Finite Group Databasebeta

← #30749404

Subgroups — #30749404

Order 17152.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#22563130C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558110C212yesmaximal
4#22558069C212yesmaximal
5C67×#1801C211yesmaximal, characteristic
6C2×Q8×Dic67—21no—
7#16863449C2212yes—
8#16863295—22no—
9#16863295—21no—
10#16863262C412yes—
11#16863262C2211yescharacteristic
12C4×C4⋊Dic67C412yes—
13C4×C4⋊Dic67C2211yescharacteristic
14#16863122—22no—
15C2×C4×Q8×C67C2211yescharacteristic
16#423×C67C2212yes—
17C2×#101×C67—21no—
18#343×C67—21no—
19Q8×Dic67—42no—
20Q8×Dic67—42no—
21C2×C4⋊Dic67—22no—
22C2×C4⋊Dic67—22no—
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67—21no—
25C2×C4×Dic67—22no—
26C2×C4×Dic67—21no—
27C4⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8—22no—
30C67⋊#95—22no—
31C67⋊#95—21no—
32C67⋊#95C2×C411yescharacteristic
33C67⋊#95C2×C412yes—
34C67⋊#95C2×C411yescharacteristic
35C22×Q8×C67D411yescharacteristic
36C2×M16×C67—22no—
37C4×Q8×C67—22no—
38C4×Q8×C67—22no—
39C2×C4⋊C4×C67D412yes—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C42×C67C2×C412yes—
42C2×C42×C67C2311yescharacteristic
43C4⋊C8×C67—22no—
44C4⋊C8×C67—22no—
45#101×C67—42no—
46C22×Dic67—21no—
47C22×Dic67—21no—
48C22×Dic67—22no—
49C2×C67⋊C8—42no—
50C4⋊Dic67—42no—
51C4⋊Dic67—44no—
52C4⋊Dic67—44no—
53C4⋊Dic67—41no—
54C4⋊Dic67—41no—
55C4×Dic67—44no—
56C4×Dic67—41no—
57C4×Dic67—41no—
58C2×Q8×C67—22no—
59C2×Q8×C67—22no—
60C2×Q8×C67D812yes—
61C2×Q8×C67SD1612yes—
62C22×C4×C67C22⋊C412yes—
63C22×C4×C67C22⋊C412yes—
64C22×C4×C67C22×C411yescharacteristic
65C22×C4×C67C2×D411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67M16×C67—42no—
68M16×C67—42no—
69C2×C8×C67—42no—
70C4⋊C4×C67—22no—
71C4⋊C4×C67—24no—
72C42×C67—22no—
73C42×C67—22no—
74C42×C67#4112yes—
75C42×C67#4112yes—
76C23×C67C2×C22⋊C411yescharacteristic
77Q8×C67—24no—
78Q8×C67—24no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—41no—
83C2×C4×C67—41no—
84C2×C4×C67—42no—
85C2×C4×C67#10214yes—
86C2×C4×C67#11711yescharacteristic
87C2×C4×C67#11711yescharacteristic
88C2×C4×C67#12312yes—
89C2×C4×C67#12312yes—
90C2×C4×C67C2×D811yescharacteristic
91C2×C4×C67C2×SD1611yescharacteristic
92C2×C4×C67#13712yes—
93C2×Dic67—22no—
94C2×Dic67—41no—
95C2×Dic67—41no—
96C2×Dic67—41no—
97C2×Dic67—42no—
98C2×Dic67—41no—
99C2×Dic67—42no—
100C2×Dic67—42no—
101C2×Dic67—41no—
102C8×C67—82no—
103C67⋊C8—82no—
104C22×C67—21no—
105C22×C67—21no—
106C22×C67#38111yescharacteristic
107C22×C67C2×#10211yescharacteristic
108C22×C67#41911yescharacteristic
109C4×C67—24no—
110C4×C67—22no—
111C4×C67—21no—
112C4×C67—21no—
113C4×C67—42no—
114C4×C67#48012yes—
115C4×C67#48312yes—
116Dic67—42no—
117Dic67—82no—
118Dic67—81no—
119#8766—671nomaximal, Sylow
120C2×C67—21no—
121C2×C67—21no—
122C2×C67#182311yescharacteristic
123C2×C67#182711yescharacteristic
124C2×C67#183011yescharacteristic
125#2351—671no—
126#1801D6711yescharacteristic
127#1461—671no—
128#1303—672no—
129#1286—672no—
130C67#876611yescharacteristic, complemented, Sylow
131C2×C4×Q8—1341no—
132C2×C4×Q8C2×D6711yescharacteristic
133#431—672no—
134#423C2×D6712yes—
135C2×#101—21no—
136#389—1342no—
137#389—1341no—
138#382—672no—
139#382—671no—
140C4×C4⋊C4—672no—
141C4×C4⋊C4—671no—
142#343—21no—
143#324—1342no—
144C22×Q8C22⋊D6711yescharacteristic
145C2×M16—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—2682no—
149C4×Q8—2682no—
150C2×C4⋊C4—1342no—
151C2×C4⋊C4—1342no—
152C2×C4⋊C4—1341no—
153C2×C4⋊C4—1341no—
154C2×C4⋊C4D26812yes—
155C2×C4⋊C4C22⋊D6711yescharacteristic
156C2×C42—1342no—
157C2×C42—1341no—
158C2×C42C4×D6712yes—
159C2×C42C22×D6711yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—1342no—
163#101—42no—
164C22⋊C8—1342no—
165C4×C8—1342no—
166#95—671no—
167#95—672no—
168#95—671no—
169#95—1342no—
170#95—1341no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1176161312yes—
174C2×Q8#1176161512yes—
175C22×C4—1341no—
176C22×C4—1342no—
177C22×C4—1341no—
178C22×C4(C2×D67)⋊C412yes—
179C22×C4(C2×D67)⋊C412yes—
180C22×C4C2×C4×D6711yescharacteristic
181C22×C4C2×C22⋊D6711yescharacteristic
182C22×C4C2×D26811yescharacteristic
183M16—42no—
184M16—42no—
185C2×C8—42no—
186C2×C8—2682no—
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—2682no—
190C4⋊C4—2684no—
191C4⋊C4—2684no—
192C4⋊C4—2681no—
193C4⋊C4—2681no—
194C42—22no—
195C42—22no—
196C42—2684no—
197C42—2681no—
198C42—2681no—
199C42#1176162712yes—
200C42#1176162712yes—
201C23C2×(C2×D67)⋊C411yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—2681no—
211C2×C4—2681no—
212C2×C4—2681no—
213C2×C4—2682no—
214C2×C4—2681no—
215C2×C4—2682no—
216C2×C4—2682no—
217C2×C4—2681no—
218C2×C4—1342no—
219C2×C4#1452396314yes—
220C2×C4#1452399911yescharacteristic
221C2×C4#1452399911yescharacteristic
222C2×C4#1452402412yes—
223C2×C4#1452402712yes—
224C2×C4C2×#1176161311yescharacteristic
225C2×C4C2×#1176161511yescharacteristic
226C2×C4#1452407312yes—
227C8—82no—
228C8—5362no—
229C22—21no—
230C22—21no—
231C22#1686325711yescharacteristic
232C22C2×#1452396311yescharacteristic
233C22#1686340411yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—2682no—
240C4—5362no—
241C4—5361no—
242C4#1686370712yes—
243C4#1686372412yes—
244C2—21no—
245C2—21no—
246C2#2256029311yescharacteristic
247C2#2256031411yescharacteristic
248C2#2256032111yescharacteristic
subgroupG/H
centerC22#16863257
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1801C2
socleC22×C67#381
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1801D67
hypocenterC67#8766
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
130. C67#8766semidirect

Maximal subgroups

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

subgroupindex
#225631302
#225585332
#225581102
#225580692
C67×#18012
#876667