Finite Group Databasebeta

← C73×#12230 #32921634

Subgroups — C73×#12230 #32921634

Order 18688.

422 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2824C211yesmaximal, characteristic
2C73×#2062C211yesmaximal, characteristic, complemented
3C73×#1602C211yesmaximal, characteristic, complemented
4C73×#1596C211yesmaximal, characteristic
5C73×#1351C211yesmaximal, characteristic, complemented
6C73×#1343C211yesmaximal, characteristic
7C73×#1326C211yesmaximal, characteristic, complemented
8#543×C73C2211yescharacteristic
9#451×C73—21no—
10#450×C73—21no—
11#400×C73—21no—
12#397×C73—21no—
13#394×C73—21no—
14#394×C73—21no—
15#393×C73—21no—
16#393×C73C2211yescharacteristic
17#392×C73C2211yescharacteristic
18#386×C73—21no—
19#385×C73—21no—
20#385×C73C2211yescharacteristic
21#342×C73—21no—
22#342×C73C2211yescharacteristic
23#328×C73—21no—
24#328×C73—21no—
25#323×C73C2211yescharacteristic
26#323×C73C2211yescharacteristic
27C22×Q8×C73D411yescharacteristic
28C22×D4×C73—21no—
29C22×D4×C73D411yescharacteristic
30C23×C4×C73D411yescharacteristic
31C2×Q16×C73—41no—
32C2×SD16×C73—41no—
33C22⋊Q8×C73—21no—
34C22⋊Q8×C73—21no—
35C22⋊Q8×C73—41no—
36C22⋊Q8×C73—41no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—41no—
40C2×C22⋊C4×C73—21no—
41C2×C22⋊C4×C73—21no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73D411yescharacteristic
49C2×C22⋊C4×C73C2311yescharacteristic
50#103×C73—41no—
51#102×C73—41no—
52C22⋊C8×C73—21no—
53C22⋊C8×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—41no—
60#95×C73D411yescharacteristic
61#95×C73D411yescharacteristic
62C24×C73—21no—
63C24×C73C2×D411yescharacteristic
64C2×Q8×C73—21no—
65C2×Q8×C73—21no—
66C2×Q8×C73—41no—
67C2×Q8×C73—81no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73—81no—
72C2×D4×C73—81no—
73C2×D4×C73—81no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73—81no—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86Q16×C73—82no—
87SD16×C73—82no—
88C2×C8×C73—41no—
89C2×C8×C73—41no—
90C4⋊C4×C73—21no—
91C4⋊C4×C73—21no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—81no—
95C4⋊C4×C73—81no—
96C4⋊C4×C73—82no—
97C22⋊C4×C73—22no—
98C22⋊C4×C73—21no—
99C22⋊C4×C73—21no—
100C22⋊C4×C73—41no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—81no—
105C22⋊C4×C73—81no—
106C22⋊C4×C73—81no—
107C22⋊C4×C73—81no—
108C22⋊C4×C73—81no—
109C22⋊C4×C73—81no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73#12011yescharacteristic
122C23×C73#13511yescharacteristic
123C23×C73#13711yescharacteristic
124Q8×C73—42no—
125Q8×C73—81no—
126Q8×C73—81no—
127Q8×C73—81no—
128D4×C73—42no—
129D4×C73—81no—
130D4×C73—82no—
131D4×C73—82no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—42no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—81no—
156C2×C4×C73—81no—
157C2×C4×C73—81no—
158C2×C4×C73—81no—
159C2×C4×C73—81no—
160C8×C73—81no—
161C8×C73—81no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—41no—
167C22×C73—41no—
168C22×C73—41no—
169C22×C73—41no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73#45211yescharacteristic
179C22×C73#45611yescharacteristic
180C22×C73B(4,𝔽2)11yescharacteristic
181C4×C73—41no—
182C4×C73—41no—
183C4×C73—81no—
184C4×C73—81no—
185C4×C73—81no—
186C4×C73—82no—
187C4×C73—81no—
188C4×C73—81no—
189C4×C73—81no—
190C4×C73—81no—
191#12230C7311yesmaximal, characteristic, direct product factor, Sylow
192C2×C73—21no—
193C2×C73—21no—
194C2×C73—41no—
195C2×C73—41no—
196C2×C73—161no—
197C2×C73#160311yescharacteristic
198C2×C73#219911yescharacteristic
199C2×C73#220011yescharacteristic
200#2824C2×C7311yescharacteristic
201#2062C2×C7311yescharacteristic, complemented
202#1602C2×C7311yescharacteristic, complemented
203#1596C2×C7311yescharacteristic
204#1351C2×C7311yescharacteristic, complemented
205#1343C2×C7311yescharacteristic
206#1326C2×C7311yescharacteristic, complemented
207C73#1223011yescharacteristic, direct product factor, Sylow
208#543C22×C7311yescharacteristic
209#451—21no—
210#450—21no—
211#400—21no—
212#397—21no—
213#394—21no—
214#394—21no—
215#393—21no—
216#393C22×C7311yescharacteristic
217#392C22×C7311yescharacteristic
218#386—21no—
219#385—21no—
220#385C22×C7311yescharacteristic
221#342—21no—
222#342C22×C7311yescharacteristic
223#328—21no—
224#328—21no—
225#323C22×C7311yescharacteristic
226#323C22×C7311yescharacteristic
227C22×Q8D4×C7311yescharacteristic
228C22×D4—21no—
229C22×D4D4×C7311yescharacteristic
230C23×C4D4×C7311yescharacteristic
231C2×Q16—41no—
232C2×SD16—41no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4D4×C7311yescharacteristic
249C2×C22⋊C4C23×C7311yescharacteristic
250#103—41no—
251#102—41no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95D4×C7311yescharacteristic
261#95D4×C7311yescharacteristic
262C24—21no—
263C24C2×D4×C7311yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—81no—
284C22×C4C2×D4×C7311yescharacteristic
285C22×C4C2×D4×C7311yescharacteristic
286Q16—82no—
287SD16—82no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—22no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#120×C7311yescharacteristic
322C23#135×C7311yescharacteristic
323C23#137×C7311yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330D4—82no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C8—81no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#452×C7311yescharacteristic
379C22#456×C7311yescharacteristic
380C22B(4,𝔽2)×C7311yescharacteristic
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2C73×#160311yescharacteristic
397C2C73×#219911yescharacteristic
398C2C73×#220011yescharacteristic
subgroupG/H
center, socleC22×C73B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C73
prime radical#392×C73C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C73 ◁ C2×C22⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2062C2semidirect
3. C73×#1602C2semidirect
5. C73×#1351C2semidirect
7. C73×#1326C2semidirect
191. #12230C73direct (N × H)
201. #2062C2×C73semidirect
202. #1602C2×C73semidirect
204. #1351C2×C73semidirect
206. #1326C2×C73semidirect
207. C73#12230direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#28242
C73×#20622
C73×#16022
C73×#15962
C73×#13512
C73×#13432
C73×#13262
#1223073