Finite Group Databasebeta

← #30749873

Subgroups — #30749873

Order 17152.

348 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 107 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#22561150C211yesmaximal, characteristic
2#22558995C212yesmaximal
3#22558533C211yesmaximal, characteristic
4#22558107C212yesmaximal
5C67×#2983C211yesmaximal, characteristic
6#16863449C2212yes—
7C2×C4⋊(C67⋊C8)C2211yescharacteristic
8C2×#14523954C2211yescharacteristic
9#16863151—21no—
10#16863151—21no—
11#16863148—22no—
12#16863146—22no—
13#16863122C412yes—
14#16863122C412yes—
15C2×C4×Q8×C67C2211yescharacteristic
16#446×C67—22no—
17#446×C67—22no—
18#432×C67—21no—
19#432×C67C2212yes—
20#431×C67—21no—
21C22×C67⋊C8—21no—
22C22×C67⋊C8—21no—
23C4⋊(C67⋊C8)—22no—
24C4⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)C2×C414yes—
26#14523954—22no—
27C4×C67⋊C8—22no—
28C22×Q8×C67Q811yescharacteristic
29C4×Q8×C67—22no—
30C4×Q8×C67—22no—
31C2×C4⋊C4×C67C2×C411yescharacteristic
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C4⋊C4×C67Q811yescharacteristic
34C2×C42×C67D411yescharacteristic
35C2×C42×C67D411yescharacteristic
36C2×C42×C67C2311yescharacteristic
37C4⋊C8×C67—21no—
38C4⋊C8×C67—21no—
39C4⋊C8×C67—21no—
40C4⋊C8×C67—21no—
41C4⋊C8×C67D412yes—
42C4⋊C8×C67D412yes—
43C22⋊C8×C67—22no—
44#97×C67—22no—
45#97×C67—22no—
46C4×C8×C67—22no—
47C2×C67⋊C8—22no—
48C2×C67⋊C8—24no—
49C2×C67⋊C8—41no—
50C2×C67⋊C8—41no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—41no—
53C2×Q8×C67—22no—
54C2×Q8×C67—22no—
55C2×Q8×C67—41no—
56C22×C4×C67C4⋊C411yescharacteristic
57C22×C4×C67C4⋊C411yescharacteristic
58C22×C4×C67C4⋊C411yescharacteristic
59C22×C4×C67C4⋊C411yescharacteristic
60C22×C4×C67C22×C411yescharacteristic
61C22×C4×C67C2×D411yescharacteristic
62C22×C4×C67C2×Q811yescharacteristic
63C2×C8×C67—22no—
64C2×C8×C67—22no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C42×C67—21no—
69C42×C67—21no—
70C42×C67—21no—
71C42×C67—21no—
72C42×C67C2×D412yes—
73C42×C67#4112yes—
74C23×C67C2×C4⋊C411yescharacteristic
75Q8×C67—44no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—41no—
88C2×C4×C67—41no—
89C2×C4×C67—41no—
90C2×C4×C67#10812yes—
91C2×C4×C67C4×D412yes—
92C2×C4×C67#12011yescharacteristic
93C2×C4×C67C22⋊Q812yes—
94C2×C4×C67#12311yescharacteristic
95C8×C67—42no—
96C8×C67—42no—
97C67⋊C8—44no—
98C67⋊C8—42no—
99C67⋊C8—81no—
100C22×C67—21no—
101C22×C67—21no—
102C22×C67#38511yescharacteristic
103C22×C67C2×#10811yescharacteristic
104C22×C67#43011yescharacteristic
105C4×C67—22no—
106C4×C67—22no—
107C4×C67—21no—
108C4×C67—22no—
109C4×C67—41no—
110C4×C67—41no—
111C4×C67—41no—
112C4×C67#45411yescharacteristic
113C4×C67#45611yescharacteristic
114#8865—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—21no—
117C2×C67#185511yescharacteristic
118C2×C67#185711yescharacteristic
119C2×C67#186811yescharacteristic
120#2983D6711yescharacteristic
121#1985—671no—
122#1573—672no—
123#1461—671no—
124#1302—672no—
125C67#886511yescharacteristic, complemented, Sylow
126C2×C4×Q8C2×D6711yescharacteristic
127#446—22no—
128#446—22no—
129#432—21no—
130#432C2×D6712yes—
131#431—21no—
132#431—672no—
133C2×C4⋊C8—671no—
134C2×#97—671no—
135#336—1341no—
136#336—1341no—
137#335—1342no—
138#334—1342no—
139#324—672no—
140#324—672no—
141C22×Q8Dic13411yescharacteristic
142C22×C8—1341no—
143C22×C8—1341no—
144C4×Q8—22no—
145C4×Q8—22no—
146C2×C4⋊C4Dic13411yescharacteristic
147C2×C4⋊C4C4×D6711yescharacteristic
148C2×C4⋊C4C4×D6711yescharacteristic
149C2×C42C22⋊D6711yescharacteristic
150C2×C42C22⋊D6711yescharacteristic
151C2×C42C22×D6711yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—1342no—
157C4⋊C8—1342no—
158C4⋊C8C22⋊D6712yes—
159C4⋊C8C22⋊D6712yes—
160C22⋊C8—22no—
161C22⋊C8—674no—
162#97—22no—
163#97—22no—
164#97—1342no—
165C4×C8—22no—
166C4×C8—1342no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—41no—
170C22×C4Dic67⋊C411yescharacteristic
171C22×C4Dic67⋊C411yescharacteristic
172C22×C4Dic67⋊C411yescharacteristic
173C22×C4Dic67⋊C411yescharacteristic
174C22×C4C2×C4×D6711yescharacteristic
175C22×C4C2×C22⋊D6711yescharacteristic
176C22×C4C2×Dic13411yescharacteristic
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—2681no—
180C2×C8—2681no—
181C2×C8—2682no—
182C2×C8—2681no—
183C2×C8—1342no—
184C2×C8—1344no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42C2×C22⋊D6712yes—
193C42#1176162812yes—
194C23C2×Dic67⋊C411yescharacteristic
195Q8—44no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4#1452397812yes—
211C2×C4C4×C22⋊D6712yes—
212C2×C4#1452401211yescharacteristic
213C2×C4#1452402212yes—
214C2×C4#1452402611yescharacteristic
215C8—42no—
216C8—42no—
217C8—2682no—
218C8—2684no—
219C8—5361no—
220C22—21no—
221C22—21no—
222C22#1686326911yescharacteristic
223C22C2×#1452397811yescharacteristic
224C22#1686344411yescharacteristic
225C4—22no—
226C4—22no—
227C4—21no—
228C4—22no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4#1686356911yescharacteristic
233C4#1686357911yescharacteristic
234C2—21no—
235C2—21no—
236C2#2256041411yescharacteristic
237C2#2256042411yescharacteristic
238C2#2256049511yescharacteristic
subgroupG/H
centerC2×C4#14524012
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2983C2
socleC22×C67#385
prime radicalC23×C67C2×C4⋊C4
cosocleC2×C42×C67C23
hypercenter#2983D67
hypocenterC67#8865
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
125. C67#8865semidirect

Maximal subgroups

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

subgroupindex
#225611502
#225589952
#225585332
#225581072
C67×#29832
#886567