Finite Group Databasebeta

← #33131105

Subgroups — #33131105

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618783C211yesmaximal, characteristic, complemented
2#22611391C212yesmaximal, complemented
3#22611201C212yesmaximal, complemented
4#22610842C211yesmaximal, characteristic, complemented
5#22610719C211yesmaximal, characteristic, complemented
6#22610420C212yesmaximal, complemented
7#22610056C211yesmaximal, characteristic, complemented
8#22610040C212yesmaximal, complemented
9#22609985C211yesmaximal, characteristic, complemented
10#22609731C211yesmaximal, characteristic, complemented
11C2×#472×C73C211yesmaximal, characteristic
12#17989062C2212yes—
13#17988699—22no—
14D8×Dic73—22no—
15#17988635C2212yes—
16C2×#14556527C2211yescharacteristic
17#17988625C2211yescharacteristic
18#17988604C2212yes—
19#17988592C2212yes—
20#17988592C2212yes—
21#17988571C2211yescharacteristic
22C2×#14556508C2212yes—
23#17988561C2211yescharacteristic
24#17988542C2211yescharacteristic
25#17988542C2211yescharacteristic
26#17988359C2212yes—
27#17988359C2212yes—
28#17988358C2212yes—
29#17988358C2211yescharacteristic
30#17988358C2211yescharacteristic
31#17988344C2212yes—
32C2×AGL(1,ℤ/8)×C73C2212yes—
33C22×D8×C73C2211yescharacteristic
34C2×#131×C73C2211yescharacteristic
35#472×C73—22no—
36#472×C73—22no—
37C2×#108×C73C2211yescharacteristic
38C2×#100×C73C2212yes—
39C2×#11924012C2311yescharacteristic
40C2×#11924012C2312yes—
41C2×#11924012C2311yescharacteristic
42D4×Dic73—24no—
43D4×Dic73—42no—
44#14556550C2311yescharacteristic
45#14556550C2312yes—
46#14556550C2311yescharacteristic
47C2×C22⋊Dic73—24no—
48#14556527—21no—
49#14556527—21no—
50#14556526—42no—
51#14556525—21no—
52#14556523—21no—
53#14556518—42no—
54#14556512—24no—
55#14556512—42no—
56#14556510—21no—
57#14556508—21no—
58#14556508—42no—
59#14556506—22no—
60#14556501—21no—
61#14556501—21no—
62C8×Dic73—21no—
63C8×Dic73—21no—
64C2×#41×C73C2311yescharacteristic
65C22×D4×C73C2312yes—
66AGL(1,ℤ/8)×C73—24no—
67AGL(1,ℤ/8)×C73—42no—
68C2×SD16×C73—22no—
69C2×D8×C73—22no—
70C2×D8×C73—21no—
71C2×D8×C73—21no—
72C2×D8×C73—21no—
73C2×D8×C73—21no—
74C2×D8×C73D412yes—
75C2×D8×C73D412yes—
76#131×C73—21no—
77#131×C73—21no—
78#131×C73—21no—
79#131×C73—21no—
80C2×M16×C73—21no—
81C2×M16×C73C2311yescharacteristic
82C2×M16×C73C2312yes—
83C22×C8×C73—21no—
84C22×C8×C73C2311yescharacteristic
85#108×C73—21no—
86#108×C73—21no—
87#100×C73—24no—
88C22×Dic73—44no—
89#11924012—21no—
90#11924012—21no—
91#11924012—22no—
92#11924012—21no—
93#11924012—22no—
94#11924012—21no—
95C2×#10596831—21no—
96C2×#10596831—21no—
97C2×#10596831—22no—
98C4⋊Dic73—21no—
99C4⋊Dic73—22no—
100C4⋊Dic73—21no—
101C22⋊Dic73—21no—
102C22⋊Dic73—22no—
103C22⋊Dic73—44no—
104C22⋊Dic73#4112yes—
105C4×Dic73—21no—
106C4×Dic73—21no—
107C4×Dic73—22no—
108C24×C73—22no—
109#41×C73—21no—
110#41×C73—21no—
111#41×C73—21no—
112#41×C73—21no—
113C2×Q8×C73#4111yescharacteristic
114C2×D4×C73—21no—
115C2×D4×C73—22no—
116C2×D4×C73—22no—
117C2×D4×C73—22no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73C2×D414yes—
121C2×D4×C73#4111yescharacteristic
122C22×C4×C73—21no—
123C22×C4×C73C2411yescharacteristic
124SD16×C73—44no—
125D8×C73—22no—
126D8×C73—22no—
127D8×C73—22no—
128D8×C73—22no—
129D8×C73—44no—
130M16×C73—21no—
131M16×C73—22no—
132M16×C73—21no—
133M16×C73—22no—
134M16×C73—41no—
135M16×C73—41no—
136C2×C8×C73—21no—
137C2×C8×C73—22no—
138C2×C8×C73—41no—
139C2×C8×C73—41no—
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73C2×D411yescharacteristic
142C2×C8×C73#4111yescharacteristic
143C2×C8×C73#4111yescharacteristic
144C2×C8×C73#4111yescharacteristic
145C2×C8×C73#4111yescharacteristic
146C23×C73—21no—
147C23×C73—24no—
148C23×C73—42no—
149C23×C73—42no—
150C23×C73—162no—
151C23×C73C2×#4111yescharacteristic
152Q8×C73—21no—
153Q8×C73—21no—
154D4×C73—24no—
155D4×C73—21no—
156D4×C73—24no—
157D4×C73—21no—
158D4×C73—41no—
159D4×C73—41no—
160D4×C73—42no—
161D4×C73—42no—
162C2×C4×C73—21no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73C22×D411yescharacteristic
166C2×C4×C73C2×#4111yescharacteristic
167C2×C4×C73C2×#4111yescharacteristic
168C2×C4×C73C2×#4111yescharacteristic
169C2×C4×C73H(5,𝔽2)11yescharacteristic
170C2×C4×C73H(5,𝔽2)11yescharacteristic
171C2×Dic73—22no—
172C2×Dic73—42no—
173C2×Dic73—41no—
174C2×Dic73—164no—
175C8×C73—41no—
176C8×C73—42no—
177C8×C73#12311yescharacteristic
178C8×C73#12311yescharacteristic
179C8×C73#12311yescharacteristic
180C8×C73#12311yescharacteristic
181#10596831—42no—
182#10596831—41no—
183#10596831—41no—
184C22×C73—41no—
185C22×C73—41no—
186C22×C73—44no—
187C22×C73—84no—
188C22×C73—81no—
189C22×C73—162no—
190C22×C73—162no—
191C22×C73#52911yescharacteristic
192C22×C73#54611yescharacteristic
193C22×C73#55511yescharacteristic
194C4×C73—41no—
195C4×C73C2×#12311yescharacteristic
196C4×C73#54011yescharacteristic
197C4×C73#54611yescharacteristic
198C4×C73#55311yescharacteristic
199Dic73—42no—
200Dic73—81no—
201Dic73—82no—
202#24988—731nomaximal, Sylow
203C2×C73—21no—
204C2×C73—21no—
205C2×C73—21no—
206C2×C73—81no—
207C2×C73—84no—
208C2×C73#242011yescharacteristic
209C2×#472D7311yescharacteristic
210#2948—731no—
211#2081—732no—
212#2048—732no—
213#1978—731no—
214#1953—731no—
215#1904—732no—
216#1833—731no—
217#1829—732no—
218#1814—731no—
219#1757—731no—
220C73#2498811yescharacteristic, complemented, Sylow
221C2×AGL(1,ℤ/8)C2×D7312yes—
222C22×D8C2×D7311yescharacteristic
223C2×#131C2×D7311yescharacteristic
224#518—732no—
225#472—22no—
226#472—22no—
227#442—1462no—
228C4×D8—1462no—
229#430—732no—
230C2×#108—731no—
231C2×#108C2×D7311yescharacteristic
232#427—731no—
233#421—732no—
234#418—732no—
235#418—732no—
236#413—731no—
237C2×#100—732no—
238C2×#100C2×D7312yes—
239#410—731no—
240#405—731no—
241#405—731no—
242#344—732no—
243#344—731no—
244#344—731no—
245#344—732no—
246#344—732no—
247H(3,ℤ/4)—732no—
248C2×#41C22×D7311yescharacteristic
249C22×D4C22×D7312yes—
250AGL(1,ℤ/8)—24no—
251AGL(1,ℤ/8)—42no—
252C2×SD16—22no—
253C2×D8—22no—
254C2×D8—21no—
255C2×D8—21no—
256C2×D8—21no—
257C2×D8—21no—
258C2×D8C22⋊D7312yes—
259C2×D8C22⋊D7312yes—
260#131—21no—
261#131—21no—
262#131—21no—
263#131—21no—
264C2×M16—21no—
265C2×M16—731no—
266C2×M16—732no—
267C2×M16—731no—
268C2×M16C22×D7311yescharacteristic
269C2×M16C22×D7312yes—
270C22×C8—21no—
271C22×C8C22×D7311yescharacteristic
272C4×D4—2922no—
273C4×D4—1464no—
274#117—731no—
275#117—732no—
276#117—731no—
277C2×C22⋊C4—1464no—
278#108—21no—
279#108—21no—
280#108—2922no—
281#108—1461no—
282#108—1461no—
283#107—1461no—
284#106—1461no—
285#104—2922no—
286#102—2922no—
287#102—1464no—
288#101—1461no—
289#100—24no—
290#100—2922no—
291#100—1461no—
292#99—1462no—
293#97—1461no—
294#97—1461no—
295C4×C8—1461no—
296C4×C8—1461no—
297C24—22no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302C2×Q8#1192402711yescharacteristic
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4C2×C22⋊D7314yes—
310C2×D4#1192402711yescharacteristic
311C22×C4—21no—
312C22×C4—2924no—
313C22×C4C23×D7311yescharacteristic
314SD16—44no—
315D8—22no—
316D8—22no—
317D8—22no—
318D8—22no—
319D8—44no—
320M16—21no—
321M16—22no—
322M16—21no—
323M16—22no—
324M16—41no—
325M16—41no—
326M16—1461no—
327M16—1461no—
328M16—1462no—
329M16—1461no—
330M16—1461no—
331M16—1462no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—1461no—
337C2×C8—1461no—
338C2×C8—1462no—
339C2×C8C2×C22⋊D7311yescharacteristic
340C2×C8C2×C22⋊D7311yescharacteristic
341C2×C8#1192402711yescharacteristic
342C2×C8#1192402711yescharacteristic
343C2×C8#1192402711yescharacteristic
344C2×C8#1192402711yescharacteristic
345C4⋊C4—1462no—
346C4⋊C4—1461no—
347C4⋊C4—1461no—
348C22⋊C4—2924no—
349C22⋊C4—732no—
350C22⋊C4—1461no—
351C22⋊C4—1462no—
352C42—1461no—
353C42—1462no—
354C42—1461no—
355C23—24no—
356C23—21no—
357C23—42no—
358C23—42no—
359C23—162no—
360C23#41×D7311yescharacteristic
361Q8—21no—
362Q8—21no—
363D4—21no—
364D4—24no—
365D4—21no—
366D4—24no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—42no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—2922no—
375C2×C4—2921no—
376C2×C4—11684no—
377C2×C4—1462no—
378C2×C4C22×C22⋊D7311yescharacteristic
379C2×C4C2×#1192402711yescharacteristic
380C2×C4C2×#1192402711yescharacteristic
381C2×C4C2×#1192402711yescharacteristic
382C2×C4#1455663611yescharacteristic
383C2×C4#1455663711yescharacteristic
384C8—42no—
385C8—41no—
386C8—2922no—
387C8—2921no—
388C8—2921no—
389C8#1455657211yescharacteristic
390C8#1455657211yescharacteristic
391C8#1455657211yescharacteristic
392C8#1455657211yescharacteristic
393C22—41no—
394C22—41no—
395C22—44no—
396C22—84no—
397C22—81no—
398C22—162no—
399C22—162no—
400C22#1798911811yescharacteristic
401C22#1798922511yescharacteristic
402C22#1798930111yescharacteristic
403C4—41no—
404C4—2922no—
405C4—5842no—
406C4—5841no—
407C4C2×#1455657211yescharacteristic
408C4#1798919011yescharacteristic
409C4#1798923111yescharacteristic
410C4#1798928911yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—84no—
415C2—81no—
416C2#2261341911yescharacteristic
subgroupG/H
centerC2#22613419
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#472×C73C2
socleC2×C73#2420
hypercenterC2×#472D73
hypocenterC73#24988
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #22618783C2semidirect
2. #22611391C2semidirect
3. #22611201C2semidirect
4. #22610842C2semidirect
5. #22610719C2semidirect
6. #22610420C2semidirect
7. #22610056C2semidirect
8. #22610040C2semidirect
9. #22609985C2semidirect
10. #22609731C2semidirect
220. C73#24988semidirect

Maximal subgroups

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

subgroupindex
#226187832
#226113912
#226112012
#226108422
#226107192
#226104202
#226100562
#226100402
#226099852
#226097312
C2×#472×C732
#2498873