Finite Group Databasebeta

← #30772742

Subgroups — #30772742

Order 17152.

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
1#22573312C211yesmaximal, characteristic, complemented
2#22561837C211yesmaximal, characteristic, complemented
3#22561837C211yesmaximal, characteristic, complemented
4#22561830C211yesmaximal, characteristic, complemented
5#22558293C211yesmaximal, characteristic, complemented
6#22558289C211yesmaximal, characteristic, complemented
7C2×#444×C67C211yesmaximal, characteristic
8#16863813—22no—
9#16863813—22no—
10C2×#14523983—22no—
11C2×#14523983C2211yescharacteristic
12C2×#14523983C2211yescharacteristic
13C2×#14523983C2211yescharacteristic
14C2×#14523983C2211yescharacteristic
15#16863180—22no—
16#16863180—22no—
17#16863178—22no—
18C2×#135×C67C411yescharacteristic
19C2×#131×C67C2211yescharacteristic
20C2×#131×C67C2211yescharacteristic
21#444×C67—22no—
22#444×C67—22no—
23C2×#108×C67C411yescharacteristic
24C2×#104×C67C411yescharacteristic
25C2×#104×C67C411yescharacteristic
26C2×#97×C67C2211yescharacteristic
27#14523983—24no—
28#14523983—21no—
29#14523983—21no—
30#14523983—21no—
31#14523983—21no—
32#14523983—21no—
33#14523983—21no—
34#14523983—21no—
35#14523983D412yes—
36C2×C67⋊C16—24no—
37C2×C67⋊C16—21no—
38C2×C67⋊C16—21no—
39C2×C67⋊C16—21no—
40C2×C67⋊C16D412yes—
41C2×#41×C67C2×C411yescharacteristic
42C2×#41×C67C2×C411yescharacteristic
43#135×C67—22no—
44#135×C67—22no—
45C2×Q16×C67C811yescharacteristic
46C2×SD16×C67C811yescharacteristic
47C2×SD16×C67C811yescharacteristic
48C2×D8×C67C811yescharacteristic
49#131×C67—21no—
50#131×C67—21no—
51#131×C67—21no—
52#131×C67—21no—
53#131×C67—21no—
54#131×C67—21no—
55#131×C67D412yes—
56#131×C67D412yes—
57C2×M16×C67—21no—
58C2×M16×C67—21no—
59C2×M16×C67C2×C411yescharacteristic
60C2×M16×C67C2×C411yescharacteristic
61C22×C8×C67—21no—
62C22×C8×C67—21no—
63C22×C8×C67C2×C411yescharacteristic
64C22×C8×C67C2311yescharacteristic
65C2×C42×C67C2×C411yescharacteristic
66#108×C67—22no—
67#104×C67—22no—
68#104×C67—22no—
69#97×C67—21no—
70#97×C67—21no—
71C67⋊C16—22no—
72C67⋊C16—22no—
73C67⋊C16—41no—
74C67⋊C16—41no—
75#41×C67—21no—
76#41×C67—21no—
77#41×C67—21no—
78#41×C67—21no—
79#41×C67—21no—
80#41×C67—21no—
81#41×C67C22⋊C412yes—
82#41×C67M1612yes—
83C2×Q8×C67C2×C811yescharacteristic
84C2×Q8×C67C2×C811yescharacteristic
85C2×D4×C67—21no—
86C2×D4×C67—21no—
87C2×D4×C67C2×C811yescharacteristic
88C2×D4×C67C2×C811yescharacteristic
89C22×C4×C67—21no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67C22×C411yescharacteristic
93Q16×C67—22no—
94SD16×C67—22no—
95SD16×C67—22no—
96D8×C67—22no—
97M16×C67—21no—
98M16×C67—22no—
99M16×C67—22no—
100M16×C67—21no—
101M16×C67—21no—
102M16×C67—41no—
103M16×C67—41no—
104M16×C67C22⋊C412yes—
105C2×C8×C67—21no—
106C2×C8×C67—21no—
107C2×C8×C67—21no—
108C2×C8×C67—22no—
109C2×C8×C67—22no—
110C2×C8×C67—21no—
111C2×C8×C67—41no—
112C2×C8×C67—41no—
113C2×C8×C67C2×C811yescharacteristic
114C2×C8×C67C2×C811yescharacteristic
115C2×C8×C67C2×D411yescharacteristic
116C2×C8×C67C2×D411yescharacteristic
117C2×C8×C67C2×D411yescharacteristic
118C2×C8×C67C2×D411yescharacteristic
119C2×C8×C67#4111yescharacteristic
120C2×C8×C67#4111yescharacteristic
121C42×C67—21no—
122C42×C67—21no—
123C23×C67—21no—
124C23×C67—21no—
125C23×C67#13111yescharacteristic
126Q8×C67—21no—
127Q8×C67—21no—
128Q8×C67—21no—
129Q8×C67C22⋊C812yes—
130D4×C67—22no—
131D4×C67—22no—
132D4×C67—21no—
133D4×C67—21no—
134D4×C67—21no—
135D4×C67—41no—
136D4×C67—41no—
137D4×C67C22⋊C812yes—
138C2×C4×C67—21no—
139C2×C4×C67—22no—
140C2×C4×C67—22no—
141C2×C4×C67—21no—
142C2×C4×C67—41no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—81no—
146C2×C4×C67C2×C22⋊C411yescharacteristic
147C2×C4×C67C4×D411yescharacteristic
148C2×C4×C67C4×D411yescharacteristic
149C2×C4×C67C22×C811yescharacteristic
150C2×C4×C67C2×M1611yescharacteristic
151C2×C4×C67#13111yescharacteristic
152C8×C67—22no—
153C8×C67—21no—
154C8×C67—21no—
155C8×C67—41no—
156C8×C67#12011yescharacteristic
157C8×C67#12111yescharacteristic
158C8×C67#12111yescharacteristic
159C8×C67#12311yescharacteristic
160C22×C67—22no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—81no—
164C22×C67—81no—
165C22×C67#40811yescharacteristic
166C22×C67C8×D411yescharacteristic
167C22×C67#43511yescharacteristic
168C4×C67—22no—
169C4×C67—41no—
170C4×C67—81no—
171C4×C67#38611yescharacteristic
172C4×C67C2×C22⋊C811yescharacteristic
173C4×C67C8×D411yescharacteristic
174C4×C67#43511yescharacteristic
175#12986—671nomaximal, Sylow
176C2×C67—21no—
177C2×C67—21no—
178C2×C67—21no—
179C2×C67—42no—
180C2×C67—81no—
181C2×C67#186911yescharacteristic
182#3407—671no—
183C2×#444D6711yescharacteristic
184#2117—671no—
185#2117—671no—
186#2115—671no—
187#1377—671no—
188#1375—671no—
189C67#1298611yescharacteristic, complemented, Sylow
190C2×#135Dic6711yescharacteristic
191C2×#131C2×D6711yescharacteristic
192C2×#131C2×D6711yescharacteristic
193#504—1342no—
194#504—1342no—
195C2×M32—671no—
196C2×M32—671no—
197C2×M32—671no—
198C2×M32—671no—
199C2×M32—1342no—
200#444—22no—
201#444—22no—
202C2×#108Dic6711yescharacteristic
203C2×#104Dic6711yescharacteristic
204C2×#104Dic6711yescharacteristic
205C2×#97C2×D6711yescharacteristic
206#350—1342no—
207#350—1342no—
208#349—1342no—
209C2×#41C2×Dic6711yescharacteristic
210C2×#41C2×Dic6711yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16C67⋊C811yescharacteristic
214C2×SD16C67⋊C811yescharacteristic
215C2×SD16C67⋊C811yescharacteristic
216C2×D8C67⋊C811yescharacteristic
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131C22⋊D6712yes—
224#131C22⋊D6712yes—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16C2×Dic6711yescharacteristic
228C2×M16C2×Dic6711yescharacteristic
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C2×Dic6711yescharacteristic
232C22×C8C22×D6711yescharacteristic
233C2×C42C2×Dic6711yescharacteristic
234M32—672no—
235M32—1341no—
236M32—1341no—
237M32—1341no—
238M32—1341no—
239M32—1344no—
240M32—1341no—
241M32—1341no—
242M32—1341no—
243C2×C16—672no—
244C2×C16—1341no—
245C2×C16—1341no—
246C2×C16—1344no—
247C2×C16—1341no—
248#108—22no—
249#104—22no—
250#104—22no—
251#97—21no—
252#97—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41C22⋊Dic6712yes—
260#41#1176161212yes—
261C2×Q8C2×C67⋊C811yescharacteristic
262C2×Q8C2×C67⋊C811yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×C67⋊C811yescharacteristic
266C2×D4C2×C67⋊C811yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C22×Dic6711yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—21no—
276M16—22no—
277M16—22no—
278M16—21no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16C22⋊Dic6712yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8C2×C67⋊C811yescharacteristic
292C2×C8C2×C67⋊C811yescharacteristic
293C2×C8D4×D6711yescharacteristic
294C2×C8D4×D6711yescharacteristic
295C2×C8C2×C22⋊D6711yescharacteristic
296C2×C8C2×C22⋊D6711yescharacteristic
297C2×C8#1176162711yescharacteristic
298C2×C8#1176162711yescharacteristic
299C42—21no—
300C42—21no—
301C16—2681no—
302C16—2681no—
303C16—1342no—
304C16—1342no—
305C23—21no—
306C23—21no—
307C23#1452405011yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C22⋊(C67⋊C8)12yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C22⋊(C67⋊C8)12yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4C2×C22⋊Dic6711yescharacteristic
329C2×C4D4×Dic6711yescharacteristic
330C2×C4D4×Dic6711yescharacteristic
331C2×C4C22×C67⋊C811yescharacteristic
332C2×C4C2×#1176161211yescharacteristic
333C2×C4#1452405011yescharacteristic
334C8—22no—
335C8—21no—
336C8—21no—
337C8—41no—
338C8#1452401111yescharacteristic
339C8#1452401511yescharacteristic
340C8#1452401511yescharacteristic
341C8#1452402411yescharacteristic
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22#1686336311yescharacteristic
348C22D4×C67⋊C811yescharacteristic
349C22#1686346811yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4#1686327611yescharacteristic
354C4C2×C22⋊(C67⋊C8)11yescharacteristic
355C4D4×C67⋊C811yescharacteristic
356C4#1686346811yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2256050211yescharacteristic
subgroupG/H
centerC4#16863276
commutatorC2×C4×C67C22×C8
FrattiniC22×C8C22×D67
FittingC2×#444×C67C2
socleC2×C67#1869
prime radicalC2×#135×C67C4
cosocleC22×C8×C67C23
hypercenterC2×#444D67
hypocenterC67#12986
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
1. #22573312C2semidirect
2. #22561837C2semidirect
3. #22561837C2semidirect
4. #22561830C2semidirect
5. #22558293C2semidirect
6. #22558289C2semidirect
189. C67#12986semidirect

Maximal subgroups

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

subgroupindex
#225733122
#225618372
#225618302
#225582932
#225582892
C2×#444×C672
#1298667