Finite Group Databasebeta

← #30753987

Subgroups — #30753987

Order 17152.

356 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14523974C211yesmaximal, characteristic
2#22560076C211yesmaximal, characteristic
3#22558050C211yesmaximal, characteristic
4#22558048C211yesmaximal, characteristic
5#22558043C211yesmaximal, characteristic
6C67⋊#1273C211yesmaximal, characteristic
7C67×#1939C211yesmaximal, characteristic
8C2×#14523974C411yescharacteristic
9C2×#14523974C411yescharacteristic
10C2×C4⋊(C67⋊C8)C2211yescharacteristic
11C2×#14523953—21no—
12C2×C4×C67⋊C8C2211yescharacteristic
13C2×C4×C67⋊C8C2211yescharacteristic
14C4×C4⋊Dic67C2211yescharacteristic
15#16863158C411yescharacteristic
16#16863158C411yescharacteristic
17#16863150—21no—
18C2×C4⋊C8×C67—21no—
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×C4×C8×C67C2211yescharacteristic
21C4×C4⋊C4×C67C2211yescharacteristic
22#336×C67—21no—
23C22×C67⋊C8—21no—
24C22×C67⋊C8—21no—
25C22×C67⋊C8C2×C411yescharacteristic
26C22×C67⋊C8C2×C411yescharacteristic
27C2×C4⋊Dic67C2×C411yescharacteristic
28C2×C4⋊Dic67C2×C411yescharacteristic
29C2×C4×Dic67—21no—
30#14523974—22no—
31#14523974—22no—
32C4⋊(C67⋊C8)—21no—
33C4⋊(C67⋊C8)Q812yes—
34#14523953—42no—
35C4×C67⋊C8—21no—
36C4×C67⋊C8—21no—
37C4×C67⋊C8—21no—
38C4×C67⋊C8—21no—
39C67⋊#95—21no—
40C22×C8×C67—21no—
41C22×C8×C67D411yescharacteristic
42C22×C8×C67D411yescharacteristic
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C42×C67—21no—
46C2×C42×C67C2311yescharacteristic
47C4⋊C8×C67—21no—
48C4⋊C8×C67—21no—
49C4⋊C8×C67—42no—
50C4×C8×C67—21no—
51C4×C8×C67—21no—
52#95×C67—21no—
53C22×Dic67—21no—
54C22×Dic67—21no—
55C2×C67⋊C8—21no—
56C2×C67⋊C8—21no—
57C2×C67⋊C8—22no—
58C2×C67⋊C8—21no—
59C2×C67⋊C8—22no—
60C2×C67⋊C8—21no—
61C2×C67⋊C8—21no—
62C2×C67⋊C8—41no—
63C2×C67⋊C8—41no—
64C2×C67⋊C8—41no—
65C2×C67⋊C8—41no—
66C2×C67⋊C8M1612yes—
67C4⋊Dic67—21no—
68C4⋊Dic67—21no—
69C4⋊Dic67—21no—
70C4⋊Dic67—21no—
71C4×Dic67—41no—
72C4×Dic67—41no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C22×C4×C67#4111yescharacteristic
78C2×C8×C67—21no—
79C2×C8×C67—21no—
80C2×C8×C67—21no—
81C2×C8×C67—21no—
82C2×C8×C67—21no—
83C2×C8×C67—41no—
84C2×C8×C67—41no—
85C2×C8×C67—41no—
86C2×C8×C67SD1612yes—
87C4⋊C4×C67—21no—
88C4⋊C4×C67—21no—
89C4⋊C4×C67—21no—
90C4⋊C4×C67M1612yes—
91C42×C67—41no—
92C42×C67—41no—
93C42×C67C2×Q811yescharacteristic
94C42×C67#4111yescharacteristic
95C42×C67#4111yescharacteristic
96C42×C67#4111yescharacteristic
97C23×C67C4×D411yescharacteristic
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—22no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67#11711yescharacteristic
107C2×C4×C67C4×Q811yescharacteristic
108C2×C4×C67C22⋊Q811yescharacteristic
109C2×C4×C67#12311yescharacteristic
110C2×C4×C67#12511yescharacteristic
111C2×C4×C67#12611yescharacteristic
112C2×C4×C67C2×M1611yescharacteristic
113C2×C4×C67C2×M1611yescharacteristic
114C2×C4×C67#13111yescharacteristic
115C2×C4×C67#13111yescharacteristic
116C2×C4×C67C2×SD1611yescharacteristic
117C2×C4×C67#13511yescharacteristic
118C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C67#13711yescharacteristic
120C2×Dic67—41no—
121C2×Dic67—41no—
122C2×Dic67—41no—
123C2×Dic67—41no—
124C2×Dic67—41no—
125C2×Dic67—41no—
126C8×C67—22no—
127C8×C67—41no—
128C8×C67—81no—
129C67⋊C8—22no—
130C67⋊C8—42no—
131C67⋊C8—41no—
132C67⋊C8—42no—
133C22×C67#38711yescharacteristic
134C22×C67#43511yescharacteristic
135C22×C67#43611yescharacteristic
136C22×C67C4×SD1611yescharacteristic
137C22×C67#44011yescharacteristic
138C22×C67#44311yescharacteristic
139C22×C67#44411yescharacteristic
140C4×C67—21no—
141C4×C67—21no—
142C4×C67—42no—
143C4×C67—81no—
144C4×C67#43211yescharacteristic
145C4×C67#43311yescharacteristic
146C4×C67#44611yescharacteristic
147C4×C67#44611yescharacteristic
148C4×C67#47611yescharacteristic
149C4×C67#47911yescharacteristic
150C4×C67#48211yescharacteristic
151C4×C67#48311yescharacteristic
152Dic67—81no—
153Dic67—81no—
154#9539—671nomaximal, Sylow
155C2×C67#158311yescharacteristic
156C2×C67#158511yescharacteristic
157C2×C67#159011yescharacteristic
158C2×C67#159211yescharacteristic
159C2×C67#191711yescharacteristic
160C2×C67#192011yescharacteristic
161C2×C67#192211yescharacteristic
162#1939D6711yescharacteristic
163C4×#106—671no—
164#1770—671no—
165#1279—671no—
166#1279—671no—
167#1277—671no—
168#1273—671no—
169C67#953911yescharacteristic, complemented, Sylow
170C2×#106—671no—
171C2×#106—671no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—671no—
174C2×C4⋊C8C2×D6711yescharacteristic
175C2×#97—1341no—
176C2×C4×C8—671no—
177C2×C4×C8—671no—
178C2×C4×C8C2×D6711yescharacteristic
179C4×C4⋊C4—671no—
180C4×C4⋊C4C2×D6711yescharacteristic
181#340—671no—
182#340—671no—
183#336—21no—
184#336—1341no—
185C22×C8—21no—
186C22×C8—671no—
187C22×C8—671no—
188C22×C8—1341no—
189C22×C8—1341no—
190C22×C8C22⋊D6711yescharacteristic
191C22×C8C22⋊D6711yescharacteristic
192C2×C4⋊C4—671no—
193C2×C4⋊C4—671no—
194C2×C4⋊C4C4×D6711yescharacteristic
195C2×C4⋊C4C4×D6711yescharacteristic
196C2×C42—21no—
197C2×C42—1341no—
198C2×C42C22×D6711yescharacteristic
199#106—1342no—
200#106—1342no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—42no—
204C4⋊C8—672no—
205C4⋊C8—1341no—
206#97—2682no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—1341no—
210C4×C8—1341no—
211C4×C8—1341no—
212C4×C8—1341no—
213#95—21no—
214#95—1341no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—1341no—
218C22×C4—1341no—
219C22×C4C2×C4×D6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221C22×C4#1176162811yescharacteristic
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—2681no—
231C2×C8—2681no—
232C2×C8—2681no—
233C2×C8—2681no—
234C2×C8—672no—
235C2×C8—1342no—
236C2×C8—1341no—
237C2×C8—1342no—
238C2×C8—1341no—
239C2×C8—1341no—
240C2×C8—1341no—
241C2×C8—1341no—
242C2×C8#1176161512yes—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—1341no—
247C4⋊C4—1341no—
248C4⋊C4—1341no—
249C4⋊C4—1341no—
250C4⋊C4#1176161112yes—
251C42—41no—
252C42—41no—
253C42—2681no—
254C42—2681no—
255C42Q8×D6711yescharacteristic
256C42#1176162711yescharacteristic
257C42#1176162711yescharacteristic
258C42#1176162911yescharacteristic
259C23C4×C22⋊D6711yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—2681no—
269C2×C4—2681no—
270C2×C4—2681no—
271C2×C4—2681no—
272C2×C4—2681no—
273C2×C4—2681no—
274C2×C4#1452400111yescharacteristic
275C2×C4#1452400811yescharacteristic
276C2×C4#1452401911yescharacteristic
277C2×C4#1452402411yescharacteristic
278C2×C4#1452403311yescharacteristic
279C2×C4#1452403811yescharacteristic
280C2×C4M16×D6711yescharacteristic
281C2×C4C2×#1176161111yescharacteristic
282C2×C4#1452404911yescharacteristic
283C2×C4#1452405111yescharacteristic
284C2×C4C2×#1176161511yescharacteristic
285C2×C4#1452406311yescharacteristic
286C2×C4#1452406911yescharacteristic
287C2×C4#1452407311yescharacteristic
288C8—22no—
289C8—41no—
290C8—81no—
291C8—2682no—
292C8—2682no—
293C8—2681no—
294C8—1342no—
295C22#1686327711yescharacteristic
296C22#1686346711yescharacteristic
297C22#1686347211yescharacteristic
298C22C4×#1176161511yescharacteristic
299C22#1686349611yescharacteristic
300C22#1686351111yescharacteristic
301C22#1686351611yescharacteristic
302C4—21no—
303C4—21no—
304C4—42no—
305C4—81no—
306C4—5361no—
307C4—5361no—
308C4#1686345211yescharacteristic
309C4#1686345611yescharacteristic
310C4#1686352311yescharacteristic
311C4#1686352511yescharacteristic
312C4#1686368311yescharacteristic
313C4#1686370011yescharacteristic
314C4#1686371711yescharacteristic
315C4#1686372411yescharacteristic
316C2#2255905211yescharacteristic
317C2#2255906611yescharacteristic
318C2#2255909911yescharacteristic
319C2#2255911111yescharacteristic
320C2#2256077011yescharacteristic
321C2#2256079111yescharacteristic
322C2#2256080511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1939C2
socle, prime radicalC23×C67C4×D4
cosocleC2×C42×C67C23
hypercenter#1939D67
hypocenterC67#9539
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1939

Complements

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

NH ≅ G/Ntype
169. C67#9539semidirect

Maximal subgroups

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

subgroupindex
C4×#145239742
#225600762
#225580502
#225580482
#225580432
C67⋊#12732
C67×#19392
#953967