Finite Group Databasebeta

← #33002897

Subgroups — #33002897

Order 18688.

456 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×#17989005C211yesmaximal, characteristic, complemented
2#22618832C211yesmaximal, characteristic
3#22611558C211yesmaximal, characteristic, complemented
4C2×#17988371C211yesmaximal, characteristic, complemented
5#22608021C211yesmaximal, characteristic, complemented
6#22608018C211yesmaximal, characteristic, complemented
7C73×#2115C211yesmaximal, characteristic, complemented
8#17989385C411yescharacteristic
9C2×#14556597C2211yescharacteristic
10#17989005—22no—
11#17989005—22no—
12C2×#14556531C2211yescharacteristic
13C2×#14556530—21no—
14C22×#11924003C2211yescharacteristic
15C2×C16×D73—21no—
16#17988707—22no—
17#17988702—22no—
18#17988635C411yescharacteristic
19#17988604C411yescharacteristic
20#17988603C411yescharacteristic
21#17988542C2211yescharacteristic
22#17988371—21no—
23#17988371—21no—
24#17988370—22no—
25C2×#131×C73C2211yescharacteristic
26C2×M32×C73C2211yescharacteristic
27C2×M32×C73C2211yescharacteristic
28#349×C73—22no—
29C2×#11924028C2×C411yescharacteristic
30#14556622C812yes—
31#14556621—21no—
32#14556617C812yes—
33#14556616—21no—
34#14556611—22no—
35#14556597—21no—
36#14556597—21no—
37#14556597D412yes—
38#14556597D412yes—
39C2×#11924012C2×C411yescharacteristic
40C2×#11924012C2×C411yescharacteristic
41C2×#11924011—21no—
42C2×C8×D73—21no—
43#14556550C2×C411yescharacteristic
44#14556531—21no—
45#14556531—21no—
46#14556530—24no—
47C2×#11924003—21no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50C2×#11924003D411yescharacteristic
51C2×#11924003D411yescharacteristic
52C16×D73—24no—
53#14556526—22no—
54#14556518—22no—
55#14556517—22no—
56#14556501—21no—
57C8×Dic73—21no—
58C2×#41×C73C2×C411yescharacteristic
59#131×C73—21no—
60#131×C73—21no—
61#131×C73—21no—
62#131×C73—21no—
63C2×M16×C73—21no—
64C2×M16×C73C2×C411yescharacteristic
65C22×C8×C73—21no—
66C22×C8×C73C2311yescharacteristic
67M32×C73—21no—
68M32×C73—21no—
69M32×C73—21no—
70M32×C73D412yes—
71C2×C16×C73—21no—
72C2×C16×C73—21no—
73#11924028—21no—
74#11924028—21no—
75#11924028C22⋊C412yes—
76#11924028C2×C812yes—
77C2×Dic146M1611yescharacteristic
78C2×D292M1611yescharacteristic
79C2×C22⋊D73—21no—
80C2×C4×D73—21no—
81#11924018—22no—
82#11924016—22no—
83#11924015—22no—
84#11924013—22no—
85#11924012—21no—
86#11924012—21no—
87#11924012C22⋊C412yes—
88#11924012C2×C812yes—
89#11924011—22no—
90#11924011—22no—
91C2×#10596831—21no—
92C2×#10596831—21no—
93C8×D73—22no—
94C8×D73—22no—
95C4⋊Dic73—21no—
96C22⋊Dic73—21no—
97C4×Dic73—21no—
98#11924003—21no—
99#11924003—21no—
100#11924003—41no—
101#41×C73—21no—
102#41×C73—21no—
103#41×C73—21no—
104#41×C73—21no—
105C2×Q8×C73C2×C811yescharacteristic
106C2×D4×C73—21no—
107C2×D4×C73C2×C811yescharacteristic
108C22×C4×C73—21no—
109C22×C4×C73C22×C411yescharacteristic
110M16×C73—21no—
111M16×C73—21no—
112M16×C73—41no—
113M16×C73—41no—
114C2×C8×C73—21no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C16×C73—22no—
124C16×C73—41no—
125C23×C73—21no—
126C23×C73#13111yescharacteristic
127C22×D73—21no—
128Q8×C73—21no—
129Q8×C73—21no—
130D4×C73—21no—
131D4×C73—21no—
132D4×C73—41no—
133D4×C73—41no—
134C2×C4×C73—21no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73C2×C22⋊C411yescharacteristic
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73C4×D411yescharacteristic
140C2×C4×C73C22×C811yescharacteristic
141C2×C4×C73C2×M1611yescharacteristic
142C2×C4×C73#13111yescharacteristic
143C22⋊D73—22no—
144C22⋊D73—22no—
145C2×Dic73—21no—
146C2×Dic73—41no—
147D292—21no—
148D292C22⋊C812yes—
149C4×D73—22no—
150C4×D73—22no—
151Dic146—21no—
152Dic146C22⋊C812yes—
153C8×C73—41no—
154C8×C73#12011yescharacteristic
155C8×C73#12111yescharacteristic
156C8×C73#12111yescharacteristic
157C8×C73#12311yescharacteristic
158#10596831—22no—
159#10596831—22no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73—81no—
163C22×C73#40811yescharacteristic
164C22×C73C8×D411yescharacteristic
165C22×C73#43511yescharacteristic
166C2×D73—22no—
167C2×D73—42no—
168C4×C73—41no—
169C4×C73#38611yescharacteristic
170C4×C73C2×C22⋊C811yescharacteristic
171C4×C73C8×D411yescharacteristic
172C4×C73#43511yescharacteristic
173Dic73—22no—
174Dic73—81no—
175#12987—731nomaximal, Sylow
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—81no—
180C2×C73#186911yescharacteristic
181D73—42no—
182C2×#504—731no—
183#2955—731no—
184#2117—731no—
185C2×#350—731no—
186#2115D7311yescharacteristic, complemented
187#1379—731no—
188#1378—731no—
189C73#1298711yescharacteristic, complemented, Sylow
190#575—731no—
191C2×#131—731no—
192C2×#131C2×D7311yescharacteristic
193#504—1462no—
194#504—1462no—
195C2×M32—731no—
196C2×M32—1461no—
197C2×M32C2×D7311yescharacteristic
198C2×M32C2×D7311yescharacteristic
199C22×C16—731no—
200C22×C16—1461no—
201#444—1462no—
202#443—1462no—
203#430—731no—
204#421—731no—
205#421—731no—
206#405—731no—
207#350—1462no—
208#350—1461no—
209#350—1461no—
210#349—22no—
211C2×#41—731no—
212C2×#41C4×D7311yescharacteristic
213#137—732no—
214#137—1461no—
215AGL(1,ℤ/8)—732no—
216AGL(1,ℤ/8)—1461no—
217#135—1462no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—732no—
223#131—732no—
224#131—1461no—
225#131—1461no—
226C2×M16—21no—
227C2×M16—731no—
228C2×M16—731no—
229C2×M16—1461no—
230C2×M16C4×D7311yescharacteristic
231C22×C8—21no—
232C22×C8—1461no—
233C22×C8C22×D7311yescharacteristic
234#117—731no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1464no—
239M32—1461no—
240M32—1461no—
241M32C22⋊D7312yes—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—731no—
245C2×C16—731no—
246C2×C16—1464no—
247C2×C16—1461no—
248C2×C16—1461no—
249C2×C16—1461no—
250#108—1462no—
251#104—1462no—
252#104—1462no—
253#97—1461no—
254C4×C8—1461no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—732no—
260#41—732no—
261#41—1461no—
262#41—1461no—
263C2×Q8—731no—
264C2×Q8C8×D7311yescharacteristic
265C2×D4—21no—
266C2×D4—731no—
267C2×D4—1461no—
268C2×D4C8×D7311yescharacteristic
269C22×C4—21no—
270C22×C4—1461no—
271C22×C4C2×C4×D7311yescharacteristic
272Q16—1462no—
273SD16—1462no—
274SD16—1462no—
275D8—1462no—
276M16—21no—
277M16—21no—
278M16—41no—
279M16—41no—
280M16—732no—
281M16—732no—
282M16—1461no—
283M16—1461no—
284M16—1462no—
285M16—1462no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1461no—
290C2×C8—1461no—
291C2×C8—1462no—
292C2×C8—1462no—
293C2×C8D4×D7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8D4×D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8#1192402711yescharacteristic
298C2×C8#1192402811yescharacteristic
299C4⋊C4—1461no—
300C22⋊C4—1461no—
301C42—1461no—
302C16—22no—
303C16—41no—
304C16—2921no—
305C16—1461no—
306C16—1461no—
307C23—21no—
308C23—1461no—
309C23#1455659711yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—732no—
313Q8—1461no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—41no—
318D4—732no—
319D4—1462no—
320D4—1462no—
321D4—1461no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—2921no—
326C2×C4—1461no—
327C2×C4—1462no—
328C2×C4—1462no—
329C2×C4C22⋊C4×D7311yescharacteristic
330C2×C4#1455655111yescharacteristic
331C2×C4C4×C22⋊D7311yescharacteristic
332C2×C4C2×C8×D7311yescharacteristic
333C2×C4M16×D7311yescharacteristic
334C2×C4#1455659911yescharacteristic
335C8—41no—
336C8—1462no—
337C8—1462no—
338C8#1455655911yescharacteristic
339C8#1455656211yescharacteristic
340C8#1455656411yescharacteristic
341C8#1455657311yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—2922no—
346C22—1462no—
347C22#1798855211yescharacteristic
348C22C8×C22⋊D7311yescharacteristic
349C22#1798866111yescharacteristic
350C4—41no—
351C4—5841no—
352C4—1462no—
353C4#1798846311yescharacteristic
354C4C22⋊C8×D7311yescharacteristic
355C4#1798865011yescharacteristic
356C4#1798865611yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—2922no—
362C2#2261022011yescharacteristic
subgroupG/H
centerC8#14556564
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#2115C2
socleC2×C73#1869
prime radical#17989385C4
cosocleC22×C8×C73C23
hypercenter#2115D73
hypocenterC73#12987
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2115

Complements

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

NH ≅ G/Ntype
1. C2×#17989005C2semidirect
3. #22611558C2semidirect
4. C2×#17988371C2semidirect
5. #22608021C2semidirect
6. #22608018C2semidirect
7. C73×#2115C2semidirect
186. #2115D73semidirect
189. C73#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#179890052
#226188322
#226115582
C2×#179883712
#226080212
#226080182
C73×#21152
#1298773