Finite Group Databasebeta

← #30749001

Subgroups — #30749001

Order 17152.

454 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
1C8×#11761628C211yesmaximal, characteristic
2#22560230C212yesmaximal
3#22558968C211yesmaximal, characteristic, complemented
4#22558112C212yesmaximal, complemented
5C4×#107×C67C211yesmaximal, characteristic, complemented
6C4×#11761628C2211yescharacteristic
7C8×Dic134C411yescharacteristic, complemented
8C8×Dic134C411yescharacteristic, complemented
9C8×D268C411yescharacteristic, complemented
10C8×D268C411yescharacteristic, complemented
11C8×C22⋊D67—21no—
12C8×C22⋊D67—21no—
13#16863449C2211yescharacteristic
14#16863447—21no—
15#16863420C2212yes—
16C2×#14523971—22no—
17C4×C8×D67—21no—
18#16863156—22no—
19#16863149—21no—
20#16863149—21no—
21C2×#107×C67—21no—
22C2×C4×C8×C67C2211yescharacteristic
23C4×C4⋊C4×C67C2212yes—
24#340×C67—21no—
25C2×#11761628D411yescharacteristic
26C2×#11761612—22no—
27C2×C8×D67—21no—
28C2×C8×D67—21no—
29C4×Dic134C2×C411yescharacteristic
30C4×Dic134Q811yescharacteristic
31C4×D268C2×C411yescharacteristic
32C4×D268Q811yescharacteristic
33C4×C22⋊D67—21no—
34C4×C22⋊D67—21no—
35#14524002D411yescharacteristic
36#14524000—21no—
37C42×D67—21no—
38C4⋊(C67⋊C8)—22no—
39C4⋊(C67⋊C8)—22no—
40C4⋊(C67⋊C8)C2×C411yescharacteristic
41C4⋊(C67⋊C8)C2×C411yescharacteristic
42Dic67⋊C8—21no—
43Dic67⋊C8—21no—
44#14523971—44no—
45C22⋊(C67⋊C8)—21no—
46(C2×D67)⋊C8—21no—
47(C2×D67)⋊C8—21no—
48C4×C67⋊C8—21no—
49C8×Dic67—21no—
50C8×Dic67—21no—
51C22×C8×C67D411yescharacteristic
52C22×C8×C67D411yescharacteristic
53C2×C4⋊C4×C67—22no—
54C2×C42×C67—22no—
55C2×C42×C67C2311yescharacteristic
56#107×C67—24no—
57C4×C8×C67C2×C411yescharacteristic
58C4×C8×C67C2×C411yescharacteristic
59C4×C8×C67D411yescharacteristic
60C4×C8×C67D411yescharacteristic
61#95×C67—22no—
62#11761628—21no—
63#11761628—21no—
64#11761628D812yes—
65#11761628Q1612yes—
66C2×Dic134C4⋊C411yescharacteristic
67C2×D268C4⋊C411yescharacteristic
68C2×C22⋊D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72#11761612—42no—
73#11761612—42no—
74C2×C67⋊C8—21no—
75C2×C67⋊C8—21no—
76C2×C67⋊C8—42no—
77C8×D67—42no—
78C8×D67—42no—
79C4⋊Dic67C4⋊C411yescharacteristic
80C4⋊Dic67C4⋊C411yescharacteristic
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83C22⋊Dic67—21no—
84(C2×D67)⋊C4—21no—
85(C2×D67)⋊C4—21no—
86C4×Dic67—21no—
87C4×Dic67—21no—
88C4×Dic67—21no—
89C22×C4×C67—42no—
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67C2×D411yescharacteristic
92C22×C4×C67#4111yescharacteristic
93C2×C8×C67—22no—
94C2×C8×C67—41no—
95C2×C8×C67C22⋊C411yescharacteristic
96C2×C8×C67C22⋊C411yescharacteristic
97C2×C8×C67C22⋊C411yescharacteristic
98C2×C8×C67C22⋊C411yescharacteristic
99C4⋊C4×C67—24no—
100C4⋊C4×C67—42no—
101C42×C67—42no—
102C42×C67—42no—
103C42×C67C22×C411yescharacteristic
104C42×C67C2×D411yescharacteristic
105C42×C67C2×Q811yescharacteristic
106C42×C67#4111yescharacteristic
107C23×C67#12111yescharacteristic
108C22×D67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—41no—
112C2×C4×C67—44no—
113C2×C4×C67—82no—
114C2×C4×C67—82no—
115C2×C4×C67C2×C22⋊C411yescharacteristic
116C2×C4×C67C2×C4⋊C411yescharacteristic
117C2×C4×C67#11711yescharacteristic
118C2×C4×C67#12111yescharacteristic
119C2×C4×C67C22⋊Q811yescharacteristic
120C2×C4×C67C22⋊Q811yescharacteristic
121C2×C4×C67C2×D811yescharacteristic
122C2×C4×C67C2×Q1611yescharacteristic
123C2×C4×C67#13511yescharacteristic
124C2×C4×C67#13511yescharacteristic
125C22⋊D67—22no—
126C22⋊D67—22no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130D268—21no—
131D268#10712yescomplemented
132C4×D67—22no—
133C4×D67—22no—
134C4×D67—42no—
135C4×D67—41no—
136C4×D67—41no—
137Dic134—21no—
138Dic134#10712yescomplemented
139C8×C67—21no—
140C8×C67—21no—
141C8×C67—21no—
142C8×C67#10412yescomplemented
143C67⋊C8—41no—
144C67⋊C8—41no—
145C67⋊C8—82no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67#38011yescharacteristic
149C22×C67#46611yescharacteristic
150C22×C67#46711yescharacteristic
151C2×D67—22no—
152C2×D67—41no—
153C2×D67—41no—
154C4×C67—21no—
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—84no—
158C4×C67C2×#10411yescharacteristic
159C4×C67#42111yescharacteristic
160C4×C67C2×#10711yescharacteristic
161C4×C67#42711yescharacteristic
162C4×C67#46611yescharacteristic
163C4×C67#46711yescharacteristic
164Dic67—22no—
165Dic67—41no—
166Dic67—41no—
167#8667—671nomaximal, Sylow
168C2×C67—21no—
169C2×C67—21no—
170C2×C67#180611yescharacteristic
171C2×C67#180811yescharacteristic
172C2×C67#181411yescharacteristic
173D67—42no—
174C8×#41—671no—
175#1806—672no—
176C4×#107D6711yescharacteristic, complemented
177#1567—671no—
178#1303—672no—
179C67#866711yescharacteristic, complemented, Sylow
180C4×#41—671no—
181C8×Q8—671no—
182C8×Q8—671no—
183C8×D4—671no—
184C8×D4—671no—
185C8×D4—1341no—
186C8×D4—1341no—
187#431—671no—
188#431—1341no—
189C2×#107—21no—
190#423—672no—
191C2×#104—1342no—
192C2×C4×C8—1341no—
193C2×C4×C8C2×D6711yescharacteristic
194C4×C4⋊C4C2×D6712yes—
195#340—21no—
196#339—1342no—
197#335—1341no—
198#335—1341no—
199C2×#41—671no—
200C2×M16—1342no—
201C22×C8—1341no—
202C22×C8—1341no—
203C22×C8C22⋊D6711yescharacteristic
204C22×C8C22⋊D6711yescharacteristic
205C4×Q8—671no—
206C4×Q8—671no—
207C4×D4—671no—
208C4×D4—671no—
209C4×D4—1341no—
210C4×D4—1341no—
211#117—671no—
212#117—1341no—
213C2×C4⋊C4—22no—
214C2×C42—22no—
215C2×C42—1341no—
216C2×C42C22×D6711yescharacteristic
217#107—24no—
218C4⋊C8—671no—
219C4⋊C8—671no—
220C4⋊C8—1341no—
221C4⋊C8—1341no—
222C4⋊C8—1342no—
223C4⋊C8—1342no—
224#104—2684no—
225C22⋊C8—1341no—
226C22⋊C8—1341no—
227C22⋊C8—1341no—
228C4×C8—1341no—
229C4×C8—1341no—
230C4×C8—1341no—
231C4×C8C4×D6711yescharacteristic
232C4×C8C4×D6711yescharacteristic
233C4×C8D26811yescharacteristic
234C4×C8D26811yescharacteristic
235#95—22no—
236#41—672no—
237#41—672no—
238#41—1341no—
239#41—1341no—
240C2×Q8—671no—
241C2×D4—671no—
242C2×D4—1341no—
243C22×C4—42no—
244C22×C4—1341no—
245C22×C4—1341no—
246C22×C4—1341no—
247C22×C4D4×D6711yescharacteristic
248C22×C4C2×C22⋊D6711yescharacteristic
249C22×C4#1176162711yescharacteristic
250M16—2682no—
251M16—2682no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—2682no—
255C2×C8—2682no—
256C2×C8—2682no—
257C2×C8—1341no—
258C2×C8—1341no—
259C2×C8(C2×D67)⋊C411yescharacteristic
260C2×C8(C2×D67)⋊C411yescharacteristic
261C2×C8(C2×D67)⋊C411yescharacteristic
262C2×C8(C2×D67)⋊C411yescharacteristic
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—671no—
266C4⋊C4—671no—
267C4⋊C4—1341no—
268C4⋊C4—1341no—
269C22⋊C4—1341no—
270C22⋊C4—1341no—
271C22⋊C4—1341no—
272C42—42no—
273C42—42no—
274C42—1341no—
275C42—1341no—
276C42—1341no—
277C42C2×C4×D6711yescharacteristic
278C42C2×D26811yescharacteristic
279C42Q8×D6711yescharacteristic
280C42#1176162911yescharacteristic
281C23—1341no—
282C23#1452401411yescharacteristic
283Q8—672no—
284Q8—1341no—
285D4—672no—
286D4—1342no—
287D4—1342no—
288D4—1341no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—2682no—
296C2×C4—2681no—
297C2×C4—2681no—
298C2×C4—1341no—
299C2×C4—1341no—
300C2×C4—1341no—
301C2×C4—1342no—
302C2×C4—1342no—
303C2×C4C2×(C2×D67)⋊C411yescharacteristic
304C2×C4C4⋊C4×D6711yescharacteristic
305C2×C4#1452400111yescharacteristic
306C2×C4#1452401811yescharacteristic
307C2×C4#1452401911yescharacteristic
308C2×C4#1452402311yescharacteristic
309C2×C4D8×D6711yescharacteristic
310C2×C4Q16×D6711yescharacteristic
311C2×C4#1452406211yescharacteristic
312C2×C4#1452406611yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—2681no—
317C8—2681no—
318C8—5362no—
319C8#1452397112yescomplemented
320C22—21no—
321C22—21no—
322C22—2681no—
323C22—2681no—
324C22—1342no—
325C22#1686325311yescharacteristic
326C22#1686362911yescharacteristic
327C22#1686363411yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—84no—
332C4—2681no—
333C4—2681no—
334C4—1342no—
335C4C2×#1452397111yescharacteristic
336C4#1686341411yescharacteristic
337C4#107×D6711yescharacteristic
338C4#1686343311yescharacteristic
339C4#1686363311yescharacteristic
340C4#1686363811yescharacteristic
341C2—21no—
342C2—21no—
343C2—2682no—
344C2#2256023011yescharacteristic
345C2#2256023511yescharacteristic
346C2#2256026011yescharacteristic
subgroupG/H
centerC2×C4#14524018
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC4×#107×C67C2
socleC22×C67#380
prime radicalC2×#11761628D4
cosocleC2×C42×C67C23
hypercenterC4×#107D67
hypocenterC67#8667
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
3. #22558968C2semidirect
4. #22558112C2semidirect
5. C4×#107×C67C2semidirect
7. C8×Dic134C4semidirect
8. C8×Dic134C4semidirect
9. C8×D268C4semidirect
10. C8×D268C4semidirect
131. D268#107semidirect
138. Dic134#107semidirect
142. C8×C67#104semidirect
176. C4×#107D67semidirect
179. C67#8667semidirect
319. C8#14523971semidirect

Maximal subgroups

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

subgroupindex
C8×#117616282
#225602302
#225589682
#225581122
C4×#107×C672
#866767