Finite Group Databasebeta

← #30765129

Subgroups — #30765129

Order 17152.

392 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 119 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#22558533C212yesmaximal
2#22558153C212yesmaximal
3#22558143C212yesmaximal
4C67×#2327C211yesmaximal, characteristic
5#16863449—22no—
6#16863122—22no—
7#16863122C2214yes—
8C2×C4⋊Q8×C67C411yescharacteristic
9C2×C4×Q8×C67C411yescharacteristic
10C2×C4×Q8×C67C2211yescharacteristic
11#391×C67C411yescharacteristic
12#391×C67C2211yescharacteristic
13#387×C67—22no—
14#387×C67—22no—
15#382×C67C2211yescharacteristic
16#376×C67C411yescharacteristic
17C4⋊(C67⋊C8)—42no—
18C22⋊(C67⋊C8)—24no—
19C4×C67⋊C8—42no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×Q8×C67C2×C411yescharacteristic
22C4⋊Q8×C67—22no—
23C4⋊Q8×C67C814yescomplemented
24C4×Q8×C67—42no—
25C4×Q8×C67—42no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67C2×C411yescharacteristic
29C2×C4⋊C4×C67D412yes—
30C2×C4⋊C4×C67C2311yescharacteristic
31C2×C42×C67—21no—
32C2×C42×C67C2×C411yescharacteristic
33C2×C42×C67C2×C411yescharacteristic
34C2×C42×C67C2×C411yescharacteristic
35C2×C42×C67D412yes—
36#95×C67—21no—
37#95×C67—21no—
38#95×C67—22no—
39#95×C67—22no—
40#95×C67—22no—
41C2×C67⋊C8—44no—
42C2×Q8×C67—21no—
43C2×Q8×C67—41no—
44C2×Q8×C67—41no—
45C2×Q8×C67—41no—
46C2×Q8×C67—41no—
47C2×Q8×C67C2×C812yes—
48C2×Q8×C67C2×C812yes—
49C2×Q8×C67M1612yes—
50C22×C4×C67—21no—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67C22⋊C412yes—
55C22×C4×C67C22⋊C411yescharacteristic
56C22×C4×C67C22⋊C411yescharacteristic
57C22×C4×C67C22×C411yescharacteristic
58C22×C4×C67C2×D412yes—
59C4⋊C4×C67—28no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—41no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—42no—
64C42×C67—22no—
65C42×C67—22no—
66C42×C67—22no—
67C42×C67—21no—
68C42×C67—41no—
69C42×C67—42no—
70C42×C67C2×C812yes—
71C23×C67C2×C22⋊C411yescharacteristic
72Q8×C67—42no—
73Q8×C67—42no—
74Q8×C67—42no—
75Q8×C67—42no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67—41no—
85C2×C4×C67—41no—
86C2×C4×C67—41no—
87C2×C4×C67—41no—
88C2×C4×C67—41no—
89C2×C4×C67—41no—
90C2×C4×C67—42no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—42no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67C22⋊C814yes—
97C2×C4×C67#9912yes—
98C2×C4×C67C22×C811yescharacteristic
99C2×C4×C67C2×M1611yescharacteristic
100C67⋊C8—84no—
101C22×C67—21no—
102C22×C67—21no—
103C22×C67C2×C22⋊C811yescharacteristic
104C22×C67C2×#9911yescharacteristic
105C22×C67#41311yescharacteristic
106C4×C67—22no—
107C4×C67—24no—
108C4×C67—21no—
109C4×C67—21no—
110C4×C67—42no—
111C4×C67—42no—
112C4×C67—41no—
113C4×C67—42no—
114C4×C67—81no—
115C4×C67—81no—
116#11482—671nomaximal, Sylow
117C2×C67—21no—
118C2×C67—21no—
119C2×C67#145911yescharacteristic
120C2×C67#212911yescharacteristic
121C2×C67#213311yescharacteristic
122#2327D6711yescharacteristic
123#1461—672no—
124#1324—672no—
125#1319—672no—
126C67#1148211yescharacteristic, complemented, Sylow
127C2×C4⋊Q8Dic6711yescharacteristic
128C2×C4×Q8Dic6711yescharacteristic
129C2×C4×Q8C2×D6711yescharacteristic
130#431—1342no—
131#391Dic6711yescharacteristic
132#391C2×D6711yescharacteristic
133#387—22no—
134#387—22no—
135#382C2×D6711yescharacteristic
136#376Dic6711yescharacteristic
137#324—674no—
138#324—1342no—
139C22×Q8C2×Dic6711yescharacteristic
140C22×Q8C2×Dic6711yescharacteristic
141C4⋊Q8—22no—
142C4⋊Q8C67⋊C814yescomplemented
143C4×Q8—42no—
144C4×Q8—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic6711yescharacteristic
148C2×C4⋊C4C22⋊D6712yes—
149C2×C4⋊C4C22×D6711yescharacteristic
150C2×C42—21no—
151C2×C42C2×Dic6711yescharacteristic
152C2×C42C2×Dic6711yescharacteristic
153C2×C42C2×Dic6711yescharacteristic
154C2×C42C22⋊D6712yes—
155C4⋊C8—2682no—
156C22⋊C8—1344no—
157C4×C8—2682no—
158#95—21no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8C2×C67⋊C812yes—
169C2×Q8C2×C67⋊C812yes—
170C2×Q8#1176161212yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C22⋊Dic6712yes—
176C22×C4C22⋊Dic6711yescharacteristic
177C22×C4C22⋊Dic6711yescharacteristic
178C22×C4C22×Dic6711yescharacteristic
179C22×C4C2×C22⋊D6712yes—
180C2×C8—2684no—
181C4⋊C4—28no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—21no—
190C42—41no—
191C42—42no—
192C42C2×C67⋊C812yes—
193C23C2×C22⋊Dic6711yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4C22⋊(C67⋊C8)14yes—
219C2×C4#1452395812yes—
220C2×C4C22×C67⋊C811yescharacteristic
221C2×C4C2×#1176161211yescharacteristic
222C8—5364no—
223C22—21no—
224C22—21no—
225C22C2×C22⋊(C67⋊C8)11yescharacteristic
226C22C2×#1452395811yescharacteristic
227C22#1686338011yescharacteristic
228C4—24no—
229C4—21no—
230C4—21no—
231C4—22no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2#2255852711yescharacteristic
241C2#2256188411yescharacteristic
242C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC2×C4×C67C22×C8
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2327C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2327D67
hypocenterC67#11482
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
23. C4⋊Q8×C67C8semidirect
126. C67#11482semidirect
142. C4⋊Q8C67⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225585332
#225581532
#225581432
C67×#23272
#1148267