Finite Group Databasebeta

← #33003031

Subgroups — #33003031

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×#17989017C211yesmaximal, characteristic
2C2×#17988703C211yesmaximal, characteristic, complemented
3C2×#17988395C211yesmaximal, characteristic, complemented
4#22610820C211yesmaximal, characteristic, complemented
5#22608024C211yesmaximal, characteristic, complemented
6#22608008C211yesmaximal, characteristic, complemented
7C2×#359×C73C211yesmaximal, characteristic, complemented
8C2×#14556613C2211yescharacteristic
9C2×#14556599—21no—
10#17989017—22no—
11#17989017—22no—
12C2×#14556537C411yescharacteristic, complemented
13C2×#14556535C411yescharacteristic, complemented
14C2×#14556534C411yescharacteristic, complemented
15C2×#14556532C411yescharacteristic, complemented
16C22×#11924003C2211yescharacteristic
17#17988703—22no—
18#17988703—22no—
19C2×#14556527C2211yescharacteristic
20C2×#14556518—21no—
21C2×#14556517—21no—
22#17988554—21no—
23C2×C8×Dic73C2211yescharacteristic
24#17988395—22no—
25C2×#135×C73C2211yescharacteristic
26C22×C16×C73C2211yescharacteristic
27C2×#108×C73C2211yescharacteristic
28#359×C73—22no—
29C2×#11924028—21no—
30#14556613—21no—
31#14556613—21no—
32#14556613D412yes—
33#14556613D412yes—
34C2×Dic292C2×C411yescharacteristic
35C2×#11924017—21no—
36C2×D584C2×C411yescharacteristic
37#14556599—24no—
38#14556599—42no—
39C2×#11924012—21no—
40C2×#11924011—41no—
41C22×#10596831D411yescharacteristic
42C2×C8×D73—41no—
43C2×C4×Dic73D411yescharacteristic
44#14556537—22no—
45#14556535—22no—
46#14556534—22no—
47#14556532—22no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50C2×#11924003C2×C411yescharacteristic
51C2×#11924003C2×C411yescharacteristic
52#14556527—21no—
53#14556527—21no—
54#14556518—42no—
55#14556517—24no—
56C22⋊#10596831—41no—
57C22⋊#10596831—41no—
58C8×Dic73—21no—
59C8×Dic73—21no—
60C2×#41×C73—21no—
61#135×C73—21no—
62#135×C73—21no—
63#135×C73—21no—
64#135×C73—21no—
65C2×Q16×C73C2×C411yescharacteristic
66C2×SD16×C73—21no—
67C2×D8×C73C2×C411yescharacteristic
68C2×M16×C73—21no—
69C22×C8×C73C2311yescharacteristic
70C2×C16×C73—21no—
71C2×C16×C73—21no—
72C2×C16×C73—21no—
73#108×C73—21no—
74#108×C73D412yes—
75#11924028—22no—
76#11924028—22no—
77#11924028—41no—
78#11924028—41no—
79C2×Dic146—21no—
80C2×D292—21no—
81C2×C22⋊D73—41no—
82C22×Dic73—21no—
83C2×C4×D73—41no—
84Dic292—21no—
85Dic292C22⋊C412yes—
86#11924017—22no—
87#11924017—22no—
88D584—21no—
89D584C22⋊C412yes—
90#11924012—41no—
91#11924012—41no—
92#11924011—44no—
93C2×#10596831—21no—
94C2×#10596831—21no—
95C2×#10596831—41no—
96C2×#10596831D811yescharacteristic
97C2×#10596831D811yescharacteristic
98C8×D73—44no—
99C4×Dic73—21no—
100C4×Dic73—21no—
101#11924003—21no—
102#11924003—21no—
103#41×C73—41no—
104#41×C73—41no—
105#41×C73—41no—
106#41×C73—41no—
107C2×Q8×C73—21no—
108C2×D4×C73—21no—
109C2×D4×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D411yescharacteristic
112Q16×C73—21no—
113Q16×C73—21no—
114SD16×C73—41no—
115SD16×C73—41no—
116D8×C73—21no—
117D8×C73—21no—
118M16×C73—22no—
119M16×C73—41no—
120C2×C8×C73—41no—
121C2×C8×C73C22×C411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73#4111yescharacteristic
126C2×C8×C73#4111yescharacteristic
127C16×C73—41no—
128C23×C73—41no—
129C23×C73#13511yescharacteristic
130C22×D73—41no—
131Q8×C73—41no—
132Q8×C73—41no—
133D4×C73—41no—
134D4×C73—41no—
135D4×C73—81no—
136D4×C73—81no—
137C2×C4×C73—41no—
138C2×C4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73#12011yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73C2×D811yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
146C22⋊D73—42no—
147C22⋊D73—42no—
148C2×Dic73—22no—
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73—41no—
152D292—22no—
153D292—41no—
154C4×D73—42no—
155C4×D73—42no—
156Dic146—22no—
157Dic146—41no—
158C8×C73—42no—
159C8×C73C2×C22⋊C411yescharacteristic
160C8×C73C4×D411yescharacteristic
161C8×C73#12111yescharacteristic
162C8×C73#12311yescharacteristic
163#10596831—21no—
164#10596831—21no—
165#10596831—81no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73C4×D811yescharacteristic
172C22×C73#44911yescharacteristic
173C22×C73#46311yescharacteristic
174C2×D73—42no—
175C2×D73—81no—
176C2×D73—81no—
177C4×C73—81no—
178C4×C73#38611yescharacteristic
179C4×C73#44211yescharacteristic
180C4×C73#44711yescharacteristic
181C4×C73#45911yescharacteristic
182Dic73—42no—
183Dic73—42no—
184#13011—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—161no—
188C2×C73#187911yescharacteristic
189C2×C73#217811yescharacteristic
190C2×C73#217911yescharacteristic
191D73—82no—
192C2×#508—731no—
193C2×#443—731no—
194C2×#359—731no—
195C2×#359D7311yescharacteristic, complemented
196#1974—731no—
197#1381—731no—
198#1374—731no—
199C73#1301111yescharacteristic, complemented, Sylow
200C2×#135—731no—
201C2×#135C2×D7311yescharacteristic
202C2×#131—1461no—
203#508—1462no—
204#508—1462no—
205C2×Q32—731no—
206C2×SD32—731no—
207C2×SD32—731no—
208C2×D16—731no—
209C22×C16—731no—
210C22×C16C2×D7311yescharacteristic
211#443—1462no—
212#443—1462no—
213C2×#108—731no—
214C2×#108C2×D7311yescharacteristic
215C2×#104—1461no—
216C2×#104—1461no—
217#408—1461no—
218C2×C4×C8—731no—
219#359—22no—
220#359—1462no—
221C2×#41—21no—
222C2×#41—1461no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—732no—
228#135—732no—
229#135—1461no—
230#135—1461no—
231C2×Q16—731no—
232C2×Q16C4×D7311yescharacteristic
233C2×SD16—21no—
234C2×SD16—1461no—
235C2×D8—731no—
236C2×D8C4×D7311yescharacteristic
237#131—2922no—
238#131—1464no—
239C2×M16—21no—
240C2×M16—2921no—
241C2×M16—1461no—
242C22×C8—2921no—
243C22×C8—731no—
244C22×C8C22×D7311yescharacteristic
245C2×C42—731no—
246Q32—1462no—
247SD32—1462no—
248SD32—1462no—
249D16—1462no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—731no—
254C2×C16—731no—
255C2×C16—1461no—
256C2×C16—1461no—
257#108—21no—
258#108—1461no—
259#108—1461no—
260#108C22⋊D7312yes—
261#104—2922no—
262#104—1464no—
263C22⋊C8—2921no—
264C22⋊C8—2921no—
265C4×C8—1461no—
266C4×C8—1461no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—2921no—
272#41—2921no—
273#41—1462no—
274#41—1462no—
275C2×Q8—21no—
276C2×Q8—1461no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—2921no—
280C2×D4—1461no—
281C22×C4—41no—
282C22×C4—2921no—
283C22×C4—1461no—
284C22×C4D4×D7311yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—732no—
288Q16—1461no—
289SD16—41no—
290SD16—41no—
291SD16—1462no—
292SD16—1462no—
293D8—21no—
294D8—21no—
295D8—732no—
296D8—1461no—
297M16—22no—
298M16—41no—
299M16—2924no—
300M16—2921no—
301M16—2921no—
302C2×C8—41no—
303C2×C8—2924no—
304C2×C8—2921no—
305C2×C8—731no—
306C2×C8—731no—
307C2×C8—1461no—
308C2×C8—1461no—
309C2×C8C2×C4×D7311yescharacteristic
310C2×C8D4×D7311yescharacteristic
311C2×C8D4×D7311yescharacteristic
312C2×C8C2×C22⋊D7311yescharacteristic
313C2×C8#1192402711yescharacteristic
314C2×C8#1192402811yescharacteristic
315C42—1461no—
316C42—1461no—
317C16—41no—
318C16—1461no—
319C16—1461no—
320C23—41no—
321C23—2921no—
322C23#1455661211yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—2921no—
326Q8—1462no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—2922no—
332D4—2922no—
333D4—2921no—
334D4—1462no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—2921no—
339C2×C4—2921no—
340C2×C4—2921no—
341C2×C4—2922no—
342C2×C4—2922no—
343C2×C4—1462no—
344C2×C4#1455655111yescharacteristic
345C2×C4#1455655911yescharacteristic
346C2×C4#1455656411yescharacteristic
347C2×C4D8×D7311yescharacteristic
348C2×C4#1455661511yescharacteristic
349C2×C4#1455661811yescharacteristic
350C8—42no—
351C8—5841no—
352C8—1461no—
353C8—1461no—
354C8C22⋊C4×D7311yescharacteristic
355C8C4×C22⋊D7311yescharacteristic
356C8#1455656211yescharacteristic
357C8#1455657311yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—2922no—
364C22—5841no—
365C22—5841no—
366C22#1798866711yescharacteristic
367C22#1798873111yescharacteristic
368C22#1798880111yescharacteristic
369C4—81no—
370C4—2922no—
371C4—2922no—
372C4#1798846311yescharacteristic
373C4#1798869611yescharacteristic
374C4#1798871911yescharacteristic
375C4#1798878111yescharacteristic
376C2—21no—
377C2—21no—
378C2—161no—
379C2—5842no—
380C2#2261027611yescharacteristic
381C2#2261180911yescharacteristic
382C2#2261181411yescharacteristic
subgroupG/H
centerC2×C4D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#359×C73C2
socleC22×C73C4×D8
prime radicalC2×#17989017C2
cosocleC22×C8×C73C23
hypercenterC2×#359D73
hypocenterC73#13011
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C2×#17988703C2semidirect
3. C2×#17988395C2semidirect
4. #22610820C2semidirect
5. #22608024C2semidirect
6. #22608008C2semidirect
7. C2×#359×C73C2semidirect
12. C2×#14556537C4semidirect
13. C2×#14556535C4semidirect
14. C2×#14556534C4semidirect
15. C2×#14556532C4semidirect
195. C2×#359D73semidirect
199. C73#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
C2×#179887032
C2×#179883952
#226108202
#226080242
#226080082
C2×#359×C732
#1301173