Finite Group Databasebeta

← #32982641

Subgroups — #32982641

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620853C211yesmaximal, characteristic
2#22610904C211yesmaximal, characteristic, complemented
3#22608617C211yesmaximal, characteristic, complemented
4#22608491C211yesmaximal, characteristic
5#22607791C211yesmaximal, characteristic, complemented
6#22607762C211yesmaximal, characteristic, complemented
7C73×#2932C211yesmaximal, characteristic
8C2×#14556591C2211yescharacteristic
9#17988959C412yescomplemented
10#17988939C412yescomplemented
11#17988792—22no—
12#17988725—22no—
13#17988640C2211yescharacteristic
14#17988640C2211yescharacteristic
15#17988453C2211yescharacteristic
16#17988350—21no—
17#17988350—21no—
18#17988332—22no—
19#17988321—22no—
20C2×#131×C73—21no—
21#532×C73C2211yescharacteristic
22#435×C73—22no—
23#435×C73—22no—
24#423×C73C2211yescharacteristic
25#408×C73—21no—
26C2×#97×C73C2211yescharacteristic
27C22×Dic146—21no—
28C2×#11924016—24no—
29#14556591—21no—
30#14556591—21no—
31#14556591—21no—
32#14556591C2×C412yes—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C4⋊#10596831—21no—
37C4⋊#10596831—21no—
38#14556514—24no—
39C22⋊#10596831—21no—
40C22⋊#10596831—21no—
41C4×#10596831—21no—
42C4×#10596831C2×C412yes—
43#14556498—21no—
44#14556498—21no—
45C2×#41×C73—21no—
46#131×C73—42no—
47#131×C73—42no—
48C2×M16×C73—22no—
49C2×M16×C73—21no—
50C22×C8×C73—22no—
51C22×C8×C73D411yescharacteristic
52C22×C8×C73D411yescharacteristic
53C4⋊Q8×C73C2×C411yescharacteristic
54#127×C73C2×C411yescharacteristic
55#124×C73—21no—
56#121×C73—21no—
57#121×C73D412yes—
58C4×D4×C73—21no—
59C4×D4×C73D412yes—
60C2×C42×C73C2311yescharacteristic
61C4⋊C8×C73—21no—
62C4⋊C8×C73D412yes—
63C22⋊C8×C73—24no—
64#97×C73—21no—
65#97×C73—21no—
66C2×Dic146—22no—
67C2×Dic146—41no—
68C2×Dic146—82no—
69C22×Dic73—41no—
70C22×Dic73—41no—
71#11924016—84no—
72C2×#10596831—22no—
73C2×#10596831—41no—
74C4⋊Dic73—22no—
75C4⋊Dic73—22no—
76C4⋊Dic73—41no—
77C4⋊Dic73—41no—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80#41×C73—41no—
81#41×C73—41no—
82#41×C73—41no—
83#41×C73—41no—
84C2×Q8×C73—21no—
85C2×D4×C73—22no—
86C2×D4×C73—21no—
87C2×D4×C73—21no—
88C2×D4×C73C22⋊C412yes—
89C22×C4×C73—22no—
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73#4111yescharacteristic
93M16×C73—44no—
94M16×C73—41no—
95M16×C73—41no—
96C2×C8×C73—21no—
97C2×C8×C73—21no—
98C2×C8×C73—21no—
99C2×C8×C73—22no—
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73—44no—
103C2×C8×C73SD1612yescomplemented
104C4⋊C4×C73—21no—
105C4⋊C4×C73C22⋊C412yes—
106C22⋊C4×C73—22no—
107C22⋊C4×C73—22no—
108C42×C73C22×C411yescharacteristic
109C42×C73C2×D411yescharacteristic
110C42×C73C2×D411yescharacteristic
111C42×C73#4111yescharacteristic
112C23×C73—22no—
113C23×C73#12111yescharacteristic
114Q8×C73—41no—
115Q8×C73—41no—
116D4×C73—42no—
117D4×C73—42no—
118D4×C73—42no—
119D4×C73—41no—
120D4×C73—41no—
121D4×C73—81no—
122C2×C4×C73—22no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73C2×C22⋊C411yescharacteristic
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73C4×D411yescharacteristic
130C2×C4×C73#12011yescharacteristic
131C2×C4×C73#12111yescharacteristic
132C2×C4×C73#12311yescharacteristic
133C2×C4×C73C2×SD1611yescharacteristic
134C2×C4×C73#13511yescharacteristic
135C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C73#13711yescharacteristic
137C2×Dic73—42no—
138C2×Dic73—82no—
139C2×Dic73—81no—
140C2×Dic73—81no—
141C2×Dic73—81no—
142Dic146—44no—
143Dic146—82no—
144C8×C73—22no—
145C8×C73—42no—
146C8×C73—41no—
147#10596831—42no—
148#10596831—81no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73—42no—
152C22×C73—42no—
153C22×C73—42no—
154C22×C73#38611yescharacteristic
155C22×C73#46111yescharacteristic
156C22×C73#46311yescharacteristic
157C4×C73—21no—
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—21no—
161C4×C73—42no—
162C4×C73C4×SD1611yescharacteristic
163C4×C73#44011yescharacteristic
164C4×C73#45011yescharacteristic
165C4×C73#45211yescharacteristic
166Dic73—82no—
167Dic73—161no—
168#9289—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—82no—
172C2×C73#187711yescharacteristic
173C2×C73#190211yescharacteristic
174C2×C73#191011yescharacteristic
175#3206—731no—
176#2932D7311yescharacteristic
177#1993—731no—
178#1549—731no—
179#1517—731no—
180#1286—731no—
181#1275—731no—
182C73#928911yescharacteristic, complemented, Sylow
183C2×#131—21no—
184#532C2×D7311yescharacteristic
185C2×C4⋊Q8—731no—
186#492—732no—
187#487—732no—
188#461—1462no—
189#448—1462no—
190#435—22no—
191#435—22no—
192#431—731no—
193#431—731no—
194#423C2×D7311yescharacteristic
195#408—21no—
196C2×#97C2×D7311yescharacteristic
197#382—731no—
198#340—1461no—
199#340—1461no—
200#332—1462no—
201#327—1462no—
202C2×#41—21no—
203C22×Q8—1461no—
204C2×SD16—1464no—
205#131—42no—
206#131—42no—
207C2×M16—22no—
208C2×M16—21no—
209C22×C8—22no—
210C22×C8C22⋊D7311yescharacteristic
211C22×C8C22⋊D7311yescharacteristic
212C4⋊Q8—732no—
213C4⋊Q8—1461no—
214C4⋊Q8—1461no—
215C4⋊Q8—1461no—
216C4⋊Q8C4×D7311yescharacteristic
217#127C4×D7311yescharacteristic
218#124—21no—
219#121—21no—
220#121D29212yes—
221C4×D4—21no—
222C4×D4C22⋊D7312yes—
223C2×C4⋊C4—1461no—
224C2×C4⋊C4—1461no—
225C2×C4⋊C4—1461no—
226C2×C42C22×D7311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—1461no—
229C4⋊C8—1461no—
230C4⋊C8C22⋊D7312yes—
231#103—1464no—
232C22⋊C8—24no—
233C22⋊C8—1461no—
234C22⋊C8—1461no—
235#97—21no—
236#97—21no—
237C4×C8—732no—
238C4×C8—1461no—
239#95—1461no—
240#95—1461no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—2921no—
247C2×Q8—5842no—
248C2×Q8—1462no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4#1192400512yes—
253C22×C4—22no—
254C22×C4—2921no—
255C22×C4—2921no—
256C22×C4C2×C22⋊D7311yescharacteristic
257C22×C4C2×D29211yescharacteristic
258C22×C4#1192402811yescharacteristic
259SD16—5844no—
260M16—44no—
261M16—41no—
262M16—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—44no—
270C2×C8—2921no—
271C2×C8—1462no—
272C2×C8#1192401612yescomplemented
273C4⋊C4—21no—
274C4⋊C4—2921no—
275C4⋊C4—2921no—
276C4⋊C4—2921no—
277C4⋊C4—2921no—
278C4⋊C4—1462no—
279C4⋊C4—1462no—
280C4⋊C4#1192400512yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C42C2×C4×D7311yescharacteristic
284C42C2×C22⋊D7311yescharacteristic
285C42C2×C22⋊D7311yescharacteristic
286C42#1192402811yescharacteristic
287C23—22no—
288C23#1455656511yescharacteristic
289Q8—41no—
290Q8—41no—
291Q8—2924no—
292Q8—5842no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298D4—81no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—2922no—
305C2×C4—5842no—
306C2×C4—5841no—
307C2×C4—5841no—
308C2×C4—5841no—
309C2×C4C2×#1192400511yescharacteristic
310C2×C4C4×C22⋊D7311yescharacteristic
311C2×C4C4×C22⋊D7311yescharacteristic
312C2×C4#1455656011yescharacteristic
313C2×C4#1455656511yescharacteristic
314C2×C4#1455657411yescharacteristic
315C2×C4C2×#1192401611yescharacteristic
316C2×C4#1455661111yescharacteristic
317C2×C4#1455661711yescharacteristic
318C2×C4#1455662111yescharacteristic
319C8—22no—
320C8—42no—
321C8—41no—
322C8—2922no—
323C8—5841no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1798846611yescharacteristic
330C22#1798879211yescharacteristic
331C22#1798880211yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—5842no—
338C4—11681no—
339C4C4×#1192401611yescharacteristic
340C4#1798868711yescharacteristic
341C4#1798873811yescharacteristic
342C4#1798874911yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2#2261026511yescharacteristic
347C2#2261040411yescharacteristic
348C2#2261045811yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2932C2
socleC22×C73#386
prime radical#532×C73C22
cosocleC2×C42×C73C23
hypercenter#2932D73
hypocenterC73#9289
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #22610904C2semidirect
3. #22608617C2semidirect
5. #22607791C2semidirect
6. #22607762C2semidirect
9. #17988959C4semidirect
10. #17988939C4semidirect
103. C2×C8×C73SD16semidirect
182. C73#9289semidirect
272. C2×C8#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208532
#226109042
#226086172
#226084912
#226077912
#226077622
C73×#29322
#928973