Finite Group Databasebeta

← C73×#10552 #32919956

Subgroups — C73×#10552 #32919956

Order 18688.

386 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3141C211yesmaximal, characteristic
2C73×#1950C211yesmaximal, characteristic, complemented
3C73×#1548C211yesmaximal, characteristic, complemented
4C73×#1547C211yesmaximal, characteristic
5C2×#332×C73C211yesmaximal, characteristic, complemented
6C73×#1480C211yesmaximal, characteristic
7C73×#1283C211yesmaximal, characteristic, complemented
8#532×C73C2211yescharacteristic
9C2×C4⋊Q8×C73C2211yescharacteristic
10#489×C73—22no—
11#487×C73—21no—
12#486×C73—21no—
13#431×C73C2211yescharacteristic
14C2×#108×C73—21no—
15#423×C73C2211yescharacteristic
16C2×C4⋊C8×C73C2211yescharacteristic
17C2×C4⋊C8×C73C2211yescharacteristic
18#417×C73—21no—
19#417×C73—21no—
20C4×M16×C73C2211yescharacteristic
21#341×C73—21no—
22#333×C73—22no—
23#332×C73—21no—
24#332×C73—22no—
25#331×C73—21no—
26D4⋊C8×C73—22no—
27C2×#41×C73—21no—
28C22×Q8×C73—21no—
29C2×M16×C73—21no—
30C2×M16×C73D411yescharacteristic
31C2×M16×C73D411yescharacteristic
32C22×C8×C73—21no—
33C22×C8×C73—21no—
34C4⋊Q8×C73—21no—
35C4⋊Q8×C73—21no—
36C4⋊Q8×C73—21no—
37C4⋊Q8×C73D411yescharacteristic
38C4⋊Q8×C73D412yes—
39#127×C73D411yescharacteristic
40#124×C73—21no—
41#121×C73—21no—
42#121×C73—21no—
43C4×D4×C73—21no—
44C4×D4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C42×C73C2311yescharacteristic
48#108×C73—42no—
49C4⋊C8×C73—21no—
50C4⋊C8×C73—21no—
51C4⋊C8×C73—21no—
52C4⋊C8×C73—21no—
53C4⋊C8×C73—21no—
54C4⋊C8×C73—21no—
55C4⋊C8×C73D412yes—
56#103×C73—22no—
57#103×C73—22no—
58#102×C73—22no—
59#102×C73—22no—
60C22⋊C8×C73—21no—
61#97×C73—21no—
62C4×C8×C73—21no—
63C4×C8×C73—21no—
64#41×C73—41no—
65#41×C73—41no—
66#41×C73—41no—
67#41×C73—41no—
68C2×Q8×C73—21no—
69C2×Q8×C73—22no—
70C2×Q8×C73—41no—
71C2×Q8×C73—81no—
72C2×Q8×C73—81no—
73C2×D4×C73—21no—
74C2×D4×C73—21no—
75C2×D4×C73—21no—
76C2×D4×C73—41no—
77C22×C4×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73C2×D411yescharacteristic
80C22×C4×C73#4111yescharacteristic
81C22×C4×C73#4111yescharacteristic
82M16×C73—21no—
83M16×C73—21no—
84M16×C73—21no—
85M16×C73—21no—
86M16×C73—41no—
87M16×C73—41no—
88C2×C8×C73—21no—
89C2×C8×C73—21no—
90C2×C8×C73—22no—
91C2×C8×C73—41no—
92C2×C8×C73—41no—
93C2×C8×C73—41no—
94C2×C8×C73—41no—
95C2×C8×C73—41no—
96C2×C8×C73—81no—
97C4⋊C4×C73—22no—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C42×C73C2×D411yescharacteristic
106C42×C73C2×D411yescharacteristic
107C42×C73C2×D411yescharacteristic
108C42×C73#4111yescharacteristic
109C23×C73—41no—
110C23×C73#12411yescharacteristic
111Q8×C73—41no—
112Q8×C73—41no—
113Q8×C73—82no—
114Q8×C73—81no—
115Q8×C73—81no—
116D4×C73—41no—
117D4×C73—41no—
118D4×C73—41no—
119D4×C73—41no—
120D4×C73—81no—
121D4×C73—81no—
122D4×C73—81no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—42no—
128C2×C4×C73—82no—
129C2×C4×C73—81no—
130C2×C4×C73—81no—
131C2×C4×C73#12011yescharacteristic
132C2×C4×C73#12111yescharacteristic
133C2×C4×C73#12111yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73#12311yescharacteristic
136C2×C4×C73#12311yescharacteristic
137C2×C4×C73#13511yescharacteristic
138C2×C4×C73#13511yescharacteristic
139C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C73#13711yescharacteristic
141C8×C73—41no—
142C8×C73—41no—
143C8×C73—42no—
144C8×C73—81no—
145C8×C73—81no—
146C8×C73—81no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—81no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73#39411yescharacteristic
153C22×C73#48811yescharacteristic
154C22×C73#48911yescharacteristic
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—21no—
158C4×C73—82no—
159C4×C73—81no—
160C4×C73#44911yescharacteristic
161C4×C73#45211yescharacteristic
162C4×C73#46311yescharacteristic
163C4×C73#46411yescharacteristic
164C4×C73#47111yescharacteristic
165C4×C73#47211yescharacteristic
166#10552C7311yesmaximal, characteristic, direct product factor, Sylow
167C2×C73—21no—
168C2×C73—21no—
169C2×C73—161no—
170C2×C73#204211yescharacteristic
171C2×C73#204911yescharacteristic
172C2×C73#205111yescharacteristic
173#3141C2×C7311yescharacteristic
174#1950C2×C7311yescharacteristic, complemented
175#1548C2×C7311yescharacteristic, complemented
176#1547C2×C7311yescharacteristic
177C2×#332C2×C7311yescharacteristic, complemented
178#1480C2×C7311yescharacteristic
179#1283C2×C7311yescharacteristic, complemented
180C73#1055211yescharacteristic, direct product factor, Sylow
181#532C22×C7311yescharacteristic
182C2×C4⋊Q8C22×C7311yescharacteristic
183#489—22no—
184#487—21no—
185#486—21no—
186#431C22×C7311yescharacteristic
187C2×#108—21no—
188#423C22×C7311yescharacteristic
189C2×C4⋊C8C22×C7311yescharacteristic
190C2×C4⋊C8C22×C7311yescharacteristic
191#417—21no—
192#417—21no—
193C4×M16C22×C7311yescharacteristic
194#341—21no—
195#333—22no—
196#332—21no—
197#332—22no—
198#331—21no—
199D4⋊C8—22no—
200C2×#41—21no—
201C22×Q8—21no—
202C2×M16—21no—
203C2×M16D4×C7311yescharacteristic
204C2×M16D4×C7311yescharacteristic
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8D4×C7312yes—
211C4⋊Q8D4×C7311yescharacteristic
212#127D4×C7311yescharacteristic
213#124—21no—
214#121—21no—
215#121—21no—
216C4×D4—21no—
217C4×D4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C42C23×C7311yescharacteristic
221#108—42no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8D4×C7312yes—
229#103—22no—
230#103—22no—
231#102—22no—
232#102—22no—
233C22⋊C8—21no—
234#97—21no—
235C4×C8—21no—
236C4×C8—21no—
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×Q8—81no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×D4×C7311yescharacteristic
253C22×C4#41×C7311yescharacteristic
254C22×C4#41×C7311yescharacteristic
255M16—21no—
256M16—21no—
257M16—21no—
258M16—21no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—81no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42C2×D4×C7311yescharacteristic
279C42C2×D4×C7311yescharacteristic
280C42C2×D4×C7311yescharacteristic
281C42#41×C7311yescharacteristic
282C23—41no—
283C23#124×C7311yescharacteristic
284Q8—41no—
285Q8—41no—
286Q8—82no—
287Q8—81no—
288Q8—81no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—81no—
294D4—81no—
295D4—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4#120×C7311yescharacteristic
305C2×C4#121×C7311yescharacteristic
306C2×C4#121×C7311yescharacteristic
307C2×C4#121×C7311yescharacteristic
308C2×C4#123×C7311yescharacteristic
309C2×C4#123×C7311yescharacteristic
310C2×C4#135×C7311yescharacteristic
311C2×C4#135×C7311yescharacteristic
312C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
313C2×C4#137×C7311yescharacteristic
314C8—41no—
315C8—41no—
316C8—42no—
317C8—81no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#394×C7311yescharacteristic
326C22#488×C7311yescharacteristic
327C22#489×C7311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—82no—
333C4#449×C7311yescharacteristic
334C4#452×C7311yescharacteristic
335C4#463×C7311yescharacteristic
336C4#464×C7311yescharacteristic
337C4#471×C7311yescharacteristic
338C4#472×C7311yescharacteristic
339C2—21no—
340C2—21no—
341C2—161no—
342C2C73×#204211yescharacteristic
343C2C73×#204911yescharacteristic
344C2C73×#205111yescharacteristic
subgroupG/H
center, socleC22×C73#394
commutator, Frattini, cosocleC2×C42C23×C73
prime radical#532×C73C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1950C2semidirect
3. C73×#1548C2semidirect
5. C2×#332×C73C2semidirect
7. C73×#1283C2semidirect
166. #10552C73direct (N × H)
174. #1950C2×C73semidirect
175. #1548C2×C73semidirect
177. C2×#332C2×C73semidirect
179. #1283C2×C73semidirect
180. C73#10552direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31412
C73×#19502
C73×#15482
C73×#15472
C2×#332×C732
C73×#14802
C73×#12832
#1055273