Finite Group Databasebeta

← #32980458

Subgroups — #32980458

Order 18688.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618675C211yesmaximal, characteristic
2#22610704C211yesmaximal, characteristic, complemented
3#22608707C211yesmaximal, characteristic, complemented
4#22608580C211yesmaximal, characteristic
5#22607768C211yesmaximal, characteristic, complemented
6#22607766C211yesmaximal, characteristic, complemented
7C73×#1492C211yesmaximal, characteristic
8C2×#14556598—21no—
9#17988665—21no—
10#17988659—22no—
11D4×#10596831—21no—
12#17988640C2211yescharacteristic
13C2×#14556526—21no—
14C2×C4⋊#10596831C2211yescharacteristic
15C2×C4⋊#10596831C2211yescharacteristic
16#17988554—21no—
17C4×#11924012C2211yescharacteristic
18#17988351—21no—
19#17988339—21no—
20#17988339—21no—
21#17988333C411yescharacteristic
22#17988333C411yescharacteristic
23#17988331C411yescharacteristic
24#17988331C411yescharacteristic
25#532×C73C2211yescharacteristic
26#423×C73C2211yescharacteristic
27C2×C4×C8×C73C2211yescharacteristic
28D4⋊C8×C73—22no—
29#14556598—42no—
30#14556598—42no—
31C2×#11924012—22no—
32C2×#11924012D411yescharacteristic
33C2×#11924012D411yescharacteristic
34C22×#10596831—22no—
35C22×#10596831—21no—
36C22×#10596831—21no—
37#14556526—42no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831C2×C411yescharacteristic
41C4⋊#10596831C2×C411yescharacteristic
42C4⋊#10596831C2×C411yescharacteristic
43C4⋊#10596831C2×C411yescharacteristic
44C4⋊#10596831D411yescharacteristic
45C4⋊#10596831D411yescharacteristic
46C22⋊#10596831—22no—
47C22⋊#10596831—22no—
48C22⋊#10596831—21no—
49#14556502—21no—
50C4×#10596831—21no—
51C4×#10596831—21no—
52C2×#41×C73—21no—
53C2×M16×C73—21no—
54C22×C8×C73—21no—
55C4⋊Q8×C73C2×C411yescharacteristic
56#127×C73C2×C411yescharacteristic
57#124×C73—21no—
58#121×C73—21no—
59#121×C73Q812yes—
60C4×D4×C73—21no—
61C4×D4×C73D412yes—
62C2×C42×C73C2311yescharacteristic
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4×C8×C73—21no—
66C4×C8×C73—21no—
67#11924012—21no—
68#11924012—21no—
69#11924012—21no—
70#11924012—44no—
71#11924012D812yes—
72C2×#10596831—21no—
73C2×#10596831—21no—
74C2×#10596831—21no—
75C2×#10596831—21no—
76C2×#10596831—21no—
77C2×#10596831—21no—
78C2×#10596831—21no—
79C2×#10596831—21no—
80C2×#10596831—44no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—41no—
84C2×#10596831—41no—
85#41×C73—41no—
86#41×C73—41no—
87#41×C73—41no—
88#41×C73—41no—
89C2×Q8×C73—21no—
90C2×D4×C73—22no—
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73C4⋊C412yes—
94C22×C4×C73—22no—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×Q811yescharacteristic
97C22×C4×C73#4111yescharacteristic
98M16×C73—41no—
99M16×C73—41no—
100C2×C8×C73—22no—
101C2×C8×C73—41no—
102C2×C8×C73—41no—
103C2×C8×C73—41no—
104C4⋊C4×C73—21no—
105C4⋊C4×C73C4⋊C412yes—
106C22⋊C4×C73—22no—
107C22⋊C4×C73—22no—
108C42×C73C22×C411yescharacteristic
109C42×C73C2×D411yescharacteristic
110C42×C73C2×D411yescharacteristic
111C42×C73#4111yescharacteristic
112C23×C73—22no—
113C23×C73C22⋊Q811yescharacteristic
114Q8×C73—41no—
115Q8×C73—41no—
116D4×C73—42no—
117D4×C73—42no—
118D4×C73—42no—
119D4×C73—41no—
120D4×C73—41no—
121D4×C73—41no—
122D4×C73—41no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—22no—
126C2×C4×C73—41no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73C2×C4⋊C411yescharacteristic
130C2×C4×C73C4×D411yescharacteristic
131C2×C4×C73C4×D411yescharacteristic
132C2×C4×C73#12011yescharacteristic
133C2×C4×C73C22⋊Q811yescharacteristic
134C2×C4×C73#12311yescharacteristic
135C2×C4×C73C2×D811yescharacteristic
136C2×C4×C73#13511yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C73#13711yescharacteristic
139C8×C73—42no—
140C8×C73—81no—
141#10596831—22no—
142#10596831—41no—
143#10596831—41no—
144#10596831—41no—
145#10596831—41no—
146#10596831—41no—
147#10596831—41no—
148#10596831—41no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73—42no—
152C22×C73—42no—
153C22×C73—42no—
154C22×C73#38511yescharacteristic
155C22×C73#47411yescharacteristic
156C22×C73#47811yescharacteristic
157C4×C73—21no—
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—42no—
161C4×C73C4×D811yescharacteristic
162C4×C73#44011yescharacteristic
163C4×C73#44311yescharacteristic
164C4×C73#44411yescharacteristic
165C4×C73#44711yescharacteristic
166C4×C73#45211yescharacteristic
167#8930—731nomaximal, Sylow
168C2×C73—21no—
169C2×C73—21no—
170C2×C73—82no—
171C2×C73#185811yescharacteristic
172C2×C73#186211yescharacteristic
173C2×C73#186611yescharacteristic
174#2935—731no—
175#1950—731no—
176#1571—731no—
177#1540—731no—
178#1492D7311yescharacteristic
179#1278—731no—
180#1277—731no—
181C73#893011yescharacteristic, complemented, Sylow
182C2×#131—1461no—
183#532C2×D7311yescharacteristic
184#436—1461no—
185#435—1462no—
186C8×D4—1461no—
187#431—731no—
188C2×#108—1461no—
189#423C2×D7311yescharacteristic
190C2×C4⋊C8—731no—
191C2×C4⋊C8—731no—
192#408—1461no—
193C4×M16—731no—
194C2×C4×C8C2×D7311yescharacteristic
195#341—1461no—
196#335—1461no—
197#335—1461no—
198#332—731no—
199#332—731no—
200#331—731no—
201#331—731no—
202D4⋊C8—22no—
203C2×#41—21no—
204#131—2922no—
205#131—2922no—
206C2×M16—21no—
207C2×M16—731no—
208C2×M16—731no—
209C2×M16—1462no—
210C22×C8—21no—
211C22×C8—1462no—
212C22×C8—1461no—
213C22×C8—1461no—
214C4⋊Q8C4×D7311yescharacteristic
215#127C4×D7311yescharacteristic
216#124—21no—
217#121—21no—
218#121Dic14612yes—
219C4×D4—21no—
220C4×D4C22⋊D7312yes—
221C2×C42C22×D7311yescharacteristic
222#108—2922no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—731no—
226C4⋊C8—731no—
227C4⋊C8—731no—
228C4⋊C8—731no—
229C4⋊C8—731no—
230C4⋊C8—731no—
231C4⋊C8—1461no—
232C4⋊C8—1461no—
233C22⋊C8—1462no—
234C22⋊C8—1462no—
235C22⋊C8—1461no—
236#97—1461no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—1461no—
240C4×C8—1461no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4#1192400712yes—
250C22×C4—22no—
251C22×C4D4×D7311yescharacteristic
252C22×C4C2×Dic14611yescharacteristic
253C22×C4#1192402711yescharacteristic
254M16—41no—
255M16—41no—
256M16—2924no—
257M16—732no—
258M16—1461no—
259M16—1461no—
260M16—1461no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—2921no—
266C2×C8—2921no—
267C2×C8—2921no—
268C2×C8—2921no—
269C2×C8—2924no—
270C2×C8—1461no—
271C2×C8—1461no—
272C2×C8—1461no—
273C2×C8—1461no—
274C2×C8—1461no—
275C2×C8—1461no—
276C2×C8—1461no—
277C2×C8—1461no—
278C4⋊C4—21no—
279C4⋊C4#1192400712yes—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C42C2×C4×D7311yescharacteristic
283C42D4×D7311yescharacteristic
284C42C2×C22⋊D7311yescharacteristic
285C42#1192402711yescharacteristic
286C23—22no—
287C23#1455656811yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4C2×#1192400711yescharacteristic
304C2×C4#1455655111yescharacteristic
305C2×C4#1455655111yescharacteristic
306C2×C4#1455655911yescharacteristic
307C2×C4#1455656811yescharacteristic
308C2×C4#1455657211yescharacteristic
309C2×C4D8×D7311yescharacteristic
310C2×C4#1455661211yescharacteristic
311C2×C4#1455661511yescharacteristic
312C2×C4#1455662011yescharacteristic
313C8—42no—
314C8—81no—
315C8—2921no—
316C8—2921no—
317C8—2921no—
318C8—2921no—
319C8—2921no—
320C8—2921no—
321C8—2921no—
322C8—1462no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22#1798845911yescharacteristic
329C22#1798886511yescharacteristic
330C22#1798888911yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4#1798866711yescharacteristic
336C4#1798868411yescharacteristic
337C4#1798870111yescharacteristic
338C4#1798870611yescharacteristic
339C4#1798871911yescharacteristic
340C4#1798874611yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2261015011yescharacteristic
345C2#2261017611yescharacteristic
346C2#2261020411yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1492C2
socleC22×C73#385
prime radical#532×C73C22
cosocleC2×C42×C73C23
hypercenter#1492D73
hypocenterC73#8930
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #22610704C2semidirect
3. #22608707C2semidirect
5. #22607768C2semidirect
6. #22607766C2semidirect
181. C73#8930semidirect

Maximal subgroups

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

subgroupindex
#226186752
#226107042
#226087072
#226085802
#226077682
#226077662
C73×#14922
#893073