Finite Group Databasebeta

← #30823583

Subgroups — #30823583

Order 17152.

938 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 395 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#22569290C211yesmaximal, characteristic
2#22563130C211yesmaximal, characteristic
3#22558773C216yesmaximal
4#22558708C216yesmaximal
5C67×#3502C211yesmaximal, characteristic
6C2×Q8×Dic67—21no—
7Q8×C67⋊C8—24no—
8#16863449C2216yes—
9#16863295—23no—
10#16863262C2213yes—
11C4×C4⋊Dic67C2213yes—
12#16863133C22112yes—
13#16863133C2214yes—
14Q82×C67C414yescomplemented
15#557×C67—24no—
16#557×C67C414yescomplemented
17#544×C67—23no—
18#533×C67C2216yes—
19#519×C67—21no—
20C2×C4×Q8×C67C2211yescharacteristic
21Q8×Dic67—44no—
22C2×C4⋊Dic67—23no—
23C2×C4⋊Dic67—23no—
24C2×C4×Dic67—23no—
25C4⋊(C67⋊C8)—26no—
26C22⋊(C67⋊C8)C2314yes—
27C4×C67⋊C8—26no—
28C67⋊#95—23no—
29C67⋊#95C2311yescharacteristic
30C67⋊#95C2313yes—
31C22×Q8×C67D411yescharacteristic
32C4⋊Q8×C67—212no—
33C4⋊Q8×C67C2×C416yes—
34C4⋊Q8×C67C2×C416yes—
35#125×C67—212no—
36#125×C67—26no—
37C22⋊Q8×C67—26no—
38C22⋊Q8×C67C2314yes—
39C4×Q8×C67—24no—
40C4×Q8×C67—24no—
41C4×Q8×C67—26no—
42C4×Q8×C67C2×C4112yes—
43C4×Q8×C67C2×C414yes—
44C4×Q8×C67D414yes—
45#117×C67—26no—
46C2×C4⋊C4×C67D413yes—
47C2×C42×C67C2313yes—
48C22×Dic67—21no—
49C22×Dic67—23no—
50C2×C67⋊C8—24no—
51C4⋊Dic67—46no—
52C4⋊Dic67—46no—
53C4×Dic67—46no—
54C2×Q8×C67—26no—
55C2×Q8×C67C22⋊C414yes—
56C2×Q8×C67SD1614yes—
57C2×Q8×C67Q1614yes—
58C2×Q8×C67C22×C414yes—
59C22×C4×C67C2×D413yes—
60C22×C4×C67C2×D413yes—
61C22×C4×C67C2411yescharacteristic
62C4⋊C4×C67—26no—
63C4⋊C4×C67—26no—
64C4⋊C4×C67—26no—
65C4⋊C4×C67—212no—
66C4⋊C4×C67—24no—
67C4⋊C4×C67C22⋊C4112yes—
68C4⋊C4×C67C22×C414yes—
69C22⋊C4×C67—24no—
70C42×C67—212no—
71C42×C67C22×C416yes—
72C42×C67C2×D416yes—
73C23×C67C22×D411yescharacteristic
74Q8×C67—44no—
75Q8×C67—86no—
76Q8×C67#103116yes—
77C2×C4×C67—24no—
78C2×C4×C67—24no—
79C2×C4×C67—43no—
80C2×C4×C67—41no—
81C2×C4×C67C2×C22⋊C416yes—
82C2×C4×C67C2×C22⋊C416yes—
83C2×C4×C67C2×SD1616yes—
84C2×C4×C67C2×Q1616yes—
85C2×C4×C67C23×C411yescharacteristic
86C2×C4×C67C22×D411yescharacteristic
87C2×Dic67—22no—
88C2×Dic67—41no—
89C2×Dic67—41no—
90C2×Dic67—43no—
91C2×Dic67—43no—
92C2×Dic67—43no—
93C67⋊C8—44no—
94C22×C67—21no—
95C22×C67—21no—
96C22×C67C22×C22⋊C411yescharacteristic
97C22×C67C22×SD1611yescharacteristic
98C22×C67C22×Q1611yescharacteristic
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—26no—
102C4×C67—44no—
103C4×C67—44no—
104C4×C67C2×#103112yes—
105Dic67—42no—
106Dic67—83no—
107#17619—671nomaximal, Sylow
108C2×C67—21no—
109C2×C67—21no—
110C2×C67#288411yescharacteristic
111C2×C67C22×#10311yescharacteristic
112C2×C67#290111yescharacteristic
113#3502D6711yescharacteristic
114#2982—671no—
115#2351—671no—
116#1518—676no—
117#1503—676no—
118C67#1761911yescharacteristic, complemented, Sylow
119Q82Dic6714yescomplemented
120#557—24no—
121#557Dic6714yescomplemented
122#544—23no—
123#533C2×D6716yes—
124#519—21no—
125C2×C4×Q8—1341no—
126C2×C4×Q8C2×D6711yescharacteristic
127C8×Q8—1344no—
128#431—676no—
129#389—1343no—
130#382—673no—
131C4×C4⋊C4—673no—
132#328—674no—
133#328—6712no—
134C22×Q8C22⋊D6711yescharacteristic
135C4⋊Q8—212no—
136C4⋊Q8C2×Dic6716yes—
137C4⋊Q8C2×Dic6716yes—
138#125—212no—
139#125—26no—
140C22⋊Q8—26no—
141C22⋊Q8C22×D6714yes—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—26no—
145C4×Q8—2684no—
146C4×Q8C2×Dic67112yes—
147C4×Q8C2×Dic6714yes—
148C4×Q8C22⋊D6714yes—
149#117—26no—
150C2×C4⋊C4—1343no—
151C2×C4⋊C4—1343no—
152C2×C4⋊C4C22⋊D6713yes—
153C2×C42—1343no—
154C2×C42C22×D6713yes—
155C4⋊C8—1346no—
156C22⋊C8—674no—
157C4×C8—1346no—
158#95—671no—
159#95—673no—
160#95—1343no—
161C2×Q8—26no—
162C2×Q8C22⋊Dic6714yes—
163C2×Q8#1176161514yes—
164C2×Q8#1176161814yes—
165C2×Q8C22×Dic6714yes—
166C22×C4—1343no—
167C22×C4—1341no—
168C22×C4C2×C22⋊D6713yes—
169C22×C4C2×C22⋊D6713yes—
170C22×C4C23×D6711yescharacteristic
171C2×C8—1344no—
172C4⋊C4—26no—
173C4⋊C4—26no—
174C4⋊C4—26no—
175C4⋊C4—212no—
176C4⋊C4—24no—
177C4⋊C4—2686no—
178C4⋊C4—2686no—
179C4⋊C4C22⋊Dic67112yes—
180C4⋊C4C22×Dic6714yes—
181C22⋊C4—24no—
182C42—212no—
183C42—2686no—
184C42C22×Dic6716yes—
185C42C2×C22⋊D6716yes—
186C23C22×C22⋊D6711yescharacteristic
187Q8—44no—
188Q8—86no—
189Q8#14523967116yes—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—43no—
193C2×C4—41no—
194C2×C4—2681no—
195C2×C4—2681no—
196C2×C4—2683no—
197C2×C4—2683no—
198C2×C4—2683no—
199C2×C4—1342no—
200C2×C4C2×C22⋊Dic6716yes—
201C2×C4C2×C22⋊Dic6716yes—
202C2×C4C2×#1176161516yes—
203C2×C4C2×#1176161816yes—
204C2×C4C23×Dic6711yescharacteristic
205C2×C4C22×C22⋊D6711yescharacteristic
206C8—2684no—
207C22—21no—
208C22—21no—
209C22C22×C22⋊Dic6711yescharacteristic
210C22C22×#1176161511yescharacteristic
211C22C22×#1176161811yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—44no—
216C4—44no—
217C4—2682no—
218C4—5363no—
219C4C2×#14523967112yes—
220C2—21no—
221C2—21no—
222C2#2256867311yescharacteristic
223C2C22×#1452396711yescharacteristic
224C2#2256875611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#3502C2
socleC22×C67C22×C22⋊C4
prime radicalC23×C67C22×D4
cosocleC22×C4×C67C24
hypercenter#3502D67
hypocenterC67#17619
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

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

NH ≅ G/Ntype
14. Q82×C67C4semidirect
16. #557×C67C4semidirect
118. C67#17619semidirect
119. Q82Dic67semidirect
121. #557Dic67semidirect

Maximal subgroups

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

subgroupindex
#225692902
#225631302
#225587732
#225587082
C67×#35022
#1761967