Finite Group Databasebeta

← #30772103

Subgroups — #30772103

Order 17152.

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

normal subgroups only

This group has 566 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#22560133C212yesmaximal
2#22558336C214yesmaximal, complemented
3C67×#2310C211yesmaximal, characteristic
4C2×#14523962—22no—
5C2×#14523962C2214yes—
6#16863360—22no—
7C23×Q8×C67C2211yescharacteristic
8C2×#117×C67C411yescharacteristic
9C2×#117×C67C411yescharacteristic
10#391×C67—22no—
11#391×C67C412yes—
12#391×C67C2212yes—
13#381×C67—22no—
14C2×#11761612—24no—
15#14523962—28no—
16#14523962—44no—
17C22⋊(C67⋊C8)—44no—
18C22×Q8×C67—24no—
19C22×Q8×C67—42no—
20C22×Q8×C67—42no—
21C22×Q8×C67C2×C411yescharacteristic
22C22×Q8×C67D412yes—
23C22×Q8×C67C2311yescharacteristic
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67D412yes—
26#117×C67—42no—
27#117×C67—42no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67C2×C412yes—
30C2×C22⋊C4×C67—21no—
31C2×C22⋊C4×C67—21no—
32C2×C42×C67—21no—
33C2×C42×C67C2×C412yes—
34#95×C67—24no—
35#95×C67—24no—
36#11761612—44no—
37#11761612—44no—
38C2×C67⋊C8—44no—
39C24×C67C22⋊C411yescharacteristic
40C2×Q8×C67—24no—
41C2×Q8×C67—22no—
42C2×Q8×C67—24no—
43C2×Q8×C67—41no—
44C2×Q8×C67—41no—
45C2×Q8×C67—42no—
46C2×Q8×C67—41no—
47C2×Q8×C67—41no—
48C2×Q8×C67—416no—
49C2×Q8×C67#4114yes—
50C22×C4×C67—22no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—24no—
54C22×C4×C67—24no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—42no—
58C22×C4×C67C22⋊C412yes—
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67C22×C411yescharacteristic
61C22×C4×C67C2×D412yes—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—42no—
64C22⋊C4×C67—42no—
65C22⋊C4×C67—42no—
66C42×C67—42no—
67C42×C67—42no—
68C23×C67—21no—
69C23×C67—21no—
70C23×C67—42no—
71C23×C67#9912yes—
72C23×C67C2×C22⋊C411yescharacteristic
73Q8×C67—44no—
74Q8×C67—48no—
75Q8×C67—44no—
76C2×C4×C67—28no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—48no—
87C2×C4×C67—44no—
88C2×C4×C67—44no—
89C2×C4×C67—42no—
90C2×C4×C67—42no—
91C2×C4×C67#9912yes—
92C2×C4×C67#11712yes—
93C2×C4×C67#12314yes—
94C67⋊C8—84no—
95C22×C67—22no—
96C22×C67—22no—
97C22×C67—24no—
98C22×C67—42no—
99C22×C67—42no—
100C22×C67#38111yescharacteristic
101C22×C67C2×#9911yescharacteristic
102C22×C67C2×#9911yescharacteristic
103C4×C67—24no—
104C4×C67—22no—
105C4×C67—22no—
106C4×C67—42no—
107C4×C67—82no—
108C4×C67—82no—
109#12862—671nomaximal, Sylow
110C2×C67—21no—
111C2×C67—21no—
112C2×C67—42no—
113C2×C67#181911yescharacteristic
114C2×C67#213312yes—
115#2310D6711yescharacteristic
116#1784—672no—
117#1394—674no—
118C67#1286211yescharacteristic, complemented, Sylow
119C23×Q8C2×D6711yescharacteristic
120C2×#117Dic6711yescharacteristic
121C2×#117Dic6711yescharacteristic
122C2×#101—674no—
123C2×#101—1342no—
124#407—1342no—
125#391—22no—
126#391Dic6712yes—
127#391C2×D6712yes—
128#381—22no—
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C2×Dic6711yescharacteristic
133C22×Q8C22⋊D6712yes—
134C22×Q8C22×D6711yescharacteristic
135C23×C4C2×Dic6711yescharacteristic
136C23×C4C22⋊D6712yes—
137C2×M16—1344no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C2×Dic6712yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42C2×Dic6712yes—
146#101—2684no—
147#101—1348no—
148C22⋊C8—2684no—
149#95—24no—
150#95—24no—
151C24C22⋊Dic6711yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—416no—
161C2×Q8#1176162814yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4C22⋊Dic6712yes—
171C22×C4C22⋊Dic6711yescharacteristic
172C22×C4C22×Dic6711yescharacteristic
173C22×C4C2×C22⋊D6712yes—
174M16—2684no—
175M16—2684no—
176C2×C8—2684no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—42no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1452395812yes—
187C23C2×C22⋊Dic6711yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—44no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1452395812yes—
207C2×C4#1452400212yes—
208C2×C4#1452402614yes—
209C8—5364no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1686325911yescharacteristic
216C22C2×#1452395811yescharacteristic
217C22C2×#1452395811yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4—82no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#2256028311yescharacteristic
228C2#2256189812yes—
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2310C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC22×Q8×C67C23
hypercenter#2310D67
hypocenterC67#12862
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #22558336C2semidirect
118. C67#12862semidirect

Maximal subgroups

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

subgroupindex
#225601332
#225583362
C67×#23102
#1286267