Finite Group Databasebeta

← #33003036

Subgroups — #33003036

Order 18688.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988704C211yesmaximal, characteristic, complemented
2C2×#17988394C211yesmaximal, characteristic, complemented
3C2×#17988394C211yesmaximal, characteristic, complemented
4#22610820C211yesmaximal, characteristic, complemented
5#22608025C211yesmaximal, characteristic, complemented
6#22608008C211yesmaximal, characteristic, complemented
7C2×#508×C73C211yesmaximal, characteristic
8C2×#14556598—21no—
9C22×#11924003C2211yescharacteristic
10#17988704—22no—
11#17988704—22no—
12C2×#14556527C2211yescharacteristic
13C2×#14556527C2211yescharacteristic
14C2×#14556518—21no—
15C2×#14556518—21no—
16#17988554—21no—
17C2×C8×Dic73C2211yescharacteristic
18#17988394—22no—
19#17988394—22no—
20C2×#135×C73C2211yescharacteristic
21C2×#135×C73C2211yescharacteristic
22#508×C73—22no—
23#508×C73—22no—
24C2×Q32×C73C411yescharacteristic, complemented
25C2×SD32×C73C411yescharacteristic, complemented
26C2×SD32×C73C411yescharacteristic, complemented
27C2×D16×C73C411yescharacteristic, complemented
28C22×C16×C73C2211yescharacteristic
29#14556598—24no—
30#14556598—42no—
31C2×#11924012—21no—
32C2×#11924012—21no—
33C2×#11924012—41no—
34C22×#10596831—41no—
35C22×#10596831D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37C2×#11924003—21no—
38C2×#11924003—21no—
39C2×#11924003—21no—
40#14556527—21no—
41#14556527—21no—
42#14556527—21no—
43#14556527D412yes—
44#14556518—24no—
45#14556518—42no—
46C22⋊#10596831—41no—
47C22⋊#10596831—41no—
48C8×Dic73—21no—
49C8×Dic73—21no—
50C2×#41×C73—21no—
51C2×#41×C73—21no—
52#135×C73—21no—
53#135×C73—21no—
54#135×C73—21no—
55#135×C73—21no—
56#135×C73—21no—
57#135×C73—21no—
58#135×C73D412yes—
59#135×C73D412yes—
60C2×Q16×C73C2×C411yescharacteristic
61C2×Q16×C73C2×C411yescharacteristic
62C2×SD16×C73—21no—
63C2×SD16×C73—21no—
64C2×D8×C73C2×C411yescharacteristic
65C2×D8×C73C2×C411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67Q32×C73—22no—
68SD32×C73—22no—
69SD32×C73—22no—
70D16×C73—22no—
71C2×C16×C73—21no—
72C2×C16×C73—21no—
73C2×C16×C73C2×C411yescharacteristic
74C2×C16×C73C2×C411yescharacteristic
75C22×Dic73—21no—
76#11924012—22no—
77#11924012—44no—
78#11924012—41no—
79#11924012—41no—
80#11924012—41no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—44no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C2×#10596831D811yescharacteristic
87C2×#10596831D811yescharacteristic
88C4×Dic73—21no—
89C4×Dic73—21no—
90#11924003—41no—
91#41×C73—22no—
92#41×C73—22no—
93#41×C73—41no—
94#41×C73—41no—
95#41×C73—41no—
96#41×C73—41no—
97#41×C73—41no—
98#41×C73—41no—
99C2×Q8×C73—21no—
100C2×Q8×C73—21no—
101C2×D4×C73—21no—
102C2×D4×C73—21no—
103C2×D4×C73—41no—
104C2×D4×C73—41no—
105C22×C4×C73—41no—
106C22×C4×C73—41no—
107C22×C4×C73C2×D411yescharacteristic
108Q16×C73—21no—
109Q16×C73—21no—
110Q16×C73—21no—
111Q16×C73C22⋊C412yes—
112SD16×C73—22no—
113SD16×C73—22no—
114SD16×C73—41no—
115SD16×C73—41no—
116D8×C73—21no—
117D8×C73—21no—
118D8×C73—21no—
119D8×C73C22⋊C412yes—
120C2×C8×C73C22×C411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73#4111yescharacteristic
125C2×C8×C73#4111yescharacteristic
126C16×C73—21no—
127C16×C73—21no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73#13511yescharacteristic
131Q8×C73—22no—
132Q8×C73—41no—
133Q8×C73—41no—
134Q8×C73—41no—
135D4×C73—22no—
136D4×C73—42no—
137D4×C73—42no—
138D4×C73—41no—
139D4×C73—41no—
140D4×C73—41no—
141D4×C73—81no—
142D4×C73—81no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73—41no—
147C2×C4×C73—81no—
148C2×C4×C73—81no—
149C2×C4×C73C4×D411yescharacteristic
150C2×C4×C73#12011yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73C2×D811yescharacteristic
153C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×Dic73—22no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C8×C73C2×C22⋊C411yescharacteristic
159C8×C73C4×D411yescharacteristic
160C8×C73#12111yescharacteristic
161C8×C73#12311yescharacteristic
162#10596831—21no—
163#10596831—21no—
164#10596831—42no—
165#10596831—81no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—42no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73C4×D811yescharacteristic
175C22×C73#44911yescharacteristic
176C22×C73#46311yescharacteristic
177C4×C73—42no—
178C4×C73—81no—
179C4×C73#38611yescharacteristic
180C4×C73#44211yescharacteristic
181C4×C73#44711yescharacteristic
182C4×C73#45911yescharacteristic
183Dic73—42no—
184#13011—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—82no—
188C2×C73—161no—
189C2×C73#187911yescharacteristic
190C2×C73#217811yescharacteristic
191C2×C73#217911yescharacteristic
192C2×#508D7311yescharacteristic
193C2×#443—731no—
194C2×#359—731no—
195C2×#359—731no—
196#1974—731no—
197#1381—731no—
198#1374—731no—
199C73#1301111yescharacteristic, complemented, Sylow
200C2×#135C2×D7311yescharacteristic
201C2×#135C2×D7311yescharacteristic
202C2×#131—1461no—
203#508—22no—
204#508—22no—
205C2×Q32Dic7311yescharacteristic, complemented
206C2×SD32Dic7311yescharacteristic, complemented
207C2×SD32Dic7311yescharacteristic, complemented
208C2×D16Dic7311yescharacteristic, complemented
209C22×C16—731no—
210C22×C16C2×D7311yescharacteristic
211#443—1462no—
212#443—1462no—
213C2×#108—731no—
214C2×#108—731no—
215C2×#104—1461no—
216C2×#104—1461no—
217#408—1461no—
218C2×C4×C8—731no—
219#359—1462no—
220#359—1462no—
221C2×#41—21no—
222C2×#41—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135C22⋊D7312yes—
230#135C22⋊D7312yes—
231C2×Q16C2×Dic7311yescharacteristic
232C2×Q16C2×Dic7311yescharacteristic
233C2×SD16—21no—
234C2×SD16—21no—
235C2×D8C2×Dic7311yescharacteristic
236C2×D8C2×Dic7311yescharacteristic
237#131—2922no—
238#131—1464no—
239C2×M16—2921no—
240C2×M16—1461no—
241C2×M16—1461no—
242C22×C8—2921no—
243C22×C8—731no—
244C22×C8C22×D7311yescharacteristic
245C2×C42—731no—
246Q32—22no—
247SD32—22no—
248SD32—22no—
249D16—22no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—1461no—
253C2×C16—1461no—
254C2×C16—1461no—
255C2×C16C2×Dic7311yescharacteristic
256C2×C16C2×Dic7311yescharacteristic
257#108—732no—
258#108—1461no—
259#108—1461no—
260#108—1461no—
261#104—2922no—
262#104—1464no—
263C22⋊C8—2921no—
264C22⋊C8—2921no—
265C4×C8—1461no—
266C4×C8—1461no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—1461no—
284C22×C4D4×D7311yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—21no—
288Q16C22⋊Dic7312yes—
289SD16—22no—
290SD16—22no—
291SD16—41no—
292SD16—41no—
293D8—21no—
294D8—21no—
295D8—21no—
296D8C22⋊Dic7312yes—
297M16—2921no—
298M16—2924no—
299M16—2921no—
300M16—2921no—
301M16—1462no—
302C2×C8—2921no—
303C2×C8—2924no—
304C2×C8—2921no—
305C2×C8—731no—
306C2×C8—731no—
307C2×C8—1461no—
308C2×C8—1461no—
309C2×C8C22×Dic7311yescharacteristic
310C2×C8D4×D7311yescharacteristic
311C2×C8C2×C22⋊D7311yescharacteristic
312C2×C8C2×C22⋊D7311yescharacteristic
313C2×C8#1192402711yescharacteristic
314C2×C8#1192402711yescharacteristic
315C42—1461no—
316C42—1461no—
317C16—21no—
318C16—21no—
319C16—2921no—
320C23—41no—
321C23—41no—
322C23#1455661011yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—41no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—2921no—
342C2×C4—2921no—
343C2×C4—1462no—
344C2×C4D4×Dic7311yescharacteristic
345C2×C4#1455655911yescharacteristic
346C2×C4#1455656311yescharacteristic
347C2×C4D8×D7311yescharacteristic
348C2×C4#1455661511yescharacteristic
349C2×C4#1455661511yescharacteristic
350C8—2922no—
351C8—5841no—
352C8—1461no—
353C8—1461no—
354C8C2×C22⋊Dic7311yescharacteristic
355C8D4×Dic7311yescharacteristic
356C8#1455656311yescharacteristic
357C8#1455657211yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22D8×Dic7311yescharacteristic
367C22#1798872911yescharacteristic
368C22#1798880011yescharacteristic
369C4—42no—
370C4—81no—
371C4—2922no—
372C4#1798846711yescharacteristic
373C4#1798869911yescharacteristic
374C4#1798871911yescharacteristic
375C4#1798878011yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—161no—
380C2#2261028111yescharacteristic
381C2#2261181211yescharacteristic
382C2#2261181711yescharacteristic
subgroupG/H
centerC2×C4D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC2×#508×C73C2
socleC22×C73C4×D8
cosocleC22×C8×C73C23
hypercenterC2×#508D73
hypocenterC73#13011
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#17988704C2semidirect
2. C2×#17988394C2semidirect
3. C2×#17988394C2semidirect
4. #22610820C2semidirect
5. #22608025C2semidirect
6. #22608008C2semidirect
24. C2×Q32×C73C4semidirect
25. C2×SD32×C73C4semidirect
26. C2×SD32×C73C4semidirect
27. C2×D16×C73C4semidirect
199. C73#13011semidirect
205. C2×Q32Dic73semidirect
206. C2×SD32Dic73semidirect
207. C2×SD32Dic73semidirect
208. C2×D16Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887042
C2×#179883942
#226108202
#226080252
#226080082
C2×#508×C732
#1301173