Finite Group Databasebeta

← #30770408

Subgroups — #30770408

Order 17152.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#22562352C211yesmaximal, characteristic
2#22562256C211yesmaximal, characteristic
3#22559803C211yesmaximal, characteristic
4#22559105C211yesmaximal, characteristic
5#22558278C211yesmaximal, characteristic
6#22558265C211yesmaximal, characteristic
7C67×#1362C211yesmaximal, characteristic
8C2×#14523989—21no—
9#16863801C2211yescharacteristic
10#16863792—21no—
11#16863748—21no—
12#16863523—21no—
13C4×Dic268C2211yescharacteristic
14#16863224—21no—
15C4⋊(C67⋊C16)C2211yescharacteristic
16#16863201—21no—
17#16863200—21no—
18#16863126—21no—
19C8×C67⋊C8C2211yescharacteristic
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×Dic268D411yescharacteristic
24#14524043—21no—
25C4×Dic134—21no—
26#14523989—42no—
27C2×C67⋊C16—21no—
28#14523977—21no—
29#14523977—21no—
30#14523977D411yescharacteristic
31#14523974—42no—
32Dic67⋊C8—41no—
33#14523968—21no—
34#14523966—41no—
35#14523966—41no—
36#14523953—41no—
37C4×C67⋊C8D411yescharacteristic
38C4×C67⋊C8D411yescharacteristic
39C2×Q16×C67—21no—
40C2×Q16×C67—21no—
41C4⋊Q8×C67—21no—
42C2×C16×C67—21no—
43C4⋊C8×C67—21no—
44C4×C8×C67C2311yescharacteristic
45C2×Dic134—21no—
46C2×Dic134—41no—
47Dic268—21no—
48Dic268—21no—
49C2×C67⋊C8—21no—
50C2×C67⋊C8D811yescharacteristic
51C2×C67⋊C8D811yescharacteristic
52C4⋊Dic67—21no—
53C4⋊Dic67—41no—
54C4⋊Dic67—41no—
55Dic67⋊C4—41no—
56C4×Dic67—41no—
57C67⋊C16—41no—
58C2×Q8×C67—41no—
59Q16×C67—41no—
60Q16×C67—41no—
61Q16×C67—81no—
62C2×C8×C67—41no—
63C2×C8×C67C2×D411yescharacteristic
64C2×C8×C67#4111yescharacteristic
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C42×C67C2×D411yescharacteristic
68C16×C67—41no—
69Q8×C67—81no—
70Q8×C67—81no—
71C2×C4×C67—81no—
72C2×C4×C67#12111yescharacteristic
73C2×C4×C67C2×D811yescharacteristic
74C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
75C2×Dic67—41no—
76C2×Dic67—41no—
77C2×Dic67—81no—
78Dic134—41no—
79Dic134—41no—
80Dic134—81no—
81Dic134—81no—
82C8×C67—21no—
83C8×C67—81no—
84C8×C67AGL(1,ℤ/8)11yescharacteristic
85C8×C67#13711yescharacteristic
86C67⋊C8—22no—
87C67⋊C8—21no—
88C67⋊C8—21no—
89C22×C67#45911yescharacteristic
90C4×C67—21no—
91C4×C67—161no—
92C4×C67#46811yescharacteristic
93C4×C67#47111yescharacteristic
94C4×C67#50811yescharacteristic
95C4×C67#51011yescharacteristic
96Dic67—81no—
97Dic67—81no—
98Dic67—161no—
99#12490—671nomaximal, Sylow
100C2×C67#168511yescharacteristic
101C2×C67#221111yescharacteristic
102C2×C67#221411yescharacteristic
103#2225—671no—
104#2209—671no—
105#1707—671no—
106#1591—671no—
107#1370—671no—
108#1364—671no—
109#1362D6711yescharacteristic
110C67#1249011yescharacteristic, complemented, Sylow
111C2×Q32—1341no—
112#500—671no—
113#498—1341no—
114#494C2×D6711yescharacteristic
115#487—1341no—
116#446—1341no—
117C4×Q16—671no—
118#367—1341no—
119C4⋊C16—671no—
120C4⋊C16C2×D6711yescharacteristic
121#358—1341no—
122#358—1341no—
123#335C2×D6711yescharacteristic
124Q8⋊C8—1341no—
125C82—671no—
126C2×Q16—21no—
127C2×Q16—21no—
128C2×Q16—671no—
129C4⋊Q8—21no—
130C4⋊Q8—1341no—
131C4×Q8—1341no—
132Q32—2682no—
133C2×C16—21no—
134C2×C16—1341no—
135#107—671no—
136#107—1341no—
137#107—1341no—
138#106—2682no—
139C4⋊C8—21no—
140C4⋊C8—2681no—
141#103—2681no—
142#103—2681no—
143#103—1341no—
144#97—2681no—
145C4×C8—671no—
146C4×C8—671no—
147C4×C8C22×D6711yescharacteristic
148C2×Q8—41no—
149C2×Q8—2681no—
150C2×Q8—1341no—
151Q16—41no—
152Q16—41no—
153Q16—81no—
154Q16—1341no—
155Q16—1341no—
156C2×C8—41no—
157C2×C8—671no—
158C2×C8—671no—
159C2×C8—1341no—
160C2×C8D4×D6711yescharacteristic
161C2×C8#1176162811yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—2681no—
165C4⋊C4—2681no—
166C4⋊C4—2681no—
167C4⋊C4—1341no—
168C42—2681no—
169C42D4×D6711yescharacteristic
170C16—41no—
171C16—2681no—
172Q8—81no—
173Q8—81no—
174Q8—2681no—
175Q8—2681no—
176Q8—5361no—
177Q8—5361no—
178C2×C4—81no—
179C2×C4—2681no—
180C2×C4—2681no—
181C2×C4—5361no—
182C2×C4#1452401611yescharacteristic
183C2×C4D8×D6711yescharacteristic
184C2×C4#1452407011yescharacteristic
185C8—21no—
186C8—81no—
187C8—1342no—
188C8—1341no—
189C8—1341no—
190C8#1452406711yescharacteristic
191C8#1452407211yescharacteristic
192C22#1686359011yescharacteristic
193C4—21no—
194C4—161no—
195C4—5361no—
196C4—5361no—
197C4—10721no—
198C4#1686364111yescharacteristic
199C4#1686365811yescharacteristic
200C4#1686382711yescharacteristic
201C4#1686383811yescharacteristic
202C2#2255967311yescharacteristic
203C2#2256227411yescharacteristic
204C2#2256228911yescharacteristic
subgroupG/H
centerC22#16863590
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socle, prime radicalC22×C67#459
hypercenter#1362D67
hypocenterC67#12490
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
110. C67#12490semidirect

Maximal subgroups

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

subgroupindex
#225623522
#225622562
#225598032
#225591052
#225582782
#225582652
C67×#13622
#1249067