Finite Group Databasebeta

← #33003127

Subgroups — #33003127

Order 18688.

466 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988709C211yesmaximal, characteristic, complemented
2#22611680C211yesmaximal, characteristic, complemented
3#22611659C211yesmaximal, characteristic, complemented
4#22610825C211yesmaximal, characteristic, complemented
5#22608043C211yesmaximal, characteristic, complemented
6#22608014C211yesmaximal, characteristic, complemented
7C73×#3414C211yesmaximal, characteristic
8C2×#14556598—21no—
9C2×#14556531C2211yescharacteristic
10#17988709—22no—
11#17988709—22no—
12C2×#14556527C2211yescharacteristic
13C2×#14556527C2211yescharacteristic
14C2×#14556518—21no—
15C2×#14556518—21no—
16#17988554—21no—
17C2×#14556501C2211yescharacteristic
18#17988403—21no—
19#17988400—21no—
20#17988394—22no—
21C2×#135×C73C2211yescharacteristic
22C2×#135×C73C2211yescharacteristic
23#510×C73—21no—
24#510×C73C412yescomplemented
25#509×C73—21no—
26#509×C73C412yescomplemented
27#508×C73—22no—
28C2×M32×C73C2211yescharacteristic
29#14556598—24no—
30#14556598—42no—
31C2×#11924012—21no—
32C2×#11924012—21no—
33C2×#11924012—41no—
34C22×#10596831—41no—
35C22×#10596831D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37#14556531—21no—
38#14556531—21no—
39C2×#11924003—21no—
40#14556527—21no—
41#14556527—21no—
42#14556527—21no—
43#14556527D412yes—
44#14556518—24no—
45#14556518—42no—
46C22⋊#10596831—42no—
47#14556501—21no—
48#14556501—21no—
49C2×#41×C73—21no—
50C2×#41×C73—21no—
51#135×C73—21no—
52#135×C73—21no—
53#135×C73—21no—
54#135×C73—21no—
55#135×C73—21no—
56#135×C73—21no—
57#135×C73C2×C412yes—
58#135×C73D412yes—
59C2×Q16×C73C2×C411yescharacteristic
60C2×Q16×C73D411yescharacteristic
61C2×SD16×C73—21no—
62C2×SD16×C73—21no—
63C2×D8×C73C2×C411yescharacteristic
64C2×D8×C73D411yescharacteristic
65C22×C8×C73C2311yescharacteristic
66Q32×C73—22no—
67SD32×C73—22no—
68SD32×C73—22no—
69D16×C73—22no—
70M32×C73—21no—
71M32×C73C2×C412yes—
72C2×C16×C73—21no—
73C22×Dic73—21no—
74#11924012—22no—
75#11924012—44no—
76#11924012—41no—
77#11924012—41no—
78#11924012—41no—
79C2×#10596831—21no—
80C2×#10596831—21no—
81C2×#10596831—44no—
82C2×#10596831—41no—
83C2×#10596831—41no—
84C2×#10596831D812yes—
85C4×Dic73—21no—
86C4×Dic73—21no—
87#11924003—41no—
88#41×C73—22no—
89#41×C73—22no—
90#41×C73—41no—
91#41×C73—41no—
92#41×C73—41no—
93#41×C73—41no—
94#41×C73—41no—
95#41×C73—41no—
96C2×Q8×C73—21no—
97C2×Q8×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—21no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73C2×D411yescharacteristic
105Q16×C73—21no—
106Q16×C73—21no—
107Q16×C73—21no—
108Q16×C73C22⋊C412yes—
109SD16×C73—22no—
110SD16×C73—22no—
111SD16×C73—41no—
112SD16×C73—41no—
113D8×C73—21no—
114D8×C73—21no—
115D8×C73—21no—
116D8×C73C22⋊C412yes—
117C2×C8×C73C22×C411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C16×C73—22no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73#13511yescharacteristic
127Q8×C73—22no—
128Q8×C73—41no—
129Q8×C73—41no—
130Q8×C73—41no—
131D4×C73—22no—
132D4×C73—42no—
133D4×C73—42no—
134D4×C73—41no—
135D4×C73—41no—
136D4×C73—41no—
137D4×C73—81no—
138D4×C73—81no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73#12011yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
151C2×Dic73—41no—
152C2×Dic73—81no—
153C8×C73C2×C22⋊C411yescharacteristic
154C8×C73C4×D411yescharacteristic
155C8×C73#12111yescharacteristic
156C8×C73#12311yescharacteristic
157#10596831—22no—
158#10596831—42no—
159#10596831—81no—
160C22×C73—41no—
161C22×C73—42no—
162C22×C73—82no—
163C22×C73—82no—
164C22×C73—81no—
165C22×C73C4×D811yescharacteristic
166C22×C73#44911yescharacteristic
167C22×C73#46311yescharacteristic
168C4×C73—42no—
169C4×C73—81no—
170C4×C73#38611yescharacteristic
171C4×C73#44211yescharacteristic
172C4×C73#44711yescharacteristic
173C4×C73#45911yescharacteristic
174Dic73—81no—
175#13024—731nomaximal, Sylow
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—82no—
180C2×C73—161no—
181C2×C73#187911yescharacteristic
182#3414D7311yescharacteristic
183C2×#444—731no—
184#2148—731no—
185#2143—731no—
186#1975—731no—
187#1388—731no—
188#1377—731no—
189C73#1302411yescharacteristic, complemented, Sylow
190C2×#135C2×D7311yescharacteristic
191C2×#135C2×D7311yescharacteristic
192C2×#131—1461no—
193#510—21no—
194#510Dic7312yescomplemented
195#509—21no—
196#509Dic7312yescomplemented
197#508—22no—
198C2×M32—731no—
199C2×M32C2×D7311yescharacteristic
200#444—1462no—
201#444—1462no—
202C2×#108—731no—
203C2×#108—731no—
204C2×#104—1461no—
205C2×#104—1461no—
206#408—1461no—
207C2×#97—731no—
208#362—1461no—
209#361—1461no—
210#359—1462no—
211C2×#41—21no—
212C2×#41—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135C2×Dic7312yes—
220#135C22⋊D7312yes—
221C2×Q16C2×Dic7311yescharacteristic
222C2×Q16C22⋊D7311yescharacteristic
223C2×SD16—21no—
224C2×SD16—21no—
225C2×D8C2×Dic7311yescharacteristic
226C2×D8C22⋊D7311yescharacteristic
227#131—2922no—
228#131—1464no—
229C2×M16—2921no—
230C2×M16—1461no—
231C2×M16—1461no—
232C22×C8—2921no—
233C22×C8—731no—
234C22×C8C22×D7311yescharacteristic
235C2×C42—731no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—21no—
241M32—1461no—
242M32—1461no—
243M32C2×Dic7312yes—
244C2×C16—21no—
245C2×C16—1461no—
246#108—732no—
247#108—1461no—
248#108—1461no—
249#108—1461no—
250#104—2922no—
251#104—1464no—
252C22⋊C8—2922no—
253#97—1461no—
254#97—1461no—
255#41—22no—
256#41—22no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—1461no—
272C22×C4D4×D7311yescharacteristic
273Q16—21no—
274Q16—21no—
275Q16—21no—
276Q16C22⋊Dic7312yes—
277SD16—22no—
278SD16—22no—
279SD16—41no—
280SD16—41no—
281D8—21no—
282D8—21no—
283D8—21no—
284D8C22⋊Dic7312yes—
285M16—2921no—
286M16—2924no—
287M16—2921no—
288M16—2921no—
289M16—1462no—
290C2×C8—2921no—
291C2×C8—2924no—
292C2×C8—2921no—
293C2×C8—732no—
294C2×C8—1461no—
295C2×C8—1461no—
296C2×C8C22×Dic7311yescharacteristic
297C2×C8D4×D7311yescharacteristic
298C2×C8C2×C22⋊D7311yescharacteristic
299C2×C8C2×C22⋊D7311yescharacteristic
300C2×C8#1192402711yescharacteristic
301C2×C8#1192402711yescharacteristic
302C42—1461no—
303C42—1461no—
304C16—22no—
305C16—2921no—
306C23—41no—
307C23—41no—
308C23#1455661011yescharacteristic
309Q8—22no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313D4—22no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—2921no—
328C2×C4—5841no—
329C2×C4D4×Dic7311yescharacteristic
330C2×C4#1455655911yescharacteristic
331C2×C4#1455656311yescharacteristic
332C2×C4D8×D7311yescharacteristic
333C2×C4#1455661511yescharacteristic
334C2×C4#1455661511yescharacteristic
335C8—2922no—
336C8—5841no—
337C8—1462no—
338C8C2×C22⋊Dic7311yescharacteristic
339C8D4×Dic7311yescharacteristic
340C8#1455656311yescharacteristic
341C8#1455657211yescharacteristic
342C22—41no—
343C22—42no—
344C22—82no—
345C22—82no—
346C22—81no—
347C22D8×Dic7311yescharacteristic
348C22#1798872911yescharacteristic
349C22#1798880011yescharacteristic
350C4—42no—
351C4—81no—
352C4—5841no—
353C4#1798846711yescharacteristic
354C4#1798869911yescharacteristic
355C4#1798871911yescharacteristic
356C4#1798878011yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—161no—
362C2#2261028111yescharacteristic
subgroupG/H
centerC4#17988719
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC73×#3414C2
socleC2×C73#1879
cosocleC22×C8×C73C23
hypercenter#3414D73
hypocenterC73#13024
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. C2×#17988709C2semidirect
2. #22611680C2semidirect
3. #22611659C2semidirect
4. #22610825C2semidirect
5. #22608043C2semidirect
6. #22608014C2semidirect
24. #510×C73C4semidirect
26. #509×C73C4semidirect
189. C73#13024semidirect
194. #510Dic73semidirect
196. #509Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887092
#226116802
#226116592
#226108252
#226080432
#226080142
C73×#34142
#1302473