Finite Group Databasebeta

← #30748596

Subgroups — #30748596

Order 17152.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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#125×Dic67C211yesmaximal, characteristic
2#22560063C212yesmaximal
3C2×#16863138C211yesmaximal, characteristic
4#22558091C212yesmaximal
5C2×#330×C67C211yesmaximal, characteristic
6C2×Dic67⋊C8—22no—
7C2×#14523954C2212yes—
8#16863295C412yes—
9#16863282—22no—
10#16863262C2211yescharacteristic
11C4⋊C4×Dic67—22no—
12C4⋊C4×Dic67C412yes—
13C42×Dic67C2211yescharacteristic
14#16863150—22no—
15#16863138—22no—
16C2×#125×C67C2211yescharacteristic
17C2×#97×C67C2212yes—
18#330×C67—22no—
19C22×C67⋊C8—22no—
20C2×C4⋊Dic67—22no—
21C2×C4⋊Dic67C2×C412yes—
22C2×C4×Dic67—22no—
23C2×C4×Dic67C2×C411yescharacteristic
24C2×C4×Dic67C2×C411yescharacteristic
25C2×C4×Dic67D412yes—
26C2×C4×Dic67D412yes—
27Dic67⋊C8—28no—
28#14523954—22no—
29#14523954—22no—
30C67⋊#95—22no—
31C67⋊#95—22no—
32C67⋊#95—22no—
33C22×C8×C67—22no—
34#125×C67—22no—
35#125×C67D412yes—
36#125×C67Q812yes—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C42×C67C2311yescharacteristic
40#97×C67—22no—
41#97×C67—22no—
42C22×Dic67—22no—
43C22×Dic67C22⋊C411yescharacteristic
44C22×Dic67C22⋊C412yes—
45C22×Dic67C22⋊C411yescharacteristic
46C2×C67⋊C8—24no—
47C2×C67⋊C8—42no—
48C2×C67⋊C8—42no—
49C4⋊Dic67—44no—
50C4⋊Dic67—42no—
51C4×Dic67—22no—
52C4×Dic67—24no—
53C4×Dic67—41no—
54C4×Dic67—42no—
55C4×Dic67—44no—
56C22×C4×C67—22no—
57C22×C4×C67C22×C411yescharacteristic
58C22×C4×C67C2×D412yes—
59C2×C8×C67—24no—
60C2×C8×C67—42no—
61C2×C8×C67—42no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—24no—
64C4⋊C4×C67—24no—
65C4⋊C4×C67C4⋊C414yes—
66C42×C67C2×D411yescharacteristic
67C42×C67C2×Q811yescharacteristic
68C42×C67#4112yes—
69C23×C67C2×C22⋊C411yescharacteristic
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—24no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67C2×C4⋊C411yescharacteristic
78C2×C4×C67#11711yescharacteristic
79C2×C4×C67#12112yes—
80C2×C4×C67C22⋊Q812yes—
81C2×Dic67—21no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—42no—
87C2×Dic67—42no—
88C2×Dic67—42no—
89C2×Dic67#10112yes—
90C2×Dic67#10414yes—
91C8×C67—44no—
92C67⋊C8—44no—
93C22×C67#38011yescharacteristic
94C22×C67C2×#10111yescharacteristic
95C22×C67#41311yescharacteristic
96C22×C67C2×#10412yes—
97C22×C67#42112yes—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—22no—
101C4×C67—22no—
102C4×C67—44no—
103Dic67—41no—
104Dic67—42no—
105Dic67—82no—
106Dic67#33014yescomplemented
107#8580—671nomaximal, Sylow
108C2×C67C2×#33011yescharacteristic
109C2×C67#152912yes—
110C2×C67#153711yescharacteristic
111C2×C67#180111yescharacteristic
112C2×C67#180612yes—
113C4×#125—671no—
114#1767—672no—
115C2×#330—671no—
116C2×#330D6711yescharacteristic
117#1295—672no—
118C67#858011yescharacteristic, complemented, Sylow
119C2×#125C2×D6711yescharacteristic
120C2×C4⋊C8—1342no—
121C2×#97—672no—
122C2×#97C2×D6712yes—
123#389—672no—
124#387—1342no—
125#382—671no—
126C4×C4⋊C4—672no—
127C4×C4⋊C4—1342no—
128C43—671no—
129#336—1342no—
130#330—22no—
131#330—1342no—
132C22×C8—22no—
133C22×C8—1342no—
134#125—22no—
135#125Dic13412yes—
136#125C22⋊D6712yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—1342no—
140C2×C4⋊C4C4×D6712yes—
141C2×C42—671no—
142C2×C42—672no—
143C2×C42—671no—
144C2×C42—672no—
145C2×C42—1342no—
146C2×C42C22×D6711yescharacteristic
147C4⋊C8—1348no—
148#97—22no—
149#97—22no—
150#97—1342no—
151#97—1342no—
152#95—1342no—
153#95—1342no—
154#95—1342no—
155C22×C4—22no—
156C22×C4—671no—
157C22×C4—671no—
158C22×C4—672no—
159C22×C4—1342no—
160C22×C4C2×C4×D6711yescharacteristic
161C22×C4D4×D6712yes—
162C2×C8—24no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—2682no—
166C2×C8—2682no—
167C2×C8—1344no—
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—2682no—
172C4⋊C4—2684no—
173C4⋊C4Dic67⋊C414yes—
174C42—2681no—
175C42—2682no—
176C42—2684no—
177C42—1344no—
178C42—1342no—
179C42C2×C22⋊D6711yescharacteristic
180C42C2×Dic13411yescharacteristic
181C42#1176162712yes—
182C23C22⋊C4×D6711yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—2682no—
191C2×C4—2682no—
192C2×C4—2682no—
193C2×C4—672no—
194C2×C4—674no—
195C2×C4—1341no—
196C2×C4—1342no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4C2×Dic67⋊C411yescharacteristic
201C2×C4#1452399911yescharacteristic
202C2×C4#1452401512yes—
203C2×C4#1452402012yes—
204C8—44no—
205C8—2684no—
206C22#1686325411yescharacteristic
207C22#101×D6711yescharacteristic
208C22#1686337911yescharacteristic
209C22#104×D6712yes—
210C22#1686341112yes—
211C4—21no—
212C4—21no—
213C4—22no—
214C4—22no—
215C4—44no—
216C4—2682no—
217C4—2681no—
218C4—5362no—
219C4—674no—
220C2#330×D6711yescharacteristic
221C2#2255881512yes—
222C2#2255884511yescharacteristic
223C2#2256020311yescharacteristic
224C2#2256022712yes—
subgroupG/H
centerC23C22⋊C4×D67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC2×#330×C67C2
socle, prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenterC2×#330D67
hypocenterC67#8580
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
106. Dic67#330semidirect
118. C67#8580semidirect

Maximal subgroups

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

subgroupindex
#125×Dic672
#225600632
C2×#168631382
#225580912
C2×#330×C672
#858067