Finite Group Databasebeta

← C73×#13037 #32922441

Subgroups — C73×#13037 #32922441

Order 18688.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C73C211yesmaximal, characteristic
2C2×#442×C73C211yesmaximal, characteristic, complemented
3C2×#361×C73C211yesmaximal, characteristic, complemented
4C2×#360×C73C211yesmaximal, characteristic, complemented
5C73×#1975C211yesmaximal, characteristic, complemented
6C73×#1383C211yesmaximal, characteristic, complemented
7C73×#1375C211yesmaximal, characteristic, complemented
8C2×#135×C73C2211yescharacteristic
9C22×D8×C73C2211yescharacteristic
10C2×C4×D4×C73—21no—
11#509×C73—22no—
12#509×C73—22no—
13C2×SD32×C73C412yescomplemented
14C2×D16×C73C412yescomplemented
15C2×M32×C73C2211yescharacteristic
16C2×M32×C73C2211yescharacteristic
17#442×C73—22no—
18#442×C73—22no—
19C2×#108×C73C2211yescharacteristic
20C2×#106×C73C2211yescharacteristic
21C2×#104×C73—21no—
22C2×#102×C73—21no—
23#408×C73—21no—
24C2×#97×C73C2211yescharacteristic
25#361×C73—22no—
26#360×C73—22no—
27C2×#41×C73—21no—
28C22×D4×C73—21no—
29C23×C4×C73—41no—
30#135×C73—21no—
31#135×C73—21no—
32#135×C73—21no—
33#135×C73—21no—
34C2×Q16×C73C2×C411yescharacteristic, complemented
35C2×SD16×C73—21no—
36C2×D8×C73—21no—
37C2×D8×C73—21no—
38C2×D8×C73—21no—
39C2×D8×C73C2×C412yes—
40C2×D8×C73C2×C411yescharacteristic, complemented
41C2×D8×C73D412yes—
42C2×D8×C73D412yes—
43C2×M16×C73—21no—
44C22×C8×C73D411yescharacteristic
45C22×C8×C73C2311yescharacteristic
46C4×D4×C73—24no—
47C4×D4×C73—42no—
48C2×C4⋊C4×C73—21no—
49C2×C22⋊C4×C73—41no—
50C2×C42×C73D411yescharacteristic
51SD32×C73—24no—
52D16×C73—24no—
53M32×C73—21no—
54M32×C73—21no—
55M32×C73—21no—
56M32×C73—21no—
57C2×C16×C73—21no—
58C2×C16×C73C2×C412yescomplemented
59#108×C73—21no—
60#108×C73—21no—
61#106×C73—21no—
62#106×C73D412yes—
63#104×C73—42no—
64#102×C73—24no—
65C22⋊C8×C73—42no—
66#97×C73—21no—
67#97×C73—21no—
68C24×C73—41no—
69#41×C73—41no—
70#41×C73—41no—
71#41×C73—41no—
72#41×C73—41no—
73C2×Q8×C73—21no—
74C2×D4×C73—22no—
75C2×D4×C73—21no—
76C2×D4×C73—22no—
77C2×D4×C73—22no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73—41no—
82C22×C4×C73—21no—
83C22×C4×C73—44no—
84C22×C4×C73—41no—
85C22×C4×C73—41no—
86C22×C4×C73—84no—
87C22×C4×C73C2×D411yescharacteristic
88Q16×C73—21no—
89Q16×C73—21no—
90SD16×C73—41no—
91SD16×C73—41no—
92D8×C73—21no—
93D8×C73—22no—
94D8×C73—21no—
95D8×C73—22no—
96D8×C73—22no—
97D8×C73C22⋊C414yes—
98M16×C73—41no—
99M16×C73—41no—
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73—21no—
103C2×C8×C73—41no—
104C2×C8×C73C22×C411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C2×C8×C73#4111yescharacteristic
110C4⋊C4×C73—22no—
111C4⋊C4×C73—41no—
112C22⋊C4×C73—44no—
113C42×C73—21no—
114C42×C73D812yes—
115C16×C73—22no—
116C16×C73—41no—
117C23×C73—42no—
118C23×C73—42no—
119C23×C73—42no—
120C23×C73—41no—
121C23×C73—82no—
122C23×C73—82no—
123C23×C73AGL(1,ℤ/8)11yescharacteristic
124Q8×C73—41no—
125Q8×C73—41no—
126D4×C73—24no—
127D4×C73—41no—
128D4×C73—42no—
129D4×C73—41no—
130D4×C73—42no—
131D4×C73—42no—
132D4×C73—81no—
133D4×C73—81no—
134C2×C4×C73—22no—
135C2×C4×C73—42no—
136C2×C4×C73—41no—
137C2×C4×C73—42no—
138C2×C4×C73—88no—
139C2×C4×C73—82no—
140C2×C4×C73—81no—
141C2×C4×C73—81no—
142C2×C4×C73C4×D411yescharacteristic
143C2×C4×C73#12011yescharacteristic
144C2×C4×C73#12111yescharacteristic
145C2×C4×C73C2×D811yescharacteristic
146C2×C4×C73#13511yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C8×C73—41no—
149C8×C73—81no—
150C8×C73C2×C22⋊C411yescharacteristic
151C8×C73C4×D411yescharacteristic
152C8×C73#12111yescharacteristic
153C8×C73#12311yescharacteristic
154C22×C73—22no—
155C22×C73—44no—
156C22×C73—82no—
157C22×C73—82no—
158C22×C73—84no—
159C22×C73—84no—
160C22×C73—81no—
161C22×C73—82no—
162C22×C73#44211yescharacteristic
163C22×C73#44711yescharacteristic
164C22×C73#46311yescharacteristic
165C4×C73—42no—
166C4×C73—82no—
167C4×C73—81no—
168C4×C73#38611yescharacteristic
169C4×C73C4×D811yescharacteristic
170C4×C73#44911yescharacteristic
171C4×C73#45911yescharacteristic
172#13037C7311yesmaximal, characteristic, direct product factor, Sylow
173C2×C73—21no—
174C2×C73—21no—
175C2×C73—84no—
176C2×C73—161no—
177C2×C73#187911yescharacteristic
178C2×C73AGL(1,ℤ/16)12yes—
179C2×#509C2×C7311yescharacteristic
180C2×#442C2×C7311yescharacteristic, complemented
181C2×#361C2×C7311yescharacteristic, complemented
182C2×#360C2×C7311yescharacteristic, complemented
183#1975C2×C7311yescharacteristic, complemented
184#1383C2×C7311yescharacteristic, complemented
185#1375C2×C7311yescharacteristic, complemented
186C73#1303711yescharacteristic, direct product factor, Sylow
187C2×#135C22×C7311yescharacteristic
188C22×D8C22×C7311yescharacteristic
189C2×C4×D4—21no—
190#509—22no—
191#509—22no—
192C2×SD32C4×C7312yescomplemented
193C2×D16C4×C7312yescomplemented
194C2×M32C22×C7311yescharacteristic
195C2×M32C22×C7311yescharacteristic
196#442—22no—
197#442—22no—
198C2×#108C22×C7311yescharacteristic
199C2×#106C22×C7311yescharacteristic
200C2×#104—21no—
201C2×#102—21no—
202#408—21no—
203C2×#97C22×C7311yescharacteristic
204#361—22no—
205#360—22no—
206C2×#41—21no—
207C22×D4—21no—
208C23×C4—41no—
209#135—21no—
210#135—21no—
211#135—21no—
212#135—21no—
213C2×Q16C2×C4×C7311yescharacteristic, complemented
214C2×SD16—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8C2×C4×C7312yes—
219C2×D8C2×C4×C7311yescharacteristic, complemented
220C2×D8D4×C7312yes—
221C2×D8D4×C7312yes—
222C2×M16—21no—
223C22×C8D4×C7311yescharacteristic
224C22×C8C23×C7311yescharacteristic
225C4×D4—24no—
226C4×D4—42no—
227C2×C4⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C42D4×C7311yescharacteristic
230SD32—24no—
231D16—24no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—21no—
236C2×C16—21no—
237C2×C16C2×C4×C7312yescomplemented
238#108—21no—
239#108—21no—
240#106—21no—
241#106D4×C7312yes—
242#104—42no—
243#102—24no—
244C22⋊C8—42no—
245#97—21no—
246#97—21no—
247C24—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—41no—
252C2×Q8—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—84no—
266C22×C4C2×D4×C7311yescharacteristic
267Q16—21no—
268Q16—21no—
269SD16—41no—
270SD16—41no—
271D8—21no—
272D8—22no—
273D8—21no—
274D8—22no—
275D8—22no—
276D8C22⋊C4×C7314yes—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8C22×C4×C7311yescharacteristic
284C2×C8C2×D4×C7311yescharacteristic
285C2×C8C2×D4×C7311yescharacteristic
286C2×C8C2×D4×C7311yescharacteristic
287C2×C8#41×C7311yescharacteristic
288C2×C8#41×C7311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C22⋊C4—44no—
292C42—21no—
293C42D8×C7312yes—
294C16—22no—
295C16—41no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—82no—
301C23—82no—
302C23AGL(1,ℤ/8)×C7311yescharacteristic
303Q8—41no—
304Q8—41no—
305D4—24no—
306D4—41no—
307D4—42no—
308D4—41no—
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—88no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4C4×D4×C7311yescharacteristic
322C2×C4#120×C7311yescharacteristic
323C2×C4#121×C7311yescharacteristic
324C2×C4C2×D8×C7311yescharacteristic
325C2×C4#135×C7311yescharacteristic
326C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
327C8—41no—
328C8—81no—
329C8C2×C22⋊C4×C7311yescharacteristic
330C8C4×D4×C7311yescharacteristic
331C8#121×C7311yescharacteristic
332C8#123×C7311yescharacteristic
333C22—22no—
334C22—44no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—84no—
339C22—81no—
340C22—82no—
341C22#442×C7311yescharacteristic
342C22#447×C7311yescharacteristic
343C22#463×C7311yescharacteristic
344C4—42no—
345C4—82no—
346C4—81no—
347C4#386×C7311yescharacteristic
348C4C4×D8×C7311yescharacteristic
349C4#449×C7311yescharacteristic
350C4#459×C7311yescharacteristic
351C2—21no—
352C2—21no—
353C2—84no—
354C2—161no—
355C2C73×#187911yescharacteristic
356C2C73×AGL(1,ℤ/16)12yes—
subgroupG/H
center, socleC22×C73#442
commutatorC2×C8C22×C4×C73
Frattini, cosocleC22×C8C23×C73
prime radicalC2×#509×C73C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×#97×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#442×C73C2semidirect
3. C2×#361×C73C2semidirect
4. C2×#360×C73C2semidirect
5. C73×#1975C2semidirect
6. C73×#1383C2semidirect
7. C73×#1375C2semidirect
13. C2×SD32×C73C4semidirect
14. C2×D16×C73C4semidirect
34. C2×Q16×C73C2×C4semidirect
40. C2×D8×C73C2×C4semidirect
58. C2×C16×C73C2×C4semidirect
172. #13037C73direct (N × H)
180. C2×#442C2×C73semidirect
181. C2×#361C2×C73semidirect
182. C2×#360C2×C73semidirect
183. #1975C2×C73semidirect
184. #1383C2×C73semidirect
185. #1375C2×C73semidirect
186. C73#13037direct (N × H)
192. C2×SD32C4×C73semidirect
193. C2×D16C4×C73semidirect
213. C2×Q16C2×C4×C73semidirect
219. C2×D8C2×C4×C73semidirect
237. C2×C16C2×C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C732
C2×#442×C732
C2×#361×C732
C2×#360×C732
C73×#19752
C73×#13832
C73×#13752
#1303773