Finite Group Databasebeta

← #30776648

Subgroups — #30776648

Order 17152.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#22562527C211yesmaximal, characteristic, complemented
2#22562524C211yesmaximal, characteristic
3#22562471C211yesmaximal, characteristic, complemented
4#22562089C211yesmaximal, characteristic, complemented
5#22558406C211yesmaximal, characteristic, complemented
6#22558398C211yesmaximal, characteristic, complemented
7C67×#1420C211yesmaximal, characteristic, complemented
8#16863828C2211yescharacteristic
9C2×#14523986—21no—
10C2×D1072C2211yescharacteristic
11#16863781—21no—
12#16863512—21no—
13#16863236—21no—
14#16863235—21no—
15#16863233—21no—
16D2144—21no—
17#16863230C2211yescharacteristic
18#16863202—21no—
19C16×Dic67C2211yescharacteristic
20C2×Q32×C67C2211yescharacteristic
21M64×C67C2211yescharacteristic
22#368×C67C2211yescharacteristic
23#14524065—21no—
24C2×#11761615—41no—
25C2×D536—21no—
26#14524051—41no—
27#14524032—41no—
28Dic536D411yescharacteristic
29#14523988—21no—
30#14523986—42no—
31D1072—21no—
32D1072—21no—
33D1072D411yescharacteristic
34C2×C67⋊C16D411yescharacteristic
35#14523969—41no—
36C8×Dic67D411yescharacteristic
37C67⋊C32—21no—
38C2×Q16×C67—21no—
39Q32×C67—21no—
40Q32×C67—21no—
41C2×C16×C67C2311yescharacteristic
42#108×C67—21no—
43C32×C67—21no—
44#11761628—41no—
45C2×D268—41no—
46Dic268—21no—
47#11761617—41no—
48#11761615—81no—
49#11761615—81no—
50D536—21no—
51D536—41no—
52D536—41no—
53#11761611—81no—
54C2×C67⋊C8D811yescharacteristic
55C8×D67—81no—
56(C2×D67)⋊C4—82no—
57C4×Dic67D811yescharacteristic
58C67⋊C16—21no—
59C2×Q8×C67—41no—
60Q16×C67—41no—
61Q16×C67—41no—
62M16×C67—41no—
63C2×C8×C67C2×D411yescharacteristic
64C16×C67C2×D411yescharacteristic
65C16×C67#4111yescharacteristic
66C22×D67—81no—
67Q8×C67—81no—
68Q8×C67—81no—
69C2×C4×C67—81no—
70C2×C4×C67C2×D811yescharacteristic
71C22⋊D67—81no—
72C2×Dic67—21no—
73D268—41no—
74D268—81no—
75D268—81no—
76C4×D67—81no—
77Dic134—41no—
78C8×C67—81no—
79C8×C67#12111yescharacteristic
80C8×C67AGL(1,ℤ/8)11yescharacteristic
81C67⋊C8D1611yescharacteristic
82C67⋊C8D1611yescharacteristic
83C22×C67#50911yescharacteristic
84C2×D67—81no—
85C2×D67—162no—
86C2×D67—161no—
87C4×C67—161no—
88C4×C67#45911yescharacteristic
89C4×C67C2×D1611yescharacteristic
90Dic67—41no—
91Dic67—81no—
92#13691—671nomaximal, Sylow
93C2×C67—21no—
94C2×C67#220611yescharacteristic
95D67—161no—
96D67—321no—
97#2264—671no—
98#2263—671no—
99#2249—671no—
100#2178—671no—
101#1422—671no—
102#1420D6711yescharacteristic, complemented
103#1419—671no—
104C67#1369111yescharacteristic, complemented, Sylow
105#508—671no—
106C2×Q32C2×D6711yescharacteristic
107C2×SD32—1341no—
108C2×D16—671no—
109#495—1341no—
110#443—1341no—
111Q64—1341no—
112SD64—1341no—
113SD64—1341no—
114D32—1341no—
115M64—671no—
116M64C2×D6711yescharacteristic
117#368C2×D6711yescharacteristic
118#359—1341no—
119C4×C16—671no—
120#135—1341no—
121C2×Q16—21no—
122C2×SD16—2681no—
123C2×D8—1341no—
124#131—2681no—
125#124—2681no—
126Q32—21no—
127Q32—21no—
128Q32—671no—
129SD32—2682no—
130SD32—1341no—
131D16—671no—
132D16—1341no—
133D16—1341no—
134C2×C16—671no—
135C2×C16C22×D6711yescharacteristic
136#108—21no—
137#104—2681no—
138C4×C8—671no—
139C32—21no—
140C32—1341no—
141#41—2681no—
142C2×Q8—41no—
143C2×D4—2681no—
144Q16—41no—
145Q16—41no—
146Q16—1341no—
147SD16—2681no—
148SD16—5361no—
149SD16—5361no—
150D8—2681no—
151D8—2681no—
152D8—1341no—
153M16—41no—
154M16—5361no—
155C2×C8—5361no—
156C2×C8—671no—
157C2×C8D4×D6711yescharacteristic
158C22⋊C4—5362no—
159C42—671no—
160C16—1341no—
161C16D4×D6711yescharacteristic
162C16#1176162811yescharacteristic
163C23—5361no—
164Q8—81no—
165Q8—81no—
166Q8—2681no—
167D4—2681no—
168D4—5361no—
169D4—5361no—
170D4—5361no—
171C2×C4—81no—
172C2×C4—5361no—
173C2×C4—1341no—
174C2×C4D8×D6711yescharacteristic
175C8—81no—
176C8—671no—
177C8—671no—
178C8#1452401611yescharacteristic
179C8#1452407011yescharacteristic
180C22—5361no—
181C22—10722no—
182C22—10721no—
183C22#1686383311yescharacteristic
184C4—2681no—
185C4—161no—
186C4—5361no—
187C4#1686359011yescharacteristic
188C4D16×D6711yescharacteristic
189C2—21no—
190C2—10721no—
191C2—21441no—
192C2#2256224311yescharacteristic
subgroupG/H
centerC2#22562243
commutator, cosocleC2×C16×C67C23
FrattiniC2×C16C22×D67
FittingC67×#1420C2
socleC2×C67#2206
prime radical#22562524C2
hypercenter#1420D67
hypocenterC67#13691
LCSG ▷ C2×C16×C67 ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1420

Complements

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

NH ≅ G/Ntype
1. #22562527C2semidirect
3. #22562471C2semidirect
4. #22562089C2semidirect
5. #22558406C2semidirect
6. #22558398C2semidirect
7. C67×#1420C2semidirect
102. #1420D67semidirect
104. C67#13691semidirect

Maximal subgroups

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

subgroupindex
#225625272
#225625242
#225624712
#225620892
#225584062
#225583982
C67×#14202
#1369167