Finite Group Databasebeta

← C73×#10386 #32919790

Subgroups — C73×#10386 #32919790

Order 18688.

602 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2607C211yesmaximal, characteristic, complemented
2C73×#1789C212yesmaximal
3C73×#1782C211yesmaximal, characteristic
4C73×#1306C211yesmaximal, characteristic, complemented
5C73×#1284C212yesmaximal, complemented
6C22×#41×C73C2211yescharacteristic
7C2×#104×C73—22no—
8C2×#104×C73—22no—
9C2×#99×C73—21no—
10C2×#99×C73—21no—
11#407×C73C2212yes—
12#399×C73—21no—
13#393×C73—21no—
14#391×C73—22no—
15#387×C73—22no—
16#381×C73C2211yescharacteristic
17#381×C73C2211yescharacteristic
18C4×C22⋊C4×C73C2212yes—
19C2×#41×C73—22no—
20C2×#41×C73—22no—
21C2×#41×C73—42no—
22C2×#41×C73—42no—
23C22×Q8×C73—21no—
24C22×Q8×C73Q811yescharacteristic, complemented
25C22×D4×C73—21no—
26C22×D4×C73Q811yescharacteristic, complemented
27C23×C4×C73—21no—
28C23×C4×C73C2311yescharacteristic
29C2×M16×C73—22no—
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73—21no—
32C2×C22⋊C4×C73—22no—
33C2×C22⋊C4×C73—21no—
34C2×C22⋊C4×C73D411yescharacteristic
35C2×C22⋊C4×C73D411yescharacteristic
36C2×C42×C73D412yes—
37C2×C42×C73D412yes—
38#104×C73—44no—
39#104×C73—44no—
40#99×C73—42no—
41#99×C73—42no—
42C22⋊C8×C73—22no—
43C22⋊C8×C73—22no—
44#95×C73—21no—
45#95×C73—22no—
46#95×C73—21no—
47#95×C73—22no—
48#95×C73—22no—
49#95×C73—22no—
50C24×C73—21no—
51C24×C73C2×D411yescharacteristic
52#41×C73—24no—
53#41×C73—24no—
54#41×C73—42no—
55#41×C73—42no—
56#41×C73—44no—
57#41×C73—44no—
58C2×Q8×C73—22no—
59C2×Q8×C73—42no—
60C2×Q8×C73—44no—
61C2×Q8×C73—81no—
62C2×D4×C73—22no—
63C2×D4×C73—22no—
64C2×D4×C73—22no—
65C2×D4×C73—42no—
66C2×D4×C73—41no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C22×C4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—22no—
74C22×C4×C73—21no—
75C22×C4×C73—41no—
76C22×C4×C73—41no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73—41no—
81C22×C4×C73C2×D412yes—
82C22×C4×C73C2×Q811yescharacteristic
83C22×C4×C73#4112yes—
84C22×C4×C73#4111yescharacteristic
85M16×C73—42no—
86M16×C73—42no—
87C2×C8×C73—42no—
88C4⋊C4×C73—44no—
89C4⋊C4×C73—81no—
90C22⋊C4×C73—44no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—41no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C42×C73—42no—
96C42×C73—42no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—22no—
100C23×C73—42no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73—82no—
104C23×C73#12011yescharacteristic
105C23×C73C22⋊Q811yescharacteristic
106C23×C73#12411yescharacteristic
107Q8×C73—24no—
108Q8×C73—42no—
109Q8×C73—84no—
110D4×C73—24no—
111D4×C73—42no—
112D4×C73—42no—
113D4×C73—42no—
114D4×C73—42no—
115D4×C73—42no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73—24no—
119C2×C4×C73—22no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—42no—
123C2×C4×C73—44no—
124C2×C4×C73—44no—
125C2×C4×C73—41no—
126C2×C4×C73—42no—
127C2×C4×C73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—81no—
137C2×C4×C73—82no—
138C2×C4×C73C22⋊Q812yes—
139C2×C4×C73#12412yes—
140C8×C73—82no—
141C22×C73—22no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—81no—
148C22×C73—81no—
149C22×C73—82no—
150C22×C73—82no—
151C22×C73—82no—
152C22×C73#39311yescharacteristic
153C22×C73B(4,𝔽2)11yescharacteristic
154C22×C73B(4,𝔽2)11yescharacteristic
155C4×C73—22no—
156C4×C73—22no—
157C4×C73—44no—
158C4×C73—42no—
159C4×C73—82no—
160C4×C73—81no—
161C4×C73—81no—
162C4×C73—161no—
163#10386C7311yesmaximal, characteristic, direct product factor, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—41no—
167C2×C73—41no—
168C2×C73—82no—
169C2×C73#202911yescharacteristic
170C2×C73#203112yes—
171#2607C2×C7311yescharacteristic, complemented
172#1789C2×C7312yes—
173#1782C2×C7311yescharacteristic
174#1306C2×C7311yescharacteristic, complemented
175#1284C2×C7312yescomplemented
176C73#1038611yescharacteristic, direct product factor, Sylow
177C22×#41C22×C7311yescharacteristic
178C2×#104—22no—
179C2×#104—22no—
180C2×#99—21no—
181C2×#99—21no—
182#407C22×C7312yes—
183#399—21no—
184#393—21no—
185#391—22no—
186#387—22no—
187#381C22×C7311yescharacteristic
188#381C22×C7311yescharacteristic
189C4×C22⋊C4C22×C7312yes—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—42no—
193C2×#41—42no—
194C22×Q8—21no—
195C22×Q8Q8×C7311yescharacteristic, complemented
196C22×D4—21no—
197C22×D4Q8×C7311yescharacteristic, complemented
198C23×C4—21no—
199C23×C4C23×C7311yescharacteristic
200C2×M16—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4D4×C7311yescharacteristic
206C2×C22⋊C4D4×C7311yescharacteristic
207C2×C42D4×C7312yes—
208C2×C42D4×C7312yes—
209#104—44no—
210#104—44no—
211#99—42no—
212#99—42no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24—21no—
222C24C2×D4×C7311yescharacteristic
223#41—24no—
224#41—24no—
225#41—42no—
226#41—42no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—81no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4C2×D4×C7312yes—
253C22×C4C2×Q8×C7311yescharacteristic
254C22×C4#41×C7312yes—
255C22×C4#41×C7311yescharacteristic
256M16—42no—
257M16—42no—
258C2×C8—42no—
259C4⋊C4—44no—
260C4⋊C4—81no—
261C22⋊C4—44no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—42no—
267C42—42no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23#120×C7311yescharacteristic
276C23C22⋊Q8×C7311yescharacteristic
277C23#124×C7311yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—84no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4C22⋊Q8×C7312yes—
310C2×C4#124×C7312yes—
311C8—82no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#393×C7311yescharacteristic
324C22B(4,𝔽2)×C7311yescharacteristic
325C22B(4,𝔽2)×C7311yescharacteristic
326C4—22no—
327C4—22no—
328C4—44no—
329C4—42no—
330C4—81no—
331C4—82no—
332C4—81no—
333C4—161no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2C73×#202911yescharacteristic
340C2C73×#203112yes—
subgroupG/H
center, socleC22×C73#393
commutator, Frattini, cosocleC23×C4C23×C73
prime radicalC22×#41×C73C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2607C2semidirect
4. C73×#1306C2semidirect
5. C73×#1284C2semidirect
24. C22×Q8×C73Q8semidirect
26. C22×D4×C73Q8semidirect
163. #10386C73direct (N × H)
171. #2607C2×C73semidirect
174. #1306C2×C73semidirect
175. #1284C2×C73semidirect
176. C73#10386direct (N × H)
195. C22×Q8Q8×C73semidirect
197. C22×D4Q8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#26072
C73×#17892
C73×#17822
C73×#13062
C73×#12842
#1038673