Finite Group Databasebeta

← #30772094

Subgroups — #30772094

Order 17152.

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

normal subgroups only

This group has 642 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×#2311C211yesmaximal, characteristic
4C2×#14523958—22no—
5C2×#14523958C2214yes—
6#16863259—22no—
7C22×#41×C67C2211yescharacteristic
8C2×#117×C67C411yescharacteristic, complemented
9C22×C4⋊C4×C67C411yescharacteristic, complemented
10#391×C67C412yescomplemented
11#390×C67C2212yes—
12#386×C67—22no—
13#380×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×C67—21no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67—42no—
31C2×C4⋊C4×C67C2×C412yes—
32C2×C22⋊C4×C67—21no—
33C2×C22⋊C4×C67—24no—
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—24no—
55C22×C4×C67—24no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—42no—
59C22×C4×C67—42no—
60C22×C4×C67—82no—
61C22×C4×C67C22⋊C412yes—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C4⋊C4×C67—42no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—42no—
67C22⋊C4×C67—42no—
68C22⋊C4×C67—84no—
69C42×C67—42no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—22no—
73C23×C67—22no—
74C23×C67—41no—
75C23×C67—41no—
76C23×C67—42no—
77C23×C67—42no—
78C23×C67#9912yes—
79C23×C67C2×C22⋊C411yescharacteristic
80C23×C67#12314yes—
81Q8×C67—44no—
82D4×C67—44no—
83D4×C67—48no—
84C2×C4×C67—28no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—22no—
92C2×C4×C67—48no—
93C2×C4×C67—42no—
94C2×C4×C67—44no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—84no—
99C2×C4×C67#9912yes—
100C2×C4×C67#11712yes—
101C2×Dic67—84no—
102C2×Dic67—84no—
103C2×Dic67—84no—
104C22×C67—21no—
105C22×C67—21no—
106C22×C67—24no—
107C22×C67—22no—
108C22×C67—42no—
109C22×C67—44no—
110C22×C67—42no—
111C22×C67—42no—
112C22×C67—41no—
113C22×C67—42no—
114C22×C67—41no—
115C22×C67—82no—
116C22×C67#38111yescharacteristic
117C22×C67C2×#9911yescharacteristic
118C22×C67C2×#9911yescharacteristic
119C4×C67—22no—
120C4×C67—22no—
121C4×C67—42no—
122C4×C67—82no—
123C4×C67—82no—
124Dic67—164no—
125#12859—671nomaximal, Sylow
126C2×C67—21no—
127C2×C67—21no—
128C2×C67—44no—
129C2×C67—42no—
130C2×C67#181911yescharacteristic
131C2×C67#212811yescharacteristic
132C2×C67#212911yescharacteristic
133#2311D6711yescharacteristic
134#1782—672no—
135#1393—674no—
136C67#1285911yescharacteristic, complemented, Sylow
137C22×#41C2×D6711yescharacteristic
138C2×#117Dic6711yescharacteristic, complemented
139C22×C4⋊C4Dic6711yescharacteristic, complemented
140C2×#99—674no—
141C2×#99—1342no—
142#391Dic6712yescomplemented
143#390C2×D6712yes—
144#386—22no—
145#381—1342no—
146#380—22no—
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8C2×Dic6711yescharacteristic
151C22×D4C22⋊D6712yes—
152C22×D4C22×D6711yescharacteristic
153C23×C4—21no—
154C23×C4C2×Dic6711yescharacteristic
155C23×C4C22⋊D6712yes—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C2×Dic6712yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—1344no—
164C2×C42C2×Dic6712yes—
165#99—2684no—
166#99—1348no—
167#95—24no—
168#95—2684no—
169C24C22⋊Dic6711yescharacteristic
170C24C2×C22⋊D6712yes—
171#41—416no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#1176162814yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—2684no—
192C22×C4C22⋊Dic6712yes—
193C22×C4C22⋊Dic6711yescharacteristic
194C22×C4C22×Dic6711yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—84no—
200C22⋊C4—2684no—
201C22⋊C4—2684no—
202C42—42no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23#1452395812yes—
212C23C2×C22⋊Dic6711yescharacteristic
213C23#1452402614yes—
214Q8—44no—
215D4—44no—
216D4—48no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4—5364no—
233C2×C4—5364no—
234C2×C4—5364no—
235C2×C4#1452395812yes—
236C2×C4#1452400212yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—82no—
249C22#1686325911yescharacteristic
250C22C2×#1452395811yescharacteristic
251C22C2×#1452395811yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—10724no—
258C2—21no—
259C2—21no—
260C2—44no—
261C2—42no—
262C2#2256028311yescharacteristic
263C2#2256188111yescharacteristic
264C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×D4C22×D67
FittingC67×#2311C2
socleC22×C67C2×#99
prime radicalC22×#41×C67C22
cosocleC22×D4×C67C23
hypercenter#2311D67
hypocenterC67#12859
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #22558334C2semidirect
8. C2×#117×C67C4semidirect
9. C22×C4⋊C4×C67C4semidirect
10. #391×C67C4semidirect
136. C67#12859semidirect
138. C2×#117Dic67semidirect
139. C22×C4⋊C4Dic67semidirect
142. #391Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601272
#225583342
C67×#23112
#1285967