Finite Group Databasebeta

← #30823585

Subgroups — #30823585

Order 17152.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#22574019C211yesmaximal, characteristic
2#22563130C211yesmaximal, characteristic
3#22558774C216yesmaximal
4#22558709C216yesmaximal
5C67×#2982C211yesmaximal, characteristic
6Q8×Dic134C414yescomplemented
7#16864125C414yescomplemented
8#16864122—24no—
9#16864027—23no—
10#16863953C2216yes—
11#16863874—21no—
12C2×Q8×Dic67—21no—
13#16863295—23no—
14#16863262C2213yes—
15C4×C4⋊Dic67C2213yes—
16#16863134C22112yes—
17#16863134C2214yes—
18C2×C4×Q8×C67C2211yescharacteristic
19C8×Q8×C67—24no—
20#431×C67C2216yes—
21#14524043C2×C416yes—
22#14524043C2×C416yes—
23#14524042—212no—
24#14524035—26no—
25#14524034—212no—
26#14524022—26no—
27#14524022C2314yes—
28Q8×Dic67—24no—
29Q8×Dic67—24no—
30Q8×Dic67—44no—
31C4×Dic134—26no—
32C4×Dic134C2×C4112yes—
33#14524002—26no—
34C2×C4⋊Dic67—23no—
35C2×C4⋊Dic67—23no—
36C2×C4×Dic67—23no—
37C67⋊#95—23no—
38C67⋊#95C2311yescharacteristic
39C67⋊#95C2313yes—
40C22×Q8×C67D411yescharacteristic
41C4×Q8×C67C2×C414yes—
42C4×Q8×C67D414yes—
43C2×C4⋊C4×C67D413yes—
44C2×C42×C67C2313yes—
45C4⋊C8×C67—26no—
46C22⋊C8×C67C2314yes—
47C4×C8×C67—26no—
48C2×Dic134—26no—
49C2×Dic134C22×C414yes—
50C22×Dic67—21no—
51C22×Dic67—23no—
52C4⋊Dic67—26no—
53C4⋊Dic67—26no—
54C4⋊Dic67—26no—
55C4⋊Dic67—46no—
56C4⋊Dic67—46no—
57C4⋊Dic67C22×C414yes—
58Dic67⋊C4—212no—
59Dic67⋊C4—24no—
60C22⋊Dic67—24no—
61C4×Dic67—212no—
62C4×Dic67—46no—
63C2×Q8×C67C22⋊C414yes—
64C2×Q8×C67SD1614yes—
65C2×Q8×C67Q1614yes—
66C22×C4×C67C2×D413yes—
67C22×C4×C67C2×D413yes—
68C22×C4×C67C2411yescharacteristic
69C2×C8×C67—24no—
70C4⋊C4×C67C22⋊C4112yes—
71C42×C67C22×C416yes—
72C42×C67C2×D416yes—
73C23×C67C22×D411yescharacteristic
74Q8×C67#103116yes—
75C2×C4×C67—43no—
76C2×C4×C67—41no—
77C2×C4×C67C2×C22⋊C416yes—
78C2×C4×C67C2×C22⋊C416yes—
79C2×C4×C67C2×SD1616yes—
80C2×C4×C67C2×Q1616yes—
81C2×C4×C67C23×C411yescharacteristic
82C2×C4×C67C22×D411yescharacteristic
83C2×Dic67—22no—
84C2×Dic67—24no—
85C2×Dic67—24no—
86C2×Dic67—41no—
87C2×Dic67—41no—
88C2×Dic67—43no—
89C2×Dic67—43no—
90C2×Dic67—43no—
91Dic134—44no—
92Dic134—86no—
93C8×C67—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×C67C2×#103112yes—
103Dic67—42no—
104Dic67—44no—
105Dic67—44no—
106Dic67—83no—
107#17619—671nomaximal, Sylow
108C2×C67—21no—
109C2×C67—21no—
110C2×C67#288411yescharacteristic
111C2×C67C22×#10311yescharacteristic
112C2×C67#290111yescharacteristic
113#3502—671no—
114#2982D6711yescharacteristic
115#2351—671no—
116#1518—676no—
117#1503—676no—
118C67#1761911yescharacteristic, complemented, Sylow
119Q82—674no—
120#557—674no—
121#557—1344no—
122#544—1343no—
123#533—676no—
124#519—1341no—
125C2×C4×Q8—1341no—
126C2×C4×Q8C2×D6711yescharacteristic
127C8×Q8—24no—
128#431C2×D6716yes—
129#389—1343no—
130#382—673no—
131C4×C4⋊C4—673no—
132#328—674no—
133#328—6712no—
134C22×Q8D26811yescharacteristic
135C4⋊Q8—676no—
136C4⋊Q8—676no—
137C4⋊Q8—13412no—
138#125—13412no—
139#125—1346no—
140C22⋊Q8—674no—
141C22⋊Q8—1346no—
142C4×Q8—2684no—
143C4×Q8—6712no—
144C4×Q8—1344no—
145C4×Q8—1344no—
146C4×Q8—1346no—
147C4×Q8C4×D6714yes—
148C4×Q8C22⋊D6714yes—
149#117—1346no—
150C2×C4⋊C4—1343no—
151C2×C4⋊C4—1343no—
152C2×C4⋊C4D26813yes—
153C2×C42—1343no—
154C2×C42C22×D6713yes—
155C4⋊C8—26no—
156C22⋊C8C22×D6714yes—
157C4×C8—26no—
158#95—671no—
159#95—673no—
160#95—1343no—
161C2×Q8—674no—
162C2×Q8—1346no—
163C2×Q8(C2×D67)⋊C414yes—
164C2×Q8#1176161714yes—
165C2×Q8Dic26814yes—
166C22×C4—1343no—
167C22×C4—1341no—
168C22×C4C2×D26813yes—
169C22×C4C2×D26813yes—
170C22×C4C23×D6711yescharacteristic
171C2×C8—24no—
172C4⋊C4—2686no—
173C4⋊C4—2686no—
174C4⋊C4—674no—
175C4⋊C4—1346no—
176C4⋊C4—1346no—
177C4⋊C4—13412no—
178C4⋊C4—1346no—
179C4⋊C4—1344no—
180C4⋊C4(C2×D67)⋊C4112yes—
181C22⋊C4—1344no—
182C42—2686no—
183C42—13412no—
184C42C2×C4×D6716yes—
185C42C2×C22⋊D6716yes—
186C23C22×D26811yescharacteristic
187Q8—2684no—
188Q8—5366no—
189Q8#14523968116yes—
190C2×C4—43no—
191C2×C4—41no—
192C2×C4—2681no—
193C2×C4—2681no—
194C2×C4—2683no—
195C2×C4—2683no—
196C2×C4—2683no—
197C2×C4—1342no—
198C2×C4—1344no—
199C2×C4—1344no—
200C2×C4C2×(C2×D67)⋊C416yes—
201C2×C4C2×(C2×D67)⋊C416yes—
202C2×C4C2×#1176161716yes—
203C2×C4C2×Dic26816yes—
204C2×C4C22×C4×D6711yescharacteristic
205C2×C4C22×C22⋊D6711yescharacteristic
206C8—44no—
207C22—21no—
208C22—21no—
209C22C22×(C2×D67)⋊C411yescharacteristic
210C22C22×#1176161711yescharacteristic
211C22C22×Dic26811yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—2682no—
216C4—2684no—
217C4—2684no—
218C4—5363no—
219C4C2×#14523968112yes—
220C2—21no—
221C2—21no—
222C2#2256867211yescharacteristic
223C2C22×#1452396811yescharacteristic
224C2#2256875411yescharacteristic
subgroupG/H
centerC22C22×(C2×D67)⋊C4
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2982C2
socleC22×C67C22×C22⋊C4
prime radicalC23×C67C22×D4
cosocleC22×C4×C67C24
hypercenter#2982D67
hypocenterC67#17619
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
6. Q8×Dic134C4semidirect
7. #16864125C4semidirect
118. C67#17619semidirect

Maximal subgroups

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

subgroupindex
#225740192
#225631302
#225587742
#225587092
C67×#29822
#1761967