Finite Group Databasebeta

← C73×#7189 #32916593

Subgroups — C73×#7189 #32916593

Order 18688.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#324×C73C212yesmaximal
2C73×C4⋊#95C211yesmaximal, characteristic
3C22×C4⋊C4×C73C411yescharacteristic
4C22×C4⋊C4×C73C2211yescharacteristic
5C22×C42×C73C411yescharacteristic
6C2×C22⋊C8×C73—22no—
7C2×#95×C73—22no—
8#324×C73—22no—
9#324×C73C414yescomplemented
10C23×C4×C73—21no—
11C23×C4×C73—21no—
12C23×C4×C73C2×C411yescharacteristic
13C23×C4×C73D412yes—
14C22×C8×C73—42no—
15C2×C4⋊C4×C73—21no—
16C2×C4⋊C4×C73—41no—
17C2×C4⋊C4×C73—41no—
18C2×C4⋊C4×C73—41no—
19C2×C4⋊C4×C73—41no—
20C2×C4⋊C4×C73C812yescomplemented
21C2×C4⋊C4×C73C2×C412yes—
22C2×C4⋊C4×C73D411yescharacteristic
23C2×C4⋊C4×C73Q811yescharacteristic
24C2×C42×C73—21no—
25C2×C42×C73—41no—
26C2×C42×C73—41no—
27C2×C42×C73C812yescomplemented
28C22⋊C8×C73—24no—
29C22⋊C8×C73—42no—
30#95×C73—44no—
31C24×C73C22⋊C411yescharacteristic
32C22×C4×C73—21no—
33C22×C4×C73—21no—
34C22×C4×C73—21no—
35C22×C4×C73—21no—
36C22×C4×C73—22no—
37C22×C4×C73—22no—
38C22×C4×C73—22no—
39C22×C4×C73—22no—
40C22×C4×C73—22no—
41C22×C4×C73—22no—
42C22×C4×C73—41no—
43C22×C4×C73—41no—
44C22×C4×C73—41no—
45C22×C4×C73—41no—
46C22×C4×C73—42no—
47C22×C4×C73—42no—
48C22×C4×C73—42no—
49C22×C4×C73—42no—
50C22×C4×C73—41no—
51C22×C4×C73—41no—
52C22×C4×C73C4211yescharacteristic
53C22×C4×C73C22⋊C412yes—
54C22×C4×C73C4⋊C411yescharacteristic
55C22×C4×C73C4⋊C412yes—
56C22×C4×C73C2×C811yescharacteristic
57C22×C4×C73C2×C811yescharacteristic
58C22×C4×C73M1611yescharacteristic
59C22×C4×C73M1611yescharacteristic
60C2×C8×C73—44no—
61C2×C8×C73—82no—
62C2×C8×C73—82no—
63C4⋊C4×C73—42no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—41no—
66C4⋊C4×C73—42no—
67C4⋊C4×C73—42no—
68C42×C73—42no—
69C42×C73—42no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73—21no—
73C23×C73—21no—
74C23×C73#9511yescharacteristic
75C23×C73C22⋊C811yescharacteristic
76C23×C73C22⋊C811yescharacteristic
77C23×C73#9911yescharacteristic
78C23×C73#9911yescharacteristic
79C23×C73#10011yescharacteristic
80C23×C73#10111yescharacteristic
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—24no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—42no—
97C2×C4×C73—42no—
98C2×C4×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—44no—
107C2×C4×C73—42no—
108C2×C4×C73—82no—
109C2×C4×C73—82no—
110C2×C4×C73—81no—
111C2×C4×C73—81no—
112C2×C4×C73C4×C811yescharacteristic
113C2×C4×C73#9711yescharacteristic
114C2×C4×C73C4⋊C811yescharacteristic
115C2×C4×C73C4⋊C811yescharacteristic
116C8×C73—84no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73#32311yescharacteristic
130C22×C73#32311yescharacteristic
131C22×C73#32411yescharacteristic
132C22×C73#32411yescharacteristic
133C22×C73#33611yescharacteristic
134C22×C73#34211yescharacteristic
135C22×C73#34311yescharacteristic
136C22×C73#35311yescharacteristic
137C22×C73#35411yescharacteristic
138C22×C73#35511yescharacteristic
139C22×C73#35611yescharacteristic
140C4×C73—21no—
141C4×C73—21no—
142C4×C73—21no—
143C4×C73—21no—
144C4×C73—44no—
145C4×C73—42no—
146C4×C73—42no—
147C4×C73—82no—
148#7189C7311yesmaximal, characteristic, direct product factor, Sylow
149C2×C73—21no—
150C2×C73—21no—
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#128011yescharacteristic
154C2×C73#131811yescharacteristic
155C2×C73#131911yescharacteristic
156C2×C73#132411yescharacteristic
157C2×C73#132511yescharacteristic
158C2×C73#139311yescharacteristic
159C2×C73#139411yescharacteristic
160C2×#324C2×C7312yes—
161C4⋊#95C2×C7311yescharacteristic
162C73#718911yescharacteristic, direct product factor, Sylow
163C22×C4⋊C4C4×C7311yescharacteristic
164C22×C4⋊C4C22×C7311yescharacteristic
165C22×C42C4×C7311yescharacteristic
166C2×C22⋊C8—22no—
167C2×#95—22no—
168#324—22no—
169#324C4×C7314yescomplemented
170C23×C4—21no—
171C23×C4—21no—
172C23×C4C2×C4×C7311yescharacteristic
173C23×C4D4×C7312yes—
174C22×C8—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C8×C7312yescomplemented
181C2×C4⋊C4C2×C4×C7312yes—
182C2×C4⋊C4D4×C7311yescharacteristic
183C2×C4⋊C4Q8×C7311yescharacteristic
184C2×C42—21no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C8×C7312yescomplemented
188C22⋊C8—24no—
189C22⋊C8—42no—
190#95—44no—
191C24C22⋊C4×C7311yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4C42×C7311yescharacteristic
213C22×C4C22⋊C4×C7312yes—
214C22×C4C4⋊C4×C7311yescharacteristic
215C22×C4C4⋊C4×C7312yes—
216C22×C4C2×C8×C7311yescharacteristic
217C22×C4C2×C8×C7311yescharacteristic
218C22×C4M16×C7311yescharacteristic
219C22×C4M16×C7311yescharacteristic
220C2×C8—44no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C42—42no—
229C42—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23#95×C7311yescharacteristic
235C23C22⋊C8×C7311yescharacteristic
236C23C22⋊C8×C7311yescharacteristic
237C23#99×C7311yescharacteristic
238C23#99×C7311yescharacteristic
239C23#100×C7311yescharacteristic
240C23#101×C7311yescharacteristic
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4C4×C8×C7311yescharacteristic
273C2×C4#97×C7311yescharacteristic
274C2×C4C4⋊C8×C7311yescharacteristic
275C2×C4C4⋊C8×C7311yescharacteristic
276C8—84no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#323×C7311yescharacteristic
290C22#323×C7311yescharacteristic
291C22#324×C7311yescharacteristic
292C22#324×C7311yescharacteristic
293C22#336×C7311yescharacteristic
294C22#342×C7311yescharacteristic
295C22#343×C7311yescharacteristic
296C22#353×C7311yescharacteristic
297C22#354×C7311yescharacteristic
298C22#355×C7311yescharacteristic
299C22#356×C7311yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—44no—
305C4—42no—
306C4—42no—
307C4—82no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—21no—
312C2C73×#128011yescharacteristic
313C2C73×#131811yescharacteristic
314C2C73×#131911yescharacteristic
315C2C73×#132411yescharacteristic
316C2C73×#132511yescharacteristic
317C2C73×#139311yescharacteristic
318C2C73×#139411yescharacteristic
subgroupG/H
center, socleC23×C73#99
commutatorC2×C4C4×C8×C73
Frattini, cosocleC22×C4⋊C4C22×C73
prime radicalC24×C73C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C24×C73 ◁ C22×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
9. #324×C73C4semidirect
20. C2×C4⋊C4×C73C8semidirect
27. C2×C42×C73C8semidirect
148. #7189C73direct (N × H)
162. C73#7189direct (N × H)
169. #324C4×C73semidirect
180. C2×C4⋊C4C8×C73semidirect
187. C2×C42C8×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#324×C732
C73×C4⋊#952
#718973