Finite Group Databasebeta

← #34021350

Subgroups — #34021350

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989602C212yesmaximal
2C2×D73⋊#108C211yesmaximal, characteristic
3C22×#14556661C211yesmaximal, characteristic
4#22626574C212yesmaximal
5#22625873C214yesmaximal, complemented
6#22625871C212yesmaximal, complemented
7#22625870C212yesmaximal, complemented
8#405×D73C211yesmaximal, characteristic
9C22×#11924038C2212yes—
10C2×D73⋊M16C2212yes—
11#17989602—28no—
12C2×#14556662C2212yes—
13C2×#14556662C2214yes—
14D73⋊#108—24no—
15C2×#14556661—24no—
16C2×#14556661C2212yes—
17C2×#14556661C2212yes—
18C2×#14556661C2211yescharacteristic
19C2×#14556661C2211yescharacteristic
20C2×#14556661C2214yes—
21#17989594—24no—
22#17989593—24no—
23#17989592C2214yes—
24#17989591C2214yes—
25C2×M16×D73C2211yescharacteristic
26C22×C8×D73C2211yescharacteristic
27#117×D73C2211yescharacteristic
28#17988545C411yescharacteristic
29#17988544C412yes—
30#17988543C411yescharacteristic
31#17988542C411yescharacteristic
32#17988541C412yes—
33#97×D73C2212yes—
34C4×C8×D73C2212yes—
35#405×C73C411yescharacteristic
36C2×#11924038—22no—
37C2×#11924038—24no—
38C2×#11924038—24no—
39C2×#11924038C2312yes—
40C2×#11924038C2312yes—
41C2×#11924038C2314yes—
42D73⋊M16—24no—
43D73⋊M16—24no—
44C2×#11924037—22no—
45C2×#11924036—22no—
46C22×#10596833—22no—
47#14556662—24no—
48#14556662—28no—
49#14556661—24no—
50#14556661—24no—
51#14556661—22no—
52#14556661—22no—
53#14556661—28no—
54#14556658—28no—
55C4⋊#10596833—28no—
56C22×C4×D73C2311yescharacteristic
57C2×#11924012C2×C411yescharacteristic
58C2×#11924011C2×C412yes—
59M16×D73D414yes—
60M16×D73Q814yes—
61C22×#10596831C2×C411yescharacteristic
62C2×C8×D73C2×C411yescharacteristic
63C2×C8×D73C2×C411yescharacteristic
64C2×C8×D73D414yes—
65C2×C8×D73Q814yes—
66C2×C8×D73C2311yescharacteristic
67C2×C8×D73C2312yes—
68C2×C8×D73C2311yescharacteristic
69#14556550C2×C411yescharacteristic
70#14556549D412yes—
71#14556548C2×C412yes—
72#14556547D412yes—
73C4⋊C4×D73Q812yes—
74C22⋊C4×D73Q812yes—
75C42×D73C2312yes—
76#14556502C2×C412yes—
77#14556501C2×C412yes—
78#14556501C2×C412yes—
79C4×#10596831C2×C412yes—
80C8×Dic73C2×C412yes—
81C8×Dic73C2×C412yes—
82C2×M16×C73C2×C411yescharacteristic
83C22×C8×C73C2×C411yescharacteristic
84#117×C73C2×C411yescharacteristic
85#97×C73C2×C412yes—
86C4×C8×C73C2×C412yes—
87#11924038—22no—
88#11924038—22no—
89#11924038—22no—
90#11924038—22no—
91#11924038—24no—
92#11924038—24no—
93#11924038—44no—
94#11924037—44no—
95#11924036—44no—
96C2×#10596833—24no—
97C2×#10596833—22no—
98C2×#10596833—22no—
99C2×#10596833—44no—
100C23×D73C2×Q811yescharacteristic
101C22×Dic73C2×D411yescharacteristic
102C2×C4×D73C22×C411yescharacteristic
103C2×C4×D73C2×D412yes—
104C2×C4×D73C2×D412yes—
105C2×C4×D73C2×D411yescharacteristic
106C2×C4×D73C2×Q814yes—
107C2×C4×D73C2×Q812yes—
108C2×C4×D73C2×Q812yes—
109C2×C4×D73C2×Q811yescharacteristic
110C2×C4×D73C2411yescharacteristic
111#11924012C4⋊C414yes—
112#11924011C4⋊C418yes—
113C2×#10596831C4⋊C414yes—
114C2×#10596831C22×C411yescharacteristic
115C2×#10596831C22×C412yes—
116C2×#10596831C22×C411yescharacteristic
117C8×D73C4⋊C414yes—
118C8×D73C4⋊C414yes—
119C8×D73C2×D412yes—
120C8×D73C2×D412yes—
121C8×D73C2×D414yes—
122C8×D73C2×Q812yes—
123C8×D73C2×Q812yes—
124C8×D73C2×Q814yes—
125C4⋊Dic73C4⋊C412yes—
126#11924007C4⋊C414yes—
127C22⋊Dic73C4⋊C412yes—
128#11924005C4⋊C414yes—
129C4×Dic73C22×C412yes—
130C4×Dic73C2×D414yes—
131C22×C4×C73C22×C411yescharacteristic
132M16×C73C4⋊C414yes—
133C2×C8×C73C4⋊C414yes—
134C2×C8×C73C22×C411yescharacteristic
135C2×C8×C73C22×C412yes—
136C2×C8×C73C22×C411yescharacteristic
137C4⋊C4×C73C4⋊C412yes—
138C22⋊C4×C73C4⋊C412yes—
139C42×C73C22×C412yes—
140C23×C73C2×C4⋊C411yescharacteristic
141C22×D73—42no—
142C22×D73C2×C4⋊C411yescharacteristic
143C22×D73C4⋊Q812yes—
144C22×D73C4⋊Q812yes—
145C22×D73C22×Q811yescharacteristic
146C2×C4×C73C2×C4⋊C414yes—
147C2×C4×C73C2×C4⋊C412yes—
148C2×C4×C73C2×C4⋊C412yes—
149C2×C4×C73C2×C4⋊C411yescharacteristic
150C2×C4×C73C23×C411yescharacteristic
151C2×Dic73C2×C4⋊C414yes—
152C2×Dic73C2×C4⋊C411yescharacteristic
153C2×Dic73#12712yes—
154C2×Dic73C4⋊Q812yes—
155C2×Dic73C22×D411yescharacteristic
156C4×D73—44no—
157C4×D73C2×C4⋊C412yes—
158C4×D73C2×C4⋊C412yes—
159C4×D73#12712yes—
160C4×D73C4⋊Q812yes—
161C4×D73C4⋊Q812yes—
162C4×D73C4⋊Q812yes—
163C4×D73C22×D411yescharacteristic
164C4×D73C22×D411yescharacteristic
165C4×D73C22×Q811yescharacteristic
166C4×D73C22×Q811yescharacteristic
167#10596833—44no—
168#10596833—42no—
169#10596833—42no—
170C8×C73C2×C4⋊C412yes—
171C8×C73C2×C4⋊C412yes—
172C8×C73C2×C4⋊C414yes—
173#10596831C2×C4⋊C412yes—
174#10596831C2×C4⋊C412yes—
175#10596831C2×C4⋊C414yes—
176C22×C73—22no—
177C22×C73#38412yes—
178C22×C73C22×C4⋊C411yescharacteristic
179C2×D73—22no—
180C2×D73—22no—
181C2×D73—42no—
182C2×D73—42no—
183C2×D73#38412yes—
184C2×D73C2×C4⋊Q811yescharacteristic
185C2×D73C2×C4⋊Q811yescharacteristic
186C4×C73—24no—
187C4×C73#38412yes—
188C4×C73C22×C4⋊C411yescharacteristic
189C4×C73C22×C4⋊C411yescharacteristic
190Dic73—24no—
191Dic73#38412yes—
192Dic73C2×#12711yescharacteristic
193Dic73C2×C4⋊Q811yescharacteristic
194#20807—731nomaximal, Sylow
195C2×C73—22no—
196C2×C73C2×#38411yescharacteristic
197C2×C73#291612yes—
198D73—21no—
199D73—21no—
200D73—22no—
201C2×#430—732no—
202C22×#108—731no—
203C22×#108—731no—
204#2905—732no—
205C2×#405—731no—
206#1849—734no—
207#1848—732no—
208#1848—732no—
209C73#2080711yescharacteristic, complemented, Sylow
210C22×M16—731no—
211C22×M16—732no—
212C22×M16—732no—
213C23×C8—731no—
214C2×#117—731no—
215#430—1468no—
216C2×#108—731no—
217C2×#108—732no—
218C2×#108—731no—
219C2×#108—734no—
220C2×#108—734no—
221C2×#108—732no—
222C2×#108—732no—
223C2×#108—1464no—
224C2×#108—1464no—
225#424—1464no—
226#424—1464no—
227#423—734no—
228#423—734no—
229#405—731no—
230#405—731no—
231#405—731no—
232#405—732no—
233#405—732no—
234#405#6440811yescharacteristic
235C2×#97—732no—
236C2×C4×C8—732no—
237C23×C4—731no—
238C2×M16—731no—
239C2×M16—734no—
240C2×M16—734no—
241C2×M16—732no—
242C2×M16—732no—
243C2×M16—732no—
244C2×M16—734no—
245C2×M16—1464no—
246C2×M16—1464no—
247C2×M16—1464no—
248C2×M16—1464no—
249C2×M16—1462no—
250C2×M16—1462no—
251C2×M16C2×#6440811yescharacteristic
252C22×C8—731no—
253C22×C8—734no—
254C22×C8—734no—
255C22×C8—731no—
256C22×C8—731no—
257C22×C8—731no—
258C22×C8—731no—
259C22×C8—732no—
260C22×C8—1462no—
261C22×C8—1462no—
262C22×C8C2×#6440811yescharacteristic
263#117—731no—
264#117—732no—
265#117—732no—
266#117—732no—
267#117C2×#6440811yescharacteristic
268C2×C4⋊C4—732no—
269C2×C22⋊C4—732no—
270C2×C42—732no—
271#108—1464no—
272#108—1462no—
273#108—1462no—
274#108—1464no—
275#108—1464no—
276#108—1468no—
277#108—1468no—
278C4⋊C8—1468no—
279C4⋊C8—1468no—
280#97—732no—
281#97—732no—
282#97—732no—
283#97C2×#6440812yes—
284C4×C8—732no—
285C4×C8—732no—
286C4×C8—732no—
287C4×C8C2×#6440812yes—
288C24—731no—
289C22×C4—734no—
290C22×C4—731no—
291C22×C4—731no—
292C22×C4—731no—
293C22×C4—731no—
294C22×C4—732no—
295C22×C4—732no—
296C22×C4—731no—
297C22×C4—732no—
298C22×C4—732no—
299C22×C4C22×#6440811yescharacteristic
300M16—2924no—
301M16—2924no—
302M16—734no—
303M16—738no—
304M16—1462no—
305M16—1462no—
306M16—1462no—
307M16—1462no—
308M16—1464no—
309M16—1464no—
310M16C4⋊#6440814yes—
311C2×C8—2924no—
312C2×C8—2924no—
313C2×C8—731no—
314C2×C8—732no—
315C2×C8—731no—
316C2×C8—734no—
317C2×C8—734no—
318C2×C8—734no—
319C2×C8—732no—
320C2×C8—732no—
321C2×C8—734no—
322C2×C8—732no—
323C2×C8—732no—
324C2×C8—734no—
325C2×C8—1464no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8C4⋊#6440814yes—
329C2×C8C22×#6440811yescharacteristic
330C2×C8C22×#6440812yes—
331C2×C8C22×#6440811yescharacteristic
332C4⋊C4—734no—
333C4⋊C4—732no—
334C4⋊C4C4⋊#6440812yes—
335C22⋊C4—732no—
336C22⋊C4—734no—
337C22⋊C4C4⋊#6440812yes—
338C42—732no—
339C42—734no—
340C42C22×#6440812yes—
341C23—2922no—
342C23—732no—
343C23—732no—
344C23—731no—
345C23—731no—
346C23C2×C4⋊#6440811yescharacteristic
347C2×C4—2924no—
348C2×C4—731no—
349C2×C4—731no—
350C2×C4—732no—
351C2×C4—732no—
352C2×C4—731no—
353C2×C4—731no—
354C2×C4—732no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—732no—
358C2×C4—732no—
359C2×C4—732no—
360C2×C4—732no—
361C2×C4—732no—
362C2×C4—734no—
363C2×C4C2×C4⋊#6440814yes—
364C2×C4C2×C4⋊#6440812yes—
365C2×C4C2×C4⋊#6440812yes—
366C2×C4C2×C4⋊#6440811yescharacteristic
367C2×C4C23×#6440811yescharacteristic
368C8—2924no—
369C8—2922no—
370C8—2922no—
371C8—732no—
372C8—732no—
373C8—734no—
374C8C2×C4⋊#6440812yes—
375C8C2×C4⋊#6440812yes—
376C8C2×C4⋊#6440814yes—
377C22—22no—
378C22—2922no—
379C22—2922no—
380C22—731no—
381C22—731no—
382C22—732no—
383C22—1462no—
384C22—1462no—
385C22#1798953312yes—
386C22C22×C4⋊#6440811yescharacteristic
387C4—24no—
388C4—731no—
389C4—731no—
390C4—732no—
391C4—1464no—
392C4#1798953312yes—
393C4C22×C4⋊#6440811yescharacteristic
394C4C22×C4⋊#6440811yescharacteristic
395C2—22no—
396C2—1461no—
397C2—1461no—
398C2—1462no—
399C2C2×#1798953311yescharacteristic
400C2#2262659212yes—
subgroupG/H
centerC22C22×C4⋊#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#405×C73C4
socleC22×C73C22×C4⋊C4
prime radicalC23×D73C2×Q8
cosocleC2×C4×D73C24
hypercenter#405#64408
hypocenterC73#20807
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
5. #22625873C2semidirect
6. #22625871C2semidirect
7. #22625870C2semidirect
209. C73#20807semidirect

Maximal subgroups

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

subgroupindex
C2×#179896022
C2×D73⋊#1082
C22×#145566612
#226265742
#226258732
#226258712
#226258702
#405×D732
#2080773