Finite Group Databasebeta

← #31013472

Subgroups — #31013472

Order 17152.

562 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×D536C211yesmaximal, characteristic, complemented
2#22562432C213yesmaximal, complemented
3#22562346C213yesmaximal, complemented
4#22562241C213yesmaximal, complemented
5Q8×C67⋊C16C211yesmaximal, characteristic, complemented
6C4×#14523984C213yesmaximal, complemented
7C67×#3391C211yesmaximal, characteristic, complemented
8Q8×D268—21no—
9C2×#14523984C2213yes—
10#16863801C2213yes—
11#16863677—23no—
12C4×D536C2213yes—
13#16863221C2213yes—
14C4⋊(C67⋊C16)C2213yes—
15#16863197C2213yes—
16#16863197C2216yes—
17#16863196C2211yescharacteristic
18C4×C67⋊C16C2213yes—
19#550×C67—21no—
20#493×C67C2213yes—
21#459×C67—23no—
22C8×Q8×C67C2211yescharacteristic
23C4×D8×C67C2213yes—
24C2×Q8×D67—41no—
25C2×D536C2311yescharacteristic
26#14524043—23no—
27#14524023—43no—
28C4×D268—23no—
29#14523984—26no—
30C2×C67⋊C16C2313yes—
31C2×C67⋊C16C2311yescharacteristic
32#14523977—23no—
33#14523977C2313yes—
34#14523965—23no—
35C2×#41×C67—41no—
36C2×D8×C67—23no—
37C2×D8×C67C2313yes—
38#127×C67—23no—
39#121×C67—43no—
40C4×Q8×C67D411yescharacteristic
41C4×D4×C67—23no—
42#107×C67C2311yescharacteristic
43C4⋊C8×C67D413yes—
44#102×C67—23no—
45C4×C8×C67C2313yes—
46C2×Dic134—43no—
47Q8×D67—84no—
48C2×D268—21no—
49C2×C4×D67—43no—
50D536—21no—
51D536#4112yescomplemented
52C4⋊Dic67—23no—
53C4⋊Dic67—43no—
54(C2×D67)⋊C4—43no—
55C67⋊C16—23no—
56C67⋊C16—21no—
57#41×C67—84no—
58C2×Q8×C67D811yescharacteristic
59C2×D4×C67—23no—
60C2×D4×C67—43no—
61C2×D4×C67—43no—
62C22×C4×C67—43no—
63D8×C67—23no—
64D8×C67—23no—
65D8×C67—83no—
66C2×C8×C67C2×D413yes—
67C2×C8×C67C2411yescharacteristic
68C4⋊C4×C67—21no—
69C4⋊C4×C67D813yes—
70C22⋊C4×C67—43no—
71C42×C67C2×D413yes—
72C23×C67—43no—
73C22×D67—41no—
74Q8×C67D1614yescomplemented
75D4×C67—43no—
76D4×C67—43no—
77D4×C67—86no—
78D4×C67—163no—
79C2×C4×C67—41no—
80C2×C4×C67—86no—
81C2×C4×C67C2×D813yes—
82C2×C4×C67C2×D813yes—
83C2×C4×C67C22×D411yescharacteristic
84C2×Dic67—43no—
85D268—22no—
86D268—41no—
87C4×D67—86no—
88Dic134—86no—
89C8×C67—23no—
90C8×C67C2×#4111yescharacteristic
91C8×C672−1+411yescharacteristic
92C22×C67—83no—
93C22×C67—83no—
94C22×C67—83no—
95C22×C67C22×D811yescharacteristic
96C2×D67—42no—
97C2×D67—81no—
98C2×D67—81no—
99C4×C67—23no—
100C4×C67—81no—
101C4×C67C2×D1616yes—
102C4×C67#54811yescharacteristic
103C4×C67#57711yescharacteristic
104Dic67—83no—
105#33925—671nomaximal, Sylow
106C2×C67—163no—
107C2×C67#335811yescharacteristic
108C2×C67C22×D1611yescharacteristic
109C2×C67#341611yescharacteristic
110D67—82no—
111#3391D6711yescharacteristic, complemented
112Q8×D8—671no—
113#2240—673no—
114#2224—673no—
115#2206—673no—
116Q8×C16—671no—
117C4×D16—673no—
118C67#3392511yescharacteristic, complemented, Sylow
119#550—21no—
120D4×Q8—1341no—
121C2×D16—673no—
122#500—673no—
123#493C2×D6713yes—
124#474—1343no—
125#459—23no—
126C8×Q8C2×D6711yescharacteristic
127C4×D8—673no—
128C4×D8C2×D6713yes—
129#366—673no—
130C4⋊C16—673no—
131#357—673no—
132#357—676no—
133#357—671no—
134C4×C16—673no—
135C2×#41—41no—
136C22×Q8—2681no—
137C2×D8—23no—
138C2×D8—671no—
139C2×D8C22×D6713yes—
140C4⋊Q8—1343no—
141#127—23no—
142C22⋊Q8—2683no—
143#121—43no—
144C4×Q8C22⋊D6711yescharacteristic
145C4×D4—23no—
146C4×D4—1343no—
147D16—1346no—
148C2×C16—671no—
149C2×C16—673no—
150#107—673no—
151#107—1343no—
152#107C22×D6711yescharacteristic
153C4⋊C8C22⋊D6713yes—
154#102—23no—
155#102—1343no—
156C4×C8C22×D6713yes—
157#41—84no—
158C2×Q8—2683no—
159C2×Q8—5364no—
160C2×Q8#1176161311yescharacteristic
161C2×D4—23no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—1341no—
165C22×C4—43no—
166C22×C4—2683no—
167D8—23no—
168D8—23no—
169D8—83no—
170D8—672no—
171D8—1341no—
172C2×C8C2×C22⋊D6713yes—
173C2×C8C23×D6711yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—2683no—
176C4⋊C4—1343no—
177C4⋊C4#1176161313yes—
178C22⋊C4—43no—
179C22⋊C4—2683no—
180C42C2×C22⋊D6713yes—
181C16—1343no—
182C16—1341no—
183C23—43no—
184C23—2681no—
185Q8—5366no—
186Q8#1452398414yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—2681no—
191D4—163no—
192D4—1342no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—2683no—
196C2×C4—5366no—
197C2×C4C2×#1176161313yes—
198C2×C4C2×#1176161313yes—
199C2×C4C22×C22⋊D6711yescharacteristic
200C8—23no—
201C8#41×D6711yescharacteristic
202C8#1452409011yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—2682no—
207C22—5361no—
208C22—5361no—
209C22C22×#1176161311yescharacteristic
210C4—23no—
211C4—81no—
212C4—5363no—
213C4C2×#1452398416yes—
214C4#1686405311yescharacteristic
215C4#1686420711yescharacteristic
216C2—163no—
217C2—5362no—
218C2#2257284011yescharacteristic
219C2C22×#1452398411yescharacteristic
220C2#2257335611yescharacteristic
subgroupG/H
centerC22C22×#11761613
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC67×#3391C2
socleC22×C67C22×D8
hypercenter#3391D67
hypocenterC67#33925
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×D536C2semidirect
2. #22562432C2semidirect
3. #22562346C2semidirect
4. #22562241C2semidirect
5. Q8×C67⋊C16C2semidirect
6. C4×#14523984C2semidirect
7. C67×#3391C2semidirect
51. D536#41semidirect
74. Q8×C67D16semidirect
111. #3391D67semidirect
118. C67#33925semidirect
186. Q8#14523984semidirect

Maximal subgroups

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

subgroupindex
Q8×D5362
#225624322
#225623462
#225622412
Q8×C67⋊C162
C4×#145239842
C67×#33912
#3392567