Finite Group Databasebeta

← #30749845

Subgroups — #30749845

Order 17152.

458 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569250C211yesmaximal, characteristic
2#22560414C212yesmaximal, complemented
3#22558527C211yesmaximal, characteristic
4#22558097C212yesmaximal, complemented
5C67×#1870C211yesmaximal, characteristic
6#16863474—22no—
7#16863474—22no—
8#16863449—21no—
9C2×#14523978—22no—
10#16863360—21no—
11#16863360C2212yes—
12C2×C22⋊(C67⋊C8)C2212yes—
13#16863160—22no—
14#16863122C412yes—
15#16863120C412yes—
16C22×M16×C67C2211yescharacteristic
17C2×C4×D4×C67C2211yescharacteristic
18C2×C22⋊C8×C67—21no—
19C2×C22⋊C8×C67C2211yescharacteristic
20#336×C67—21no—
21C2×#11761612—21no—
22C2×#11761612—21no—
23C2×#11761612D412yes—
24C2×#11761612D412yes—
25C22×C67⋊C8—22no—
26#14523978—44no—
27C4⋊(C67⋊C8)—21no—
28C4⋊(C67⋊C8)—21no—
29C22⋊(C67⋊C8)—22no—
30C22⋊(C67⋊C8)—22no—
31C22⋊(C67⋊C8)—22no—
32C22⋊(C67⋊C8)—22no—
33C22⋊(C67⋊C8)C2×C414yes—
34C4×C67⋊C8—22no—
35C22×D4×C67Q811yescharacteristic
36C23×C4×C67D411yescharacteristic
37C23×C4×C67C2311yescharacteristic
38C2×M16×C67—22no—
39C2×M16×C67—42no—
40C22×C8×C67—21no—
41C22×C8×C67—21no—
42C4×D4×C67—22no—
43C4×D4×C67—22no—
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67C2×C411yescharacteristic
46C2×C22⋊C4×C67Q811yescharacteristic
47C2×C42×C67D411yescharacteristic
48C22⋊C8×C67—22no—
49C22⋊C8×C67—42no—
50#11761612—42no—
51#11761612—42no—
52#11761612—81no—
53#11761612—81no—
54C2×C67⋊C8—24no—
55C2×C67⋊C8—22no—
56C2×C67⋊C8—22no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C24×C67C4⋊C411yescharacteristic
60C24×C67C2×Q811yescharacteristic
61C2×D4×C67—22no—
62C2×D4×C67—22no—
63C2×D4×C67—41no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—41no—
67C22×C4×C67—41no—
68C22×C4×C67—41no—
69C22×C4×C67—41no—
70C22×C4×C67C4⋊C411yescharacteristic
71C22×C4×C67C4⋊C411yescharacteristic
72C22×C4×C67C4⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D412yes—
75C22×C4×C67C2×D411yescharacteristic
76C22×C4×C67#4112yes—
77M16×C67—24no—
78M16×C67—42no—
79C2×C8×C67—22no—
80C2×C8×C67—41no—
81C2×C8×C67—41no—
82C2×C8×C67—41no—
83C2×C8×C67—41no—
84C2×C8×C67—41no—
85C4⋊C4×C67—22no—
86C22⋊C4×C67—22no—
87C22⋊C4×C67—22no—
88C42×C67—21no—
89C42×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67C2×C4⋊C411yescharacteristic
100C23×C67C22⋊Q812yes—
101D4×C67—44no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—22no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—41no—
111C2×C4×C67—42no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—81no—
117C2×C4×C67#10812yescomplemented
118C2×C4×C67C4×D412yes—
119C2×C4×C67#12011yescharacteristic
120C2×C4×C67#12311yescharacteristic
121C8×C67—42no—
122C8×C67—81no—
123C67⋊C8—44no—
124C67⋊C8—42no—
125C67⋊C8—42no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—22no—
129C22×C67—22no—
130C22×C67—22no—
131C22×C67—42no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—81no—
136C22×C67—81no—
137C22×C67—81no—
138C22×C67#38511yescharacteristic
139C22×C67C2×#10811yescharacteristic
140C22×C67#43011yescharacteristic
141C4×C67—22no—
142C4×C67—22no—
143C4×C67—21no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67B(4,𝔽2)11yescharacteristic
147C4×C67#45811yescharacteristic
148#8860—671nomaximal, Sylow
149C2×C67—21no—
150C2×C67—21no—
151C2×C67—42no—
152C2×C67—81no—
153C2×C67#185511yescharacteristic
154C2×C67#185811yescharacteristic
155C2×C67#186711yescharacteristic
156#2975—671no—
157#1870D6711yescharacteristic
158#1855—672no—
159#1459—671no—
160#1298—672no—
161C67#886011yescharacteristic, complemented, Sylow
162C22×M16C2×D6711yescharacteristic
163C2×C4×D4C2×D6711yescharacteristic
164#436—1342no—
165#436—1342no—
166#431—1341no—
167C2×#108—1342no—
168#407—672no—
169#407—1341no—
170C2×C22⋊C8—21no—
171C2×C22⋊C8—672no—
172C2×C22⋊C8C2×D6711yescharacteristic
173#341—1342no—
174#336—21no—
175#324—672no—
176#323—672no—
177C22×D4Dic13411yescharacteristic
178C23×C4C22⋊D6711yescharacteristic
179C23×C4C22×D6711yescharacteristic
180C2×M16—22no—
181C2×M16—42no—
182C2×M16—672no—
183C2×M16—672no—
184C2×M16—1341no—
185C2×M16—1341no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—1342no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4C4×D6711yescharacteristic
192C2×C22⋊C4Dic13411yescharacteristic
193C2×C22⋊C4C4×D6711yescharacteristic
194C2×C42C22⋊D6711yescharacteristic
195#108—2684no—
196C4⋊C8—1341no—
197C4⋊C8—1341no—
198C22⋊C8—22no—
199C22⋊C8—42no—
200C22⋊C8—674no—
201C22⋊C8—1342no—
202C22⋊C8—1342no—
203C22⋊C8—1342no—
204C22⋊C8—1342no—
205C4×C8—1342no—
206C24Dic67⋊C411yescharacteristic
207C24C2×Dic13411yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4Dic67⋊C411yescharacteristic
218C22×C4Dic67⋊C411yescharacteristic
219C22×C4Dic67⋊C411yescharacteristic
220C22×C4C2×C4×D6711yescharacteristic
221C22×C4D4×D6712yes—
222C22×C4C2×C22⋊D6711yescharacteristic
223C22×C4#1176162712yes—
224M16—24no—
225M16—42no—
226M16—2682no—
227M16—2682no—
228M16—5361no—
229M16—5361no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—2682no—
237C2×C8—2682no—
238C2×C8—1344no—
239C2×C8—1342no—
240C2×C8—1342no—
241C4⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C2×Dic67⋊C411yescharacteristic
256C23#1452402012yes—
257D4—44no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4#1452397812yescomplemented
274C2×C4(C22⋊D67)⋊C412yes—
275C2×C4#1452401111yescharacteristic
276C2×C4#1452402411yescharacteristic
277C8—42no—
278C8—81no—
279C8—2684no—
280C8—2682no—
281C8—2682no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#1686326811yescharacteristic
295C22C2×#1452397811yescharacteristic
296C22#1686344411yescharacteristic
297C4—22no—
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4#1686358211yescharacteristic
303C4#1686358511yescharacteristic
304C2—21no—
305C2—21no—
306C2—42no—
307C2—81no—
308C2#2256041311yescharacteristic
309C2#2256042811yescharacteristic
310C2#2256048911yescharacteristic
subgroupG/H
centerC2×C4#14524011
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#1870C2
socleC22×C67#385
prime radicalC22×D4×C67Q8
cosocleC23×C4×C67C23
hypercenter#1870D67
hypocenterC67#8860
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. #22560414C2semidirect
4. #22558097C2semidirect
117. C2×C4×C67#108semidirect
161. C67#8860semidirect
273. C2×C4#14523978semidirect

Maximal subgroups

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

subgroupindex
#225692502
#225604142
#225585272
#225580972
C67×#18702
#886067