Finite Group Databasebeta

← #33003414

Subgroups — #33003414

Order 18688.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989004C211yesmaximal, characteristic, complemented
2#22611667C211yesmaximal, characteristic, complemented
3C2×#17988394C211yesmaximal, characteristic, complemented
4#22609985C211yesmaximal, characteristic, complemented
5#22608044C211yesmaximal, characteristic, complemented
6#22608028C211yesmaximal, characteristic, complemented
7C73×#3067C211yesmaximal, characteristic
8#17989004—22no—
9#17989004—22no—
10C2×#14556531—21no—
11C2×#14556531C2211yescharacteristic
12C22×#11924003—21no—
13C22×#11924003C2211yescharacteristic
14C2×#14556527C2211yescharacteristic
15#17988625C2211yescharacteristic
16#17988604—21no—
17#17988397—22no—
18#17988394—21no—
19#17988394—21no—
20#17988359—21no—
21#575×C73—21no—
22C2×#135×C73C2211yescharacteristic
23C2×#131×C73C2211yescharacteristic
24#473×C73C412yescomplemented
25#472×C73C412yescomplemented
26#471×C73—22no—
27C2×#108×C73C2211yescharacteristic
28#413×C73—21no—
29C2×#11924012—21no—
30#14556550—21no—
31#14556531—24no—
32#14556531—21no—
33#14556531D412yes—
34C2×#11924003—24no—
35C2×#11924003—21no—
36C2×#11924003—21no—
37C2×#11924003—21no—
38C2×#11924003D411yescharacteristic
39C2×#11924003D411yescharacteristic
40#14556527—21no—
41#14556527—21no—
42#14556525—21no—
43#14556523—21no—
44#14556518—42no—
45C2×#41×C73—21no—
46C2×#41×C73D411yescharacteristic
47#137×C73—22no—
48#137×C73—41no—
49AGL(1,ℤ/8)×C73—22no—
50AGL(1,ℤ/8)×C73—41no—
51#135×C73—21no—
52#135×C73—21no—
53#135×C73—21no—
54#135×C73—21no—
55#135×C73—42no—
56C2×Q16×C73C2×C411yescharacteristic
57C2×SD16×C73—21no—
58C2×D8×C73C2×C411yescharacteristic
59#131×C73—21no—
60#131×C73—21no—
61#131×C73C2×C412yes—
62#131×C73D412yes—
63C2×M16×C73—21no—
64C2×M16×C73—21no—
65C2×M16×C73D411yescharacteristic
66C22×C8×C73—21no—
67C22×C8×C73C2311yescharacteristic
68#108×C73—21no—
69#108×C73C2×C412yes—
70#101×C73—22no—
71#100×C73—22no—
72#11924012—41no—
73#11924012—41no—
74C2×#10596831—41no—
75C4⋊Dic73—41no—
76C22⋊Dic73—41no—
77C4×Dic73—41no—
78#11924003—21no—
79#11924003—22no—
80#11924003—21no—
81#41×C73—21no—
82#41×C73—21no—
83#41×C73—41no—
84#41×C73—41no—
85#41×C73—41no—
86#41×C73—41no—
87#41×C73C22⋊C412yes—
88#41×C73D812yes—
89C2×Q8×C73—21no—
90C2×Q8×C73SD1611yescharacteristic
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—41no—
94C2×D4×C73SD1611yescharacteristic
95C22×C4×C73—21no—
96C22×C4×C73—41no—
97C22×C4×C73C2×D411yescharacteristic
98Q16×C73—21no—
99Q16×C73—21no—
100Q16×C73—42no—
101SD16×C73—42no—
102SD16×C73—42no—
103SD16×C73—41no—
104SD16×C73—41no—
105D8×C73—21no—
106D8×C73—21no—
107D8×C73—42no—
108M16×C73—21no—
109M16×C73—22no—
110M16×C73—22no—
111M16×C73—22no—
112M16×C73—41no—
113M16×C73C22⋊C412yes—
114C2×C8×C73—21no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73—41no—
118C2×C8×C73C22×C411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C2×C8×C73#4111yescharacteristic
124C23×C73—21no—
125C23×C73—41no—
126C23×C73#13511yescharacteristic
127Q8×C73—21no—
128Q8×C73—41no—
129Q8×C73—41no—
130Q8×C73#10212yes—
131D4×C73—22no—
132D4×C73—22no—
133D4×C73—21no—
134D4×C73—41no—
135D4×C73—41no—
136D4×C73—81no—
137D4×C73—81no—
138D4×C73#10212yes—
139C2×C4×C73—21no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—41no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73C2×C22⋊C411yescharacteristic
146C2×C4×C73#12111yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73C2×SD1611yescharacteristic
150C2×C4×C73#13511yescharacteristic
151C2×Dic73—81no—
152C8×C73—22no—
153C8×C73—42no—
154C8×C73C4×D411yescharacteristic
155C8×C73C4×D411yescharacteristic
156C8×C73#12011yescharacteristic
157C8×C73#12311yescharacteristic
158#10596831—81no—
159C22×C73—22no—
160C22×C73—41no—
161C22×C73—42no—
162C22×C73—81no—
163C22×C73—161no—
164C22×C73#41611yescharacteristic
165C22×C73#46511yescharacteristic
166C22×C73#46611yescharacteristic
167C4×C73—22no—
168C4×C73—81no—
169C4×C73#38611yescharacteristic
170C4×C73C2×#10211yescharacteristic
171C4×C73#46511yescharacteristic
172C4×C73#46611yescharacteristic
173Dic73—161no—
174#13071—731nomaximal, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—42no—
179C2×C73—161no—
180C2×C73#189311yescharacteristic
181C2×#504—731no—
182#3067D7311yescharacteristic
183#2145—731no—
184C2×#359—731no—
185#1814—731no—
186#1388—731no—
187#1382—731no—
188C73#1307111yescharacteristic, complemented, Sylow
189#575—21no—
190C2×#135C2×D7311yescharacteristic
191C2×#131C2×D7311yescharacteristic
192#504—1462no—
193#504—1462no—
194C2×M32—731no—
195C2×M32—1461no—
196C22×C16—731no—
197C22×C16—1461no—
198#473Dic7312yescomplemented
199#472Dic7312yescomplemented
200#471—22no—
201C2×#108—731no—
202C2×#108C2×D7311yescharacteristic
203#427—731no—
204#421—1461no—
205#413—21no—
206#360—1462no—
207#359—1461no—
208#359—1461no—
209#344—1461no—
210C2×#41—21no—
211C2×#41C22⋊D7311yescharacteristic
212#137—22no—
213#137—41no—
214AGL(1,ℤ/8)—22no—
215AGL(1,ℤ/8)—41no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135—42no—
221C2×Q16C2×Dic7311yescharacteristic
222C2×SD16—21no—
223C2×D8C2×Dic7311yescharacteristic
224#131—21no—
225#131—21no—
226#131C2×Dic7312yes—
227#131C22⋊D7312yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1461no—
231C2×M16C22⋊D7311yescharacteristic
232C22×C8—21no—
233C22×C8C22×D7311yescharacteristic
234#117—1461no—
235M32—732no—
236M32—1464no—
237M32—1461no—
238C2×C16—731no—
239C2×C16—731no—
240C2×C16—1464no—
241C2×C16—1461no—
242C2×C16—1461no—
243C2×C16—1461no—
244#108—21no—
245#108—1461no—
246#108—1461no—
247#108C2×Dic7312yes—
248#107—1461no—
249#106—1461no—
250#104—2922no—
251#101—22no—
252#100—22no—
253#41—21no—
254#41—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41C22⋊Dic7312yes—
260#41#1192401312yes—
261C2×Q8—21no—
262C2×Q8#1192401611yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#1192401611yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×C22⋊D7311yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—42no—
273SD16—42no—
274SD16—42no—
275SD16—41no—
276SD16—41no—
277D8—21no—
278D8—21no—
279D8—42no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—41no—
285M16—2921no—
286M16—2921no—
287M16C22⋊Dic7312yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—2921no—
293C2×C8C22×Dic7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8D4×D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8#1192402711yescharacteristic
298C2×C8#1192402711yescharacteristic
299C4⋊C4—2921no—
300C22⋊C4—2921no—
301C42—2921no—
302C16—1461no—
303C16—1461no—
304C16—1462no—
305C23—21no—
306C23—41no—
307C23#1455661111yescharacteristic
308Q8—21no—
309Q8—41no—
310Q8—41no—
311Q8#1455651212yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4#1455651212yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—5841no—
327C2×C4C2×C22⋊Dic7311yescharacteristic
328C2×C4#1455656311yescharacteristic
329C2×C4#1455656311yescharacteristic
330C2×C4C2×#1192401311yescharacteristic
331C2×C4C2×#1192401611yescharacteristic
332C2×C4#1455661111yescharacteristic
333C8—22no—
334C8—42no—
335C8—5841no—
336C8D4×Dic7311yescharacteristic
337C8D4×Dic7311yescharacteristic
338C8#1455655911yescharacteristic
339C8#1455657211yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22#1798858311yescharacteristic
346C22#1798881411yescharacteristic
347C22#1798882111yescharacteristic
348C4—22no—
349C4—81no—
350C4—11681no—
351C4#1798846711yescharacteristic
352C4C2×#1455651211yescharacteristic
353C4#1798881411yescharacteristic
354C4#1798882111yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2#2261036011yescharacteristic
subgroupG/H
centerC4#17988821
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC73×#3067C2
socleC2×C73#1893
cosocleC22×C8×C73C23
hypercenter#3067D73
hypocenterC73#13071
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#17989004C2semidirect
2. #22611667C2semidirect
3. C2×#17988394C2semidirect
4. #22609985C2semidirect
5. #22608044C2semidirect
6. #22608028C2semidirect
24. #473×C73C4semidirect
25. #472×C73C4semidirect
188. C73#13071semidirect
198. #473Dic73semidirect
199. #472Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890042
#226116672
C2×#179883942
#226099852
#226080442
#226080282
C73×#30672
#1307173