Finite Group Databasebeta

← C73×#13295 #32922699

Subgroups — C73×#13295 #32922699

Order 18688.

488 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2487C211yesmaximal, characteristic
2C73×#1800C211yesmaximal, characteristic
3C4×#99×C73C211yesmaximal, characteristic
4C73×#1390C214yesmaximal
5C2×C4×D4×C73C2211yescharacteristic
6C22×C4⋊C4×C73C411yescharacteristic, complemented
7#423×C73—21no—
8C2×#100×C73C2212yes—
9C2×#99×C73C2212yes—
10#400×C73—21no—
11#399×C73—21no—
12#389×C73—21no—
13#386×C73C2212yes—
14#385×C73C412yescomplemented
15#380×C73—21no—
16#380×C73C411yescharacteristic, complemented
17C4×C22⋊C4×C73—21no—
18#343×C73—21no—
19#342×C73—21no—
20C22×D4×C73C2311yescharacteristic
21C23×C4×C73C2×C411yescharacteristic
22C23×C4×C73C2×C412yes—
23C23×C4×C73D411yescharacteristic
24C2×M16×C73—22no—
25C4×D4×C73—41no—
26C4×D4×C73—41no—
27C2×C4⋊C4×C73—21no—
28C2×C4⋊C4×C73—21no—
29C2×C4⋊C4×C73—21no—
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73—41no—
32C2×C4⋊C4×C73—41no—
33C2×C4⋊C4×C73D411yescharacteristic
34C2×C22⋊C4×C73—22no—
35C2×C22⋊C4×C73—22no—
36C2×C22⋊C4×C73C2×C412yes—
37C2×C22⋊C4×C73C2×C411yescharacteristic
38C2×C22⋊C4×C73D411yescharacteristic
39C2×C42×C73—41no—
40C2×C42×C73D411yescharacteristic
41C4⋊C8×C73—41no—
42C4⋊C8×C73—41no—
43#100×C73—24no—
44#99×C73—22no—
45#99×C73Q814yes—
46#95×C73—22no—
47#95×C73—21no—
48#95×C73—22no—
49#95×C73—41no—
50C24×C73C22×C411yescharacteristic
51C24×C73C2×D411yescharacteristic
52C2×D4×C73—41no—
53C2×D4×C73—41no—
54C2×D4×C73C2×Q812yes—
55C2×D4×C73#4112yes—
56C22×C4×C73—21no—
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—42no—
63C22×C4×C73—42no—
64C22×C4×C73—42no—
65C22×C4×C73—42no—
66C22×C4×C73—82no—
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C2×D411yescharacteristic
72M16×C73—42no—
73M16×C73—42no—
74C2×C8×C73—42no—
75C4⋊C4×C73—41no—
76C4⋊C4×C73—42no—
77C4⋊C4×C73—42no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C4⋊C4×C73—82no—
81C22⋊C4×C73—24no—
82C22⋊C4×C73—41no—
83C22⋊C4×C73—41no—
84C22⋊C4×C73—42no—
85C22⋊C4×C73—42no—
86C22⋊C4×C73—82no—
87C42×C73—21no—
88C42×C73—21no—
89C42×C73—82no—
90C23×C73—21no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73C2×C22⋊C411yescharacteristic
99C23×C73C4×Q812yes—
100C23×C73C22⋊Q812yes—
101D4×C73—42no—
102D4×C73—42no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—44no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—81no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×C4×C73—84no—
126C2×C4×C73#9911yescharacteristic
127C2×C4×C73#9911yescharacteristic
128C2×C4×C73#12411yescharacteristic
129C2×C4×C73C4⋊Q811yescharacteristic
130C8×C73—82no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—22no—
134C22×C73—22no—
135C22×C73—42no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—81no—
143C22×C73#39111yescharacteristic
144C22×C73C2×#9911yescharacteristic
145C22×C73#41011yescharacteristic
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—41no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—82no—
153C4×C73—84no—
154C4×C73—82no—
155C4×C73#35212yes—
156#13295C7311yesmaximal, characteristic, direct product factor, Sylow
157C2×C73—21no—
158C2×C73—21no—
159C2×C73—42no—
160C2×C73—42no—
161C2×C73#199611yescharacteristic
162C2×C73C2×#35211yescharacteristic
163C2×C73#212211yescharacteristic
164#2487C2×C7311yescharacteristic
165#1800C2×C7311yescharacteristic
166C4×#99C2×C7311yescharacteristic
167#1390C2×C7314yes—
168C73#1329511yescharacteristic, direct product factor, Sylow
169C2×C4×D4C22×C7311yescharacteristic
170C22×C4⋊C4C4×C7311yescharacteristic, complemented
171#423—21no—
172C2×#100C22×C7312yes—
173C2×#99C22×C7312yes—
174#400—21no—
175#399—21no—
176#389—21no—
177#386C22×C7312yes—
178#385C4×C7312yescomplemented
179#380—21no—
180#380C4×C7311yescharacteristic, complemented
181C4×C22⋊C4—21no—
182#343—21no—
183#342—21no—
184C22×D4C23×C7311yescharacteristic
185C23×C4C2×C4×C7311yescharacteristic
186C23×C4C2×C4×C7312yes—
187C23×C4D4×C7311yescharacteristic
188C2×M16—22no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4D4×C7311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C2×C4×C7312yes—
201C2×C22⋊C4C2×C4×C7311yescharacteristic
202C2×C22⋊C4D4×C7311yescharacteristic
203C2×C42—41no—
204C2×C42D4×C7311yescharacteristic
205C4⋊C8—41no—
206C4⋊C8—41no—
207#100—24no—
208#99—22no—
209#99Q8×C7314yes—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—41no—
214C24C22×C4×C7311yescharacteristic
215C24C2×D4×C7311yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4C2×Q8×C7312yes—
219C2×D4#41×C7312yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4C22⋊C4×C7311yescharacteristic
232C22×C4C22⋊C4×C7311yescharacteristic
233C22×C4C22⋊C4×C7311yescharacteristic
234C22×C4C22⋊C4×C7311yescharacteristic
235C22×C4C2×D4×C7311yescharacteristic
236M16—42no—
237M16—42no—
238C2×C8—42no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C22⋊C4—24no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—82no—
251C42—21no—
252C42—21no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊C4×C7311yescharacteristic
263C23C4×Q8×C7312yes—
264C23C22⋊Q8×C7312yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4#99×C7311yescharacteristic
291C2×C4#99×C7311yescharacteristic
292C2×C4#124×C7311yescharacteristic
293C2×C4C4⋊Q8×C7311yescharacteristic
294C8—82no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#391×C7311yescharacteristic
308C22C2×#99×C7311yescharacteristic
309C22#410×C7311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—84no—
318C4—82no—
319C4#352×C7312yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2C73×#199611yescharacteristic
325C2C2×#352×C7311yescharacteristic
326C2C73×#212211yescharacteristic
subgroupG/H
center, socleC22×C73C2×#99
commutatorC24C22×C4×C73
Frattini, cosocleC22×D4C23×C73
prime radicalC22×D4×C73C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×C4×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊C4×C73C4semidirect
14. #385×C73C4semidirect
16. #380×C73C4semidirect
156. #13295C73direct (N × H)
168. C73#13295direct (N × H)
170. C22×C4⋊C4C4×C73semidirect
178. #385C4×C73semidirect
180. #380C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#24872
C73×#18002
C4×#99×C732
C73×#13902
#1329573