Finite Group Databasebeta

← #30884956

Subgroups — #30884956

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568799C211yesmaximal, characteristic
2#22568798C211yesmaximal, characteristic
3C2×#16863432C211yesmaximal, characteristic
4C2×#16863424C211yesmaximal, characteristic
5#22561723C214yesmaximal, complemented
6#22561718C214yesmaximal, complemented
7#22559947C211yesmaximal, characteristic
8C2×#16863160C211yesmaximal, characteristic
9C67×#1742C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C2×#14524002C2211yescharacteristic
14#16863439—22no—
15#16863438—22no—
16#16863438C2214yes—
17#16863436C2214yes—
18#16863434—22no—
19#16863432—22no—
20#16863432—22no—
21#16863432—22no—
22C2×#14523976C2212yes—
23C2×#14523974C2212yes—
24#16863424—22no—
25#16863424C2212yes—
26#16863424C2212yes—
27C2×Dic67⋊C8—22no—
28#16863164C2214yes—
29#16863160C2212yes—
30#16863160C2212yes—
31C22×M16×C67C2211yescharacteristic
32C22×M16×C67C2211yescharacteristic
33C2×#117×C67C2211yescharacteristic
34#343×C67C2214yes—
35C2×#11761612—21no—
36C2×#11761612—22no—
37C2×#11761612Q814yes—
38C2×#11761612C2312yes—
39C22×C67⋊C8—21no—
40C22×C67⋊C8—22no—
41C22×C67⋊C8—22no—
42C22×C67⋊C8D412yes—
43C22×C67⋊C8D412yes—
44C22×C67⋊C8C2311yescharacteristic
45C22×C67⋊C8C2311yescharacteristic
46#14524002—21no—
47#14524002—21no—
48#14524002—21no—
49#14524002—21no—
50#14524002—21no—
51#14524002—21no—
52#14524002D412yes—
53#14524002C2312yes—
54C2×C4⋊Dic67D412yes—
55C2×C4⋊Dic67C2312yes—
56C2×C22⋊Dic67—21no—
57C2×C22⋊Dic67—21no—
58C2×C4×Dic67—21no—
59C2×C4×Dic67—21no—
60#14523977—24no—
61#14523976—24no—
62#14523976—24no—
63#14523975—24no—
64#14523974—24no—
65#14523974—24no—
66Dic67⋊C8—24no—
67Dic67⋊C8—24no—
68C23×C4×C67C2311yescharacteristic
69C2×M16×C67—21no—
70C2×M16×C67—22no—
71C2×M16×C67—21no—
72C2×M16×C67—22no—
73C2×M16×C67—22no—
74C2×M16×C67Q814yes—
75C2×M16×C67C2312yes—
76C2×M16×C67C2312yes—
77C22×C8×C67—21no—
78C22×C8×C67—21no—
79#117×C67—21no—
80#117×C67—21no—
81#117×C67—21no—
82#117×C67—21no—
83C2×C4⋊C4×C67C2312yes—
84C2×C22⋊C4×C67—21no—
85C2×C42×C67—21no—
86C22×Dic67—22no—
87C22×Dic67—22no—
88#11761612—24no—
89#11761612—22no—
90#11761612C2×Q814yes—
91C2×C67⋊C8—24no—
92C2×C67⋊C8—24no—
93C2×C67⋊C8—24no—
94C2×C67⋊C8—22no—
95C2×C67⋊C8—21no—
96C2×C67⋊C8—21no—
97C2×C67⋊C8—21no—
98C2×C67⋊C8—21no—
99C2×C67⋊C8C2×D412yes—
100C2×C67⋊C8C2×D412yes—
101C4⋊Dic67—22no—
102C4⋊Dic67—22no—
103C4⋊Dic67—22no—
104C4⋊Dic67C2×D414yes—
105C22⋊Dic67—22no—
106C22⋊Dic67—22no—
107C22⋊Dic67—41no—
108C22⋊Dic67—41no—
109C4×Dic67—22no—
110C4×Dic67—22no—
111C4×Dic67—41no—
112C4×Dic67—41no—
113C24×C67#4111yescharacteristic
114C22×C4×C67—22no—
115C22×C4×C67C2×D411yescharacteristic
116C22×C4×C67C2×D411yescharacteristic
117C22×C4×C67C2×D411yescharacteristic
118C22×C4×C67C2×D411yescharacteristic
119C22×C4×C67C2×Q812yes—
120C22×C4×C67C2×Q812yes—
121C22×C4×C67#4112yes—
122C22×C4×C67#4111yescharacteristic
123C22×C4×C67#4112yes—
124C22×C4×C67C2411yescharacteristic
125M16×C67—24no—
126M16×C67—22no—
127M16×C67—22no—
128M16×C67—22no—
129M16×C67—42no—
130M16×C67C2×Q814yes—
131C2×C8×C67—24no—
132C2×C8×C67—22no—
133C2×C8×C67—22no—
134C2×C8×C67—42no—
135C4⋊C4×C67—22no—
136C4⋊C4×C67—22no—
137C22⋊C4×C67—41no—
138C22⋊C4×C67—41no—
139C42×C67—41no—
140C42×C67—41no—
141C23×C67—42no—
142C23×C67#12111yescharacteristic
143C23×C67#12111yescharacteristic
144C23×C67#12512yes—
145C23×C67#12512yes—
146C23×C67C2×#4111yescharacteristic
147C2×C4×C67—42no—
148C2×C4×C67—82no—
149C2×C4×C67#12111yescharacteristic
150C2×C4×C67#12111yescharacteristic
151C2×C4×C67C22⋊Q812yes—
152C2×C4×C67C22⋊Q812yes—
153C2×C4×C67C22⋊Q812yes—
154C2×C4×C67C22⋊Q812yes—
155C2×C4×C67#12312yes—
156C2×C4×C67#12312yes—
157C2×C4×C67#12512yes—
158C2×C4×C67#12512yes—
159C2×C4×C67C4⋊Q812yes—
160C2×C4×C67C4⋊Q812yes—
161C2×C4×C67C22×D411yescharacteristic
162C2×C4×C67C22×D411yescharacteristic
163C2×C4×C67C22×Q811yescharacteristic
164C2×C4×C67C22×Q811yescharacteristic
165C2×C4×C67C2×#4111yescharacteristic
166C2×C4×C67C2×#4111yescharacteristic
167C2×Dic67—24no—
168C2×Dic67—44no—
169C2×Dic67—42no—
170C2×Dic67—82no—
171C8×C67—24no—
172C8×C67—42no—
173C67⋊C8—24no—
174C67⋊C8—22no—
175C67⋊C8—22no—
176C22×C67—22no—
177C22×C67—22no—
178C22×C67—22no—
179C22×C67—42no—
180C22×C67—42no—
181C22×C67#39812yes—
182C22×C67#39812yes—
183C22×C67C2×#12111yescharacteristic
184C22×C67C2×#12511yescharacteristic
185C22×C67C2×#12511yescharacteristic
186C4×C67—82no—
187C4×C67#39812yes—
188C4×C67#39812yes—
189C4×C67C2×C22⋊Q811yescharacteristic
190C4×C67C2×C22⋊Q811yescharacteristic
191C4×C67C2×#12311yescharacteristic
192C4×C67C2×C4⋊Q811yescharacteristic
193Dic67—44no—
194Dic67—82no—
195#23674—671nomaximal, Sylow
196C2×C67—21no—
197C2×C67—21no—
198C2×C67—22no—
199C2×C67—22no—
200C2×C67C2×#39811yescharacteristic
201C2×C67#306712yes—
202#2911—671no—
203#2911—671no—
204C2×#427—671no—
205C2×#424—671no—
206#2090—674no—
207#2089—674no—
208#1742—671no—
209#1742D6711yescharacteristic
210C2×#341—671no—
211C67#2367411yescharacteristic, complemented, Sylow
212C22×M16—671no—
213C22×M16C2×D6711yescharacteristic
214C22×M16C2×D6711yescharacteristic
215C23×C8—671no—
216C2×#117—671no—
217C2×#117—671no—
218C2×#117C2×D6711yescharacteristic
219#428—674no—
220#428—674no—
221#428—1342no—
222#428—1342no—
223#427—1342no—
224#427—1342no—
225#427—1342no—
226#427—1342no—
227C2×#107—672no—
228C2×#106—672no—
229#424—672no—
230#424—672no—
231#424—1342no—
232C2×C4⋊C8—1342no—
233#343—674no—
234#343C2×D6714yes—
235#341—672no—
236#341—672no—
237C23×C4C22×D6711yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—674no—
244C2×M16—672no—
245C2×M16—1341no—
246C2×M16—1342no—
247C2×M16Dic13414yes—
248C2×M16C22×D6712yes—
249C2×M16C22×D6712yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—672no—
253C22×C8—672no—
254C22×C8—671no—
255C22×C8—671no—
256C22×C8—1341no—
257C22×C8—1342no—
258C22×C8—1342no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—672no—
264#117—672no—
265#117—1341no—
266#117—1341no—
267#117—1341no—
268#117—1341no—
269#117—1341no—
270#117—1341no—
271C2×C4⋊C4—672no—
272C2×C4⋊C4—672no—
273C2×C4⋊C4C22×D6712yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1341no—
276C2×C22⋊C4—1341no—
277C2×C42—21no—
278C2×C42—1341no—
279C2×C42—1341no—
280#107—1344no—
281#107—1344no—
282#107—1344no—
283#106—1344no—
284#106—1344no—
285#106—1344no—
286C4⋊C8—1344no—
287C4⋊C8—1344no—
288C24#1176162811yescharacteristic
289C22×C4—22no—
290C22×C4—1342no—
291C22×C4—1342no—
292C22×C4D4×D6711yescharacteristic
293C22×C4D4×D6711yescharacteristic
294C22×C4D4×D6711yescharacteristic
295C22×C4D4×D6711yescharacteristic
296C22×C4Q8×D6712yes—
297C22×C4C2×Dic13412yes—
298C22×C4#1176162712yes—
299C22×C4#1176162811yescharacteristic
300C22×C4#1176162812yes—
301C22×C4C23×D6711yescharacteristic
302M16—24no—
303M16—22no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—674no—
308M16—1344no—
309M16—1342no—
310M16C2×Dic13414yes—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—672no—
316C2×C8—672no—
317C2×C8—1344no—
318C2×C8—1344no—
319C2×C8—1344no—
320C2×C8—1342no—
321C2×C8—1341no—
322C2×C8—1341no—
323C2×C8—1341no—
324C2×C8—1341no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—674no—
328C4⋊C4—1342no—
329C4⋊C4—1342no—
330C4⋊C4—1342no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—2681no—
334C22⋊C4—2681no—
335C22⋊C4—1342no—
336C22⋊C4—1342no—
337C42—41no—
338C42—41no—
339C42—2681no—
340C42—2681no—
341C42—1342no—
342C42—1342no—
343C23—42no—
344C23#1452401611yescharacteristic
345C23#1452401611yescharacteristic
346C23#1452403312yes—
347C23#1452403512yes—
348C23C2×#1176162811yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—2684no—
352C2×C4—2682no—
353C2×C4—5362no—
354C2×C4—1344no—
355C2×C4#1452401611yescharacteristic
356C2×C4#1452401611yescharacteristic
357C2×C4#1452402012yes—
358C2×C4#1452402012yes—
359C2×C4#1452402112yes—
360C2×C4#1452402112yes—
361C2×C4#1452402512yes—
362C2×C4#1452402512yes—
363C2×C4#1452403312yes—
364C2×C4#1452403512yes—
365C2×C4#1452404212yes—
366C2×C4#1452404212yes—
367C2×C4C2×D4×D6711yescharacteristic
368C2×C4C2×D4×D6711yescharacteristic
369C2×C4C2×Q8×D6711yescharacteristic
370C2×C4C22×Dic13411yescharacteristic
371C2×C4C2×#1176162711yescharacteristic
372C2×C4C2×#1176162811yescharacteristic
373C8—24no—
374C8—42no—
375C8—1344no—
376C8—1342no—
377C8—1342no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1686333012yes—
384C22#1686333012yes—
385C22C2×#1452401611yescharacteristic
386C22C2×#1452403311yescharacteristic
387C22C2×#1452403511yescharacteristic
388C4—82no—
389C4—2684no—
390C4—5362no—
391C4#1686333012yes—
392C4#1686333012yes—
393C4C2×#1452402011yescharacteristic
394C4C2×#1452402111yescharacteristic
395C4C2×#1452402511yescharacteristic
396C4C2×#1452404211yescharacteristic
397C2—22no—
398C2—22no—
399C2—21no—
400C2—21no—
401C2C2×#1686333011yescharacteristic
402C2#2256992812yes—
subgroupG/H
centerC2×C4#14524016
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC67×#1742C2
socleC22×C67C2×#121
prime radicalC24×C67#41
hypercenter#1742D67
hypocenterC67#23674
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #22561723C2semidirect
6. #22561718C2semidirect
211. C67#23674semidirect

Maximal subgroups

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

subgroupindex
#225687992
#225687982
C2×#168634322
C2×#168634242
#225617232
#225617182
#225599472
C2×#168631602
C67×#17422
#2367467