Finite Group Databasebeta

← #30753671

Subgroups — #30753671

Order 17152.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#22569233C211yesmaximal, characteristic
2#22560201C211yesmaximal, characteristic
3#22558638C212yesmaximal, complemented
4#22558067C212yesmaximal, complemented
5C67×#2580C211yesmaximal, characteristic
6D4×C67⋊C8—24no—
7#16863449C2212yes—
8#16863420C2212yes—
9C2×#14523960—21no—
10C2×C22⋊(C67⋊C8)—22no—
11#16863165—21no—
12Q8⋊(C67⋊C8)—24no—
13C2×C4×Q8×C67C2211yescharacteristic
14C2×C4×D4×C67C2211yescharacteristic
15#399×C67C412yes—
16#393×C67C412yes—
17#390×C67—21no—
18#389×C67—21no—
19#385×C67—22no—
20#382×C67C2211yescharacteristic
21C2×#11761612—22no—
22C22×C67⋊C8—24no—
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26#14523960—42no—
27C22⋊(C67⋊C8)—24no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8D414yes—
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67D411yescharacteristic
32C23×C4×C67D412yes—
33C4×Q8×C67—22no—
34C4×Q8×C67—22no—
35C4×D4×C67—22no—
36C4×D4×C67—22no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67C2×C412yes—
45C2×C42×C67—21no—
46C2×C42×C67C2311yescharacteristic
47#95×C67—22no—
48#95×C67—21no—
49#95×C67C2×C412yes—
50#11761612—42no—
51#11761612—42no—
52C2×C67⋊C8—24no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—84no—
55C24×C67C22⋊C412yes—
56C2×Q8×C67—22no—
57C2×Q8×C67—42no—
58C2×D4×C67—22no—
59C2×D4×C67—22no—
60C2×D4×C67SD1612yes—
61C2×D4×C67Q1612yes—
62C22×C4×C67—21no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67—82no—
68C22×C4×C67C22⋊C412yes—
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—22no—
73C4⋊C4×C67—22no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—82no—
78C22⋊C4×C67—24no—
79C22⋊C4×C67—44no—
80C22⋊C4×C67—82no—
81C42×C67—42no—
82C42×C67C2×D412yes—
83C42×C67#4112yes—
84C23×C67—22no—
85C23×C67—22no—
86C23×C67—42no—
87C23×C67—42no—
88C23×C67#10314yes—
89C23×C67C2×C22⋊C411yescharacteristic
90Q8×C67—24no—
91Q8×C67—42no—
92D4×C67—24no—
93D4×C67—24no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—41no—
97C2×C4×C67—41no—
98C2×C4×C67—44no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—41no—
104C2×C4×C67—42no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—42no—
108C2×C4×C67—82no—
109C2×C4×C67C4×D412yes—
110C2×C4×C67#12011yescharacteristic
111C2×C4×C67#12112yes—
112C2×C4×C67#12311yescharacteristic
113C2×C4×C67C2×SD1611yescharacteristic
114C2×C4×C67C2×Q1611yescharacteristic
115C2×C4×C67#13512yes—
116C67⋊C8—44no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67—24no—
121C22×C67—44no—
122C22×C67—42no—
123C22×C67—42no—
124C22×C67—82no—
125C22×C67#38611yescharacteristic
126C22×C67C2×#10311yescharacteristic
127C22×C67#41611yescharacteristic
128C4×C67—22no—
129C4×C67—21no—
130C4×C67—21no—
131C4×C67—42no—
132C4×C67—42no—
133C4×C67—82no—
134C4×C67—81no—
135C4×C67#46512yes—
136C4×C67#46712yes—
137#9483—671nomaximal, Sylow
138C2×C67—21no—
139C2×C67—21no—
140C2×C67—44no—
141C2×C67#189111yescharacteristic
142C2×C67#189211yescharacteristic
143C2×C67#191111yescharacteristic
144#2972—671no—
145#2580D6711yescharacteristic
146#1800—671no—
147#1487—672no—
148#1286—672no—
149C67#948311yescharacteristic, complemented, Sylow
150C2×C4×Q8C2×D6711yescharacteristic
151C2×C4×D4C2×D6711yescharacteristic
152C8×D4—1344no—
153#431—672no—
154#423—672no—
155C2×#100—1341no—
156C2×C22⋊C8—1342no—
157#399Dic6712yes—
158#393Dic6712yes—
159#390—21no—
160#389—21no—
161#385—22no—
162#382C2×D6711yescharacteristic
163#343—1341no—
164Q8⋊C8—1344no—
165C22×Q8C2×Dic6711yescharacteristic
166C22×D4C22⋊D6711yescharacteristic
167C23×C4C22⋊D6712yes—
168C2×M16—1342no—
169C22×C8—1344no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic6711yescharacteristic
178C2×C4⋊C4C22⋊D6711yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic6712yes—
182C2×C42—21no—
183C2×C42C22×D6711yescharacteristic
184C4⋊C8—1342no—
185C4⋊C8—1342no—
186C4⋊C8—1342no—
187#100—2682no—
188C22⋊C8—1344no—
189C22⋊C8—1342no—
190C4×C8—674no—
191#95—22no—
192#95—21no—
193#95C2×Dic6712yes—
194C24C22⋊Dic6712yes—
195C2×Q8—22no—
196C2×Q8—42no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1176161512yes—
200C2×D4#1176161812yes—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—82no—
207C22×C4C22⋊Dic6712yes—
208C22×C4C22×Dic6711yescharacteristic
209C22×C4C2×C22⋊D6711yescharacteristic
210C22×C4C2×C22⋊D6711yescharacteristic
211M16—2682no—
212M16—2682no—
213C2×C8—2682no—
214C2×C8—5364no—
215C2×C8—1344no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—42no—
226C42D4×D6712yes—
227C42#1176162712yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1452396714yes—
233C23C2×C22⋊Dic6711yescharacteristic
234Q8—24no—
235Q8—42no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4D4×Dic6712yes—
254C2×C4#1452401111yescharacteristic
255C2×C4#1452401512yes—
256C2×C4#1452402411yescharacteristic
257C2×C4C2×#1176161511yescharacteristic
258C2×C4C2×#1176161811yescharacteristic
259C2×C4#1452406312yes—
260C8—2684no—
261C8—5362no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1686327611yescharacteristic
270C22C2×#1452396711yescharacteristic
271C22#1686339211yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#1686362712yes—
280C4#1686363512yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2256062811yescharacteristic
285C2#2256063311yescharacteristic
286C2#2256074211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2580C2
socleC22×C67#386
prime radicalC22×D4×C67D4
cosocleC2×C42×C67C23
hypercenter#2580D67
hypocenterC67#9483
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #22558638C2semidirect
4. #22558067C2semidirect
149. C67#9483semidirect

Maximal subgroups

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

subgroupindex
#225692332
#225602012
#225586382
#225580672
C67×#25802
#948367