C7311yescharacteristic
188C23×C4C2×C4×C7311yescharacteristic
189C23×C4C23×C7311yescharacteristic
190C22×C8—24no—
191#126—232no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4D4×C7314yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4D4×C7314yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42D4×C7314yes—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C24—21no—
216C24—21no—
217C24—24no—
218C24—42no—
219C24—42no—
220C24C22⋊C4×C7312yes—
221C24C22×C4×C7311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊C4×C7312yes—
239C22×C4C22⋊C4×C7312yes—
240C22×C4C22⋊C4×C7312yes—
241C22×C4C22⋊C4×C7312yes—
242C22×C4C22⋊C4×C7312yes—
243C22×C4C2×D4×C7312yes—
244C22×C4C2×D4×C7312yes—
245C22×C4C2×D4×C7312yes—
246C2×C8—48no—
247C2×C8—44no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C42—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—88no—
268C23C2×C22⋊C4×C7311yescharacteristic
269C23C2×C22⋊C4×C7311yescharacteristic
270C23C2×C22⋊C4×C7311yescharacteristic
271C23#131×C7314yes—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4#120×C7314yes—
296C8—84no—
297C22—28no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—84no—
306C22—84no—
307C22—84no—
308C22#379×C7311yescharacteristic
309C22#408×C7312yes—
310C22#408×C7312yes—
311C22#408×C7312yes—
312C22#410×C7312yes—
313C22#413×C7312yes—
314C4—24no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—42no—
320C2—22no—
321C2—22no—
322C2—82no—
323C2C73×#147314yes—
324C2C73×#178011yescharacteristic
325C2C73×#178211yescharacteristic
326C2C73×#178511yescharacteristic
subgroupG/H
center, socleC23×C73C2×C22⋊C4
commutatorC24C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC25×C73C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C22×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1817C2semidirect
3. C2×#324×C73C2semidirect
156. #8309C73direct (N × H)
165. #1817C2×C73semidirect
166. C2×#324C2×C73semidirect
168. C73#8309direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#126×C732
C73×#18172
C2×#324×C732
C2×#323×C732
#830973