Finite Group Databasebeta

← C73×#7215 #32916619

Subgroups — C73×#7215 #32916619

Order 18688.

422 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C73×#1821C211yesmaximal, characteristic
2C73×#1463C211yesmaximal, characteristic
3C73×#1436C211yesmaximal, characteristic
4C22×C4⋊C4×C73C2211yescharacteristic
5C22×C42×C73C411yescharacteristic
6#407×C73—21no—
7#407×C73—21no—
8C2×#95×C73—21no—
9C2×#95×C73—21no—
10C2×#95×C73C411yescharacteristic
11#343×C73—21no—
12#343×C73C412yescomplemented
13#324×C73—21no—
14#324×C73C412yescomplemented
15C23×C4×C73—21no—
16C23×C4×C73—21no—
17C23×C4×C73C2×C411yescharacteristic
18C23×C4×C73D411yescharacteristic
19C23×C4×C73D411yescharacteristic
20C2×M16×C73—22no—
21C2×M16×C73—41no—
22C2×C4⋊C4×C73—41no—
23C2×C4⋊C4×C73—41no—
24C2×C4⋊C4×C73C2×C411yescharacteristic
25C2×C4⋊C4×C73C2×C411yescharacteristic
26C2×C4⋊C4×C73D411yescharacteristic
27C2×C4⋊C4×C73Q811yescharacteristic
28C2×C42×C73—41no—
29C2×C42×C73—42no—
30C22⋊C8×C73—22no—
31C22⋊C8×C73—41no—
32C22⋊C8×C73—41no—
33C22⋊C8×C73—41no—
34#95×C73—24no—
35#95×C73—42no—
36#95×C73—41no—
37C24×C73C22⋊C411yescharacteristic
38C22×C4×C73—22no—
39C22×C4×C73—22no—
40C22×C4×C73—22no—
41C22×C4×C73—21no—
42C22×C4×C73—21no—
43C22×C4×C73—21no—
44C22×C4×C73—21no—
45C22×C4×C73—21no—
46C22×C4×C73—21no—
47C22×C4×C73—22no—
48C22×C4×C73—22no—
49C22×C4×C73—41no—
50C22×C4×C73—41no—
51C22×C4×C73—41no—
52C22×C4×C73—41no—
53C22×C4×C73—41no—
54C22×C4×C73—41no—
55C22×C4×C73—41no—
56C22×C4×C73—41no—
57C22×C4×C73—41no—
58C22×C4×C73—41no—
59C22×C4×C73—41no—
60C22×C4×C73—41no—
61C22×C4×C73—41no—
62C22×C4×C73—41no—
63C22×C4×C73—41no—
64C22×C4×C73C4211yescharacteristic
65C22×C4×C73C22⋊C411yescharacteristic
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C4⋊C411yescharacteristic
68C22×C4×C73C4⋊C411yescharacteristic
69C22×C4×C73C4⋊C411yescharacteristic
70M16×C73—42no—
71M16×C73—42no—
72M16×C73—81no—
73M16×C73—81no—
74C2×C8×C73—42no—
75C2×C8×C73—42no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C42×C73—42no—
81C42×C73—42no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—21no—
87C23×C73—21no—
88C23×C73#9511yescharacteristic
89C23×C73#9911yescharacteristic
90C23×C73#9911yescharacteristic
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—24no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73—81no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73#10412yes—
127C8×C73—82no—
128C8×C73—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73H(3,ℤ/4)11yescharacteristic
144C22×C73#33911yescharacteristic
145C22×C73#34211yescharacteristic
146C22×C73AΣL(1,𝔽16)11yescharacteristic
147C22×C73#35211yescharacteristic
148C22×C73#35511yescharacteristic
149C22×C73#35611yescharacteristic
150C4×C73—22no—
151C4×C73—44no—
152C4×C73—41no—
153C4×C73—41no—
154C4×C73—41no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—82no—
158#7215C7311yesmaximal, characteristic, direct product factor, Sylow
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—21no—
163C2×C73—21no—
164C2×C73—21no—
165C2×C73#128411yescharacteristic
166C2×C73#139011yescharacteristic
167C2×C73#139411yescharacteristic
168#1821C2×C7311yescharacteristic
169#1463C2×C7311yescharacteristic
170#1436C2×C7311yescharacteristic
171C73#721511yescharacteristic, direct product factor, Sylow
172C22×C4⋊C4C22×C7311yescharacteristic
173C22×C42C4×C7311yescharacteristic
174#407—21no—
175#407—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95C4×C7311yescharacteristic
179#343—21no—
180#343C4×C7312yescomplemented
181#324—21no—
182#324C4×C7312yescomplemented
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C2×C4×C7311yescharacteristic
186C23×C4D4×C7311yescharacteristic
187C23×C4D4×C7311yescharacteristic
188C2×M16—22no—
189C2×M16—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×C4×C7311yescharacteristic
193C2×C4⋊C4C2×C4×C7311yescharacteristic
194C2×C4⋊C4D4×C7311yescharacteristic
195C2×C4⋊C4Q8×C7311yescharacteristic
196C2×C42—41no—
197C2×C42—42no—
198C22⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#95—24no—
203#95—41no—
204#95—42no—
205C24C22⋊C4×C7311yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4C42×C7311yescharacteristic
233C22×C4C22⋊C4×C7311yescharacteristic
234C22×C4C22⋊C4×C7311yescharacteristic
235C22×C4C4⋊C4×C7311yescharacteristic
236C22×C4C4⋊C4×C7311yescharacteristic
237C22×C4C4⋊C4×C7311yescharacteristic
238M16—42no—
239M16—42no—
240M16—81no—
241M16—81no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—42no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#95×C7311yescharacteristic
257C23#99×C7311yescharacteristic
258C23#99×C7311yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4#104×C7312yes—
295C8—82no—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22H(3,ℤ/4)×C7311yescharacteristic
312C22#339×C7311yescharacteristic
313C22#342×C7311yescharacteristic
314C22AΣL(1,𝔽16)×C7311yescharacteristic
315C22#352×C7311yescharacteristic
316C22#355×C7311yescharacteristic
317C22#356×C7311yescharacteristic
318C4—22no—
319C4—44no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2C73×#128411yescharacteristic
333C2C73×#139011yescharacteristic
334C2C73×#139411yescharacteristic
subgroupG/H
center, socleC22×C73#342
commutatorC22×C4C42×C73
Frattini, cosocleC22×C4⋊C4C22×C73
prime radicalC24×C73C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C24×C73 ◁ C22×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
12. #343×C73C4semidirect
14. #324×C73C4semidirect
158. #7215C73direct (N × H)
171. C73#7215direct (N × H)
180. #343C4×C73semidirect
182. #324C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#18212
C73×#14632
C73×#14362
#721573