Finite Group Databasebeta

← #30765169

Subgroups — #30765169

Order 17152.

330 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16863520C211yesmaximal, characteristic
2#22562009C212yesmaximal
3#22558149C211yesmaximal, characteristic
4#22558148C212yesmaximal
5C67×#1322C211yesmaximal, characteristic
6C2×C4×Dic134C411yescharacteristic
7#16863520—22no—
8#16863520—22no—
9C2×Dic67⋊C8C411yescharacteristic
10C2×Dic67⋊C8C2211yescharacteristic
11C2×C4×C67⋊C8C411yescharacteristic
12C2×C8×Dic67C411yescharacteristic
13C2×C8×Dic67C2211yescharacteristic
14Dic67⋊C16—22no—
15C22⋊(C67⋊C16)C2212yes—
16#16863171—22no—
17C2×C4⋊C8×C67C2211yescharacteristic
18C22⋊C16×C67C2212yes—
19C22×Dic134C811yescharacteristic
20C22×C67⋊C8C2×C411yescharacteristic
21C22×C67⋊C8C2×C411yescharacteristic
22C4×Dic134—22no—
23C4×Dic134—22no—
24C2×C4⋊Dic67C811yescharacteristic
25C2×Dic67⋊C4C811yescharacteristic
26C2×Dic67⋊C4C811yescharacteristic
27C2×C4×Dic67C2×C411yescharacteristic
28C2×C4×Dic67C2×C411yescharacteristic
29C2×C67⋊C16—22no—
30Dic67⋊C8—22no—
31Dic67⋊C8—22no—
32C4×C67⋊C8—21no—
33C4×C67⋊C8—21no—
34C8×Dic67—21no—
35C8×Dic67—21no—
36C8×Dic67D412yes—
37C8×Dic67D412yes—
38C22×C8×C67C2×C411yescharacteristic
39C22×C8×C67C2311yescharacteristic
40C2×C42×C67C2×C411yescharacteristic
41C2×C16×C67—22no—
42C4⋊C8×C67—22no—
43C2×Dic134—22no—
44C2×Dic134—22no—
45C2×Dic134—22no—
46C22×Dic67C2×C811yescharacteristic
47C22×Dic67C2×C811yescharacteristic
48C22×Dic67C2×C811yescharacteristic
49C22×Dic67C2×C811yescharacteristic
50C2×C67⋊C8—22no—
51C2×C67⋊C8—21no—
52C2×C67⋊C8—41no—
53C2×C67⋊C8C22⋊C411yescharacteristic
54C2×C67⋊C8C22⋊C411yescharacteristic
55C4⋊Dic67—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58C4×Dic67—21no—
59C4×Dic67—21no—
60C4×Dic67C22⋊C412yes—
61C4×Dic67M1612yes—
62C67⋊C16—42no—
63C22×C4×C67C2×C811yescharacteristic
64C22×C4×C67C2×C811yescharacteristic
65C22×C4×C67C22×C411yescharacteristic
66C2×C8×C67—21no—
67C2×C8×C67—41no—
68C2×C8×C67—41no—
69C2×C8×C67C2×D412yes—
70C42×C67—21no—
71C42×C67—21no—
72C16×C67—42no—
73C23×C67C22×C811yescharacteristic
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—41no—
80C2×C4×C67M3212yes—
81C2×C4×C67C2×C22⋊C411yescharacteristic
82C2×C4×C67C2×M1611yescharacteristic
83C2×Dic67—21no—
84C2×Dic67—21no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—21no—
88C2×Dic67—41no—
89C2×Dic67—41no—
90C2×Dic67C22⋊C812yes—
91C2×Dic67C22⋊C812yes—
92C2×Dic67M3212yes—
93Dic134—28no—
94C8×C67—22no—
95C8×C67—41no—
96C67⋊C8—21no—
97C67⋊C8—41no—
98C67⋊C8#10112yes—
99C22×C67—21no—
100C22×C67—21no—
101C22×C67C2×C22⋊C811yescharacteristic
102C22×C67C2×M3211yescharacteristic
103C22×C67C2×M3211yescharacteristic
104C4×C67—22no—
105C4×C67—21no—
106C4×C67—41no—
107C4×C67#41011yescharacteristic
108C4×C67C2×#10111yescharacteristic
109Dic67—22no—
110Dic67—22no—
111Dic67—22no—
112Dic67—41no—
113#11494—671nomaximal, Sylow
114C2×C67—21no—
115C2×C67—21no—
116C2×C67#146111yescharacteristic
117C2×C67#211211yescharacteristic
118C2×C67#211511yescharacteristic
119C2×C8×Q8—671no—
120#2163—672no—
121#1322—671no—
122#1322—672no—
123#1322D6711yescharacteristic
124C67#1149411yescharacteristic, complemented, Sylow
125C2×C4×Q8—671no—
126C8×Q8—1342no—
127C8×Q8—1342no—
128C2×C4⋊C8—671no—
129C2×C4⋊C8—671no—
130C2×C4⋊C8C2×D6711yescharacteristic
131C2×C4×C8—671no—
132C2×C4×C8—671no—
133C2×C4×C8—671no—
134C4⋊C16—1342no—
135C22⋊C16—672no—
136C22⋊C16C2×D6712yes—
137#346—1342no—
138C22×Q8—671no—
139C22×C8—671no—
140C22×C8—671no—
141C22×C8C4×D6711yescharacteristic
142C22×C8C22×D6711yescharacteristic
143C4×Q8—1342no—
144C4×Q8—1342no—
145C2×C4⋊C4—671no—
146C2×C4⋊C4—671no—
147C2×C4⋊C4—671no—
148C2×C42—671no—
149C2×C42—671no—
150C2×C42C4×D6711yescharacteristic
151C2×C16—22no—
152C2×C16—1342no—
153C4⋊C8—22no—
154C4⋊C8—1342no—
155C4⋊C8—1342no—
156C4×C8—672no—
157C4×C8—672no—
158C4×C8—1341no—
159C4×C8—1341no—
160C4×C8—1341no—
161C4×C8—1341no—
162C2×Q8—1342no—
163C2×Q8—1342no—
164C2×Q8—1342no—
165C22×C4—671no—
166C22×C4—671no—
167C22×C4—671no—
168C22×C4—671no—
169C22×C4C8×D6711yescharacteristic
170C22×C4C8×D6711yescharacteristic
171C22×C4C2×C4×D6711yescharacteristic
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—2681no—
176C2×C8—671no—
177C2×C8—671no—
178C2×C8—1341no—
179C2×C8—1342no—
180C2×C8D4×D6712yes—
181C4⋊C4—1342no—
182C4⋊C4—1342no—
183C4⋊C4—1342no—
184C42—21no—
185C42—21no—
186C42—672no—
187C42—672no—
188C42—1341no—
189C42—1341no—
190C16—42no—
191C16—2682no—
192C23C2×C8×D6711yescharacteristic
193Q8—1348no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—2681no—
201C2×C4—2681no—
202C2×C4—672no—
203C2×C4—672no—
204C2×C4—672no—
205C2×C4—1341no—
206C2×C4—1341no—
207C2×C4—1341no—
208C2×C4—1341no—
209C2×C4—1341no—
210C2×C4#1452398212yes—
211C2×C4C22⋊C4×D6711yescharacteristic
212C2×C4M16×D6711yescharacteristic
213C8—22no—
214C8—41no—
215C8—2681no—
216C8—672no—
217C8—1341no—
218C22—21no—
219C22—21no—
220C22C22⋊C8×D6711yescharacteristic
221C22M32×D6711yescharacteristic
222C22C2×#1452398211yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—2681no—
227C4—1342no—
228C4—1342no—
229C4—1342no—
230C4#1686336811yescharacteristic
231C4#101×D6711yescharacteristic
232C2—21no—
233C2—21no—
234C2#2255853111yescharacteristic
235C2#2256181811yescharacteristic
236C2#2256182811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D67
commutator, prime radicalC23×C67C22×C8
FrattiniC22×C8C22×D67
FittingC67×#1322C2
socleC22×C67C2×C22⋊C8
cosocleC22×C8×C67C23
hypercenter#1322D67
hypocenterC67#11494
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
124. C67#11494semidirect

Maximal subgroups

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

subgroupindex
C2×#168635202
#225620092
#225581492
#225581482
C67×#13222
#1149467