Finite Group Databasebeta

← #30744962

Subgroups — #30744962

Order 17152.

366 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 87 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#22561752C211yesmaximal, characteristic
2#22561680C212yesmaximal
3#22558035C211yesmaximal, characteristic
4#22558032C212yesmaximal
5C67×#2809C211yesmaximal, characteristic
6C2×#14523976—22no—
7C2×#14523974—21no—
8C2×#14523967—21no—
9#16863266C2211yescharacteristic
10#16863158—22no—
11#16863151C2211yescharacteristic
12#16863151C2212yes—
13#400×C67—21no—
14#398×C67—22no—
15#391×C67—21no—
16#391×C67C2211yescharacteristic
17#389×C67C2212yes—
18#387×C67—22no—
19C22×C67⋊C8—21no—
20C22×C67⋊C8—22no—
21C2×C4⋊Dic67—21no—
22C2×C4⋊Dic67—22no—
23#14523976—44no—
24#14523974—42no—
25#14523967—42no—
26C22×Q8×C67D411yescharacteristic
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67D411yescharacteristic
33C2×C42×C67—21no—
34C2×C42×C67C2311yescharacteristic
35#95×C67—21no—
36#95×C67—22no—
37#95×C67—22no—
38#95×C67—22no—
39#95×C67—22no—
40C22×Dic67—41no—
41C2×C67⋊C8—22no—
42C2×C67⋊C8—24no—
43C2×C67⋊C8—42no—
44C2×C67⋊C8—41no—
45C2×C67⋊C8—42no—
46C2×C67⋊C8—41no—
47C4⋊Dic67—41no—
48C4⋊Dic67—41no—
49C4⋊Dic67—42no—
50C4⋊Dic67—42no—
51C2×Q8×C67—21no—
52C2×Q8×C67—41no—
53C2×Q8×C67—41no—
54C2×Q8×C67Q1612yes—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—21no—
59C22×C4×C67C2×D411yescharacteristic
60C22×C4×C67C2×D412yes—
61C4⋊C4×C67—22no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—42no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—42no—
67C4⋊C4×C67—81no—
68C4⋊C4×C67—81no—
69C42×C67—41no—
70C42×C67—81no—
71C23×C67#12011yescharacteristic
72Q8×C67—42no—
73Q8×C67—81no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—41no—
80C2×C4×C67—41no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—41no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67C2×Q1611yescharacteristic
90C2×C4×C67#13512yes—
91C2×C4×C67AGL(1,ℤ/8)12yes—
92C2×C4×C67#13711yescharacteristic
93C2×Dic67—82no—
94C2×Dic67—81no—
95C67⋊C8—44no—
96C67⋊C8—42no—
97C22×C67#44912yes—
98C22×C67#45111yescharacteristic
99C22×C67#45311yescharacteristic
100C22×C67#45612yes—
101C22×C67#45811yescharacteristic
102C4×C67—22no—
103C4×C67—22no—
104C4×C67—21no—
105C4×C67—21no—
106C4×C67—42no—
107C4×C67—82no—
108C4×C67—82no—
109C4×C67—81no—
110Dic67—161no—
111#7802—671nomaximal, Sylow
112C2×C67#160912yes—
113C2×C67#161411yescharacteristic
114C2×C67#162411yescharacteristic
115C2×C67#164812yes—
116C2×C67#166011yescharacteristic
117#2809D6711yescharacteristic
118#2096—671no—
119#2083—672no—
120#1272—671no—
121#1271—672no—
122C67#780211yescharacteristic, complemented, Sylow
123C2×#107—1342no—
124C2×#106—1341no—
125C2×#103—1341no—
126#400—21no—
127#398—22no—
128#391—21no—
129#391C2×D6711yescharacteristic
130#389C2×D6712yes—
131#387—22no—
132#384—671no—
133#340—1342no—
134#336—672no—
135#336—671no—
136C22×Q8C22⋊D6711yescharacteristic
137C22×C8—1342no—
138C22×C8—1341no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—1342no—
143C2×C4⋊C4—1341no—
144C2×C4⋊C4C22⋊D6712yes—
145C2×C4⋊C4C22⋊D6712yes—
146C2×C4⋊C4C22⋊D6711yescharacteristic
147C2×C42—21no—
148C2×C42C22×D6711yescharacteristic
149#107—2684no—
150#106—2682no—
151#103—2682no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#1176161812yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—2681no—
166C22×C4C2×C22⋊D6711yescharacteristic
167C22×C4C2×C22⋊D6712yes—
168C2×C8—2682no—
169C2×C8—2681no—
170C2×C8—2682no—
171C2×C8—2681no—
172C2×C8—1342no—
173C2×C8—1344no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—2681no—
183C4⋊C4—2681no—
184C4⋊C4—2682no—
185C4⋊C4—2682no—
186C42—41no—
187C42—81no—
188C23#1452401211yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—5362no—
207C2×C4—5361no—
208C2×C4C2×#1176161811yescharacteristic
209C2×C4#1452406312yes—
210C2×C4#1452406812yes—
211C2×C4#1452407411yescharacteristic
212C8—2682no—
213C8—2684no—
214C22#1686354112yes—
215C22#1686355211yescharacteristic
216C22#1686356411yescharacteristic
217C22#1686357912yes—
218C22#1686358611yescharacteristic
219C4—21no—
220C4—21no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4—10721no—
228C2#2255920612yes—
229C2#2255923411yescharacteristic
230C2#2255929211yescharacteristic
231C2#2255945112yes—
232C2#2255952011yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2809C2
socle, prime radicalC23×C67#120
hypercenter#2809D67
hypocenterC67#7802
LCSG ▷ C2×C42×C67 ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
122. C67#7802semidirect

Maximal subgroups

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

subgroupindex
#225617522
#225616802
#225580352
#225580322
C67×#28092
#780267