Finite Group Databasebeta

← C73×#12695 #32922099

Subgroups — C73×#12695 #32922099

Order 18688.

1098 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#532×C73C211yesmaximal, characteristic
2C2×#354×C73C212yesmaximal, complemented
3C2×#353×C73C212yesmaximal, complemented
4C73×#1782C212yesmaximal, complemented
5C22×#41×C73C411yescharacteristic, complemented
6C22×#41×C73C2211yescharacteristic
7#532×C73—24no—
8#532×C73—24no—
9C2×C4⋊Q8×C73C411yescharacteristic, complemented
10C2×#127×C73C2211yescharacteristic
11C2×#124×C73C411yescharacteristic, complemented
12C2×#124×C73C2211yescharacteristic
13C2×#121×C73—22no—
14C2×C4×D4×C73—22no—
15C22×C42×C73C411yescharacteristic, complemented
16C2×#99×C73—22no—
17C2×#99×C73C2214yes—
18#381×C73—22no—
19#354×C73—24no—
20#353×C73—24no—
21C2×#41×C73—22no—
22C2×#41×C73—22no—
23C2×#41×C73—22no—
24C2×#41×C73—22no—
25C2×#41×C73—42no—
26C2×#41×C73—42no—
27C2×#41×C73—42no—
28C2×#41×C73—42no—
29C22×Q8×C73C2×C411yescharacteristic
30C22×Q8×C73C2×C411yescharacteristic
31C22×D4×C73—21no—
32C22×D4×C73—22no—
33C22×D4×C73C2×C411yescharacteristic
34C22×D4×C73D412yes—
35C22×D4×C73C2311yescharacteristic
36C23×C4×C73—21no—
37C23×C4×C73—21no—
38C23×C4×C73C2×C411yescharacteristic
39C23×C4×C73D412yes—
40C4⋊Q8×C73—22no—
41C4⋊Q8×C73—22no—
42#127×C73—22no—
43#127×C73D414yes—
44#124×C73—22no—
45#124×C73—22no—
46#124×C73—22no—
47#124×C73D414yes—
48#121×C73—216no—
49C4×D4×C73—216no—
50C2×C4⋊C4×C73—22no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73—24no—
54C2×C42×C73—21no—
55C2×C42×C73—22no—
56C2×C42×C73—22no—
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73C2×C411yescharacteristic
59#99×C73—24no—
60#99×C73—24no—
61#99×C73—44no—
62#95×C73—44no—
63C24×C73—21no—
64C24×C73C22⋊C411yescharacteristic
65C24×C73C2×D412yes—
66#41×C73—48no—
67#41×C73—48no—
68#41×C73—48no—
69#41×C73—48no—
70C2×Q8×C73—21no—
71C2×Q8×C73—21no—
72C2×Q8×C73—21no—
73C2×Q8×C73—41no—
74C2×Q8×C73—41no—
75C2×Q8×C73—41no—
76C2×Q8×C73—41no—
77C2×Q8×C73C22⋊C412yes—
78C2×D4×C73—24no—
79C2×D4×C73—24no—
80C2×D4×C73—28no—
81C2×D4×C73—24no—
82C2×D4×C73—21no—
83C2×D4×C73—42no—
84C2×D4×C73—41no—
85C2×D4×C73—41no—
86C2×D4×C73—41no—
87C2×D4×C73—41no—
88C2×D4×C73—48no—
89C2×D4×C73—42no—
90C2×D4×C73—42no—
91C2×D4×C73C22⋊C412yes—
92C2×D4×C73C2×D412yes—
93C2×D4×C73C2×D412yes—
94C22×C4×C73—22no—
95C22×C4×C73—21no—
96C22×C4×C73—22no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—24no—
100C22×C4×C73—24no—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—24no—
104C22×C4×C73—24no—
105C22×C4×C73—42no—
106C22×C4×C73—44no—
107C22×C4×C73—42no—
108C22×C4×C73—42no—
109C22×C4×C73—44no—
110C22×C4×C73C22⋊C412yes—
111C22×C4×C73C22⋊C411yescharacteristic
112C22×C4×C73C22×C411yescharacteristic
113C4⋊C4×C73—28no—
114C22⋊C4×C73—28no—
115C22⋊C4×C73—28no—
116C22⋊C4×C73—44no—
117C22⋊C4×C73—44no—
118C42×C73—24no—
119C42×C73—21no—
120C42×C73—21no—
121C42×C73C22⋊C412yes—
122C42×C73C22⋊C412yes—
123C23×C73—21no—
124C23×C73—22no—
125C23×C73—21no—
126C23×C73—22no—
127C23×C73—22no—
128C23×C73—42no—
129C23×C73—44no—
130C23×C73—44no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73#9912yes—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73#12014yes—
136Q8×C73—42no—
137Q8×C73—42no—
138Q8×C73—42no—
139Q8×C73—42no—
140D4×C73—42no—
141D4×C73—42no—
142D4×C73—42no—
143D4×C73—416no—
144D4×C73—42no—
145D4×C73—48no—
146D4×C73—48no—
147C2×C4×C73—24no—
148C2×C4×C73—24no—
149C2×C4×C73—24no—
150C2×C4×C73—21no—
151C2×C4×C73—22no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—22no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73—22no—
159C2×C4×C73—22no—
160C2×C4×C73—48no—
161C2×C4×C73—48no—
162C2×C4×C73—48no—
163C2×C4×C73—42no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—84no—
167C2×C4×C73—88no—
168C2×C4×C73#9912yes—
169C2×C4×C73C2×C22⋊C411yescharacteristic
170C2×C4×C73C2×C22⋊C411yescharacteristic
171C22×C73—22no—
172C22×C73—22no—
173C22×C73—24no—
174C22×C73—44no—
175C22×C73—42no—
176C22×C73—42no—
177C22×C73—44no—
178C22×C73—44no—
179C22×C73—42no—
180C22×C73—42no—
181C22×C73—82no—
182C22×C73—82no—
183C22×C73—82no—
184C22×C73#37911yescharacteristic
185C22×C73C2×#9911yescharacteristic
186C22×C73C2×#9911yescharacteristic
187C4×C73—24no—
188C4×C73—22no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—42no—
192C4×C73—42no—
193C4×C73—42no—
194C4×C73—164no—
195#12695C7311yesmaximal, characteristic, direct product factor, Sylow
196C2×C73—21no—
197C2×C73—21no—
198C2×C73—44no—
199C2×C73—42no—
200C2×C73—82no—
201C2×C73#178111yescharacteristic
202C2×C73#212612yes—
203C2×#532C2×C7311yescharacteristic
204C2×#354C2×C7312yescomplemented
205C2×#353C2×C7312yescomplemented
206#1782C2×C7312yescomplemented
207C73#1269511yescharacteristic, direct product factor, Sylow
208C22×#41C4×C7311yescharacteristic, complemented
209C22×#41C22×C7311yescharacteristic
210#532—24no—
211#532—24no—
212C2×C4⋊Q8C4×C7311yescharacteristic, complemented
213C2×#127C22×C7311yescharacteristic
214C2×#124C4×C7311yescharacteristic, complemented
215C2×#124C22×C7311yescharacteristic
216C2×#121—22no—
217C2×C4×D4—22no—
218C22×C42C4×C7311yescharacteristic, complemented
219C2×#99—22no—
220C2×#99C22×C7314yes—
221#381—22no—
222#354—24no—
223#353—24no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—42no—
229C2×#41—42no—
230C2×#41—42no—
231C2×#41—42no—
232C22×Q8C2×C4×C7311yescharacteristic
233C22×Q8C2×C4×C7311yescharacteristic
234C22×D4—21no—
235C22×D4—22no—
236C22×D4C2×C4×C7311yescharacteristic
237C22×D4D4×C7312yes—
238C22×D4C23×C7311yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×C4×C7311yescharacteristic
242C23×C4D4×C7312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245#127—22no—
246#127D4×C7314yes—
247#124—22no—
248#124—22no—
249#124—22no—
250#124D4×C7314yes—
251#121—216no—
252C4×D4—216no—
253C2×C4⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C2×C4×C7311yescharacteristic
261C2×C42C2×C4×C7311yescharacteristic
262#99—24no—
263#99—24no—
264#99—44no—
265#95—44no—
266C24—21no—
267C24C22⋊C4×C7311yescharacteristic
268C24C2×D4×C7312yes—
269#41—48no—
270#41—48no—
271#41—48no—
272#41—48no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8C22⋊C4×C7312yes—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—48no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C22⋊C4×C7312yes—
295C2×D4C2×D4×C7312yes—
296C2×D4C2×D4×C7312yes—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4C22⋊C4×C7312yes—
314C22×C4C22⋊C4×C7311yescharacteristic
315C22×C4C22×C4×C7311yescharacteristic
316C4⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C42—24no—
322C42—21no—
323C42—21no—
324C42C22⋊C4×C7312yes—
325C42C22⋊C4×C7312yes—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—41no—
335C23—41no—
336C23#99×C7312yes—
337C23C2×C22⋊C4×C7311yescharacteristic
338C23#120×C7314yes—
339Q8—42no—
340Q8—42no—
341Q8—42no—
342Q8—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—416no—
348D4—48no—
349D4—48no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4—88no—
371C2×C4#99×C7312yes—
372C2×C4C2×C22⋊C4×C7311yescharacteristic
373C2×C4C2×C22⋊C4×C7311yescharacteristic
374C22—22no—
375C22—22no—
376C22—24no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22#379×C7311yescharacteristic
388C22C2×#99×C7311yescharacteristic
389C22C2×#99×C7311yescharacteristic
390C4—24no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—164no—
398C2—21no—
399C2—21no—
400C2—44no—
401C2—42no—
402C2—82no—
403C2C73×#178111yescharacteristic
404C2C73×#212612yes—
subgroupG/H
center, socleC22×C73C2×#99
commutatorC22×C4C22×C4×C73
Frattini, cosocleC22×D4C23×C73
prime radicalC2×#532×C73C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C22×#41×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#354×C73C2semidirect
3. C2×#353×C73C2semidirect
4. C73×#1782C2semidirect
5. C22×#41×C73C4semidirect
9. C2×C4⋊Q8×C73C4semidirect
11. C2×#124×C73C4semidirect
15. C22×C42×C73C4semidirect
195. #12695C73direct (N × H)
204. C2×#354C2×C73semidirect
205. C2×#353C2×C73semidirect
206. #1782C2×C73semidirect
207. C73#12695direct (N × H)
208. C22×#41C4×C73semidirect
212. C2×C4⋊Q8C4×C73semidirect
214. C2×#124C4×C73semidirect
218. C22×C42C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#532×C732
C2×#354×C732
C2×#353×C732
C73×#17822
#1269573