Finite Group Databasebeta

← #30772088

Subgroups — #30772088

Order 17152.

670 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560127C212yesmaximal
2#22558334C214yesmaximal, complemented
3C67×#2297C211yesmaximal, characteristic
4C2×#14523958—22no—
5C2×#14523958C2214yes—
6#16863259—22no—
7C22×#41×C67C2211yescharacteristic
8C2×#117×C67C411yescharacteristic, complemented
9C22×C42×C67C411yescharacteristic, complemented
10#391×C67C412yescomplemented
11#390×C67C2212yes—
12#386×C67—22no—
13C4×C22⋊C4×C67—22no—
14C2×C22⋊Dic67—24no—
15#14523958—28no—
16#14523958—44no—
17C67⋊#95—44no—
18C2×#41×C67—24no—
19C2×#41×C67—42no—
20C2×#41×C67—42no—
21C22×Q8×C67C2×C411yescharacteristic
22C22×D4×C67D412yes—
23C22×D4×C67C2311yescharacteristic
24C23×C4×C67—21no—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67D412yes—
27#117×C67—42no—
28C2×C4⋊C4×C67C2×C412yes—
29C2×C22⋊C4×C67—21no—
30C2×C22⋊C4×C67—24no—
31C2×C42×C67—21no—
32C2×C42×C67—21no—
33C2×C42×C67—42no—
34C2×C42×C67C2×C412yes—
35#95×C67—24no—
36C22×Dic67—44no—
37C22⋊Dic67—44no—
38C22⋊Dic67—44no—
39C24×C67C22⋊C411yescharacteristic
40C24×C67C2×D412yes—
41#41×C67—416no—
42C2×Q8×C67—22no—
43C2×Q8×C67—41no—
44C2×Q8×C67—41no—
45C2×D4×C67—24no—
46C2×D4×C67—24no—
47C2×D4×C67—42no—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C2×D4×C67#4114yes—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—22no—
55C22×C4×C67—24no—
56C22×C4×C67—24no—
57C22×C4×C67—22no—
58C22×C4×C67—42no—
59C22×C4×C67—44no—
60C22×C4×C67—42no—
61C22×C4×C67C22⋊C412yes—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C4⋊C4×C67—42no—
65C22⋊C4×C67—48no—
66C22⋊C4×C67—42no—
67C42×C67—42no—
68C42×C67—42no—
69C42×C67—42no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—22no—
73C23×C67—22no—
74C23×C67—42no—
75C23×C67—44no—
76C23×C67#9912yes—
77C23×C67C2×C22⋊C411yescharacteristic
78C23×C67#12314yes—
79Q8×C67—44no—
80D4×C67—44no—
81D4×C67—48no—
82C2×C4×C67—28no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—24no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—48no—
92C2×C4×C67—42no—
93C2×C4×C67—48no—
94C2×C4×C67—42no—
95C2×C4×C67—44no—
96C2×C4×C67—42no—
97C2×C4×C67#9912yes—
98C2×C4×C67#11712yes—
99C2×Dic67—84no—
100C2×Dic67—88no—
101C22×C67—22no—
102C22×C67—24no—
103C22×C67—22no—
104C22×C67—42no—
105C22×C67—44no—
106C22×C67—42no—
107C22×C67—44no—
108C22×C67—42no—
109C22×C67—82no—
110C22×C67#38111yescharacteristic
111C22×C67C2×#9911yescharacteristic
112C22×C67C2×#9911yescharacteristic
113C4×C67—22no—
114C4×C67—22no—
115C4×C67—44no—
116C4×C67—42no—
117C4×C67—82no—
118Dic67—164no—
119#12857—671nomaximal, Sylow
120C2×C67—21no—
121C2×C67—21no—
122C2×C67—44no—
123C2×C67—42no—
124C2×C67#181911yescharacteristic
125C2×C67#212612yes—
126#2297D6711yescharacteristic
127#1782—672no—
128#1393—674no—
129C67#1285711yescharacteristic, complemented, Sylow
130C22×#41C2×D6711yescharacteristic
131C2×#117Dic6711yescharacteristic, complemented
132C22×C42Dic6711yescharacteristic, complemented
133C2×#99—674no—
134C2×#99—1342no—
135#391Dic6712yescomplemented
136#390C2×D6712yes—
137#386—22no—
138#381—1342no—
139C4×C22⋊C4—22no—
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8C2×Dic6711yescharacteristic
144C22×D4C22⋊D6712yes—
145C22×D4C22×D6711yescharacteristic
146C23×C4—21no—
147C23×C4C2×Dic6711yescharacteristic
148C23×C4C22⋊D6712yes—
149#117—42no—
150C2×C4⋊C4C2×Dic6712yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—1344no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C2×Dic6712yes—
158#99—2684no—
159#99—1348no—
160#95—24no—
161#95—2684no—
162C24C22⋊Dic6711yescharacteristic
163C24C2×C22⋊D6712yes—
164#41—416no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4#1176162814yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—2684no—
185C22×C4C22⋊Dic6712yes—
186C22×C4C22⋊Dic6711yescharacteristic
187C22×C4C22×Dic6711yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—2684no—
192C22⋊C4—2684no—
193C42—42no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—42no—
201C23—44no—
202C23#1452395812yes—
203C23C2×C22⋊Dic6711yescharacteristic
204C23#1452402614yes—
205Q8—44no—
206D4—44no—
207D4—48no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—5364no—
224C2×C4—5368no—
225C2×C4#1452395812yes—
226C2×C4#1452400212yes—
227C22—22no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22#1686325911yescharacteristic
237C22C2×#1452395811yescharacteristic
238C22C2×#1452395811yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—42no—
243C4—82no—
244C4—10724no—
245C2—21no—
246C2—21no—
247C2—44no—
248C2—42no—
249C2#2256028311yescharacteristic
250C2#2256187412yes—
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×D4C22×D67
FittingC67×#2297C2
socleC22×C67C2×#99
prime radicalC22×#41×C67C22
cosocleC22×D4×C67C23
hypercenter#2297D67
hypocenterC67#12857
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #22558334C2semidirect
8. C2×#117×C67C4semidirect
9. C22×C42×C67C4semidirect
10. #391×C67C4semidirect
129. C67#12857semidirect
131. C2×#117Dic67semidirect
132. C22×C42Dic67semidirect
135. #391Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601272
#225583342
C67×#22972
#1285767