Finite Group Databasebeta

← #30745131

Subgroups — #30745131

Order 17152.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#22561772C212yesmaximal
2#22561747C211yesmaximal, characteristic
3#22558035C211yesmaximal, characteristic
4#22558032C212yesmaximal
5C67×#2734C211yesmaximal, characteristic
6C2×#14523976—21no—
7C2×#14523967—21no—
8#16863266C2211yescharacteristic
9#16863158—22no—
10#16863158—22no—
11#16863151C2211yescharacteristic
12#16863151C2212yes—
13C2×C4⋊Q8×C67—21no—
14#398×C67—21no—
15#395×C67—22no—
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—42no—
24#14523967—42no—
25C22×Q8×C67—21no—
26C22×Q8×C67D411yescharacteristic
27C4⋊Q8×C67—42no—
28C4⋊Q8×C67—42no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67D412yes—
33C2×C4⋊C4×C67D412yes—
34C2×C4⋊C4×C67D411yescharacteristic
35C2×C42×C67—21no—
36C2×C42×C67C2311yescharacteristic
37#95×C67—21no—
38#95×C67—22no—
39#95×C67—22no—
40#95×C67—22no—
41C22×Dic67—41no—
42C2×C67⋊C8—22no—
43C2×C67⋊C8—41no—
44C2×C67⋊C8—41no—
45C2×C67⋊C8—42no—
46C2×C67⋊C8—42no—
47C2×C67⋊C8—42no—
48C4⋊Dic67—41no—
49C4⋊Dic67—41no—
50C4⋊Dic67—44no—
51C2×Q8×C67—21no—
52C2×Q8×C67—42no—
53C2×Q8×C67—41no—
54C2×Q8×C67—41no—
55C2×Q8×C67—81no—
56C2×Q8×C67—81no—
57C2×Q8×C67SD1612yes—
58C22×C4×C67—22no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—21no—
62C22×C4×C67C2×D411yescharacteristic
63C22×C4×C67C2×D412yes—
64C4⋊C4×C67—22no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—44no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—81no—
71C4⋊C4×C67Q1614yes—
72C42×C67—41no—
73C42×C67—42no—
74C23×C67#12011yescharacteristic
75Q8×C67—42no—
76Q8×C67—81no—
77Q8×C67—84no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—21no—
82C2×C4×C67—22no—
83C2×C4×C67—24no—
84C2×C4×C67—42no—
85C2×C4×C67—41no—
86C2×C4×C67—42no—
87C2×C4×C67—41no—
88C2×C4×C67—42no—
89C2×C4×C67—41no—
90C2×C4×C67—42no—
91C2×C4×C67—42no—
92C2×C4×C67—42no—
93C2×C4×C67C2×SD1611yescharacteristic
94C2×C4×C67C2×Q1612yes—
95C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C67#13712yes—
97C2×Dic67—81no—
98C2×Dic67—82no—
99C67⋊C8—42no—
100C67⋊C8—82no—
101C22×C67#45011yescharacteristic
102C22×C67#45112yes—
103C22×C67#45311yescharacteristic
104C22×C67#45412yes—
105C22×C67#45811yescharacteristic
106C4×C67—21no—
107C4×C67—21no—
108C4×C67—22no—
109C4×C67—22no—
110C4×C67—42no—
111C4×C67—44no—
112C4×C67—82no—
113C4×C67—81no—
114Dic67—161no—
115#7829—671nomaximal, Sylow
116C2×C67#161311yescharacteristic
117C2×C67#161712yes—
118C2×C67#163311yescharacteristic
119C2×C67#163712yes—
120C2×C67#165711yescharacteristic
121#2734D6711yescharacteristic
122#2100—672no—
123#2095—671no—
124#1272—671no—
125#1271—672no—
126C67#782911yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—21no—
128C2×#107—1341no—
129C2×#103—1341no—
130#398—21no—
131#395—22no—
132#391C2×D6711yescharacteristic
133#389C2×D6712yes—
134#387—22no—
135#384—671no—
136#340—1342no—
137#340—1342no—
138#336—671no—
139#336—672no—
140C22×Q8—21no—
141C22×Q8C22⋊D6711yescharacteristic
142C22×C8—1342no—
143C22×C8—1341no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—1342no—
150C2×C4⋊C4—1341no—
151C2×C4⋊C4C22⋊D6712yes—
152C2×C4⋊C4C22⋊D6712yes—
153C2×C4⋊C4C22⋊D6711yescharacteristic
154C2×C42—21no—
155C2×C42C22×D6711yescharacteristic
156#107—2682no—
157#103—2682no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8#1176161612yes—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—2681no—
174C22×C4C2×C22⋊D6711yescharacteristic
175C22×C4C2×C22⋊D6712yes—
176C2×C8—2681no—
177C2×C8—2681no—
178C2×C8—2682no—
179C2×C8—2682no—
180C2×C8—2682no—
181C2×C8—1342no—
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—2681no—
190C4⋊C4—2681no—
191C4⋊C4—2684no—
192C4⋊C4#1176161814yes—
193C42—41no—
194C42—42no—
195C23#1452401211yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—84no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—5361no—
215C2×C4—5362no—
216C2×C4C2×#1176161611yescharacteristic
217C2×C4C2×#1176161812yes—
218C2×C4#1452406911yescharacteristic
219C2×C4#1452407412yes—
220C8—2682no—
221C8—5362no—
222C22#1686354711yescharacteristic
223C22#1686355212yes—
224C22#1686356411yescharacteristic
225C22#1686356912yes—
226C22#1686358611yescharacteristic
227C4—22no—
228C4—22no—
229C4—21no—
230C4—21no—
231C4—44no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—10721no—
236C2#2255922911yescharacteristic
237C2#2255925112yes—
238C2#2255934911yescharacteristic
239C2#2255937412yes—
240C2#2255950711yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2734C2
socle, prime radicalC23×C67#120
hypercenter#2734D67
hypocenterC67#7829
LCSG ▷ C2×C42×C67 ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
126. C67#7829semidirect

Maximal subgroups

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

subgroupindex
#225617722
#225617472
#225580352
#225580322
C67×#27342
#782967